Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
IntechOpen is proud to announce that 191 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\n
Throughout the years, the list has named a total of 261 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
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\r\n\tIn March 2022, another book on human migration seems important when the events or tragedies unfolding in Eastern Europe are considered. People have always migrated and have moved, but, specifically looking at the last three hundred years, involuntary migration is on the rise. Involuntary migration does not only affect Europe; Asia, Africa, and North as well as South America, have had their fair share of natural catastrophes, invasions, and wars. \r\n\tThis book will intend to look at different migrant patterns, voluntary and involuntary migration, over the last three centuries. What influenced people to leave their home countries, family, and friends and settle somewhere else? The book may include histories of the 19th century, consider tragedies and movements activated by political events in the 20th century, and/or look at recent events of the 21st century. Push and pull factors are important points. While most of us may be influenced in a negative way by the current happenings in Eastern Europe, the Russian invasion and resulting tragedies also demonstrate some very positive human traits – the preparedness of Ukraine’s surrounding countries to help those in need and to provide a safe place for the present. \r\n\tWhether one looks at voluntary or involuntary migration into any country, after a period of adjustment, migrants do play a positive role. The research found that migrants contribute to the economy (food, shelter, employment, tax) and enrich a country’s cultural norms. Prerequisites for successful settlements are that the host society adopts a tolerant approach and that the migrants recognize the law and the language of the host country. Nothing is ever easy or without controversy, but I am a migrant (German Australian), and life in Australia has been relatively harmonious. Issues that could be considered in the book are multicultural societies (do monocultural societies still exist?) and theories of acculturation versus integration (settlement processes). \r\n\tTwo further issues are very important in relation to human migration. There is climate change, global warming, and the environment, which clearly affect people’s movement. Small island populations are very concerned about rising sea levels. 2021 has also seen floods costing human lives: Turkey (August 2021), Brazil (December 2021), Chile (January 2021), and South India (November 2021), to name but a few. In Australia (March 2022), farms and whole townships in New South Wales and Queensland have been flooded for the second time in five years, and plans to resettle these towns are considered. Official and social media provide ample coverage of the events, which leads me to the next issue. There is today’s very important role of the media, of the official and social media. We are constantly bombarded with images of human war tragedies and flood victims. People in industrialized, western countries must be the best-informed populace. How far do the images and up-to-date TV news influence us, make us change our behavior, and perhaps even consider us more generous than we have been? \r\n\tClimate change and the media are relatively new to the human migration debate, but both issues play important parts, and some interesting discussions are appreciated. \r\n\t
",isbn:"978-1-80356-618-4",printIsbn:"978-1-80356-617-7",pdfIsbn:"978-1-80356-619-1",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,isSalesforceBook:!1,isNomenclature:!1,hash:"9836df9e82aa9f82e3852a60204909a8",bookSignature:"Dr. Ingrid Muenstermann",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/11433.jpg",keywords:"Voluntary Migration, Involuntary Migration, Push Factors, Pull Factors, Receiving Countries, Human Rights Violations, Migrants' Acculturation, Migrants' Integration, Young People's Movement, Climate Change, War, Psychological Consequences",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 13th 2022",dateEndSecondStepPublish:"June 10th 2022",dateEndThirdStepPublish:"August 9th 2022",dateEndFourthStepPublish:"October 28th 2022",dateEndFifthStepPublish:"December 27th 2022",dateConfirmationOfParticipation:null,remainingDaysToSecondStep:"14 days",secondStepPassed:!1,areRegistrationsClosed:!1,currentStepOfPublishingProcess:2,editedByType:null,kuFlag:!1,biosketch:"Dr. Ingrid Muenstermann is a Casual Academic at the College of Nursing and Health Sciences, Flinders University of South Australia, with a rich research background in relation to migration. 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\n\t\t\t
1. Introduction
\n\t\t\t
Finite element analysis (FEA) is a well-established numerical simulation method for structural dynamics. It serves as the main computational tool for Noise, Vibration and Harshness (NVH) analysis in the low-frequency range. Because of developments in numerical methods and advances in computer software and hardware, FEA can now handle much more complex models far more efficiently than even a few years ago. However, the demand for computational capabilities increases in step with or even beyond the pace of these improvements. For example, automotive companies are constructing more detailed models with millions of degrees of freedom (DOFs) to study vibro-acoustic problems in higher frequency ranges. Although these tasks can be performed with FEA, the computational cost can be prohibitive even for high-end workstations with the most advanced software.
\n\t\t\t
For large finite element (FE) models, a modal reduction is commonly used to obtain the system response. An eigenanalysis is performed using the system stiffness and mass matrices and a smaller in size modal model is formed which is solved more efficiently for the response. The computational cost is also reduced using substructuring (superelement analysis). Modal reduction is applied to each substructure to obtain the component modes and the system level response is obtained using Component Mode Synthesis (CMS).
\n\t\t\t
When design changes are involved, the FEA analysis must be repeated many times in order to obtain the optimum design. Furthermore in probabilistic analysis where parameter uncertainties are present, the FEA analysis must be repeated for a large number of sample points. In such cases, the computational cost is even higher, if not prohibitive. Reanalysis methods are intended to analyze efficiently structures that are modified due to various changes. They estimate the structural response after such changes without solving the complete set of modified analysis equations. Several reviews have been published on reanalysis methods [1-3] which are usually based on local and global approximations. Local approximations are very efficient but they are effective only for small structural changes. Global approximations are preferable for large changes, but they are usually computationally expensive especially for cases with many design parameters. The well-known Rayleigh-Ritz reanalysis procedure [4, 5] belongs to the category of local approximation methods. The mode shapes of a nominal design are used to form a Ritz basis for predicting the response in a small parametric zone around the nominal design point. However, it is incapable of capturing relatively large design changes.
\n\t\t\t
A parametric reduced-order modeling (PROM) method, developed by Balmes [6, 7], expands on the Rayleigh-Ritz method by using the mode shapes from a few selected design points to predict the response throughout the design space. The PROM method belongs to the category of local approximation methods and can handle relatively larger parameter changes because it uses multiple design points. An improved component-based PROM method has been introduced by Zhang et al. [8, 9] for design changes at the component level. The PROM method using a ‘parametric’ approach has been successfully applied to design optimization and probabilistic analysis of vehicle structures. However, the ‘parametric’ approach is only applicable to problems where the mass and stiffness matrices can be approximated by a polynomial function of the design parameters and their powers. A new ‘parametric’ approach using Kriging interpolation [10] has been recently proposed [11]. It improves efficiency by interpolating the reduced system matrices without needing to assume a polynomial relationship of the system matrices with respect to the design parameters as in [6, 7].
\n\t\t\t
The Combined Approximations (CA) method [12-14] combines the strengths of both local and global approximations and can be accurate even for large design changes. It uses a combination of binomial series (local) approximations, called Neumann expansion approximations, and reduced basis (global) approximations. The CA method is developed for linear static reanalysis and eigen-problem reanalysis [15-19]. Accurate results and significant computational savings have been reported. All reported studies on the CA method [12-19] use relatively simple frame or truss systems for static or dynamics analysis with a relatively small number of DOF and/or modes. For these problems, the computational efficiency was improved by a factor of 5 to 10. Such an improvement is beneficial for a single design change evaluation or even for gradient-based design optimization where only a limited number of reanalyses (e.g. less than 50) is performed. However, the computational efficiency of the CA method may not be adequate in simulation-based (e.g. Monte-Carlo) probabilistic dynamic analysis of large finite-element models where reanalysis must be performed hundreds or thousands of times in order to estimate accurately the reliability of a design.
\n\t\t\t
A large number of modes must be calculated and used in a dynamic analysis of a large finite-element model with a high modal density, even if a reduced-order modeling approach (Section 2) is used. In such a case, the implicit assumption of the CA method that the cost of solving a linear system is dominated by the cost of matrix decomposition way no be longer valid (see Section 3.4) and the computational savings from using the CA method may not be substantial. For this reason, we developed a modified combined approximation (MCA) and integrated it with the PROM method to improve accuracy and computational efficiency. The computational savings can be substantial for problems with a large number of design parameters. Examples in this Chapter demonstrate the benefits of this reanalysis methodology.
\n\t\t\t
The Chapter presents methodologies
\n\t\t\t
for accurate and efficient vibration analysis methods of large-scale, finite-element models,
for efficient and yet accurate reanalysis methods for dynamic response and optimization, and
for efficient design optimization methods to optimize structures for best vibratory response.
\n\t\t\t
The optimization is able to handle a large number of design variables and identify local and global optima. It is organized as follows. Section 2 presents an overview of reduced-order modeling and substructuring methods including modal reduction and component mode synthesis (CMS). Improvements to the CMS method are presented using interface modes and filtration of constraint modes. The section also overviews two Frequency Response Function (FRF) substructuring methods where two substructures, represented by FRFs or FE models, are assembled to form an efficient reduced-order model to calculate the dynamic response. Section 3 presents four reanalysis methods: the CDH/VAO method, the Parametric Reduced Order Modeling (PROM) method, the Combined Approximation (CA) method, and the Modified Combined Approximation (MCA) method. It also points out their strong and weak points in terms of efficiency and accuracy. Section 4 demonstrates how the reanalysis methods are used in vibration and optimization of large-scale structures. It also presents a new reanalysis method in Craig-Bampton substructuring with interface modes which is very useful for problems with many interface DOFs where the FRF substructuring methods cannot be used. Section 5 presents a vibration and optimization case study of a large-scale vehicle model demonstrating the value of reduced-order modeling and reanalysis methods in practice. Finally, Section 6 summarizes and concludes.
\n\t\t
\n\t\t
\n\t\t\t
2. Reduced-Order Modeling for Dynamic Analysis
\n\t\t\t
Computational efficiency is of paramount importance in vibration analysis of large-scale, finite-element models. Reduced-order modeling (or substructuring) is a common approach to reduce the computational effort. Substructuring methods can be classified in “mathematical” and “physical” methods. The “mathematical” substructuring methods include the Automatic Multi-level Substructuring (AMLS) and the Automatic Component Mode Synthesis (ACMS) in NASTRAN. The “physical” substructuring methods include the well known fixed-interface Craig-Bampton method. Both the AMLS and ACMS methods use graph theory to obtain an abstract (mathematical) substructuring using matrix partitioning of the entire finite-element model. The computational savings from the “mathematical” and “physical” methods can be comparable depending on the problem at hand.
\n\t\t\t
\n\t\t\t\t
2.1. Modal Reduction
\n\t\t\t\t
For an undamped structure with stiffness and mass matrices \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t respectively, under the excitation force vector\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, the equations of motion (EOM) for frequency response are
where the displacement d is calculated at the forcing frequency\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\n\t\t\t\t\t. If the response is required at multiple frequencies, the repeated direct solution of Equation (1) is computationally very expensive and therefore, impractical for large scale finite-element models.
\n\t\t\t\t
A reduced order model (ROM) is a subspace projection method. Instead of solving the original response equations, it is assumed that the solution can be approximated in a subspace spanned by the dominant mode shapes. A modal response approach can be used to calculate the response more efficiently. A set of eigen-frequencies \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and corresponding eigenvectors (mode shapes) \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tφ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tare first obtained. Then, the displacement d is approximated in the reduced space formed by the first \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\n\t\t\t\t\t dominant modes as
\n\t\t\t\t
\n\t\t\t\t\t\n\t\t\t\tE2
\n\t\t\t\t
where is the modal basis and U is the vector of principal coordinates or modal degrees of freedom (DOF). Using the approximation of Equation (2), the EOM of Equation (1) can be transformed from the original physical to the modal degrees of freedom as
\n\t\t\t\t
\n\t\t\t\t\t\n\n\t\t\t\tE3
\n\t\t\t\t
The response d can be recovered by solving Equation (3) for the modal response U and projecting it back to the physical coordinates using Equation (2). If \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the maximum excitation frequency, it is common practice to retain the mode shapes in the frequency range of\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t÷\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The system modes in the high frequency range can be safely truncated with minimal loss of accuracy.
\n\t\t\t\t
Due to the modal truncation, the size of the ROM is reduced considerably, compared to the original model. However, the size increases with the maximum excitation frequency. An added benefit of the ROM is that Equations (3) are decoupled because of the orthogonality of the mode shapes and can be therefore, solved separately reducing further the overall computational effort.
\n\t\t\t\t
Note that for a damped structure with a damping matrix\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, Equation (1) becomes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand Equation (3) is modified as by adding the modal damping term . For proportional (structural or Rayleigh) damping, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis a linear combination of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t; i.e. \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\twhere \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\n\t\t\t\t\tare constants. In this case, the reduced Equations (3) of the modal model are decoupled. Otherwise, they are not. In this Chapter for simplicity, we present all theoretical concepts for undamped systems. However for forced vibrations of damped systems, the addition of damping is straightforward.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
2.2. Substructuring with Component Mode Synthesis (CMS)
\n\t\t\t\t
To model the dynamics of a complex structure, a finite-element analysis of the entire structure can be very expensive, and sometimes infeasible, due to computer hardware and/or software constraints. This is especially true in the mid-frequency range, where a fine finite element mesh must be used in order to capture the shorter wavelengths of vibration. Component mode synthesis (CMS) was developed as a practical and efficient approach to modeling and analyzing the dynamics of a structure in such circumstances [20–23]. The structure is partitioned in component structures and the dynamics are described by the normal modes of the individual components and a set of modes that couple all component. Besides the significant computational savings, this component-based approach also facilitates distributed design. Components may be modified or redesigned individually without re-doing the entire analysis.
\n\t\t\t\t
One of the most accurate and widely-used CMS methods is the Craig-Bampton method [22] where the component normal modes are calculated with the interface between connected component structures held fixed. Attachment at the interface is achieved by a set of “constraint modes.” A constraint mode shape is the static deflection induced in the structure by applying a unit displacement to one interface DOF while all other interface DOF are held fixed. The motion at the interface is thus completely described by the constraint modes. The Craig-Bampton reduced-order model (CBROM) results in great model size reduction by including only component normal modes within a certain frequency range. However, there is no size reduction for constraint modes because CBROM must have one DOF for each interface DOF. If the finite element mesh is sufficiently fine, the constraint-mode DOFs will dominate the CBROM mass and stiffness matrices, and increase the computational cost.
\n\t\t\t\t
We address this problem by using interface modes (also called characteristic constraint –CC- modes) in order to reduce the number of retained interface DOFs of the Craig-Bampton approach. For that, a secondary eigenvalue analysis is performed using the constraint-mode partitions of the CMS mass and stiffness matrices. The CC modes are the resultant eigenvectors. The basic formulation is described in Sections 2.2.1 and 2.2.2. The interface modes represent more “natural” physical motion at the interface. Because they capture the characteristic motion of the interface, they may be truncated as if they were traditional modes of vibration, leading to a highly compact CC-mode-based reduced order model (CCROM). In addition, the CC modes provide a significant insight into the physical mechanisms of vibration transmission between the component structures. This information could be used, for example, to determine the design parameters that have a critical impact on power flow. Figure 1 compares a conventional constraint mode used in Craig-Bampton analysis with an interface mode, for a simple cantilever plate which is subdivided in two substructures.
\n\t\t\t\t
It should be noted that the calculation of the CC modes is essentially a secondary modal analysis. Therefore, the benefits are the same as those of a traditional modal analysis. For instance, refining the finite-element mesh increases the accuracy of a CCROM without introducing any additional degrees of freedom. The ability of the CC mode approach to produce CCROM whose size does not depend on the original level of discretization makes it especially well suited for finite-element based analysis of mid-frequency vibration.
\n\t\t\t\t
Figure 1.
Illustration of interface modes.
\n\t\t\t\t
\n\t\t\t\t\t
2.2.1. Craig-Bampton Fixed Interface CMS
\n\t\t\t\t\t
This section provides the basics of Craig-Bampton method using the fixed-interface assumption. The method is commonly used in CMS algorithms. The finite-element model of the entire structure is partitioned into a group of substructures. The DOFs in each substructure are divided into interface (\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΓ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t) DOF and interior (\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t) DOF. The equations of motion for the i\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tth\n\t\t\t\t\t\t substructure are then expressed as
\n\t\t\t\t\t
\n\t\t\t\t\t\n\t\t\t\t\t\t\n\n\t\t\t\t\tE4
\n\t\t\t\t\t
The fixed-interface Craig-Bampton CMS method utilizes two sets of modes to represent the substructure motion; substructure normal (N) modes\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and constraint (C) modes\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, where i denotes the i\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tth\n\t\t\t\t\t\t substructure. The size reduction of the Craig-Bampton method comes from the truncation of the normal modes \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tφ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tφ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t⋯\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tφ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t which are calculated by fixing all interface DOFs and solving the following eigenvalue problem
The static constraint modes\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tare calculated by enforcing a set of static unit constraints to the interface DOFs as
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\t\t\t\t\tE6
\n\t\t\t\t\t
The original physical DOFs \n \n\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tcan be thus represented by the constraint-mode DOFs \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tand the normal-mode DOFs \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tas
where the superscripts C and N are used to indicate partition related to static constraint mode DOFs and fixed-interface normal mode DOFs, respectively. The matrix partitions of Equation (8) are
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\t\t\t\t\tE9
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\n\t\t\t\t\tE10
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\n\n\t\t\t\t\tE11
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\n\n\n\t\t\t\t\tE12
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\t\t\t\t\tE13
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\n\t\t\t\t\tE14
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\t\t\t\t\tE15
\n\t\t\t\t\t
\n\t\t\t\t\t\t\n\n\n\t\t\t\t\tE16
\n\t\t\t\t\t
The matrices of each substructure are then assembled by applying displacement continuity and force balance along the interface to obtain the EOM of the reduced system. A secondary modal analysis is finally carried out using the mass and stiffness matrices of the reduced system to obtain the eigenvalues and eigenvectors.
\n\t\t\t\t\t
Note that constraint mode matrix \n\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t is usually a full matrix. Therefore Equation (9) can be computationally expensive due to the triple-product \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tinvolving constraint modes. The computational cost of the Craig-Bamtpon method is mostly related to
In Craig-Bampton CMS, the matrices from all substructures are assembled into a global CBROM with substructures coupled at interfaces by enforcing displacement compatibility. This synthesis yields the modal displacement vector \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t of the synthesized system to be partitioned as
where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t is the number of substructures in the global structure. The corresponding synthesized CMS mass and stiffness matrices are as follows
where the component modal matrices \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t are diagonalized and their sizes depend on the number of selected modes for the frequency range of interest. However, the number of constraint-mode DOFs, or the size of matrices \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tand\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t, is equal to the number of DOFs of the interfaces between components and is therefore, determined by the finite-element mesh. If the mesh is fine in the interface regions, or if there are many substructures, the constraint-mode partitions of the CMS matrices may be relatively large. For this reason, we further reduce the CMS matrices by performing a modal analysis on the constraint-mode DOFs as follows
The eigenvectors \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t are transformed into the finite-element DOFs for the i\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tth\n\t\t\t\t\t\t component structure using the following transformation
is a selected set of \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t interface eigenvectors which are few compared to the number of the constraint-mode DOFs. The matrix \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t maps the global (system) interface DOFs \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tback to the local (subsystem i) DOF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t. The vectors in \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t are referred to as the interface modes or characteristic constraint (CC) modes, because they represent the characteristic physical motion associated with the constraint modes. Relatively few CC-mode DOFs are used compared to the number of interface DOFs.
\n\t\t\t\t\t
Finally, the CMS matrices are transformed using the CC modes and the reduced-order CMS matrices are obtained similarly to Equations (18). Now, the unknown displacement vector \n\t\t\t\t\t\t dROM is partitioned as
where superscript CC indicates the partition associated with the CC modes. The equations of motion of the reduced order CMS model (ROM) are expressed by
\n\t\t\t\t\t\tFigure 2 shows a typical constraint mode for a plate substructure. The non-zero displacement field (indicated by red color) is usually limited to a very small region close to the perturbed interface DOF.
\n\t\t\t\t\t
Figure 2.
Illustration of a “filtered” constraint mode.
