InTechOpen uses cookies to offer you the best online experience. By continuing to use our site, you agree to our Privacy Policy.

Computer and Information Science » Numerical Analysis and Scientific Computing » "Uncertainty Quantification and Model Calibration", book edited by Jan Peter Hessling, ISBN 978-953-51-3280-6, Print ISBN 978-953-51-3279-0, Published: July 5, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 3

State‐of‐the‐Art Nonprobabilistic Finite Element Analyses

By Wang Lei, Qiu Zhiping and Zheng Yuning
DOI: 10.5772/intechopen.68154

Article top

State-of-the-Art Nonprobabilistic Finite Element Analyses

Wang Lei, Qiu Zhiping and Zheng Yuning
Show details


The finite element analysis of a mechanical system is conventionally performed in the context of deterministic inputs. However, uncertainties associated with material properties, geometric dimensions, subjective experiences, boundary conditions, and external loads are ubiquitous in engineering applications. The most popular techniques to handle these uncertain parameters are the probabilistic methods, in which uncertainties are modeled as random variables or stochastic processes based on a large amount of statistical information on each uncertain parameter. Nevertheless, subjective results could be obtained if insufficient information unavailable and nonprobabilistic methods can be alternatively employed, which has led to elegant procedures for the nonprobabilistic finite element analysis. In this chapter, each nonprobabilistic finite element analysis method can be decomposed as two individual parts, i.e., the core algorithm and preprocessing procedure. In this context, four types of algorithms and two typical preprocessing procedures as well as their effectiveness were described in detail, based on which novel hybrid algorithms can be conceived for the specific problems and the future work in this research field can be fostered.

Keywords: interval finite element method, fuzzy finite element method, arithmetic approach, perturbation approach, sampling approach, optimization approach, subinterval technique, surrogate model

1. Introduction

The traditional finite element analysis (FEA) was performed in the context of deterministic parameters. However, uncertainties associated with material properties, geometric dimensions, and external loads are always unavoidable in engineering. The ability to include uncertainties is of great value for a design engineer. In the last decade, criticism has arisen regarding the general application of the probabilistic concept. Especially when the statistical information on uncertainties is limited [1], the subjective probabilistic analysis result may be obtained by the probabilistic method [2, 3], which proves to be of little value and does not justify the high computational cost [35]. Consequently, nonprobabilistic concepts have been introduced.

In this context, interval and fuzzy approaches are gaining more and more momentum for the uncertainty analysis and optimization of numerical models in their descriptions. In the interval approach, uncertainties are considered to be contained within a predefined range and only the lower and upper bounds are necessary for each uncertain parameter. The fuzzy approach further extends this methodology by the αlevel technique, where α stands for the extent that a specific value is member of the range of possible input values. From this viewpoint, a fuzzy analysis requires the consecutive solution for a number of interval analysis based on the αlevel technique [6]. For this reason, current researches on nonprobabilistic uncertainty propagation mainly focus on the solution and implementation of the interval analysis. In the past decades, the interval and fuzzy concepts in FEA have been studied extensively and some typical solution schemes for the interval FEA (IFEA) and fuzzy FEA (FFEA) were developed. This chapter is to give an overview of state‐of‐the‐art numerical implementations of IFEA and FFEA in applied mechanics.

FFEA aims to obtain a fuzzy description of an FEA result, starting from fuzzy descriptions of all uncertainties. The αlevel technique subdivides the membership function range into a number of discrete αlevels. The αcuts of the input quantities are defined as xiα={xiXi, μx˜i(xi)α} where μx˜(x) is the membership function. This means that an αcut is the interval resulting from intersecting the membership function at μx˜i(xi)=α. The αlevel interval describes the grade of membership to the fuzzy set for each element in the domain and enables the representation of a value that is only to a certain degree member of the set. However, the confidence interval defined in statistics is the range of likely values for a population parameter, such as the population mean. The selection of a confidence level for an interval determines the probability that confidence interval produced will contain the true parameter value. The intersection with the membership function of the input uncertainties on each αlevel results in an interval and IFEA is formulated, resulting in an interval for the output on the considered αlevel. The fuzzy solution is finally assembled from the resulting intervals on each sublevel. The IFEA is based on the interval description of uncertainties and its goal is to capture the range of specific output quantities of interest that corresponds to a given interval description of input uncertainties. For the sake of simplicity, the static analysis of a mechanical system is adopted in this chapter to explain current IFEA schemes. The FEA equation can be expressed in a general form as follows:

where K and F stand for the stiffness matrix and load vector, respectively; U represents the static response vector; and p is the input parameter vector of the mechanical system. In the IFEA, p is quantified as an interval input vector pI and shown in Figure 1.


Figure 1.

The diagram of interval variable p.

where pic is the nominal