\n\t\t\t\t\t
If the small-displacement part of the constraint mode shape is explicitly replaced by zero, the density of the resulting “filtered” constraint mode will be significantly reduced. Consequently, the computational cost in Equation (9) will be considerably reduced. To filter the constraint modes, the following criterion is used
where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tφ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t is the p\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tth\n\t\t\t\t\t\t element of the q\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tth\n\t\t\t\t\t\t constraint mode\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t. If the ratio of an element of the constraint mode vector to the maximum value in the vector is smaller than a defined small\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, the element of the constraint mode is truncated to zero. For the constraint mode of Figure 4, the constraint mode density reduces from 100% to 16% if\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t0.03\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
\n\t\t\t\t
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
2.3. Frequency Response Function (FRF) Substructuring and Assembling
\n\t\t\t\t
If the number of interface nodes (or DOFs) between connected substructures is small, a reduced-order model of small size can be developed using an FRF representation of each substructure. This is known as FRF substructuring. The FRF representation can be easily obtained from a finite element (FE) model or even experimentally. If the FE model of one substructure is very small (e.g. a vehicle suspension model), it can be easily coupled directly to another substructure which is represented using FRFs. This section provides the fundamentals of FRF substructuring for both FRF-FE and FRF-FRF coupling.
\n\t\t\t\t
\n\t\t\t\t\t
2.3.1. Algorithm for FRF/FE Coupling
\n\t\t\t\t\t
The numerical algorithm is explained using the two-substructure example of Figure 3. Sub1 is an FRF type substructure, meaning that its dynamic behavior is described using FRF matrices which are denoted by H (see Equation 30 for notation). The elements of H are frequency dependent and complex if damping is present. A bold letter indicates a matrix or vector. According to Equation (30), HAC for example, represents the displacement XA of DOF A due to a unit force FC on DOF C. Sub2 is a finite element (FE) type substructure. Its dynamic behavior is described using the stiffness K, mass M and damping B matrices.
\n\t\t\t\t\t
Figure 3.
Two-substructure example and notation.
\n\t\t\t\t\t
The FRF matrix of Sub1 can be calculated by either a direct frequency response method or a modal response method. In the former case, the original FE equations are used in the frequency domain while in the latter a modal model is first developed and then used to calculate the FRF matrix. The size of the FRF matrix is small and depends on the number of DOFs of the excitation, response and interface DOFs. Usually FRFs are calculated between excitation and response DOFs. However in order to assemble two substructures, FRFs are also calculated between interface DOFs and excitation/response DOFs. The Sub1 FRF matrix H in Equation (30) is thus partitioned into interior (A) DOFs and interface/coupling (C) DOFs. The interior DOFs include all excitation and all response DOFs (Figure 3).
\n\t\t\t\t\t
The second substructure Sub2 is expressed in FE format. The system FE matrices K, M, and B form the frequency dependent dynamic matrix \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t which is then partitioned according to the interior (B) and interface (C) DOFs. The interface DOFs for Sub1 and Sub2 have the same node IDs so that they can be assembled to obtain the system FRF matrix. The procedure to assemble the H matrix of Sub1 with the Z matrix of Sub2 and calculate (solve for) the system matrix H is described below.
where subscript A indicates the interior (excitation plus response) DOFs of Sub1, and subscript C indicates the connection/common/coupling DOFs between Sub1 and Sub2. The equations of motion for Sub2 are expressed as
where subscript B indicates the interior DOFs of Sub2, and subscript C indicates the connection/common/coupling DOFs between Sub2 and Sub1. Because of displacement compatibility at the interface, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tappears on the left-hand side of Equation (30) for Sub1 and on the right-hand side of Equation (31) for Sub2. Superscripts 1 and 2 are used to differentiate the interface forces FC at Sub1 and Sub2.
\n\t\t\t\t\t
To couple Sub1 and Sub2, compatibility of forces at the interface is applied as\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t where the force vector \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t with \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\tis obtained from the second row of Equation (30) and the second row of Equation (31) provides the force vector\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t \n \n\t\t\t\t\t\t. We thus have
To couple Sub1 and Sub2, we enforce displacement compatibility at the interface and apply the interface force relationship\n\t\t\t\t\t\t\t\n\n\t\t\t\t\t\t. In this case, the assembled system equations can be re-arranged in matrix form as
The FRF/FRF coupling is a special case of the FRF/FE coupling.
\n\t\t\t\t
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
3. Reanalysis Methods for Dynamic Analysis
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\n\t\t\t\t
3.1. CDH/VAO Method
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The CDH/VAO method, developed by CDH AG for vibro-acoustic analysis, is a Rayleigh-Ritz type of approximation. If the stiffness and mass matrices of the baseline design structure are K0 and M0, the exact mode shapes in Φ0 are obtained by solving the eigen-problem
\n\t\t\t\t
\n\t\t\t\t\t\n\n\t\t\t\tE44
\n\t\t\t\t
where Λ0 is the diagonal matrix of the baseline eigenvalues. A new design (subscript p) has the following stiffness and mass matrices
For a modest design change where ΔK and ΔM are small, it is assumed that the change in mode shapes is small and the new response can be therefore, captured in the sub-space spanned by the mode shapes Φ0 of the initial design. The new stiffness and mass matrices are then condensed as\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand the following reduced eigen-value problem is solved to calculate the eigen-vector \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΘ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t
where\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Thus, the modal response of the modified structure can be easily obtained and the actual response can be recovered using the eigenvectors\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
The PROM method approximates the mode shapes of a new design in the subspace spanned by the dominant mode shapes of some representative designs, which are selected such that the formed basis captures the dynamic characteristics in each dimension of the parameter space. Balmes et al. [6, 7] suggested that these representative designs should correspond to the middle points on the faces of a box in the parameter space representing the range of design parameters. For a structure with \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t design variables, Zhang [9] suggested that the representative designs include a baseline design for which all parameters are at their lower limits plus m designs obtained by perturbing the design variables from their lower limits to their upper limits, one at a time. The points representing these designs in the space of the design variables are called corner points (see Figure 5). This selection of representative designs results in a more accurate PROM algorithm.
\n\t\t\t\t
Figure 5.
Design space of three parameters.
\n\t\t\t\t
The mode shapes of a new design are approximated in the space of the mode shapes of the corner points as
where the modal matrix P includes the basis vectors as in Equation (49) and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΘ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t represents the participation factors of these vectors. The columns of P are the dominant mode shapes of the above \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t designs,
where\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the modal matrix composed of the dominant mode shapes of the baseline design, and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the modal matrix of the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t corner point. The mode shapes of the new design satisfy the following eigenvalue problem,
where Λ is a diagonal matrix of the first s eigenvalues.
\n\t\t\t\t
A reduced eigenvalue problem is obtained by pre-multiplying both sides of Equation (50) by \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tas
Thus, the matrix \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΘ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in Equation (48) consists of the eigenvectors of the reduced stiffness and mass matrices \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
For \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t design variables, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\teigenvalue problems must be solved in order to form the basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of Equation (49). Therefore, both the cost of obtaining the modal matrices \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and the size of matrix \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t increase linearly with m. The PROM approach uses the following algorithm to compute the mode shapes of a new design:
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Calculate the mode shapes of the baseline design and the designs corresponding to the \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t corner points in the design space, and form subspace basis\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.
Calculate the reduced stiffness and mass matrices \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t from Equation (52).
Solve eigenproblem (51) for matrix\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΘ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.
Reconstruct the approximated eigenvectors in \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t using Equation (48).
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Step 1 is performed only once. A reanalysis requires only steps 2 to 4. For a small number of mode shapes and a small number of design variables, the cost of steps 2 to 4 is much smaller than the cost of a full analysis. The computational cost of PROM consists of
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the cost of performing \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t full eigen-analyses to form subspace basis \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tin Equation (49), and
the cost of reanalysis of each new design in steps 2 to 4.
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The former is the fixed cost of PROM because it does not depend on the numbers of reanalyses and the latter is the variable cost of PROM because it is proportional to the number of reanalyses. The fixed cost is not attributed to the calculation of the response for a particular design. It is simply required to obtain the information needed to apply PROM. The variable cost (cost of reanalysis of a new design in part b) is small compared to the fixed cost. The fixed cost of PROM is proportional to the number of design variables m because it consists of the dominant eigenvectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of the baseline design and the dominant eigenvectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of the m corner design points (see Equation 49). When the size of basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t increases so does the fixed cost because more eigenvalue problems and mode shapes must be calculated to form basis\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The PROM method results in significant cost savings when applied to problems that involve few design variables (less than 10) and require many analyses (e.g. Monte Carlo simulation or gradient-free optimization using genetic algorithms).
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3.3. Combined Approximations (CA) Method
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The PROM method requires an eigenvalue analysis for multiple designs (corner points) to form a basis for approximating the eigenvectors at other designs. It is efficient only when the number of design parameters is relatively small. On the contrary, the CA method of this section does not have such a restriction because the reanalysis cost is not proportional to the number of design parameters. The CA method is thus more suitable than the PROM method, when the number of reanalyses is less than or comparable to the number of design parameters, such as in gradient-based design optimization.
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The fundamentals of the combined approximations (CA) method [15-19] are given below. A subspace basis is formed through a recursive process for calculating the natural frequencies and mode shapes of a system. If \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are the stiffness and mass matrices of the original (baseline) design, the exact mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are obtained by solving the eigen-problem\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. We want to approximate the mode shapes of a modified design (subscript p) with stiffness and mass matrices \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\twhere \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t represent large perturbations. The CA method estimates the new eigenvalues \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and eigenvectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t without performing an exact eigenvalue analysis.
\n\t\t\t\t
The eigen-problem for the modified design can be expressed as
which produces a sequence of approximations of the mode shapes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t…\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The CA method uses the changes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t to form a subspace basis to approximate the modes of the new design. In order to simplify the calculations, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tin Equation (53) is replaced with \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and Equation (54) becomes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t showing that the basis vectors satisfy the following recursive equation
where the first basis vector is assumed to be\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\n\t\t\t\t\t is usually between 3 and 6 [16-18, 23] and the mode shapes of the new (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t) design are then approximated in the subspace spanned by \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t using the following algorithm:
Solve the reduced eigen-problem (using matrices KR and MR) to calculate the eigenvector matrix\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΘ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.
\n\t\t\t\t
Reconstruct the approximate eigenvectors of the new design\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t as
The eigenvalues of the new design are approximated by the eigenvalues \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of the reduced eigen-problem.
\n\t\t\t\t
The CA method has three main advantages:
\n\t\t\t\t
it only requires a single matrix decomposition of stiffness matrix \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t in Equation (55) to calculate the subspace basis\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,
it is accurate because the basis is updated for every new design, and
the eigenvectors of a new design are efficiently approximated by Equation (58) where the eigenvectors \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΘ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t correspond to a much smaller reduced eigen-problem.
\n\t\t\t\t
However for a large number of reanalyses, the computational cost can increase substantially because a new basis and the condensed mass and stiffness matrices in Equation (57) must be calculated for every reanalysis. Examples where many analyses are needed are optimization problems in which a Genetic Algorithm is employed to search for the optimum, and probabilistic analysis problems using Monte-Carlo simulation. The PROM method can be more suitable for these problems because the subspace basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t does not change for every new design point. Note that steps 1 and 3 (Equations 57and 58) are similar to steps 2 and 4 of PROM. CA uses basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and PROM uses basis\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
The CA method is more efficient than PROM for design problems where few reanalyses are required for two reasons. First, it does not require calculation of the eigenvectors\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t⋯\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, of the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t corner design points, and second the cost of matrix condensation of Equation (57) is much lower than that of Equation (52), because the size (number of columns) of basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is not proportional to the number of parameters \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t as in basis P\n\t\t\t\t\t. For problems with a large number of design parameters, the PROM approach is efficient only when a ‘parametric’ relationship is established [7] because a large overhead cost, proportional to the number of design parameters, is required. In contrast, the CA method does not require such an overhead cost because the reanalysis cost is not proportional to the number of design parameters. The CA method is therefore, more suitable than PROM, if the number of reanalyses is less than or comparable to the number of design parameters.
In the literature, the accuracy and efficiency of the CA method has been mostly tested on problems involving structures with up to few thousands of DOFs, such as frames or trusses [12-19]. We have tested the CA method using, among others, the structural dynamics response of a medium size (65,000 DOFs) finite-element model (Figure 7 of Section 4.3). Due to its high modal density, there were more than two hundred dominant modes in the frequency range of zero to 50 Hz. It was observed that the computational savings of the CA method, using the recursive process of Equation (55), were not substantial. For this reason, we developed a modified combined approximations method (MCA) by modifying the recursive process of Equation (55) which is much more efficient than the original CA method for large size models.
\n\t\t\t\t
The cost of calculating the subspace basis in Equation (55) consists of one matrix decomposition (DCMP) and one forward-backward substitution (FBS). The DCMP cost is only related to the size and density of the symmetric stiffness matrix, while the FBS cost depends on both the size and density of the stiffness matrix and the number of columns of\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. As the frequency range of interest increases, more modes are needed to predict the structural response accurately. In such a case, although a single DCMP is needed in Equation (55), the number of columns in \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t may increase considerably, thereby increasing the cost of the repeated FBS. When the number of dominant modes becomes very large, the cost of performing the calculations in Equation (55) can be dominated by the FBS cost. For example, the vehicle model of Section 4.3 (Figure 7) has 65,000 degrees of freedom and 1050 modes in the frequency range of 0-300 Hz. The cost of one DCMP is 1.1 seconds (using a SUN ULTRA workstation and NASTRAN v2001) and the cost of one FBS is less than 0.1 seconds if \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t has only one mode. In this case, the total cost is dominated by the DCMP, and the CA method reduces the cost of one reanalysis considerably. However, if \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t has 1050 modes, the cost of FBS increases to 29 seconds dominating the cost of the DCMP. The CA method can therefore, improve the efficiency only when the number of retained modes is small. Otherwise, the computational savings do not compensate for the loss of accuracy from using \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t (stiffness matrix of baseline design) instead of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t (stiffness matrix of new design). The modified combined approximations (MCA) method of this section addresses this issue.
\n\t\t\t\t
The MCA method uses a subspace basis T whose columns are constructed using the recursive process
instead of that of Equation (55). The selection of the appropriate number of basis vectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\n\t\t\t\t\t is discussed later in this section. The only difference between Equations (55) and (59) is that matrix \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is inverted in the former while matrix \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is inverted in the latter. The DCMP of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t must be repeated for every new design. However, the cost of the repeated DCMP does not significantly increase the overall cost in Equation (59) because the latter is dominated by the FBS cost. The iterative process of Equation (59) provides a continuous mode shape updating of the new design. If the process converges in s iterations, the mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t will be the exact mode shapes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Equation (55) does not have the same property. The vectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t provide therefore, a more accurate approximation of the exact mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t than the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t vectors of the original CA method. This is an important advantage of MCA.
\n\t\t\t\t
Because the mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in Equation (59) can quickly converge to the exact mode shapes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, for many practical problems only one iteration (i.e. s = 1) may be needed, resulting in
For better accuracy, the above basis also includes the mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of the baseline design. Because the approximate modes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are more accurate than the CA vectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in approximating the exact mode shapes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, MCA can achieve similar accuracy to the CA method using fewer modes. The example of Section 4.3 demonstrates that the MCA method achieves good accuracy with only 1 basis vector whereas the CA method requires 3 to 6 basis vectors [13-17].
\n\t\t\t\t
In summary, the proposed MCA method involves four steps in calculating the approximate eigenvectors \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tas follows
Solve the following reduced eigen-problem to calculate the eigenvalues and the projections of the modes in the reduced space spanned by T \n\t\t\t\t\t\t
Reconstruct the approximate eigenvectors \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t as
The slightly increased cost of using Equation (61) instead of Equation (60) is usually smaller than the realized savings in steps 2 through 4 of Equations (62) through (64) due to the smaller size of the reduced basis\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The bases of Equations (60) and (61) are smaller in size than the CA basis of Equation (56) for comparable accuracy. The MCA method requires therefore, less computational effort for steps 2 through 4. The computational savings compensate for the increased cost of DCMP for dynamic reanalysis of large finite-element models with a large number of dominant modes.
\n\t\t\t\t
All mode shapes in Equation (63) must be calculated simultaneously in order to ensure that the approximate mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are orthogonal with respect to the mass and stiffness matrices. However, the cost of estimating the mode shapes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tusing Equations (62) to (64) may increase quickly (quadratically) with the number of modes, and as a result, the MCA method may become more expensive than a direct eigen-solution when the number of dominant modes exceeds a certain limit. One way to circumvent this problem is to divide the frequency response into smaller frequency bands and calculate the frequency response in each band separately instead of solving for the frequency response in one step. The modal basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tin Equation (61) is divided into k groups as
Each group \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tcontains roughly \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t original modes \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t from\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and their corresponding improved modes. The eigenvectors of the new design are calculated using \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tinstead of T in Equations (62) to (64). The process is repeated \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\n\t\t\t\t\t times using a modal basis that is \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of the size of the original modal basis. All \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\n\t\t\t\t\t groups of eigenvectors are then collected together to calculate the frequency response of the new design. As demonstrated in Section 4.3.2, this reduces the cost considerably with minimal loss of accuracy.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
3.5. Comparison of CA/MCA and PROM Methods
\n\t\t\t\t
As we have discussed, a large overhead cost which is proportional to the number of design parameters is required before the PROM reanalysis is carried out. However, the CA/MCA method does not require this overhead cost because it does not need the basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of Equation (49) (see Section 3.2). Therefore, the CA/MCA method is more suitable, when the number of reanalyses is comparable to the number of design parameters. This is usually true in gradient-based design optimization. The CA and MCA methods can become expensive however, when many reanalyses are needed because, for each reanalysis, they require a new basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t or \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t (see Equations 56and 61, respectively) and new condensed mass and stiffness matrices in Equations (57) and (62). This is the case in gradient-free optimization problems employing a Genetic Algorithm for example, and in simulation-based probabilistic analysis problems employing the Monte-Carlo method. For these problems, the PROM method is more suitable because the subspace basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t does not change for every new design point. Table 1 summarizes the main characteristics, advantages and application areas of the CA/MCA and PROM methods.
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\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\tCA/MCA Method\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\tPROM Method\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Overhead Cost
\n\t\t\t\t\t\t\t
None
\n\t\t\t\t\t\t\t
Required cost to construct \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Cost proportional to the number of design parameters \n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t.
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Basis Vector
\n\t\t\t\t\t\t\t
Variable basis \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Size proportional to \n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t.
\n\t\t\t\t\t\t\t
Constant basis \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Size proportional to \n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t.
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Reanalysis Cost
\n\t\t\t\t\t\t\t
Moderate Relatively small size of \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Must recalculate \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t at every new design.
\n\t\t\t\t\t\t\t
High if no parametric relationship exists due to the condensation of large size and dense \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Low if a parametric relationship exists.
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Best Application
\n\t\t\t\t\t\t\t
Small number of reanalyses compared to the number of design parameters. Evaluation of few design alternatives and gradient-based optimization.
\n\t\t\t\t\t\t\t
Very large number of reanalyses. Gradient-free optimization (e.g. genetic algorithms) and probabilistic analysis.
\n\t\t\t\t\t\t
\n\t\t\t\t\t
Table 1.
Comparison of the CA/MCA and PROM methods.
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4. Reanalysis Methods in Dynamic Analysis and Optimization
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The reanalysis methods of Section 3 can be used in different dynamic analyses such as modal or direct frequency response and free or forced vibration in time domain. Depending on the problem and the type of analysis, a particular reanalysis method may be preferred considering how many times it will be performed and how many design parameters will be allowed to change. This section demonstrates the computational efficiency and accuracy of reanalysis methods in dynamic analysis and optimization. It also introduces a new reanalysis method in Craig-Bampton substructuring with interface modes which is very useful for problems with many interface DOFs where FRF substructuring is not practically applicable.
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4.1. Integration of MCA Method in Optimization
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We have mentioned that the MCA method provides a good balance between accuracy and efficiency for problems that require a moderate number of reanalyses, as in gradient-based optimization. For problems where a large number of reanalyses is necessary, as in probabilistic analysis and gradient-free (e.g. genetic algorithms) optimization, a combination of the MCA and PROM methods is more suitable.
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\n\t\t\t\t\tFigure 6 shows a flowchart of the optimization process for modal frequency response problems. The DMAP (Direct Matrix Abstraction Program) capabilities in NASTRAN have been used to integrate the MCA method and the NASTRAN modal dynamic response and optimization (SOL 200). The highlighted boxes indicate modifications to the NASTRAN optimizer. Starting from the original design, the code first calculates the design parameter sensitivities in order to establish a local search direction and determine an improved design along the local direction. At the improved design, an eigen-solution is obtained to calculate a modal model and the corresponding modal response. The dynamic response at certain physical DOFs is then recovered from the modal response. At this point, a convergence check is performed to decide if the optimal design is obtained. If not, further iterations are needed and the above procedure is repeated. Many iterations are usually needed for practical problems to obtain the final optimal design. Section 4.3.2 demonstrates how this process was used to optimize the vibro-acoustic behavior of a 65,000 DOF, finite-element model of a truck. Using the MCA method, the computational cost of the entire optimization process was reduced in half compared with the existing NASTRAN approach.
\n\t\t\t\t\n\t\t\t\t
As for a stand alone modal frequency response, the eigen-solution accounts for a large part of the overall optimization cost for vibratory problems where a modal model is used. A reanalysis method can be inserted into the procedure as shown in Figure 6 to provide an approximate eigen-solution saving therefore, substantial computational cost. Other reanalysis methods such as the CDH/VAO, CA or PROM can also be used depending on the number of design variables and the number of expected iterations.
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Figure 6.
Flowchart for mca-enhanced nastran optimization.
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4.2. Integration of MCA and PROM Methods
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The PROM method requires exact calculation of the mode shapes for all designs corresponding to the corner points of the parameter space in order to calculate the subspace basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tof Equation (49). The required computational effort can be prohibitive for a large number of parameters (optimization design variables). This effort can be reduced substantially if the modes of each corner point are approximated by the MCA method. In this case, an exact eigen-solution is required only for the baseline design. The following steps describe an algorithm to integrate the MCA and PROM methods:
\n\t\t\t\t
Perform an exact eigen-analysis at the baseline design point p0 all parameters are at their lower limit, to obtain the baseline mode shapes\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.
Use the MCA method at design point pi with all parameters at their low limit except the \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t parameter which is set at its upper limit. Obtain approximate mode shapes for the \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t corner point using the following recursive process
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t is the total number of parameters.
\n\t\t\t\t
Obtain the approximate mode shapes \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t using the subspace projection procedure of Equations (50) through (52) where \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t is used instead of\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.
\n\t\t\t\t
The modal basis \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t can be subsequently used in a modal dynamic response solution. Only step 4 is repeated in reanalysis. The computational savings can be substantial especially for problems where many reanalyses are needed.
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4.3. Combined MCA and PROM Methods: Vibro-Acoustic Response of a Vehicle
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The pickup truck vehicle model with 65,000 DOFs of Figure 7 is used in this section to demonstrate the advantages of the combined MCA and PROM method in optimizing the vibro-acoustic response of a vehicle. The model has 78 components and roughly 11,000 nodes and elements. The example is performed on a SUN ULTRA workstation using NASTRAN v2001. The MCA and PROM methods have been implemented in NASTRAN DMAP.
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Figure 7.
FE model of a pickup truck.
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The sound pressure level at the driver’s ear location is calculated using a vibro-acoustic analysis. The structural forced vibration response due to unit harmonic forces in x, y, and z directions at the engine mount locations, is coupled with an interior acoustic analysis. The first and second eigen-frequencies of the acoustic volume inside the cabin are 95.9 Hz and 128.3 Hz. The sound pressure level is calculated in the 80 to 140 Hz frequency range. The structure and fluid domains are coupled through boundary conditions ensuring continuity of vibratory displacement and acoustic pressure. A finite-element formulation of the coupled undamped problem yields the following system equations of motion [24].
where the vibratory displacement \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and the acoustic pressure \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are the primary variables. The finite-element representation of the two domains consists of stiffness and mass matrix pairs \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, respectively. The air density and wave speed are \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The right hand side of Equation (69) denotes the external forces.
\n\t\t\t\t
The spatial coupling matrix \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t indicates coupling between the two domains which is usually referred to as “two-way coupling.” Due to this coupling, the combined structural-acoustic system of equations is not symmetric. If the acoustic effect on the structural response is small, the coupling term can be omitted, resulting in the so-called “one-way coupling,” where the structural response is first calculated and then used as input (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tin Equation 69) to solve for the acoustic response. The coupled structure-acoustic system can be solved either by a direct method, or more efficiently by a modal response method which can be applied to both the structural and acoustic domains.
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\n\t\t\t\t\t
4.3.1. Combined MCA and PROM Methods
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To demonstrate the computational effectiveness and accuracy of integrating MCA in PROM, a reanalysis was performed for a modified design where five plate thickness parameters vary; chassis and its cross links, cabin, truck bed, left door, and right door. All parameters were increased by 100% from their baseline values. The sound pressure at the driver’s ear was calculated using “two-way” coupling. A structural modal frequency response was used. The acoustic response was calculated using a direct method because the size of the acoustic model is relatively small. For the structural analysis, 1050 modes were retained in the 0 to 300 Hz frequency range. The combined MCA and PROM approach was compared against the NASTRAN direct solution for a modified design where all five parameters were at their upper limits. Only one iteration was used in Equation (59) in order to get the set of once-updated mode shapes for each corner design point. The subspace basis, which includes information for all five design parameters, is therefore, represented by
The maximum error in natural frequencies as predicted by the combined MCA and PROM method and NASTRAN, is less than 0.45% in the entire frequency range. Figure 8 indicates that the sound pressures calculated by both methods are almost identical. The computational effort for the MCA method to obtain approximate mode shapes at each corner design point is about 30 seconds. In contrast, it takes about 180 seconds for an exact eigen-solution using NASTRAN. The computational cost to construct the reduced basis (\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tin PROM and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t in PROM+MCA) is compared in Table 2. The total cost was reduced from 1080 seconds to 330 seconds. The computational saving is more significant if the number of design parameter increases.
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Figure 8.
Comparison of sound pressure at driver’s ear between combined MCA and PROM method and NASTRAN.
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\n\t\t\t\t\t\t\t\t\tMethod\n\t\t\t\t\t\t\t\t
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\n\t\t\t\t\t\t\t\t\tSolving for mode shape \n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t at baseline design\n\t\t\t\t\t\t\t\t
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\n\t\t\t\t\t\t\t\t\tSolving for mode shapes at 5 corner design points\n\t\t\t\t\t\t\t\t
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\n\t\t\t\t\t\t\t\t\tTotal Cost\n\t\t\t\t\t\t\t\t
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PROM
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180 sec
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180*5=900 sec
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1080 sec
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PROM+MCA
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180 sec
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30*5=150 sec
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330 sec
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Table 2.
CPU time to construct reduced basis.
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4.3.2. Optimization using MCA Method
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The goal here is to minimize the sound pressure at the driver’s ear. A total of 41 design parameters are used representing the thickness of all vehicle components modeled with plate elements. All thicknesses are allowed to change by 100% from their baseline values. Table 3 describes all design parameters. At the initial point of the optimization process, all parameters are at their low bound.
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\n\t\t\t\t\t\t\t\tPrm.#
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Description(thickness of)
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Prm.#
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Description(thickness of)
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Prm.#
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Description(thickness of)
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1
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Bumper
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15
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Radiator mtg.
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29
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Tire, front right
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2
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Rails
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16
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Radiator mtg., mid.
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30
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Tire, rear left
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3
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A-arm, low left
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17
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Fan cover, low
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31
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Tire, rear right
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4
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A-arm, low right
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18
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Fan cover, up
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32
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Engine outer
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5
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A-arm, up left
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19
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Cabin
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33
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A-arm conn., up left
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6
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A-arm, up right
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20
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Cabin mtg. reinf.
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34
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A-arm conn., up right
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7
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Tire rim
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21
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Door, left
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35
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A-arm conn., low left
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8
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Engine Oil-box
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22
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Door, right
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36
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A-arm conn., low right
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9
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Fan
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23
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Bed
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37
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Glass, left
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10
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Hood
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24
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Brake, front left
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38
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Glass, right
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11
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Fender, left
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25
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Brake, front right
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39
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Glass, rear
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12
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Fender, right
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26
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Rail conn., rear
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40
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Glass, front
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13
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Wheel house, left
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27
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Rail mount
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41
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Rail conn., front
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14
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Wheel house, right
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28
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Tire, front left
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Table 3.
Description of design parameters.
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Because of the large number of design parameters, the combined MCA and PROM approach Section 4.3.1 is not computationally efficient because the size of the PROM basis is very large (see Equation 70). For this reason, we use the MCA reanalysis method and demonstrate its capability to handle a large number of parameters. It approximates the mode shapes at intermediate design points using only \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t in Equation (59). The subspace basis at each optimization step is thus\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Because 1050 modes exist in the frequency range of 0 to 300 Hz of the initial design, the size of the MCA modal basis is 2*1050 = 2100.
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Eq. (59)
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31 sec
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31 sec
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Eq. (62)
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258 sec
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50 sec
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Eq. (63)
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48 sec
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6 sec
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Eq. (64)
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67 sec
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10 sec
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Total Cost
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404 sec
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97 sec
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Table 4.
CPU time of the MCA method.
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The cost of solving for 1050 modes directly from NASTRAN is 180 seconds (see Table 2). In the MCA method, the cost of solving the linear system of equations in Equation (59) is 31 seconds, and the additional combined cost of Equations (62) to (64) is 373 seconds, resulting in a total cost of 404 seconds (see Table 4). To reduce this cost, the 1050 modes are divided into 21 groups and the modes in each group are obtained separately as explained in the last paragraph of Section 3.4. This reduces the cost of Equations (62) to (64) to 66 seconds for a total cost of 97 seconds, which is about half the cost of the direct NASTRAN method.
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Figure 9.
Comparison of sound pressure at driver’s ear between initial and optimal designs.
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Figure 10.
Percent increase of optimal design parameters relative to baseline design parameters.
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The gradient-based optimizer in NASTRAN (SOL 200) using the optimization process of Figure 6 needed three iterations to calculate the optimal design. Figure 9 compares the sound pressure at the driver’s ear between the optimal and initial designs. Figure 10 shows the percentage increase of optimal values relative to the initial values for all 41 design parameters. In the frequency range of 80-140Hz, the maximum sound pressure is slightly reduced from 7.9E-7 to 7.2E-7 Pascal. Most parameters are minimally changed. The largest increase is 20% for the rail mount thickness (parameter #27).
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Figure 11.
Comparison of sound pressure at driver’s ear between baseline and optimal designs with 20 initial populations and 4 generations.
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Figure 12.
Comparison of sound pressure at driver’s ear between baseline and optimal designs with 100 initial populations and 6 generations.
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The Sequential Quadratic Programming (SQP) algorithm of NASTRAN can only find a local optimum. To obtain a more significant design improvement, two additional studies were performed using a Genetic Algorithm with the MCA method. The first study used 20 initial populations and 4 generations, and the second study used 100 initial populations and 6 generations. Figures 11and 12 show that the number of initial populations and the number of generations, affect the optimization results. While a higher number of initial populations and generations results in a slightly better result, both studies produced a much better optimum than the SQP algorithm. In the case of 100 initial populations and 6 generations, the sound pressure is reduced from 7.9E-7 Pascal to 2.0E-7 Pascal, which is equivalent to about 15 dB in sound pressure level (SPL).
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To verify the accuracy of the MCA approximation, the sound pressure response at the optimal design from MCA+GA with 100 initial populations and 6 generations was evaluated by both direct NASTRAN and MCA. Figure 13 shows that the MCA method is very accurate. For a similar to MCA accuracy, the original CA method needed three sets of mode shapes to form the subspace basis, requiring 90 seconds to solve the linear equations. The much larger mode basis R in CA increases the computational cost to calculate the triple matrix products of Equation (57). Therefore for large scale, finite-element models with a high modal density, the proposed MCA method can be more efficient compared to either a complete NASTRAN analysis or the original CA method.
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Figure 13.
Comparison of sound pressure at driver’s ear between direct nastran and mca.
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4.4. Reanalysis in Craig-Bampton Substructuring with Interface Modes
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The FRF substructuring of Section 2.3 couples two structures using FRF information between the coupling (interface) DOFs, and the excitation and/or response DOFs. Although this approach is very efficient, it is practical only if we have a few coupling DOFs; e.g. connection of a vehicle suspension to chassis or connection of the exhaust system to body through a few hangers. If the physical substructures have interfaces with many DOFs, a different reduced-order modeling (ROM) approach must be used such as the Craig-Bampton ROM of Section 2.2.1. The Craig-Bampton ROM can be large however, if the number of retained interface DOFs is large. We address this problem by performing a secondary eigenvalue analysis which yields the so-called interface modes (see Section 2.2.2). The following section describes a reanalysis methodology for physical substructuring with Craig-Bampton ROMs using interface modes. We show that its accuracy is very good and the computational savings are substantial.
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4.4.1. Craig-Bampton with Interface Modes and Reanalysis
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In the Craig-Bampton CMS method (Craig-Bampton reduced-order model or CBROM), the mass and stiffness matrices of each substructure are partitioned into interface sub-matrices, interior (omitted DOF) sub-matrices, and their coupling sub-matrices. The dynamics of a structure are then described by the normal modes of its individual components, plus a set of modes called constraint modes that couple the components. In CBROM, there is no size reduction for constraint modes since all of them are kept in the reduced equations. If the finite element mesh is sufficiently fine, the constraint-mode DOFs will dominate the size of CBROM mass and stiffness matrices and result in a large computational cost. This issue is addressed by using interface modes (also called characteristic constraint –CC- modes). For that, a secondary eigenvalue analysis is performed using the constraint-mode partitions of the CMS mass and stiffness matrices. The CC modes are the resultant eigenvectors. Details are provided in Sections 2.2.1 and 2.2.2.
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The number of constraint modes \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t equals to the number of interface DOF. For many FE models of large structures, the number of interface DOF can be rather large. The calculation of constraint modes in Equation (6) involves a decomposition step and a FBS step. The cost of FBS is proportional to\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. For any matrix multiplication that involves\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, the cost is proportional to\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. For any triple-product that involves \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t the cost is proportional to\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
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The matrices from all substructures are assembled into a global CBROM with substructures coupled at interfaces by enforcing displacement compatibility. If \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t are the component (substructure) matrices, the global matrices \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tare assembled as
and a secondary eigenvalue analysis is performed to calculate the interface modes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t as
where the matrices miCC\n\t\t\t\t\t, miCCN\n\t\t\t\t\t and kiCC\n\t\t\t\t\t are of much smaller size than matrices miC\n\t\t\t\t\t, miCN\n\t\t\t\t\tand kiC\n\t\t\t\t\t.
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The interface modes reduce the interface size producing a smaller reduced order model (ROM) compared with the traditional Craig-Bampton ROM (CBROM). However, they are calculated from the assembled interface K and M matrices. Thus, the calculation of constraint modes and all matrix multiplications related to constraint modes are still necessary. The interface mode method reduces the size of ROM but it does not reduce the computational cost related to the constraint modes.
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If the interface modes \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t were known before hand, the calculations in Equations (6), (9), (10) and (12) and Equation (72) could be performed much more efficiently as follows55
In Equations (74) to (76), the computation involves \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t and does not involve\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Therefore, the calculation of original constraint modes \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t is no longer needed.
In Equation (73), the number of columns of matrix \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\t\t\t\t\t\t\tΓ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t is equal to the number of interface modes \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\twhich is usually smaller than\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Therefore, the FBS cost of solving for \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t is proportional to\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tand it is much smaller than the FBS cost of solving for\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t.
Because both \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t are of size \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tthe cost of matrix multiplication and triple-product in Equations (74) to (76) are now proportional to \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t and\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t. Therefore, the cost is much smaller than the corresponding cost in Equations (9), (10) and (12).
\n\t\t\t\t\t
In this CCROM method which is based on CBROM, the interface modes \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t are obtained using the assembled interface partitions of the CBROM formulation. Thus, it is impossible to know \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t before hand for a new design. For this reason, Equations (73) to (76) can not be theoretically implemented to improve efficiency. For this reason, we propose a reanalysis approach where the calculated interface modes \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t for original (baseline) design can be used as an approximation of the new interface modes at any modified design. In this case, Equations (73) to (76) are applied to improve the computational efficiency.
\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t
4.4.2. A Car Door Example
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The car door model of Figures 14 and 15 is used to demonstrate the proposed reanalysis method for substructuring with Craig-Bampton method using interface modes. It has 25,800 nodes and 25,300 elements and is divided into two substructures. The first substructure includes the outer door shell and a bar attached to it. The second substructure includes the rest of the door. There are 293 nodes (1758 DOFs) on the interface. Therefore, the CBROM or CCROM method must calculate 1758 constraint modes according to Equation (6) for both substructures. The 1758 constraint modes are involved in matrix multiplication or triple-products in Equations (9), (10) and (12). Figure 16 shows the interface nodes.
\n\t\t\t\t\t
For the initial design using the CCROM method, 52 interface modes are calculated below 600 Hz. A modified design is created where the shell thicknesses for the outer door (substructure 1) and inner door (substructure 2) are doubled. To provide baseline numbers, the CCROM method is used on the new design to solve for the system natural frequencies. The new reanalysis approach is used on the new design to calculate approximate natural frequencies which are then compared with the baseline numbers. The interface modes calculated at the original design are used as an initial guess for the interface modes of the new design.
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Figure 14.
Outside and inside views of car door model.
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Figure 15.
Substructure 1 (outer door shell) and substructure 2 (rest of door).
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Figure 16.
Interface nodes indicated by white dots.
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Figure 17.
Comparison of natural frequencies between original CCROM method and CCROM with reanalysis for the car door example.
\n\t\t\t\t\t
\n\t\t\t\t\t\tFigure 17 compares the natural frequencies of the new (modified) design between the original CCROM (Craig-Bampton with Interface modes) method and the new approach where reanalysis is used in CCROM to approximate the interface modes. We observe that the natural frequencies of the modified design are very different from those of the original design. Also, the accuracy of the proposed reanalysis method is excellent. The frequencies for the modified design calculated by the original CCROM and the proposed new approach are almost identical. The percentage error of the new approach versus the original CCROM approach is less than 1% on average. The computation cost is summarized in Table 5.
Summary of computational cost for the car door example.
\n\t\t\t\t\t
In the new approach to reduce the cost related to constraint modes, the total remaining cost is dominated by the cost of calculating the normal modes for each substructure. For example, the calculation of the normal modes for Substructure 2 took 110 seconds out of a total of 139 seconds (see Table 5). It should be noted that the normal modes cost can be further reduced by applying another reanalysis method such as CDH/VAO, CA or MCA to approximate the normal modes. Therefore, the overall cost of substructuring based on Craig-Bampton with interface modes, can be drastically reduced by using the proposed reanalysis to approximate the constraint modes and a CDH/VAO or MCA reanalysis to approximate the normal modes at a new design.
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5. Optimization of a Vehicle Model
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A detailed optimization study is presented using a large-scale FE model of a vehicle. For simplicity, we call it “BETA” car model. It is composed of approximately 7.1 million DOFs and 1.1 million elements. Figure 18 shows all modeling details.
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Figure 18.
Details of “BETA” car model.
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Figure 19.
Ten response locations on two front doors.
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We form an optimization problem in terms of the maximum vibratory displacement at any location of the outer shell of the two front doors by minimizing the maximum displacement among ten locations of the two front doors (Figure 19) due to a hypothetical engine excitation in the vertical (up-down) direction. The engine is represented by a lumped mass connected rigidly to the engine mounts (Figure 20). The powertrain-exhaust model has about 1.3 million DOFs and is composed of 29 PSHELL components and 12 PSOLID components. There are also some RBE2 and PBUSH elements which are used as connectors. The maximum displacement at each of the ten door locations is observed in the y direction (lateral direction – perpendicular to the door plane).
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Figure 20.
Description of the fifteen design variables.
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Figure 21.
Description of the five design variables on the doors.
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Fifteen design variables are chosen; five structural elements of each door (thickness of door shell, front frame, rear frame, top panel, middle pipe), vertical stiffness of each of the four engine mounts, and vertical stiffness of each of the six exhaust system supports. All design variables are schematically indicated in Figures 20 and 21.
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The optimization problem is stated as follows:
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\n
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The optimal value of each of the fifteen design variables is calculated in order to minimize the maximum response among the ten locations on the doors while the mass of the vehicle remains less or equal to the mass of the initial (nominal) vehicle. The response is calculated in the 100 Hz to 200 Hz frequency range and a 3% structural damping is used. The optimization problem is numerically very challenging because of
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the many local optima and
the computational cost of each dynamic analysis.
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The former was handled by using a hybrid optimization algorithm which first explores the entire design space using a Niching Genetic Algorithm (GA) [25] and then switches to a gradient-based optimizer (fmincon in MATLAB) using the best estimate of the optimal point from the GA as initial point. This ensures a rapid convergence to the final optimum because although all GA optimizers can move quickly to the vicinity of the final optimum, they have a very slow convergence rate in pinpointing the final optimum.
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FRF substructuring is used to assemble all components of the vehicle (body, doors, and engine-exhaust) into a small reduced-order model. This keeps the computational cost of each dynamic analysis low (4 minutes per analysis). A modal model is created only once for the body subsystem and then used to generate an FRF representation. This modal model does not change during the optimization because the chosen design variables are not associated with the body. However, the modal models of the doors change during the optimization. The final model for the entire vehicle is created by assembling the FRF models of each component. The FRF assembly operation is repeated during optimization because the FRF models of the two doors keep changing.
\n\t\t\t
The Niching GA optimizer maximizes a fitness function by modifying all design variables. A proper fitness function which minimizes the maximum response among the ten door locations while satisfying the vehicle mass constraint is chosen as follows
The ratio of the nominal maximum response over the actual maximum response is used so that the fitness value increases when the actual response is reduced. This ratio is multiplied by \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t where p = 10 is a penalty value and\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. \n\t\t\tThus, c is positive if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is less than \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t satisfying the constraint and the value of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is equal to one.
\n\t\t\t\t
\nOtherwise, c becomes negative if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is greater than\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and the term \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t assumes a large negative value which reduces the fitness value considerably. As a result, the GA optimizer always satisfies the mass constraint while maximizing the value of the fitness function.
\n\t\t\t
\n\t\t\t\tFigure 22 summarizes the optimization results by comparing the maximum door response between the optimal and initial designs. The optimizer determined that the maximum response occurs at location 9 (center of left front door of Figure 19) at approximately 105 Hz. Figure 23 shows that this represents a vehicle local mode involving motion of the doors only. At the optimal design the maximum response was reduced from the initial 10-3 m to 0.47*10-3 m (Table 6).
\n\t\t\t
Figure 22.
Comparison of optimal and initial designs.
\n\t\t\t
Figure 23.
Vehicle local mode at 105 Hz indicating door deformation.
\n\t\t\t\n\t\t\t
\n\t\t\t\tTable 6 compares the value of each design variable between the initial (nominal) and final optimal designs. It also indicates that all designed variables were allowed to vary within a lower and upper bound. The values of the five door design variables changed considerably between the initial and optimal designs. This is expected because the optimizer tried to suppress the local door mode. The stiffness of the four engine mounts and the six exhaust supports also changed. Although we intuitively expect the stiffness of the engine mounts to change but not the stiffness of the exhaust supports, this is not the case in this example. Table 6 also indicates that at the optimum we not only reduced the maximum response from 10-3 m to 0.47*10-3 m but the vehicle mass was also reduced from the initial 55.12 units to the final 51.92 units.
\n\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\tDesignVariables
\n\t\t\t\t\t\t
Thickness
\n\t\t\t\t\t\t
LowerBound
\n\t\t\t\t\t\t
UpperBound
\n\t\t\t\t\t\t
NominalDesign
\n\t\t\t\t\t\t
OptimalDesign
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X1\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Door Shell
\n\t\t\t\t\t\t
0.1
\n\t\t\t\t\t\t
1
\n\t\t\t\t\t\t
0.7
\n\t\t\t\t\t\t
0.6638
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X2\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Front Frame
\n\t\t\t\t\t\t
0.1
\n\t\t\t\t\t\t
1
\n\t\t\t\t\t\t
0.7
\n\t\t\t\t\t\t
0.3084
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X3\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Rear Frame
\n\t\t\t\t\t\t
0.1
\n\t\t\t\t\t\t
1
\n\t\t\t\t\t\t
0.7
\n\t\t\t\t\t\t
0.2019
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X4\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Top Panel
\n\t\t\t\t\t\t
0.1
\n\t\t\t\t\t\t
2
\n\t\t\t\t\t\t
0.7
\n\t\t\t\t\t\t
0.2019
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X5\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Middel Pipe
\n\t\t\t\t\t\t
0.5
\n\t\t\t\t\t\t
4
\n\t\t\t\t\t\t
2.4
\n\t\t\t\t\t\t
1.3722
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X6\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Engine mount
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
110.6
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X7\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Engine mount
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
161.8
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X8\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Engine mount
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
108.8
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X9\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Engine mount
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
132.1
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X10\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Exhaust support
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
213.9
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X11\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Exhaust support
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
218.1
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X12\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Exhaust support
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
345.9
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X13\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Exhaust support
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
286.1
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X14\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Exhaust support
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
118.7
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
X15\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
Exhaust support
\n\t\t\t\t\t\t
19.5
\n\t\t\t\t\t\t
370.5
\n\t\t\t\t\t\t
195
\n\t\t\t\t\t\t
233.4
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
Max Resp.
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
1*10-3\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
0.47*10-3\n\t\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
Door Mass
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
55.12
\n\t\t\t\t\t\t
51.92
\n\t\t\t\t\t
\n\t\t\t\t
Table 6.
Summary of optimal design.
\n\t\t\t
\n\t\t\t\tFigure 24 shows the actual function evaluations (design points where the vehicle dynamic response was calculated) in the X1-X2-X3 space and indicates the vicinity of the optimal design point. The GA optimizer needed only 359 function evaluations and used a population size of 5*(15+1) = 80 and a maximum of 10 generations. The population size and the number of allowed generations were kept at a minimum in order to locate the vicinity of the optimum quickly without “wasting” valuable computational effort.
\n\t\t\t
\n\t\t\t
Figure 24.
Function evaluations of the Niching GA in the X1-X2-X3 space.
\n\t\t\t
Considering that the computational cost for each function evaluation was 4 minutes, the total computational time was (359 function evaluations) * (4 minutes per evaluation) = 1436 minutes or 23.9 hours. This is acceptable considering the size and type of performed analysis. The computational cost, in terms of number of function evaluations, was kept low by coupling the Niching GA with a Lazy Learning metamodeling technique [26, 27]. The latter estimates the value of the fitness function from existing values at close by designs without calculating the actual response. It uses an error measure to figure out if the estimation is accurate. The error is small if enough previous designs, for which the fitness value was evaluated, are close to the new design. In this case, the metamodel estimates the current fitness value without running an actual dynamic response. If the error is large, an actual response is calculated and the fitness value of this new design is added to the “pool” of previous designs the Lazy Learning metamodeling technique will use downstream.
\n\t\t
\n\t\t
\n\t\t\t
6. Conclusions and Future Work
\n\t\t\t
Reduced-order models and reanalysis methodologies were presented for accurate and efficient vibration analysis of large-scale, finite element models, and for efficient design optimization of structures for best vibratory response. The optimization is able to handle a large number of design variables and identify local and global optima.
\n\t\t\t
For large FE models, it is common to solve for the system response through modal reduction in order to improve computational efficiency. An eigenanalysis is performed using the system stiffness and mass matrices and a modal model is formed which is then solved for the response. The computational cost can be also reduced using substructuring (or reduced-order modeling) methods. A modal reduction is applied to each substructure to obtain the component modes and the system level response is then obtained using component mode synthesis. In optimization of dynamic systems involving design changes (e.g. thicknesses, material properties, etc) the FEA analysis must be repeated many times in order to obtain the optimum design. Also in probabilistic analysis where parameter uncertainties are present, the FEA analysis must be repeated for a large number of sample points. In such cases, the computational cost is very high, if not prohibitive.
\n\t\t\t
To drastically reduce the computational cost without compromising accuracy beyond an acceptable level, we developed and used various reanalysis methods in conjunction with reduced-order modeling, in optimization of vibratory systems. Reanalysis methods are intended to efficiently calculate the structural response of a modified structure without solving the complete set of modified analysis equations. We presented a variety of reanalysis methods including the CDH/VAO method, the Combined Approximations (CA) and Modified Combined Approximations (MCA) method, and the Parametric Reduced-Order Modeling (PROM) method. Their advantages and limitations were fully described and demonstrated with practical examples.
\n\t\t\t
Future work will concentrate on developing reanalysis methodologies for shape and topology optimization of vibratory systems and extend the presented work in optimization under uncertainty where efficient deterministic reanalysis methods will be combined with efficient probabilistic reanalysis methods.
\n\t\t
\n\t\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/39413.pdf",chapterXML:"https://mts.intechopen.com/source/xml/39413.xml",downloadPdfUrl:"/chapter/pdf-download/39413",previewPdfUrl:"/chapter/pdf-preview/39413",totalDownloads:2828,totalViews:167,totalCrossrefCites:0,totalDimensionsCites:1,totalAltmetricsMentions:0,impactScore:0,impactScorePercentile:15,impactScoreQuartile:1,hasAltmetrics:0,dateSubmitted:"March 3rd 2012",dateReviewed:"July 8th 2012",datePrePublished:null,datePublished:"October 2nd 2012",dateFinished:"September 25th 2012",readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/39413",risUrl:"/chapter/ris/39413",book:{id:"3121",slug:"advances-in-vibration-engineering-and-structural-dynamics"},signatures:"Zissimos P. Mourelatos, Dimitris Angelis and John Skarakis",authors:[{id:"152513",title:"Prof.",name:"Zissimos P.",middleName:null,surname:"Mourelatos",fullName:"Zissimos P. Mourelatos",slug:"zissimos-p.-mourelatos",email:"mourelat@oakland.edu",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",institution:null},{id:"163427",title:"Mr.",name:"Dimitrios",middleName:null,surname:"Angelis",fullName:"Dimitrios Angelis",slug:"dimitrios-angelis",email:"dangelis@beta-cae.gr",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",institution:null},{id:"163428",title:"Mr.",name:"John",middleName:null,surname:"Skarakis",fullName:"John Skarakis",slug:"john-skarakis",email:"jskar@ansa-usa.com",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Reduced-Order Modeling for Dynamic Analysis",level:"1"},{id:"sec_2_2",title:"2.1. Modal Reduction",level:"2"},{id:"sec_3_2",title:"2.2. Substructuring with Component Mode Synthesis (CMS)",level:"2"},{id:"sec_3_3",title:"2.2.1. Craig-Bampton Fixed Interface CMS",level:"3"},{id:"sec_4_3",title:"2.2.2. Craig-Bampton CMS with Interface Modes",level:"3"},{id:"sec_5_3",title:"2.2.3. Filtration of Constraint Modes",level:"3"},{id:"sec_7_2",title:"2.3. Frequency Response Function (FRF) Substructuring and Assembling ",level:"2"},{id:"sec_7_3",title:"2.3.1. Algorithm for FRF/FE Coupling",level:"3"},{id:"sec_8_3",title:"2.4.2. Algorithm for FRF/FRF Coupling",level:"3"},{id:"sec_11",title:"3. Reanalysis Methods for Dynamic Analysis",level:"1"},{id:"sec_11_2",title:"3.1. CDH/VAO Method",level:"2"},{id:"sec_12_2",title:"3.2. Parametric Reduced-Order Modeling (PROM) Method ",level:"2"},{id:"sec_13_2",title:"3.3. Combined Approximations (CA) Method",level:"2"},{id:"sec_14_2",title:"3.4. Modified Combined Approximations (MCA) Method",level:"2"},{id:"sec_15_2",title:"3.5. Comparison of CA/MCA and PROM Methods",level:"2"},{id:"sec_17",title:"4. Reanalysis Methods in Dynamic Analysis and Optimization",level:"1"},{id:"sec_17_2",title:"4.1. Integration of MCA Method in Optimization",level:"2"},{id:"sec_18_2",title:"4.2. Integration of MCA and PROM Methods",level:"2"},{id:"sec_19_2",title:"4.3. Combined MCA and PROM Methods: Vibro-Acoustic Response of a Vehicle",level:"2"},{id:"sec_19_3",title:"Table 2.",level:"3"},{id:"sec_20_3",title:"Table 3.",level:"3"},{id:"sec_22_2",title:"4.4. Reanalysis in Craig-Bampton Substructuring with Interface Modes",level:"2"},{id:"sec_22_3",title:"4.4.1. Craig-Bampton with Interface Modes and Reanalysis",level:"3"},{id:"sec_23_3",title:"Table 5.",level:"3"},{id:"sec_26",title:"5. Optimization of a Vehicle Model",level:"1"},{id:"sec_27",title:"6. Conclusions and Future Work",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAbu Kasim\n\t\t\t\t\t\t\tA. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tTopping\n\t\t\t\t\t\t\tB. H. 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W.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1997\n\t\t\t\t\tEditorial- Lazy Learning\n\t\t\t\t\tArtificial Intelligence Review\n\t\t\t\t\t11\n\t\t\t\t\t105\n\t\t\t\t\t1\n\t\t\t\t\t6\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Zissimos P. Mourelatos",address:"mourelat@oakland.edu",affiliation:'
Mechanical Engineering Department, Oakland University, U. S. A.
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1. Introduction
Food emulsions are produced from two immiscible liquids (usually oil and water), which in the presence of an emulsifier and by applying an emulsification method, can be dispersed one into another. Some typical examples include mayonnaise, salad dressings, sauces, milk, ice cream, and sausages. These systems can be used to encapsulate, protect, and deliver biocompounds, including vitamins, flavors, colorants, and nutraceuticals [1]. Emulsifiers are food additives acting by forming a physical barrier between the oil and water, enabling their compatibilization. Effective emulsifiers must be quickly adsorbed at the oil–water interfaces leading to a rapid decrease in the interfacial tension, preventing droplets aggregation. Moreover, they must generate strong repulsive interactions promoting emulsion stability [2, 3].
Synthetic emulsifiers (e.g., Tweens and Spans) are well-known for their ability to form highly stable emulsions. However, consumers’ preferences for healthy, sustainable and natural lifestyle habits have increased worldwide. Moreover, some studies have reported intestinal dysfunctions caused by synthetic emulsifiers [4, 5]. In this context, natural emulsifiers have emerged as great alternatives to replace their conventional counterparts, namely proteins [6], polysaccharides [7], phospholipids [8] and saponins [9]. Concerning protein-based natural emulsifiers, the most use ones come from animal sources (e.g., whey proteins, caseins, egg protein, gelatin) [10]. However, plant-based proteins have demonstrated to be good alternatives for their replacement in products with dietary restrictions (e.g., lactose-free) and in vegetarian and vegan foods. Moreover, plant-based proteins are more sustainable as they have a lower carbon footprint [11, 12]. Examples include pea [13, 14] and soy proteins [15], which have been reported for emulsions production.
Aligned with natural emulsifiers, Pickering stabilizers (in particular organic-based colloidal particles) are emerging as promising solutions. Pickering emulsions or particle-stabilized emulsions present high resistance to coalescence and Oswald ripening due to the tight fixation of the particles to the droplets surface [16]. Several food-grade particles have been studied, namely particles based on proteins [17], polysaccharides [18], and protein/polysaccharide complexes [19]. Furthermore, natural emulsifiers from microbial origin such as biosurfactants and bioemulsifiers are also potential alternatives to be explored in food emulsions [20, 21].
This chapter covers a bibliographic review focused on the last 10-years on natural emulsifiers and emulsion technology field. Research and market trends are also highlighted, showing the most relevant natural emulsifier families. Basic concepts concerning emulsion production, classification, and stabilization methods are introduced. A special emphasis is given to Pickering emulsions regarding novel trends in food emulsion systems.
1.1 Bibliometric and market trend analysis
According to the Research and Markets report, amidst the Covid-19 crisis, the global emulsifiers’ market is projected to reach US$ 6.1 Billion by 2027, growing at a Compound Annual Growth Rate (CAGR) of 4.8% over the forecast period (2020–2027). Particularly, natural emulsifiers’ area is estimated to get US$ 3.3 Billion, recording a 5.4% CAGR [22]. In agreement, the “Global Food Emulsifiers Market 2020-2027 report” from MarketResearch, foresees a high potential for the plant-based emulsifiers in the global food emulsifiers market [23].
Concurrently, scientific literature corroborates the global food emulsifiers report’s projections. More than 8,000 documents were found using the terms “natural emulsifier*” OR “bioemulsifier*” OR “bio-emulsifier*” OR “biosurfactant*” OR “bio-surfactant*” OR “Pickering emulsion*” searched in title, abstract, keywords and Keywords plus sections using the Web of Science Core Collection (SCI-EXPANDED), in the 2010–2020 period. Excluding documents with early publication and applying the “Food Science and Technology” filter from WOS, 792 documents were found. By removing 4 documents from 2021 in a final manual screening, 788 documents were analyzed using Biblioshiny app from the Bibliometrix-R package (RStudio) [24] and VosViewer software [25]. The survey was performed on April 25th, 2021.
Table 1 presents some of the retrieved 788 documents concerning the application of natural emulsifiers or Pickering stabilizers in emulsion formation/stability, including their use in biocompound delivery systems. Some works regarding the production of bioemulsifiers or biosurfactants by microorganisms were also found [31, 32]. Several studies addressing Pickering emulsions and the use of high-pressure homogenization were identified.
Studies reporting the use of natural molecules and Pickering stabilizers selected from the retrieved 788 documents of the bibliometric search.
Figure 1a shows the wordcloud from Author’s Keyword. The higher font size indicates an increased frequency of the keyword. Figure 1b also illustrates keyword co-occurrence network analysis; the terms distributed in the same cluster present the higher similarity, in comparison with the terms distributed in different clusters.
Figure 1.
(a) Wordcloud from Author’s keywords (100 keywords; minimum frequency of 5); (b) keyword co-occurrence network (9 clusters; Author’s keywords; number of occurrences 5).
“Pickering emulsions” is the most frequent keyword, followed by biosurfactant (Figure 1). Other keywords (e.g., whey protein, sodium caseinate, glycolipid, sophorolipids, rhamnolipids, Quillaja saponin) appeared in the wordcloud.
These findings substantiate the keyword co-occurrence analysis (Figure 1). 93 keywords (Author’s keywords) were organized in 9 clusters. The number of occurrences indicates the number of documents where the keyword appears. Each circle represents a keyword with at least 5 occurrences, being their areas proportional to the number of occurrences. The clusters are characterized by different colors and their words can be related.
Some clusters present words associated to recent trends in the area of natural emulsifiers. Clusters 1, 6, 8 and 9 refer to “Pickering emulsions” and other inter-related words, including nanoparticles, Pickering stabilization, and some commonly used Pickering stabilizers such as starch granules, cellulose nanocrystals and kafirin. Clusters 1 and 2 comprise terms related to the rheological properties of emulsions, an important parameter in food applications. The words included in clusters 4 and 5 are associated with microorganisms (e.g., Pseudomonas aeruginosa; Starmerella bombicola; Bacillus subtilis) and the biosurfactants they produce (e.g., rhamnolipids; sufactin; sophorolipids). P. aeruginosa is a food-borne pathogen and a source of rhamnolipids [33]. Starmerella bombicola is a non-pathogenic yeast producing sophorolipids, whereas B. subtilis is a non-pathogenic bacteria yielding surfactin [31, 34]. Some of these biosurfactants have been applied in food emulsion systems (e.g., surfactin in O/W food emulsions [20]) due to their high ability to stabilize emulsions, and present antioxidant and antimicrobial properties. However, some aspects, especially safety, require attention as biosurfactants may be produced by pathogenic bacteria [35].
Cluster 7 and 9 are centered in words related to the biocompounds delivery systems, namely bioavailability/bioaccessibility, controlled release, encapsulation and examples of used biocompounds, such as beta-carotene, curcumin, and vitamin E. Clusters 8 and 9 refer to proteins, phospholipids, saponins and polysaccharides, such as whey protein isolate, soy lecithin, Quillaja saponin, and pectin, respectively. Some of these natural emulsifiers also appeared in the wordcloud analysis, being the most used ones in the food science and technological fields.
In a general overview, the analysis showed the progressive interest in natural emulsifiers due to their relevance for the scientific and industrial communities, as well as for the global market. Moreover, Pickering emulsions are emerging as advanced emulsion technologies within future trends in the food industry.
2. Principal natural-based emulsifier groups
Natural emulsifiers belong to a broad range of chemical families and some main examples are shown in Figure 2. Within each family, aspects such as the used natural source or extraction method can lead to different properties. Therefore, the next sections summarize the most relevant families in the area of natural emulsifiers and their contextualization in the field of food applications.
Figure 2.
Representative chemical structures for each emulsifier family.
2.1 Phospholipids
Phospholipids are amphiphilic molecules, and a main constituent of natural membranes. Their structure comprises a hydrophilic head holding a phosphoric acid (H3PO4), combined with a hydrophobic tail composed by one or two non-polar fatty acids. They comprise groups as glycerophospholipids or sphingolipids, with lecithins (glycerophospholipid) assuming an important role. Phospholipids can be obtained from diverse natural sources, including milk, vegetable oils (soybean, rapeseed or sunflower), egg yolk, meat and fish [36, 37]. Specifically, lecithins are known to be good stabilizers for food emulsions, for example the ones derived from soy or egg yolk are applied in mayonnaise, creams, or sauces [38]. Other phospholipid examples include phosphatidylcholine, phosphatidylglycerol, phosphatidylethanolamine, phosphatidylinositol, phosphatidylserine, phosphatidic acid, sphingomyelin. The amphiphilic character of these compounds supports their capacity to stabilize emulsions. Concurrently to their ability to stabilize emulsions they can act as texturizing agents, thus influencing the organoleptic attributes of the final product [39].
2.2 Saponins
Saponins are a complex family derived from plants, constituted by triterpenes or steroid aglycones linked to glycosyl derived sugar structures [40]. Usually the aglycones involve pentacyclic triterpenoids with oleanolic acid and the sugars moieties comprise rhamnose, xylose, glucose or galactose [41]. Factors conditioning the composition of saponins are their botanical origin and extraction method. Quillaja saponaria Molina is the principal source of saponins, named Quillaja saponins, characterized by high contents of quillaic acid groups (hydrophobic) and rhamnose, galactose or glucuronic acids (hydrophylic) [42]. Saponins produce highly stable emulsions, including at the nanoscale, and at relatively low surfactant contents, with promising stability in terms of pH, ionic strength or temperature conditions [40]. Their promising properties avail their use in diverse applications, with examples of food-grade saponins applied in beverages added with flavors or bioactives such as vitamins [43]. More recently, the utilization of saponins was extended to the use of saponin-rich extracts obtained from plant sources [44], by-products and food wastes [40]. Besides the increased costs of using highly pure saponins, they provide weaker functional properties comparatively with extracts, due to the lack of additional bioactive compounds, e.g. polyphenols [45].
2.3 Proteins
Proteins are molecules resulting from the combination of 21 different amino acids, having diverse properties, including water solubility, which varies depending on their composition [46]. Structurally, the presence of both hydrophobic and hydrophilic amino acids confer an amphiphilic character, allowing them to be absorbed at oil/water interfaces, leading to emulsion stabilization [47]. However, proteins have low surface activity in comparison with conventional emulsifiers. This is attributed to the random distribution of the hydrophilic and hydrophobic groups within the peptide chains, limiting their adsorption. This effect is balanced by the protein film formation around the droplets, leading to stabilization through molecular interactions [48]. Diverse proteins (e.g., whey, casein, soy or faba bean proteins) have been tested in food applications, e.g., emulsions for the controlled release of lutein [49], ѡ-3 oil [50], bioactive hydrophobic compounds [51], fish oil [52, 53] or β-carotene [54]. Their application in final products is still hindered by environmental conditions such as pH, temperature and ionic strength [48]. However, these drawbacks can be surpassed by using more complex formulations, namely by combining proteins with polysaccharides [48] or by chemically modifying the proteins trough grafting with other compounds such as polyphenols [54].
2.4 Polysaccharides
Polysaccharides are biopolymers composed of monosaccharide units such as glucose, fructose, mannose or galactose, bonded by glycosidic bonds. Their structural rearrangement, i.e., type and number of monosaccharides, type of glycosidic bonds, molecular weight, electrical charge, branching degree, hydrophobicity and the presence of other groups (carboxylate, sulfate or phosphate), rule the polysaccharides functional properties such as solubility, rheology, and amphiphilic character, among others [10]. Their amphiphilicity depends on the presence of hydrophobic (glycolipids) and hydrophilic (hydroxyls) groups, being adsorbed at the interface, forming a thick stabilizing layer (e.g., pectins, gum Arabic) [55]. Moreover, non-amphiphilic polysaccharides can contribute to emulsion stabilization due to their thickener role, increasing the viscosity and decreasing oil droplets’ motion (e.g., alginates, carrageenan) [56]. Despite the high number of polysaccharides available in nature, only few are authorized as food emulsifiers in EU, namely alginic acid (E400), gum Arabic (E414), pectin (E440), cellulose and chemically modified celluloses (E460 to E469) [57]. Polysaccharides can be obtained from animal, vegetal, microbial fermentation or marine sources (algae), being their properties mostly dependent on the source and extraction process [10].
2.5 Natural based emulsifiers from microbial sources
Microbial synthetic routes are emerging as valuable sustainable and green alternatives to produce emulsifiers. They generate compounds with low ecotoxicity, biodegradability, stability (pH and salinity) and low critical micellar concentration (CMC), in addition to biological activity, biocompatibility and digestibility [58]. Emulsifiers produced by microorganisms are classified according to their molecular weight. Low molecular weight family includes glycolipids (e.g., rhamnolipids, sophorolipids, trehalose lipids) and lipopeptides (e.g., surfactin, iturin, fengycin) and are referred as biosurfactants. Polysaccharides, proteins, lipoproteins, and lipopolysaccharides belong to the high molecular weight family and are referred as bioemulsifiers [59, 60]. Glycolipids like rhamnolipids and trehalose lipids are mostly produced by bacterial strains like Pseudomonas aeruginosa, Pseudomonas fluorescens, Rhodococcus erythropolis, Nocardia erythropolis, Arthrobacter sp. or Mycobacterium sp. while sophorolipids are generally produced by yeasts (e.g. Candida bombicola or Candida antartica) or by filamentous fungi like Aspergillus flavus or Rhizopus oryzae [58, 61]. Lipopeptides can be produced by Bacillus sp. Serratia marcescens and P. fluorescens [58]. Bioemulsifiers such as Emulsan/Biodispersan are commercially available products, being produced by Acinetobacter spp., while mannoproteins are commonly obtained from the yeasts Saccharomyces cerevisiae or Kluyveromyces marxianus [61]. The excellent properties of both microbial derived biosurfactants and bioemulsifiers make them appealing as natural based emulsifiers for foods. Several studies reported the use of glycolipids for fat emulsions stabilization [62, 63], and glycolipids and lipopeptides as rheology modifiers in cookies and muffins dough [64, 65, 66]. Other works refer bioemulsifiers (e.g., exopolysaccharides, mannoproteins) as having high potential in aromas emulsification [67]. Nevertheless, the practical application in foods is still limited due to two main factors, their high production costs narrowing the commercial profit, together with the legal regulations that limit the use of compounds produced by microbial strains classified as pathogenic in food applications. Examples include the bacteria genera like Pseudomonas and Bacillus. In opposition, yeasts like S. cerevisiae and Kluyveromyces lactis, which are classified as GRAS organisms, are authorized for food applications [68].
3. Emulsion technology
Emulsions are colloidal systems constituted by two immiscible liquids (oil and water), formed in the presence of an emulsifier, and, usually, by applying an energy input. The emulsifier selection is therefore an important step to reach stability. They can be classified based on the hydrophilic region that correspond to ionic structures (anionic or cationic surfactants), change charge with pH (amphoteric surfactants) or present no charged centers (nonionic surfactants) [69]. Among them, nonionic surfactants are often used in food applications because they are less toxic and less affected by pH and ionic strength changes [70, 71]. The choice of a nonionic surfactant can be based on the hydrophilic–lipophilic balance (HLB) index [72]. This scale (0–20), reflects the changing from hydrophobic to hydrophilic character, that is, a lower HLB value corresponds to a lipophilic surfactant being appropriate to stabilize water-in-oil (W/O) emulsions, whereas a high HLB indicates the ability to stabilize oil-in-water (O/W) emulsions, due to the strong hydrophilic balance [72].
3.1 Emulsion classification
Emulsions can be classified according to their typology and structure. The first refers to the relative distribution of the immiscible phases (oil and water), and the latter refers to the arrangement of the emulsified entities [73]. Considering the typology, they can be classified as simple (O/W and W/O) or double (oil-in-water-in-oil (O/W/O), and water-in-oil-in-water (W/O/W)) emulsions (Figure 3). Examples of O/W emulsions in food systems include products such as milk, sauces, beverages, yogurts, ice-creams, and mayonnaise [74]. W/O emulsions are not so frequent but can be found in butter and margarine [73, 75]. For double emulsions, W/O/W are the most used systems due to their ability to generate reduced-fat products, when compared to O/W emulsions. Moreover, they can serve as base systems to encapsulate and control the release of sensitive water-soluble compounds, such as flavors or bioactive ingredients [16, 75, 76].
Figure 3.
Typology of simple and double emulsions.
Regarding structure, emulsions can be classified as macroemulsions (usually called emulsions), nanoemulsions, or microemulsions. These systems present specific physicochemical properties that influence their range of applications [71]. Emulsions and nanoemulsions are thermodynamically unstable systems because their free energy is higher than the one of the individual phases [74, 77]. Thus, considering that all systems tend to their lowest energy state, phase separation will occur. However, due to their kinetic stability, they may remain in a metastable state for a considerable period of time, delaying the phase separation phenomenon. The kinetic stability is governed by two mechanisms, namely the energy barriers between the two states (emulsified and separated phases) and mass transfer between the phases. Therefore, high energy barriers and slow mass transfer processes delay phase separation [78]. By contrast, microemulsions are thermodynamically stable systems because their free energy is lower than the one of separate phases. Thus, they can be formed spontaneously under particular compositions and temperature conditions. In practice, some energy input is needed due to the existence of kinetic energy barriers [71]. Regarding the droplet size, nanoemulsions and microemulsions present droplet sizes <200 nm, whereas emulsions hold sizes between 200 nm and 100 μm [16, 71].
Nanoemulsions and microemulsions are optically transparent or slightly turbid due to their small droplet size, being valuable for applications requiring transparency, such as soft drinks [79]. Comparatively with nanoemulsions, microemulsions require a higher emulsifier content, have a lower particle size, and droplets can assume a non-spherical shape, feature that can be used to differentiate the two systems. Emulsions are typically turbid to opaque and are used in creamy systems such as dairy products [80]. Table 2 provides some application examples for each system addressing natural emulsifiers.
Food applications of emulsions, nanoemulsions and microemulsions using natural emulsifiers.
3.2 Stabilization mechanisms
Emulsions are thermodynamically unstable mixtures, characterized by the presence of at least two immiscible phases and an emulsifier that, when provided with enough mixing energy, are able to maintain stability over time [98]. The role of the emulsifier is essential to assure stable long-term properties. In general, emulsifiers are active surface substances, enabling their positioning at the oil–water interface, reducing the interfacial tension, hindering (or delaying) aggregation phenomena [99]. Typically, the hydrophilic part of the emulsifier is located in the aqueous phase, while the hydrophobic tail remains enclosed in the oil phase [82, 100]. During emulsion formation, the surfactant molecules require time to move to the interface, forming a layer to reach the interfacial tension equilibrium, a phenomenon related with their adsorption kinetics [82]. This pattern is dependent on emulsifiers’ nature, taking from minutes (e.g., some saponins) to hours (e.g., some proteins), besides being dependent on environmental conditions (e.g., pH, temperature) [82]. To note that, even emulsions are commonly stabilized by a monolayer structure around the droplets, multilayer structures can also be formed. The multilayer pattern favors the electrostatic and steric repulsion of the droplets, improving stability while providing additional protection to the internal phase [16].
The emulsion stabilization mechanism can differ depending on the nature of the used surfactant. In this context four principal stabilization mechanisms are known, namely electrostatic repulsion, steric repulsion, Marangoni-Gibbs effect, and thin film stabilization mechanisms [101]. The electrostatic repulsion is related to ionic emulsifiers and consists on the formation of an electrical double layer at the droplet’s interface, hindering their approximation. Steric repulsion is characteristic of nonionic and/or polymeric emulsifiers, and droplet’s distance is kept due to the adsorption of the hydrophobic segment by the oil phase [101]. The Marangoni-Gibbs effect preserve emulsions’ structure through the deformation of adjacent droplet’s surface, avoiding their outflow, whereas the thin film stabilization mechanism avail the stability of the emulsion by generating a rigid and viscoelastic film, preventing droplets from destabilization effects [101].
Other factors can condition emulsion’s stabilization mechanism, including the emulsifier content, the oil to water ratio or the preparation conditions (pH or temperature). For example, some phospholipids can have no charge at neutral pH, turning into anionic at acidic media, promoting molecule’s swelling at the interface [100]. Moreover, the surfactant concentration can have also impact, e.g., sunflower lecithin in O/W emulsions, at low contents, create a layer surrounding the oil droplets, while at higher concentrations, the stabilization mechanism changes, producing, concurrently, liposomes that might destabilize the emulsion [10]. Considering amphiphilic polymers, when they are used as emulsifiers, they become positioned at the interface, just like the small molecules, but their ability to create intermolecular interactions can provide additional stabilization effects. Their effect on viscosity can also provide a positive stabilization effect [102]. The high hydrophilicity of most polysaccharides can difficult their emulsifier role, if considering the importance of the emulsifiers’ hydrophilic/lipophilic character to interact with both phases. This constraint can be overcome by either chemical or physical strategies [103]. Namely, the suitability of anchoring hydrophobic groups into the polysaccharide structure can equilibrate the hydrophilic/lipophilic balance, that is the hydrophobisation of emulsifier’s surface. Otherwise, alternative approaches imply the mixture of the polysaccharides with other polymers (co-surfactants) to favor the hydrophilic/lipophilic equilibrium and stabilization role.
3.3 Production methods
Food emulsions can be produced using several methods, classified as low-energy and high-energy processes, as represented schematically in Figure 4. The selection of the most appropriate method and respective equipment is based on the volume to process, characteristics of the initial mixture, emulsion’s physicochemical properties, droplet size, and process costs [104]. In Table 3 a survey of recent works dealing with emulsion production trough different methods and using natural emulsifiers in their pure form or compounded with synthetic emulsifiers is presented. Moreover, their potential to encapsulate bioactives for food industry applications is also described.
Figure 4.
Schematic representation of the emulsification process through high- and low-energy methodologies.
Studies applying different productive methods using natural emulsifiers or natural/synthetic blends to form emulsions and/or to encapsulate biocompounds for food industry applications.
WPI: Whey protein isolate; WPC: Whey protein concentrate; MCT: medium-chain triglycerides; SMP: Sucrose monopalmitate.
Low-energy methods comprise, spontaneous emulsion, and emulsion phase inversion (e.g., phase inversion composition and phase inversion temperature), which occur due to environmental or composition changes namely temperature, pH, and ionic strength of the formulation [104]. Low-energy approaches are more cost effective than high-energy methods. However, they are limited to certain oils and emulsifiers, requiring also large amounts of surfactants, which is not desirable for many food applications [71]. In the work reported by Komaiko et al. [36], spontaneous emulsification lead to emulsions with large droplet size (>10 μm), comparatively with those produced by high-energy methods (<10 μm). The authors concluded that natural emulsifiers can be used in SE emulsions for applications where fine droplets are not essential (Table 3). By contrast, Mayer et al. [105] concluded that it was not possible to produce nanoemulsions using natural emulsifiers by the emulsion phase inversion method. These limitations imply that even natural-based emulsions can be prepared through low-energy methods, high-energy approaches are needed when natural emulsifiers are used.
High-energy methods generate intensive forces promoting the water and oil phases disruption and their subsequent mixture. High-shear homogenizers are the most used equipment’s for producing emulsions in the food industry. They consist on a rotor-stator or stirrer device able to mix the components at high speeds. Usually, large droplets are produced using this approach (1–10 μm) in comparison to alternative high-energy methods. High-pressure homogenization is also widely used in the food industry, being more effective to reduce the droplet size of a pre-emulsion. Generally, this coarse pre-emulsion is produced by high-shear homogenizers, then subjected to the high-pressure homogenization process. The equipment consists of a high-pressure pump (3–500 MPa) to pass the coarse emulsion through a narrow homogenizing valve, generating intensive disruptive forces (shear and cavitation), breaking down the droplets into smaller ones [80, 81].
Many studies have been conducted using two high-energy sequential methods (high-shear and -pressure homogenizers) to produce emulsions/nanoemulsions with natural emulsifiers [111, 112, 113]. Flores-Andrade and co-workers performed a study with soy lecithin, whey protein concentrate (WPC) and gum Arabic as natural emulsifiers, and paprika oleoresin as the oil phase. The coarse emulsion was produced by a high-speed homogenizer, then treated in a high-pressure homogenizer. O/W nanoemulsions were produced, being WPC more effective to form small droplets (d < 150 nm) than the other tested emulsifiers [92].
Microfluidization is the most effective method for producing emulsions with fine droplets (d < 100 nm). This approach is based on feeding the fluid into the homogenizer, which consists of a mixture chamber with two channels. Intensive disruptive forces are generated when these two fluid streams collide at high speed, breaking larger droplets and intermingling the fluids [3]. As the high-pressure homogenizers, microfluidizers were used after preparing a pre-emulsion by high-shear mixers [42, 114]. Ultrasound technique uses high-intense ultrasonic waves, generating intense shear and pressure gradients. The droplets are disrupted mainly by cavitation and turbulent effects [99, 115].
Currently, high-energy approaches are commonly used in the food industry due to their large-scale production capacity and the possibility to process a wide range of raw ingredients [71]. Although several high-energy emulsification devices are available, high-shear and pressure homogenizers, microfluidizers and ultrasound equipment’s are the most used in the production of natural emulsifiers-based emulsions.
3.4 Emulsion stability
Emulsion stability is an important parameter indicating its ability to resist physicochemical changes over time [116]. For food emulsions, the required stability varies according to the intended final application. For example, short-term stability of minutes to hours, is enough for intermediate food emulsions such as cake batter and ice cream mixtures, while long-term stability is required for long shelf-life products, including mayonnaise and salad dressings [117]. For the latter ones, the development of effective strategies to retard emulsion destabilization implies the identification of the main mechanisms leading to this effect [73].
Emulsion instability can occur due to physical and/or chemical processes. The physical instability is responsible for modifying the emulsion droplets spatial distribution and structure, including gravitational separation (creaming/sedimentation), flocculation, coalescence, and Ostwald ripening phenomena (Figure 5). These effects depend on the emulsion composition and structure, besides the storage conditions, namely temperature variation and mechanical stirring [74, 116]. Moreover, the physical phenomena are interrelated and can influence each other during emulsion storage [77].
Figure 5.
Common types of instability phenomena in emulsions.
Gravitational separation is driven by density differences between the droplets and the continuous phase. The droplets are subjected to gravitational forces tending to accumulate in the top (creaming) or in the bottom (sedimentation) of the system. Most edible oils present densities lower than water, favoring creaming in O/W emulsions, whereas sedimentation is usually observed in W/O emulsions [116]. Considering the impact of gravitational forces in the large droplets, the separation usually occurs for emulsions with droplet sizes higher than 100 nm or in a final stage of a sequence of instability phenomena [116]. By contrast, for lower droplet sizes, e.g., nanoemulsions, Brownian motion dominates over gravitational forces. Thus, reducing the droplet size is a suitable strategy to retard gravitational separation, with the emulsifier playing an important role to effectively reduce droplets’ size [2, 74]. Furthermore, the emulsifier’ layers tend to minimize the density difference between the emulsion phases, thus reducing the velocity of gravitational separation. Other strategies include modifying the rheology of the continuous phase or increasing the concentration of the droplets [74, 116].
Ostwald Ripening consists of the increase of the droplets size due to the diffusion of small droplets into larger ones, effect driven by their solubility in the continuous phase. This effect is promoted when the droplet’s size decreases [73], being also influenced by the emulsifiers’ properties. Namely, Ostwald Ripening can be retarded by decreasing the interfacial tension of the phases, favored when small-molecule surfactants are used or when using emulsifiers able to form rigid shell around the droplets. By contrast, emulsifiers prone to solubilize the oil and water phases through the formation of colloidal structures (e.g., micelles) accelerate the Ostwald Ripening [2].
Flocculation and coalescence mechanisms are related to droplets aggregation, effect leading to droplet size increase [74]. In flocculation the association of at least two droplets in an aggregate occurs, whereas in the coalescence, the droplets merge into a larger one [77]. Both phenomena are highly dependent on the selected emulsifier [77, 116], namely their nature and colloidal interactions’ capacity [2].
4. Pickering emulsions
Pickering emulsions are defined as systems stabilized by solid colloidal particles adsorbed at the oil–water interface in a practically irreversible process, creating a coating around the droplets, either in the form of a single or multiple layer, generating a strong steric barrier providing high stability [118]. In the context of Pickering emulsions, the search for natural-based particles is currently a hot topic to face market demands for novel clean label products (absent of emulsifiers) [119]. Pickering emulsions (Figure 6) are raising high interest in the recent years. They are characterized by a long-term stability and have green connotations due to the absence of conventional emulsifiers. These attributes comply with the recent trends of food industry towards the use of sustainable and healthy technologies [16]. The stability of Pickering emulsions is related with the intrinsic properties of the oil and water phases (e.g., type, oil/water ratio, pH, ionic strength) and of the particle stabilizers (e.g., wettability, particle morphology, size and concentration). Particles presenting a contact angle (θ) below 90° are generally suitable for preparing O/W emulsions, whereas θ values greater than 90° indicate good stabilizers for W/O emulsions. At 90°, the particle is immersed equally in both phases [120].
Figure 6.
Schematic representation of a Pickering emulsion putting in evidence the particle stabilizers where θ represent the contact angle.
Regarding natural-based particles, three main typologies of stabilizers can be used, namely nanoparticles, microgels and fibrils. Examples include protein derived stabilizers, namely nanoparticles based on corn zein, and colloidal particles of kafirin and gliadin [118, 121, 122, 123]. Although many polysaccharides have high hydrophilic character, some can include hydrophobic side groups (e.g., beet pectin and modified starch) or even active proteins attached to the surface (e.g., gum Arabic) [120], offering potential to act as Pickering stabilizers. Other polysaccharides widely used to produce Pickering bionanoparticles include chitin, chitosan and cellulose. To overcome particle’s limitations as Pickering stabilizers, the formation of complexes has been also proposed, namely complexes such as polysaccharide-polysaccharide, protein–protein, and polysaccharide-protein [124]. Examples include zein-xanthan [125], and tea water insoluble proteins/κ-carrageenan complexes [126].
In the context of the recent trends in Pickering emulsions, research aiming at finding new biological particles, the use of high internal phase emulsions (HIPPE), and the development of bio-based films from Pickering emulsions are becoming topics of high interest for the development of novel food applications. Table 4 presents an overview of recent works dealing with the preparation of Pickering emulsions based on novel biological particles together with the description of the main results envisaging potential food applications.
Particle materials
Main results
Reference
Apple pomace
Smaller particles led to emulsions with smaller droplet size, showing higher stability over time (30 days), in addition to improved physical properties (gel-like samples) and antioxidant activity.
The emulsion with higher particles content (2%) showed suitable physical stability for 30 days, elastic-solid characteristics and good thermal stability (20–80°C).
The emulsions were reached a cream index of 17%, presenting stability for 7 weeks. They presented improved oxidative stability and resistance to in vitro digestion conditions.
Cream index values were very low, and the emulsion presented good stability to a broad range of ionic strength and mild temperature conditions (4–60°C).
The emulsions maintained excellent physical stability for 1 month. In addition, they demonstrated good performance as delivery systems of essential oils.
Examples of bionanoparticles as Pickering stabilizers. All the systems are of O/W type.
HIPPEs are characterized by having a high volume fraction of internal phase (generally higher than 74%), together with relatively low particles concentration resulting in an extremely compacted droplet’s structure [140]. HIPPEs are becoming a novel approach of increasing interest in the food industry, since it combines diverse advantages, namely a semi-solid texture with the ability to encapsulate high amounts of bioactive compounds [141]. HIPPEs allow to control the droplet size distribution, manipulate the morphology and rheological properties, generally presenting enhanced stability against physical, chemical and microbiological stresses [142]. They are positioned as extremely promising substitutes for foods such as margarine, mayonnaise or ice creams [143, 144]. For example, Liu et al. studied wheat gluten as stabilizer in a HIPPE to develop a novel mayonnaise substitute [145]. They obtained excellent results concerning texture and sensory attributes when compared with commercial products.
Bio-based films made from hydrophilic particles added with hydrophobic compounds is another emerging approach in the scope of new applications developed from Pickering emulsions [146]. These strategies provide the ability to improve the stability of the base materials (hydrophilic), in addition to facilitate the combination with hydrophobic materials (e.g., waxes, fatty oils and oils) leading to systems with enhanced moisture barrier properties [147].
5. New trends in food emulsion systems
The wide variety of emulsion-based systems using natural emulsifiers makes their applicability attractive for various products, particularly in the food industry. The nature and function of emulsifiers, and the formed emulsion type (e.g., nano/micro-scale, simple or double character) can tailor appearance, sensorial characteristics, and attractiveness of foods. Among their diverse functions, the increasing use of emulsions as functionality carriers should be highlighted. In fact, recent works have demonstrated their potential and versatility for the encapsulation of flavors, and to protect and deliver specific bioactives in foods or beverages, helping to strengthen nutritional balances, and enabling the production of reduced-fat products. A summary of examples addressing new trends of emulsion-based products with potential in the food industry are included in Table 5, with some highlighs provided next.
Applications of natural-based emulsifiers in food industry.
Lopes Francisco et al. [149] reported an emulsifying system with encapsulation potential based on commercial pea and soy proteins. The work involved the encapsulation of an orange essential oil rich in d-limonene using a O/W emulsion followed by spray drying to obtain powder microparticles. It was demonstrated the ability of pea and soy proteins to act as emulsifiers in the encapsulation of orange essential oil, getting a slightly higher efficiency if using soy protein as the natural emulsifier. These promising results can help consolidate a platform aiming at developing new protective systems to encapsulate flavors for foods, complying with the increasing demand from this industrial sector for natural-based systems.
At the nanoscale, Flores-Andrade et al. [92] reported the preparation of O/W nanoemulsions by high-pressure homogenization, using amphiphilic biopolymers to stabilize paprika oleoresin, namely whey protein, gum Arabic, phospholipids, and soy lecithin. The results demonstrated the effective oil encapsulation, preserving carotenoids (e.g., lipophilic colorants) from chemical degradation, besides positioning this strategy as an attractive route to design new protective and delivery carriers for bioactive compounds aimed at food and/or beverage products.
The potential of double emulsions was also demonstrated by Cetinkaya et al. [152] that evidenced the reduction of the saturated fat content in O1/W/O2 emulsions prepared by fat crystallization according to a two-stage process. Firstly, the primary O1/W emulsion was prepared using sunflower oil and xanthan gum and gelatin as emulsifiers, which was then stabilized in a second oil phase (palm oil), resulting in a structured O1/W/O2 system. Microstructure examination revealed that the accumulation of fat crystals at the interface contributed to stabilize the internal water phase containing the encapsulated sunflower phase. These complex structures showed potential to directly encapsulate hydrophobic oils and act as texturizing reduced-fat agents, which might be of particular interest for the edible oils industry.
6. Conclusions
This chapter presents an up-to-date overview of current trends in natural emulsifiers and their application in emulsion technology directed to food applications. For this purpose, first, the evolution of food emulsifiers’ scenario over the last 10 years was analyzed through the Bibliometrix-R package (RStudio) and VosViewer software. This analysis indicated a clear driving force towards using natural emulsifiers and the re-emerging importance of the Pickering emulsions. These facts are expected to impact the market growth following the prospectus of available market analysis. The six main identified families of natural emulsifiers were phospholipids, saponins, proteins, polysaccharides, emulsifiers from microbial sources and Pickering stabilizers. Some of them already find extensive use in practical food applications. However, others, mainly natural-based emulsifiers from microbial sources and Pickering stabilizers, despite their high potential, are still needing research investment and regulation clarification (e.g., related to the use of nanoparticles and the use of microbial strains classified as pathogenic in foods). From a technological perspective, the main concepts related to the typology, production methods, stabilization mechanisms, and instability phenomena were presented. Highlighting the increasing interest in Pickering emulsions, a summary of the most recent applications of these systems, including the so-called HIPPEs and their advantages in reduced-fat products development, was provided. To conclude, an analysis of current trends in food emulsion-based products was discussed, putting in evidence the emulsions increasing role as delivery systems of bioactives to support innovative fortified foods advances and the increasing interest in systems based on double emulsions, which provide the opportunity to combine bioactives of different nature. Overall, the field of natural-based emulsifiers combined with the new trends in emulsion technology can, hopefully, be the basis of a new generation of healthy and nutritious food products.
Acknowledgments
CIMO (UIDB/00690/2020) and AL LSRE-LCM (UIDB/50020/2020) funded by FCT/MCTES (PIDDAC). National funding by FCT, P.I., through the institutional scientific employment program-contract for Arantzazu Santamaria-Echart and Isabel P. Fernandes. FCT for the Research grants (SFRH/BD/148281/2019 of Samara C. da Silva, and SFRH/BD/147326/2019 of Stephany C. de Rezende). GreenHealth project (Norte-01-0145-FEDER-000042).
Conflict of interest
The authors declare no conflict of interest.
\n',keywords:"natural emulsifiers, biosurfactants, emulsion technology, Pickering emulsions, food applications, market and product trends",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/78387.pdf",chapterXML:"https://mts.intechopen.com/source/xml/78387.xml",downloadPdfUrl:"/chapter/pdf-download/78387",previewPdfUrl:"/chapter/pdf-preview/78387",totalDownloads:308,totalViews:0,totalCrossrefCites:0,dateSubmitted:"February 12th 2021",dateReviewed:"August 11th 2021",datePrePublished:"September 1st 2021",datePublished:null,dateFinished:"September 1st 2021",readingETA:"0",abstract:"The food industry depends on using different additives, which increases the search for effective natural or natural-derived solutions, to the detriment of the synthetic counterparts, a priority in a biobased and circular economy scenario. In this context, different natural emulsifiers are being studied to create a new generation of emulsion-based products. Among them, phospholipids, saponins, proteins, polysaccharides, biosurfactants (e.g., compounds derived from microbial fermentation), and organic-based solid particles (Pickering stabilizers) are being used or start to gather interest from the food industry. This chapter includes the basic theoretical fundamentals of emulsions technology, stabilization mechanisms, and stability. The preparation of oil-in-water (O/W) and water-in-oil (W/O) emulsions, the potential of double emulsions, and the re-emerging Pickering emulsions are discussed. Moreover, the most relevant natural-derived emulsifier families (e.g., origin, stabilization mechanism, and applications) focusing food applications are presented. The document is grounded in a bibliographic review mainly centered on the last 10-years, and bibliometric data was rationalized and used to better establish the hot topics in the proposed thematic.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/78387",risUrl:"/chapter/ris/78387",signatures:"Arantzazu Santamaria-Echart, Isabel P. Fernandes, Samara C. Silva, Stephany C. Rezende, Giovana Colucci, Madalena M. Dias and Maria Filomena Barreiro",book:{id:"10362",type:"book",title:"Food Additives",subtitle:null,fullTitle:"Food Additives",slug:null,publishedDate:null,bookSignature:"Dr. Miguel Ángel Ángel Prieto Lage and Dr. Paz Otero",coverURL:"https://cdn.intechopen.com/books/images_new/10362.jpg",licenceType:"CC BY 3.0",editedByType:null,isbn:"978-1-83968-960-4",printIsbn:"978-1-83968-959-8",pdfIsbn:"978-1-83968-961-1",isAvailableForWebshopOrdering:!0,editors:[{id:"317263",title:"Dr.",name:"Miguel Ángel",middleName:"Ángel",surname:"Prieto Lage",slug:"miguel-angel-prieto-lage",fullName:"Miguel Ángel Prieto Lage"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_1_2",title:"1.1 Bibliometric and market trend analysis",level:"2"},{id:"sec_3",title:"2. Principal natural-based emulsifier groups",level:"1"},{id:"sec_3_2",title:"2.1 Phospholipids",level:"2"},{id:"sec_4_2",title:"2.2 Saponins",level:"2"},{id:"sec_5_2",title:"2.3 Proteins",level:"2"},{id:"sec_6_2",title:"2.4 Polysaccharides",level:"2"},{id:"sec_7_2",title:"2.5 Natural based emulsifiers from microbial sources",level:"2"},{id:"sec_9",title:"3. Emulsion technology",level:"1"},{id:"sec_9_2",title:"3.1 Emulsion classification",level:"2"},{id:"sec_10_2",title:"3.2 Stabilization mechanisms",level:"2"},{id:"sec_11_2",title:"3.3 Production methods",level:"2"},{id:"sec_12_2",title:"3.4 Emulsion stability",level:"2"},{id:"sec_14",title:"4. Pickering emulsions",level:"1"},{id:"sec_15",title:"5. New trends in food emulsion systems",level:"1"},{id:"sec_16",title:"6. Conclusions",level:"1"},{id:"sec_17",title:"Acknowledgments",level:"1"},{id:"sec_20",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'McClements DJ, Bai L, Chung C. 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DOI: 10.1016/j.foodhyd.2020.106117'},{id:"B145",body:'Liu X, Guo J, Wan ZL, Liu YY, Ruan QJ, Yang XQ. Wheat gluten-stabilized high internal phase emulsions as mayonnaise replacers. Food Hydrocoll. 2018;77:168-175. DOI: 10.1016/j.foodhyd.2017.09.032'},{id:"B146",body:'Niro CM, Medeiros JA, Freitas JAM, Azeredo HMC. Advantages and challenges of Pickering emulsions applied to bio-based films: a mini-review. J Sci Food Agric. 2021;101:3535-3540. DOI: 10.1002/jsfa.11029'},{id:"B147",body:'Li Q, Ma Q, Wu Y, Li Y, Li B, Luo X, et al. Oleogel Films through the Pickering Effect of Bacterial Cellulose Nanofibrils Featuring Interfacial Network Stabilization. J Agric Food Chem. 2020;68:9150-9157. DOI: 10.1021/acs.jafc.0c03214'},{id:"B148",body:'Huang L, Liu J, Addy M, Ding B, Cheng Y, Peng P, et al. Physicochemical and emulsifying properties of orange fibers stabilized oil-in-water emulsions. LWT - Food Sci Technol. 2020;133:110054. DOI:10.1016/j.lwt.2020.110054'},{id:"B149",body:'Francisco CRL, de Oliveira Júnior FD, Marin G, Alvim ID, Hubinger MD. Plant proteins at low concentrations as natural emulsifiers for an effective orange essential oil microencapsulation by spray drying. Colloids Surfaces A Physicochem Eng Asp. 2020;607:125470. DOI:10.1016/j.colsurfa.2020.125470'},{id:"B150",body:'Jarzębski M, Smułek W, Siejak P, Rezler R, Pawlicz J, Trzeciak T, et al. Aesculus hippocastanum L. as a stabilizer in hemp seed oil nanoemulsions for potential biomedical and food applications. Int J Mol Sci. 2021;22:1-18. DOI: 10.3390/ijms22020887'},{id:"B151",body:'Gaur VK, Regar RK, Dhiman N, Gautam K, Srivastava JK, Patnaik S, et al. Biosynthesis and characterization of sophorolipid biosurfactant by Candida spp.: Application as food emulsifier and antibacterial agent. Bioresour TechnoL. 2019;285:121314. DOI:10.1016/j.biortech.2019.121314'},{id:"B152",body:'Cetinkaya T, Altay F, Ceylan Z. A new application with characterized oil-in-water-in-oil double emulsions: Gelatin-xanthan gum complexes for the edible oil industry. Lwt. 2021;138:110773. DOI:10.1016/j.lwt.2020.110773'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Arantzazu Santamaria-Echart",address:null,affiliation:'
Centro de Investigação de Montanha (CIMO), Instituto Politécnico de Bragança, Campus de Santa Apolónia, Portugal
'},{corresp:null,contributorFullName:"Isabel P. Fernandes",address:null,affiliation:'
Centro de Investigação de Montanha (CIMO), Instituto Politécnico de Bragança, Campus de Santa Apolónia, Portugal
'},{corresp:null,contributorFullName:"Samara C. Silva",address:null,affiliation:'
Centro de Investigação de Montanha (CIMO), Instituto Politécnico de Bragança, Campus de Santa Apolónia, Portugal
Laboratory of Separation and Reaction Engineering – Laboratory of Catalysis and Materials (LSRE-LCM), Faculdade de Engenharia, Universidade do Porto, Portugal
'},{corresp:null,contributorFullName:"Stephany C. Rezende",address:null,affiliation:'
Centro de Investigação de Montanha (CIMO), Instituto Politécnico de Bragança, Campus de Santa Apolónia, Portugal
Laboratory of Separation and Reaction Engineering – Laboratory of Catalysis and Materials (LSRE-LCM), Faculdade de Engenharia, Universidade do Porto, Portugal
Centro de Investigação de Montanha (CIMO), Instituto Politécnico de Bragança, Campus de Santa Apolónia, Portugal
'},{corresp:null,contributorFullName:"Madalena M. Dias",address:null,affiliation:'
Laboratory of Separation and Reaction Engineering – Laboratory of Catalysis and Materials (LSRE-LCM), Faculdade de Engenharia, Universidade do Porto, Portugal
Centro de Investigação de Montanha (CIMO), Instituto Politécnico de Bragança, Campus de Santa Apolónia, Portugal
'}],corrections:null},book:{id:"10362",type:"book",title:"Food Additives",subtitle:null,fullTitle:"Food Additives",slug:null,publishedDate:null,bookSignature:"Dr. Miguel Ángel Ángel Prieto Lage and Dr. Paz Otero",coverURL:"https://cdn.intechopen.com/books/images_new/10362.jpg",licenceType:"CC BY 3.0",editedByType:null,isbn:"978-1-83968-960-4",printIsbn:"978-1-83968-959-8",pdfIsbn:"978-1-83968-961-1",isAvailableForWebshopOrdering:!0,editors:[{id:"317263",title:"Dr.",name:"Miguel Ángel",middleName:"Ángel",surname:"Prieto Lage",slug:"miguel-angel-prieto-lage",fullName:"Miguel Ángel Prieto Lage"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},profile:{item:{id:"240832",title:"Dr.",name:"Siska",middleName:null,surname:"Fitrianie",email:"s.fitrianie@gmail.com",fullName:"Siska Fitrianie",slug:"siska-fitrianie",position:null,biography:null,institutionString:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",totalCites:0,totalChapterViews:"0",outsideEditionCount:0,totalAuthoredChapters:"1",totalEditedBooks:"0",personalWebsiteURL:null,twitterURL:null,linkedinURL:null,institution:null},booksEdited:[],chaptersAuthored:[{id:"59739",title:"Public Awareness and Education for Flooding Disasters",slug:"public-awareness-and-education-for-flooding-disasters",abstract:"In recent years, dramatic river flooding occurred in the city of Prague causing the raising of the river water level more than 8 m and the inundation of the lower parts of the city. The disaster resulted in numerous casualties and damages of buildings and infrastructures. Prior disaster analysis showed that the city was not well prepared for facing the disasters. A digital training could be a powerful tool for increasing knowledge and awareness of the citizens and providing training facilities about response to such disasters. A special Massive Open Online Course (MOOC), therefore, has been developed focused on flooding in Prague offering educational material in crisis flooding. Dedicated technologies have been developed such as a flooding alert and a nonverbal communication language based on icons. The MOOC enables participants to train the new technologies. Participants can also play roles in the crisis management team to increase the sensitivity of the complexity of a flooding crisis and its measurements. The first prototype of the MOOC was tested on a group of students of the Technical Universities of Prague and Delft. The test results will be reported.",signatures:"Leon J. M. Rothkrantz and Siska Fitrianie",authors:[{id:"230761",title:"Prof.",name:"Leon",surname:"Rothkrantz",fullName:"Leon Rothkrantz",slug:"leon-rothkrantz",email:"ljm.rothkrantz@gmail.com"},{id:"240832",title:"Dr.",name:"Siska",surname:"Fitrianie",fullName:"Siska Fitrianie",slug:"siska-fitrianie",email:"s.fitrianie@gmail.com"}],book:{id:"6620",title:"Crisis Management",slug:"crisis-management-theory-and-practice",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"145558",title:"Associate Prof.",name:"Aida",surname:"Alvinius",slug:"aida-alvinius",fullName:"Aida Alvinius",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/145558/images/2643_n.jpg",biography:"Aida Alvinius (Ph.D sociolog., Karlstad University, Sweden) is associate professor and university lecturer at the Department of Security, Strategy and Leadership, Swedish Defence University. She has published articles, chapters in books, and research reports within the field of organisation, collaboration, gender studies and leadership, sociology of disasters, and military sociology.",institutionString:null,institution:{name:"National Defence College Kenya",institutionURL:null,country:{name:"Kenya"}}},{id:"147004",title:"Prof.",name:"Gerry",surname:"Larsson",slug:"gerry-larsson",fullName:"Gerry Larsson",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"229241",title:"Dr.",name:"Eszter",surname:"Solt",slug:"eszter-solt",fullName:"Eszter Solt",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Budapest University of Technology and Economics",institutionURL:null,country:{name:"Hungary"}}},{id:"230543",title:"Prof.",name:"Patrizia",surname:"Riva",slug:"patrizia-riva",fullName:"Patrizia Riva",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:"Piemonte Orientale University, Italy",institution:null},{id:"230761",title:"Prof.",name:"Leon",surname:"Rothkrantz",slug:"leon-rothkrantz",fullName:"Leon Rothkrantz",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"230981",title:"Associate Prof.",name:"Xiaolei",surname:"Li",slug:"xiaolei-li",fullName:"Xiaolei Li",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"235866",title:"Dr.",name:"Qing",surname:"Song",slug:"qing-song",fullName:"Qing Song",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/235866/images/6471_n.jpg",biography:"Qing Song received the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2012. In 2013-2014, she was a Postdoctoral Researcher with the Department of Computer Science, Czech Technical University in Prague, Prague, Czech Republic. She is currently a lecturer with the School of Electrical Engineering, University of Jinan. Her research interests include optimization of complex systems, big-data analytics, and machine learning.",institutionString:null,institution:{name:"University of Jinan",institutionURL:null,country:{name:"China"}}},{id:"236187",title:"M.Sc.",name:"Jean-Pierre",surname:"Himpler",slug:"jean-pierre-himpler",fullName:"Jean-Pierre Himpler",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"240637",title:"Prof.",name:"Martin",surname:"Hromada",slug:"martin-hromada",fullName:"Martin Hromada",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"303533",title:"Dr.",name:"Kateřina",surname:"Víchová",slug:"katerina-vichova",fullName:"Kateřina Víchová",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null}]},generic:{page:{slug:"scientific-advisors",title:"Scientific Advisory Boards",intro:"
IntechOpen’s team of Scientific Advisors supports the publishing team by providing editorial and academic input and ensuring the highest quality output of free peer-reviewed articles. The Boards consist of independent external collaborators who assist us on a voluntary basis. Their input includes advising on new topics within their field, proposing potential expert collaborators and reviewing book publishing proposals if required. Board members are experts who cover major STEM and HSS fields. All are trusted IntechOpen collaborators and Academic Editors, ensuring that the needs of the scientific community are met.
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Physical Sciences, Technology and Engineering Board
\\n\\n
Chemistry
\\n\\n
\\n\\t
Ayben Kilislioglu - Department of Chemical Engineering Istanbul University, İstanbul, Turkey
\\n\\t
Goran Nikolic - Faculty of Technology, University of Nis, Leskovac, Serbia
\\n\\t
Mark T. Stauffer - Associate Professor of Chemistry, The University of Pittsburgh, USA
\\n\\t
Margarita Stoytcheva - Autonomous University of Baja California Engineering Institute Mexicali, Baja California, Mexico
Joao Luis Garcia Rosa - Associate Professor Bio-inspired Computing Laboratory (BioCom) Department of Computer Science University of Sao Paulo (USP) at Sao Carlos, Brazil
\\n\\t
Jan Valdman - Institute of Mathematics and Biomathematics, University of South Bohemia, České Budějovice, Czech Republic Institute of Information Theory and Automation of the ASCR, Prague, Czech Republic
\\n
\\n\\n
Earth and Planetary Science
\\n\\n
\\n\\t
Jill S. M. Coleman - Department of Geography, Ball State University, Muncie, IN, USA
\\n\\t
İbrahim Küçük Erciyes - Üniversitesi Department of Astronomy and Space Sciences Melikgazi, Kayseri, Turkey
\\n\\t
Pasquale Imperatore - Electromagnetic Environmental Sensing (IREA), Italian National Council of Research (CNR), Naples, Italy
\\n\\t
Mohammad Mokhtari - Director of National Center for Earthquake Prediction International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran
\\n
\\n\\n
Engineering
\\n\\n
\\n\\t
Narottam Das - University of Southern Queensland, Australia
\\n\\t
Jose Ignacio Huertas - Energy and Climate Change Research Group; Instituto Tecnológico y Estudios Superiores de Monterrey, Mexico
Likun Pan - Engineering Research Center for Nanophotonics and Advanced Instrument, Ministry of Education, Department of Physics, East China Normal University, China
\\n\\t
Mukul Chandra Paul - Central Glass & Ceramic Research Institute Jadavpur, Kolkata, India
\\n\\t
Stephen E. Saddow - Electrical Engineering Department, University of South Florida, USA
\\n\\t
Ali Demir Sezer - Marmara University, Faculty of Pharmacy, Department of Pharmaceutical Biotechnology, İstanbul, Turkey
\\n\\t
Krzysztof Zboinski - Warsaw University of Technology, Faculty of Transport, Warsaw, Poland
\\n
\\n\\n
Materials Science
\\n\\n
\\n\\t
Vadim Glebovsky - Senior Researcher, Institute of Solid State Physics, Chernogolovka, Russia Expert of the Russian Fund for Basic Research, Moscow, Russia
\\n\\t
Jianjun Liu - State Key Laboratory of High Performance Ceramics and Superfine Microstructure of Shanghai Institute of Ceramics, Chinese Academy of Sciences, China
\\n\\t
Pietro Mandracci - Department of Applied Science and Technology, Politecnico di Torino, Italy
\\n\\t
Waldemar Alfredo Monteiro - Instituto de Pesquisas Energéticas e Nucleares Materials Science and Technology Center (MSTC) São Paulo, SP, Brazil
Toshio Ogawa - Department of Electrical and Electronic Engineering, Shizuoka Institute of Science and Technology, Toyosawa, Fukuroi, Shizuoka, Japan
\\n
\\n\\n
Mathematics
\\n\\n
\\n\\t
Paul Bracken - Department of Mathematics University of Texas, Edinburg, TX, USA
\\n
\\n\\n
Nanotechnology and Nanomaterials
\\n\\n
\\n\\t
Muhammad Akhyar - Farrukh Nano-Chemistry Lab. Registrar, GC University Lahore, Pakistan
\\n\\t
Khan Maaz - Chinese Academy of Sciences, China & The Pakistan Institute of Nuclear Science and Technology, Pakistan
\\n
\\n\\n
Physics
\\n\\n
\\n\\t
Izabela Naydenova - Lecturer, School of Physics Principal Investigator, IEO Centre College of Sciences and Health Dublin Institute of Technology Dublin, Ireland
\\n\\t
Mitsuru Nenoi - National Institute of Radiological Sciences, Japan
\\n\\t
Christos Volos - Physics Department, Aristotle University of Thessaloniki, Greece
\\n
\\n\\n
Robotics
\\n\\n
\\n\\t
Alejandra Barrera - Instituto Tecnológico Autónomo de México, México
\\n\\t
Dusan M. Stipanovic - Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign
\\n\\t
Andrzej Zak - Polish Naval Academy Faculty of Navigation and Naval Weapons Institute of Naval Weapons and Computer Science, Gdynia, Poland
Petr Konvalina - Faculty of Agriculture, University of South Bohemia in České Budějovice, Czech Republic
\\n
\\n\\n
Biochemistry, Genetics and Molecular Biology
\\n\\n
\\n\\t
Chunfa Huang - Saint Louis University, Saint Louis, USA
\\n\\t
Michael Kormann - University Children's Clinic Department of Pediatrics I, Pediatric Infectiology & Immunology, Translational Genomics and Gene Therapy in Pediatrics, University of Tübingen, Tübingen, Germany
\\n\\t
Bin WU - Ph.D. HCLD Scientific Laboratory Director, Assisted Reproductive Technology Arizona Center for Reproductive Endocrinology and Infertility Tucson, Arizona , USA
\\n
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Environmental Sciences
\\n\\n
\\n\\t
Juan A. Blanco - Senior Researcher & Marie Curie Research Fellow Dep. Ciencias del Medio Natural, Universidad Publica de Navarra Campus de Arrosadia, Pamplona, Navarra, Spain
\\n\\t
Mikkola Heimo - University of Eastern Finland, Kuopio, Finland
\\n\\t
Bernardo Llamas Moya - Politechnical University of Madrid, Spain
\\n\\t
Toonika Rinken - Department of Environmental Chemistry, University of Tartu, Estonia
\\n
\\n\\n
Immunology and Microbiology
\\n\\n
\\n\\t
Dharumadurai Dhanasekaran - Department of Microbiology, School of Life Sciences, Bharathidasan University, India
Isabel Gigli - Facultad de Agronomia-UNLPam, Argentina
\\n\\t
Milad Manafi - Department of Animal Science, Faculty of Agricultural Sciences, Malayer University, Malayer, Iran
\\n\\t
Rita Payan-Carreira - Universidade de Trás-os-Montes e Alto Douro, Departamento de Zootecnia, Portugal
\\n
\\n\\n
Medicine
\\n\\n
\\n\\t
Mazen Almasri - King Abdulaziz University, Faculty of Dentistry Jeddah, Saudi Arabia Dentistry
\\n\\t
Craig Atwood - University of Wisconsin-Madison, USA Stem Cell Research, Tissue Engineering and Regenerative Medicine
\\n\\t
Oreste Capelli - Clinical Governance, Local Health Authority, Modena, Italy Public Health
\\n\\t
Michael Firstenberg - Assistant Professor of Surgery and Integrative Medicine NorthEast Ohio Medical University, USA & Akron City Hospital - Summa Health System, USA Surgery
\\n\\t
Parul Ichhpujani - MD Government Medical College & Hospital, Department of Ophthalmology, India
Amidou Samie - University of Venda, SA Infectious Diseases
\\n\\t
Shailendra K. Saxena - CSIR-Centre for Cellular and Molecular Biology, Hyderabad, India Infectious Diseases
\\n\\t
Dan T. Simionescu - Department of Bioengineering, Clemson University, Clemson SC, USA Stem Cell Research, Tissue Engineering and Regenerative Medicine
\\n\\t
Ke Xu - Tianjin Lung Cancer Institute Tianjin Medical University General Hospital Tianjin, China Oncology
\\n
\\n\\n
Ophthalmology
\\n\\n
\\n\\t
Hojjat Ahmadzadehfar - University Hospital Bonn Department of Nuclear Medicine Bonn, Germany Medical Diagnostics, Engineering Technology and Telemedicine
\\n\\t
Miroslav Blumenberg - Department of Ronald O. Perelman Department of Dermatology; Department of Biochemistry and Molecular Pharmacology, Dermatology, NYU School of Medicine, NY, USA Dermatology
\\n\\t
Wilfred Bonney - University of Dundee, Scotland, UK Medical Diagnostics, Engineering Technology and Telemedicine
\\n\\t
Christakis Constantinides - Department of Cardiovascular Medicine University of Oxford, Oxford, UK Medical Diagnostics, Engineering Technology and Telemedicine
\\n\\t
Atef Mohamed Mostafa Darwish - Department of Obstetrics and Gynecology , Faculty of Medicine, Assiut University, Egypt Gynecology
\\n\\t
Ana Polona Mivšek - University of Ljubljana, Ljubljana, Slovenia Midwifery
\\n\\t
Gyula Mozsik - First Department of Medicine, Medical and Health Centre, University of Pécs, Hungary
\\n\\t
Shimon Rumelt - Western Galilee-Nahariya Medical Center, Nahariya, Israel Ophthalmology
\\n\\t
Marcelo Saad - S. Paulo Medical College of Acupuncture, SP, Brazil Complementary and Alternative Medicine
\\n\\t
Minoru Tomizawa - National Hospital Organization Shimoshizu Hospital, Japan Gastroenterology
\\n\\t
Pierre Vereecken - Centre Hospitalier Valida and Cliniques Universitaires Saint-Luc, Belgium Dermatology
\\n
\\n\\n
Gastroenterology
\\n\\n
\\n\\t
Hany Aly - Director, Division of Newborn Services The George Washington University Hospital Washington, USA Pediatrics
\\n\\t
Yannis Dionyssiotis - National and Kapodistrian University of Athens, Greece Orthopedics, Rehabilitation and Physical Medicine
\\n\\t
Alina Gonzales- Quevedo Instituto de Neurología y Neurocirugía Havana, Cuba Mental and Behavioural Disorders and Diseases of the Nervous System
\\n\\t
Margarita Guenova - National Specialized Hospital for Active Treatment of Haematological Diseases, Bulgaria
\\n\\t
Eliska Potlukova - Clinic of Medicine, University Hospital Basel, Switzerland Edocrinology
\\n\\t
Raymond L. Rosales -The Royal and Pontifical University of Santo Tomas, Manila, Philippines & Metropolitan Medical Center, Manila, Philippines & St. Luke's Medical Center International Institute in Neuroscience, Quezon City, Philippines Mental and Behavioural Disorders and Diseases of the Nervous System
\\n\\t
Alessandro Rozim - Zorzi University of Campinas, Departamento de Ortopedia e Traumatologia, Campinas, SP, Brazil Orthopedics, Rehabilitation and Physical Medicine
\\n\\t
Dieter Schoepf - University of Bonn, Germany Mental and Behavioural Disorders and Diseases of the Nervous System
\\n
\\n\\n
Hematology
\\n\\n
\\n\\t
Hesham Abd El-Dayem - National Liver Institute, Menoufeyia University, Egypt Hepatology
\\n\\t
Fayez Bahmad - Health Science Faculty of the University of Brasilia Instructor of Otology at Brasilia University Hospital Brasilia, Brazil Otorhinolaryngology
\\n\\t
Peter A. Clark - Saint Joseph's University Philadelphia, Pennsylvania, USA Bioethics
\\n\\t
Celso Pereira - Coimbra University, Coimbra, Portugal Immunology, Allergology and Rheumatology
\\n\\t
Luis Rodrigo - Asturias Central University Hospital (HUCA) School of Medicine, University of Oviedo, Oviedo, Spain Hepatology & Gastroenterology
\\n\\t
Dennis Wat - Liverpool Heart and Chest Hospital NHS Foundation Trust, UK Pulmonology
\\n
\\n\\n
Social Sciences and Humanities Board
\\n\\n
Business, Management and Economics
\\n\\n
\\n\\t
Vito Bobek - University of Applied Sciences, FH Joanneum, Graz, Austria
Joao Luis Garcia Rosa - Associate Professor Bio-inspired Computing Laboratory (BioCom) Department of Computer Science University of Sao Paulo (USP) at Sao Carlos, Brazil
\n\t
Jan Valdman - Institute of Mathematics and Biomathematics, University of South Bohemia, České Budějovice, Czech Republic Institute of Information Theory and Automation of the ASCR, Prague, Czech Republic
\n
\n\n
Earth and Planetary Science
\n\n
\n\t
Jill S. M. Coleman - Department of Geography, Ball State University, Muncie, IN, USA
\n\t
İbrahim Küçük Erciyes - Üniversitesi Department of Astronomy and Space Sciences Melikgazi, Kayseri, Turkey
\n\t
Pasquale Imperatore - Electromagnetic Environmental Sensing (IREA), Italian National Council of Research (CNR), Naples, Italy
\n\t
Mohammad Mokhtari - Director of National Center for Earthquake Prediction International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran
\n
\n\n
Engineering
\n\n
\n\t
Narottam Das - University of Southern Queensland, Australia
\n\t
Jose Ignacio Huertas - Energy and Climate Change Research Group; Instituto Tecnológico y Estudios Superiores de Monterrey, Mexico
Likun Pan - Engineering Research Center for Nanophotonics and Advanced Instrument, Ministry of Education, Department of Physics, East China Normal University, China
\n\t
Mukul Chandra Paul - Central Glass & Ceramic Research Institute Jadavpur, Kolkata, India
\n\t
Stephen E. Saddow - Electrical Engineering Department, University of South Florida, USA
\n\t
Ali Demir Sezer - Marmara University, Faculty of Pharmacy, Department of Pharmaceutical Biotechnology, İstanbul, Turkey
\n\t
Krzysztof Zboinski - Warsaw University of Technology, Faculty of Transport, Warsaw, Poland
\n
\n\n
Materials Science
\n\n
\n\t
Vadim Glebovsky - Senior Researcher, Institute of Solid State Physics, Chernogolovka, Russia Expert of the Russian Fund for Basic Research, Moscow, Russia
\n\t
Jianjun Liu - State Key Laboratory of High Performance Ceramics and Superfine Microstructure of Shanghai Institute of Ceramics, Chinese Academy of Sciences, China
\n\t
Pietro Mandracci - Department of Applied Science and Technology, Politecnico di Torino, Italy
\n\t
Waldemar Alfredo Monteiro - Instituto de Pesquisas Energéticas e Nucleares Materials Science and Technology Center (MSTC) São Paulo, SP, Brazil
Toshio Ogawa - Department of Electrical and Electronic Engineering, Shizuoka Institute of Science and Technology, Toyosawa, Fukuroi, Shizuoka, Japan
\n
\n\n
Mathematics
\n\n
\n\t
Paul Bracken - Department of Mathematics University of Texas, Edinburg, TX, USA
\n
\n\n
Nanotechnology and Nanomaterials
\n\n
\n\t
Muhammad Akhyar - Farrukh Nano-Chemistry Lab. Registrar, GC University Lahore, Pakistan
\n\t
Khan Maaz - Chinese Academy of Sciences, China & The Pakistan Institute of Nuclear Science and Technology, Pakistan
\n
\n\n
Physics
\n\n
\n\t
Izabela Naydenova - Lecturer, School of Physics Principal Investigator, IEO Centre College of Sciences and Health Dublin Institute of Technology Dublin, Ireland
\n\t
Mitsuru Nenoi - National Institute of Radiological Sciences, Japan
\n\t
Christos Volos - Physics Department, Aristotle University of Thessaloniki, Greece
\n
\n\n
Robotics
\n\n
\n\t
Alejandra Barrera - Instituto Tecnológico Autónomo de México, México
\n\t
Dusan M. Stipanovic - Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign
\n\t
Andrzej Zak - Polish Naval Academy Faculty of Navigation and Naval Weapons Institute of Naval Weapons and Computer Science, Gdynia, Poland
Petr Konvalina - Faculty of Agriculture, University of South Bohemia in České Budějovice, Czech Republic
\n
\n\n
Biochemistry, Genetics and Molecular Biology
\n\n
\n\t
Chunfa Huang - Saint Louis University, Saint Louis, USA
\n\t
Michael Kormann - University Children's Clinic Department of Pediatrics I, Pediatric Infectiology & Immunology, Translational Genomics and Gene Therapy in Pediatrics, University of Tübingen, Tübingen, Germany
\n\t
Bin WU - Ph.D. HCLD Scientific Laboratory Director, Assisted Reproductive Technology Arizona Center for Reproductive Endocrinology and Infertility Tucson, Arizona , USA
\n
\n\n
Environmental Sciences
\n\n
\n\t
Juan A. Blanco - Senior Researcher & Marie Curie Research Fellow Dep. Ciencias del Medio Natural, Universidad Publica de Navarra Campus de Arrosadia, Pamplona, Navarra, Spain
\n\t
Mikkola Heimo - University of Eastern Finland, Kuopio, Finland
\n\t
Bernardo Llamas Moya - Politechnical University of Madrid, Spain
\n\t
Toonika Rinken - Department of Environmental Chemistry, University of Tartu, Estonia
\n
\n\n
Immunology and Microbiology
\n\n
\n\t
Dharumadurai Dhanasekaran - Department of Microbiology, School of Life Sciences, Bharathidasan University, India
Isabel Gigli - Facultad de Agronomia-UNLPam, Argentina
\n\t
Milad Manafi - Department of Animal Science, Faculty of Agricultural Sciences, Malayer University, Malayer, Iran
\n\t
Rita Payan-Carreira - Universidade de Trás-os-Montes e Alto Douro, Departamento de Zootecnia, Portugal
\n
\n\n
Medicine
\n\n
\n\t
Mazen Almasri - King Abdulaziz University, Faculty of Dentistry Jeddah, Saudi Arabia Dentistry
\n\t
Craig Atwood - University of Wisconsin-Madison, USA Stem Cell Research, Tissue Engineering and Regenerative Medicine
\n\t
Oreste Capelli - Clinical Governance, Local Health Authority, Modena, Italy Public Health
\n\t
Michael Firstenberg - Assistant Professor of Surgery and Integrative Medicine NorthEast Ohio Medical University, USA & Akron City Hospital - Summa Health System, USA Surgery
\n\t
Parul Ichhpujani - MD Government Medical College & Hospital, Department of Ophthalmology, India
Amidou Samie - University of Venda, SA Infectious Diseases
\n\t
Shailendra K. Saxena - CSIR-Centre for Cellular and Molecular Biology, Hyderabad, India Infectious Diseases
\n\t
Dan T. Simionescu - Department of Bioengineering, Clemson University, Clemson SC, USA Stem Cell Research, Tissue Engineering and Regenerative Medicine
\n\t
Ke Xu - Tianjin Lung Cancer Institute Tianjin Medical University General Hospital Tianjin, China Oncology
\n
\n\n
Ophthalmology
\n\n
\n\t
Hojjat Ahmadzadehfar - University Hospital Bonn Department of Nuclear Medicine Bonn, Germany Medical Diagnostics, Engineering Technology and Telemedicine
\n\t
Miroslav Blumenberg - Department of Ronald O. Perelman Department of Dermatology; Department of Biochemistry and Molecular Pharmacology, Dermatology, NYU School of Medicine, NY, USA Dermatology
\n\t
Wilfred Bonney - University of Dundee, Scotland, UK Medical Diagnostics, Engineering Technology and Telemedicine
\n\t
Christakis Constantinides - Department of Cardiovascular Medicine University of Oxford, Oxford, UK Medical Diagnostics, Engineering Technology and Telemedicine
\n\t
Atef Mohamed Mostafa Darwish - Department of Obstetrics and Gynecology , Faculty of Medicine, Assiut University, Egypt Gynecology
\n\t
Ana Polona Mivšek - University of Ljubljana, Ljubljana, Slovenia Midwifery
\n\t
Gyula Mozsik - First Department of Medicine, Medical and Health Centre, University of Pécs, Hungary
\n\t
Shimon Rumelt - Western Galilee-Nahariya Medical Center, Nahariya, Israel Ophthalmology
\n\t
Marcelo Saad - S. Paulo Medical College of Acupuncture, SP, Brazil Complementary and Alternative Medicine
\n\t
Minoru Tomizawa - National Hospital Organization Shimoshizu Hospital, Japan Gastroenterology
\n\t
Pierre Vereecken - Centre Hospitalier Valida and Cliniques Universitaires Saint-Luc, Belgium Dermatology
\n
\n\n
Gastroenterology
\n\n
\n\t
Hany Aly - Director, Division of Newborn Services The George Washington University Hospital Washington, USA Pediatrics
\n\t
Yannis Dionyssiotis - National and Kapodistrian University of Athens, Greece Orthopedics, Rehabilitation and Physical Medicine
\n\t
Alina Gonzales- Quevedo Instituto de Neurología y Neurocirugía Havana, Cuba Mental and Behavioural Disorders and Diseases of the Nervous System
\n\t
Margarita Guenova - National Specialized Hospital for Active Treatment of Haematological Diseases, Bulgaria
\n\t
Eliska Potlukova - Clinic of Medicine, University Hospital Basel, Switzerland Edocrinology
\n\t
Raymond L. Rosales -The Royal and Pontifical University of Santo Tomas, Manila, Philippines & Metropolitan Medical Center, Manila, Philippines & St. Luke's Medical Center International Institute in Neuroscience, Quezon City, Philippines Mental and Behavioural Disorders and Diseases of the Nervous System
\n\t
Alessandro Rozim - Zorzi University of Campinas, Departamento de Ortopedia e Traumatologia, Campinas, SP, Brazil Orthopedics, Rehabilitation and Physical Medicine
\n\t
Dieter Schoepf - University of Bonn, Germany Mental and Behavioural Disorders and Diseases of the Nervous System
\n
\n\n
Hematology
\n\n
\n\t
Hesham Abd El-Dayem - National Liver Institute, Menoufeyia University, Egypt Hepatology
\n\t
Fayez Bahmad - Health Science Faculty of the University of Brasilia Instructor of Otology at Brasilia University Hospital Brasilia, Brazil Otorhinolaryngology
\n\t
Peter A. Clark - Saint Joseph's University Philadelphia, Pennsylvania, USA Bioethics
\n\t
Celso Pereira - Coimbra University, Coimbra, Portugal Immunology, Allergology and Rheumatology
\n\t
Luis Rodrigo - Asturias Central University Hospital (HUCA) School of Medicine, University of Oviedo, Oviedo, Spain Hepatology & Gastroenterology
\n\t
Dennis Wat - Liverpool Heart and Chest Hospital NHS Foundation Trust, UK Pulmonology
\n
\n\n
Social Sciences and Humanities Board
\n\n
Business, Management and Economics
\n\n
\n\t
Vito Bobek - University of Applied Sciences, FH Joanneum, Graz, Austria
Denis Erasga - De La Salle University, Phillippines
\n\t
Rosario Laratta - Associate Professor of Social Policy and Development Graduate School of Governance Studies, Meiji University, Japan
\n
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The most typical simulation and ground thermal response test models for the vertical ground heat exchangers (GHEs) currently available are summarized. Also, a new GWHP using a heat exchanger with special construction, tested in laboratory, is well presented. The second objective of the chapter is to compare the main performance parameters (energy efficiency and CO2 emissions) of radiator and radiant floor heating systems connected to a GCHP. These performances were obtained with site measurements in an office room. Furthermore, the thermal comfort for these systems is compared using the ASHRAE Thermal Comfort program. Additionally, two numerical simulation models of useful thermal energy and the system coefficient of performance (COPsys) in heating mode are developed using the TRNSYS (Transient Systems Simulation) software. Finally, the simulations obtained in TRNSYS program are analysed and compared to experimental measurements.",book:{id:"5084",slug:"advances-in-geothermal-energy",title:"Advances in Geothermal Energy",fullTitle:"Advances in Geothermal Energy"},signatures:"Ioan Sarbu and Calin Sebarchievici",authors:[{id:"173440",title:"Prof.",name:"Ioan",middleName:null,surname:"Sarbu",slug:"ioan-sarbu",fullName:"Ioan Sarbu"},{id:"176508",title:"Dr.",name:"Calin",middleName:null,surname:"Sebarchievici",slug:"calin-sebarchievici",fullName:"Calin Sebarchievici"}]},{id:"49547",doi:"10.5772/61651",title:"Airborne Magnetic Surveys to Investigate High Temperature Geothermal Reservoirs",slug:"airborne-magnetic-surveys-to-investigate-high-temperature-geothermal-reservoirs",totalDownloads:2657,totalCrossrefCites:4,totalDimensionsCites:6,abstract:"Airborne magnetic survey is an effective geophysical exploration method in terms of coverage, resolution and cost, particularly for area with restricted or difficult ground access. Research studies in New Zealand have shown airborne magnetic surveys can indicate the regions of high reservoir permeability and thermal up-flow of active geothermal systems. However, the method has not been extensively used in the geothermal investigations, probably because the interpretation of airborne magnetic data has so far been seen as difficult and requires a complex quantitative 3D modelling of subsurface magnetisation.",book:{id:"5084",slug:"advances-in-geothermal-energy",title:"Advances in Geothermal Energy",fullTitle:"Advances in Geothermal Energy"},signatures:"Supri Soengkono",authors:[{id:"176580",title:"Dr.",name:"Supri",middleName:null,surname:"Soengkono",slug:"supri-soengkono",fullName:"Supri Soengkono"}]},{id:"64812",doi:"10.5772/intechopen.81157",title:"Geothermal Explorations on the Slate Formation of Taiwan",slug:"geothermal-explorations-on-the-slate-formation-of-taiwan",totalDownloads:1318,totalCrossrefCites:3,totalDimensionsCites:4,abstract:"Currently, over 90% operated geothermal power plants are distributed in the volcanic- or magmatic intrusion-related geological systems. Only a few cases are done in metamorphic terranes, especially on the slate formation. Taiwan is located at the ring of fire and is famous for the young orogenic belt, which has wide distributions of rapid uplifting terranes with few active volcanoes. The metamorphic rocks, for example, schist and slate formations with high geothermal gradients, are occurring in the major mountain range. This chapter introduces the techniques or methods we used for geothermal exploration in the slate formation of the Chingshui geothermal field of Taiwan, where a 3-MW pilot geothermal power plant had been installed in 1983 and operated for 12 years.",book:{id:"7504",slug:"renewable-geothermal-energy-explorations",title:"Renewable Geothermal Energy Explorations",fullTitle:"Renewable Geothermal Energy Explorations"},signatures:"Sheng-Rong Song and Yi-Chia Lu",authors:[{id:"253615",title:"Prof.",name:"Sheng-Rong",middleName:null,surname:"Song",slug:"sheng-rong-song",fullName:"Sheng-Rong Song"},{id:"253623",title:"Dr.",name:"Yi-Chia",middleName:null,surname:"Lu",slug:"yi-chia-lu",fullName:"Yi-Chia Lu"}]},{id:"63548",doi:"10.5772/intechopen.81062",title:"Geothermal Potential of the Global Oil Industry",slug:"geothermal-potential-of-the-global-oil-industry",totalDownloads:1212,totalCrossrefCites:2,totalDimensionsCites:3,abstract:"There are around 40 new geothermal power projects commissioned in each of the last few years. Growth of the market is around 5% annually and current installed capacity is about 13,300 MW with about the same in development in 24 countries. These figures are impressive, but they do not bear comparison with any of the fossil fuels. However, few will realise that the global oil industry has a cryptic geothermal power potential that is equal to the entire current output of the geothermal industry. The oil industry is ageing. Many areas still produce copious quantities of oil, but the oil comes with an unwanted by-product, water. The volume of water produced is typically is 10–20 times that of the oil; and the water is hot—in some places very hot (>100°C). In a recent study we showed that the power depleted oil production platforms of the North Sea’s North Viking Graben produce sufficient hot water to deliver around 60% of the power requirement for each field. A review of global oil and hence water production has enabled us to calculate that power production alone from waste water from producing oilfields could be at least 15,000 MW.",book:{id:"7504",slug:"renewable-geothermal-energy-explorations",title:"Renewable Geothermal Energy Explorations",fullTitle:"Renewable Geothermal Energy Explorations"},signatures:"Jon Gluyas, Alison Auld, Charlotte Adams, Catherine Hirst, Simon Hogg\nand Jonathan Craig",authors:[{id:"258666",title:"Dr.",name:"Jon",middleName:null,surname:"Gluyas",slug:"jon-gluyas",fullName:"Jon Gluyas"},{id:"262369",title:"Dr.",name:"Alison",middleName:null,surname:"Auld",slug:"alison-auld",fullName:"Alison Auld"},{id:"262370",title:"Dr.",name:"Charlotte",middleName:null,surname:"Adams",slug:"charlotte-adams",fullName:"Charlotte Adams"},{id:"262371",title:"Dr.",name:"Catherine",middleName:null,surname:"Hirst",slug:"catherine-hirst",fullName:"Catherine Hirst"},{id:"262372",title:"Prof.",name:"Simon",middleName:null,surname:"Hogg",slug:"simon-hogg",fullName:"Simon Hogg"},{id:"262373",title:"Prof.",name:"Jonthan",middleName:null,surname:"Craig",slug:"jonthan-craig",fullName:"Jonthan Craig"}]},{id:"64027",doi:"10.5772/intechopen.81017",title:"Stages of a Integrated Geothermal Project",slug:"stages-of-a-integrated-geothermal-project",totalDownloads:4230,totalCrossrefCites:2,totalDimensionsCites:3,abstract:"A geothermal project constitutes two big stages: the exploration and the exploitation. Each one has a single task whose results allow defining the feasibility of a geothermal project, until achieving the construction and operation stage of the power generation plant. The first stage contains the area recognition, its limitation to the target, and elimination of external factors until defining a geothermal zone with characteristics to be commercially exploited. The main studies and analysis that can be applied during the exploration stage are listed, and the major indicator to continue with the project or suspend is the prefeasibility report. The major risks in the exploration stage are due to studies that are carried out on the surface; at this stage, the costs can be considered low. The main results of the exploration are the selection of sites to drill three or four initial wells. Each well provides a direct overview of the reservoir: depth, production thicknesses, thermodynamic parameters, and production characteristics. The drilling of three to four exploratory wells is recommended, as far as there is certainty of the feasibility of the project, and the development of the field begins with drilling of sufficient wells to feed the plant. In this stage, the cost increases, but the risks decrease.",book:{id:"7504",slug:"renewable-geothermal-energy-explorations",title:"Renewable Geothermal Energy Explorations",fullTitle:"Renewable Geothermal Energy Explorations"},signatures:"Alfonso Aragón-Aguilar, Georgina Izquierdo-Montalvo,\nDaniel Octavio Aragón-Gaspar and Denise N. Barreto-Rivera",authors:[{id:"258358",title:"Dr.",name:"Alfonso",middleName:null,surname:"Aragón-Aguilar",slug:"alfonso-aragon-aguilar",fullName:"Alfonso Aragón-Aguilar"}]}],mostDownloadedChaptersLast30Days:[{id:"64027",title:"Stages of a Integrated Geothermal Project",slug:"stages-of-a-integrated-geothermal-project",totalDownloads:4236,totalCrossrefCites:2,totalDimensionsCites:3,abstract:"A geothermal project constitutes two big stages: the exploration and the exploitation. Each one has a single task whose results allow defining the feasibility of a geothermal project, until achieving the construction and operation stage of the power generation plant. The first stage contains the area recognition, its limitation to the target, and elimination of external factors until defining a geothermal zone with characteristics to be commercially exploited. The main studies and analysis that can be applied during the exploration stage are listed, and the major indicator to continue with the project or suspend is the prefeasibility report. The major risks in the exploration stage are due to studies that are carried out on the surface; at this stage, the costs can be considered low. The main results of the exploration are the selection of sites to drill three or four initial wells. Each well provides a direct overview of the reservoir: depth, production thicknesses, thermodynamic parameters, and production characteristics. The drilling of three to four exploratory wells is recommended, as far as there is certainty of the feasibility of the project, and the development of the field begins with drilling of sufficient wells to feed the plant. In this stage, the cost increases, but the risks decrease.",book:{id:"7504",slug:"renewable-geothermal-energy-explorations",title:"Renewable Geothermal Energy Explorations",fullTitle:"Renewable Geothermal Energy Explorations"},signatures:"Alfonso Aragón-Aguilar, Georgina Izquierdo-Montalvo,\nDaniel Octavio Aragón-Gaspar and Denise N. Barreto-Rivera",authors:[{id:"258358",title:"Dr.",name:"Alfonso",middleName:null,surname:"Aragón-Aguilar",slug:"alfonso-aragon-aguilar",fullName:"Alfonso Aragón-Aguilar"}]},{id:"64812",title:"Geothermal Explorations on the Slate Formation of Taiwan",slug:"geothermal-explorations-on-the-slate-formation-of-taiwan",totalDownloads:1320,totalCrossrefCites:3,totalDimensionsCites:4,abstract:"Currently, over 90% operated geothermal power plants are distributed in the volcanic- or magmatic intrusion-related geological systems. Only a few cases are done in metamorphic terranes, especially on the slate formation. Taiwan is located at the ring of fire and is famous for the young orogenic belt, which has wide distributions of rapid uplifting terranes with few active volcanoes. The metamorphic rocks, for example, schist and slate formations with high geothermal gradients, are occurring in the major mountain range. This chapter introduces the techniques or methods we used for geothermal exploration in the slate formation of the Chingshui geothermal field of Taiwan, where a 3-MW pilot geothermal power plant had been installed in 1983 and operated for 12 years.",book:{id:"7504",slug:"renewable-geothermal-energy-explorations",title:"Renewable Geothermal Energy Explorations",fullTitle:"Renewable Geothermal Energy Explorations"},signatures:"Sheng-Rong Song and Yi-Chia Lu",authors:[{id:"253615",title:"Prof.",name:"Sheng-Rong",middleName:null,surname:"Song",slug:"sheng-rong-song",fullName:"Sheng-Rong Song"},{id:"253623",title:"Dr.",name:"Yi-Chia",middleName:null,surname:"Lu",slug:"yi-chia-lu",fullName:"Yi-Chia Lu"}]},{id:"49252",title:"Using Ground-Source Heat Pump Systems for Heating/Cooling of Buildings",slug:"using-ground-source-heat-pump-systems-for-heating-cooling-of-buildings",totalDownloads:3857,totalCrossrefCites:4,totalDimensionsCites:15,abstract:"This chapter mainly presents a detailed theoretical study and experimental investigations of ground-source heat pump (GSHP) technology, concentrating on the ground-coupled heat pump (GCHP) systems. A general introduction on the GSHPs and its development, and a description of the surface water (SWHP), ground-water (GWHP), and ground-coupled heat pumps are briefly performed. The most typical simulation and ground thermal response test models for the vertical ground heat exchangers (GHEs) currently available are summarized. Also, a new GWHP using a heat exchanger with special construction, tested in laboratory, is well presented. The second objective of the chapter is to compare the main performance parameters (energy efficiency and CO2 emissions) of radiator and radiant floor heating systems connected to a GCHP. These performances were obtained with site measurements in an office room. Furthermore, the thermal comfort for these systems is compared using the ASHRAE Thermal Comfort program. Additionally, two numerical simulation models of useful thermal energy and the system coefficient of performance (COPsys) in heating mode are developed using the TRNSYS (Transient Systems Simulation) software. Finally, the simulations obtained in TRNSYS program are analysed and compared to experimental measurements.",book:{id:"5084",slug:"advances-in-geothermal-energy",title:"Advances in Geothermal Energy",fullTitle:"Advances in Geothermal Energy"},signatures:"Ioan Sarbu and Calin Sebarchievici",authors:[{id:"173440",title:"Prof.",name:"Ioan",middleName:null,surname:"Sarbu",slug:"ioan-sarbu",fullName:"Ioan Sarbu"},{id:"176508",title:"Dr.",name:"Calin",middleName:null,surname:"Sebarchievici",slug:"calin-sebarchievici",fullName:"Calin Sebarchievici"}]},{id:"49620",title:"Radiogenic Heat Generation in Western Australia — Implications for Geothermal Energy",slug:"radiogenic-heat-generation-in-western-australia-implications-for-geothermal-energy",totalDownloads:2044,totalCrossrefCites:1,totalDimensionsCites:2,abstract:"The chapter reviews heat generation in crystalline rocks and influences on overlying sedimentary basins in Western Australia (WA). Regions of elevated thorium and uranium will cause elevated heat generation, which in turn can cause elevated heat flow. Western Australia hosts several large sedimentary basins with the potential for hot sedimentary aquifers (HSAs). These include the Perth, Carnarvon, and Canning basins. Parts of these basins are underlain by crystalline rocks that contain high levels of heat-generating elements, such as uranium, thorium, and potassium. Also, the Pilbara Craton, which contains both sedimentary and crystalline rocks, that entertains a number of active mines, which may benefit from geothermal energy, is investigated. Further, the southern part of the Perth Basin (Vasse Shelf), which is underlain by crystalline rocks with elevated concentrations of thorium and uranium, is shown to possess higher than usual temperatures. From observations, and geothermal modeling, it is concluded that the Perth Basin has a high potential for medium- to low-temperature geothermal energy developments. In other parts of Western Australia, the Carnarvon Basin has elevated temperatures in artesian groundwater. Heat flow in the Canning Basin is briefly reviewed; this basin has some geothermal potential, but it is far from the major population centers.",book:{id:"5084",slug:"advances-in-geothermal-energy",title:"Advances in Geothermal Energy",fullTitle:"Advances in Geothermal Energy"},signatures:"Mike F. 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\r\n\tThe Business and Management series topic focuses on the most pressing issues confronting organizations today and in the future. Businesses are trying to figure out how to lead in a time of global uncertainty. In emerging markets, issues such as ill-defined or unstable policies, as well as corrupt practices, can be hugely problematic. Changes in governments can result in new policy, regulations, and interest rates, all of which can be detrimental to foreign businesses and investments. A growing trend towards economic nationalism also makes the current global political landscape potentially hostile towards international businesses.
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\r\n\tThe demographic shifts are creating interesting challenges. People are living longer, resulting to an aging demographic. We have a large population of older workers and retirees who are living longer lives, combined with a declining birthrate in most parts of the world. Businesses of all types are looking at how technology is affecting their operations. Several questions arise, such as: How is technology changing what we do? How is it transforming us internally, how is it influencing our clients and our business strategy? It is about leveraging technology to improve efficiency, connect with customers more effectively, and drive innovation. The majority of innovative companies are technology-driven businesses. Realizing digital transformation is today’s top issue and will remain so for the next five years. Improving organizational agility, expanding portfolios of products and services, creating, and maintaining a culture of innovation, and developing next -generation leaders were also identified as top challenges in terms of both current and future issues.
\r\n
\r\n\tThe most sustained profitable growth occurs when a company expands its core business into an adjacent space. This has significant implications for management because innovation in business ecosystems differs from traditional, vertically integrated firms. Every organization in the ecosystem must be aware of the bigger picture. Innovation in ecosystems necessitates collaborative action to invent and appraise, efficient, cross-organizational knowledge flows, modular architectures, and good stewardship of legacy systems. It is built on multiple, interconnected platforms. Environmental factors have already had a significant impact in the West and will continue to have an impact globally. Businesses must take into account the environmental impact of their daily operations. The advantage of this market is that it is expected to grow more rapidly than the overall economy. Another significant challenge is preparing the next generation of leaders to elevate this to the number one priority within the next five years. There can be no culture of innovation unless there is diverse leadership or development of the next generation of leaders; and these diverse, next-generation leaders are the ones who will truly understand the digital strategies that will drive digital transformation.
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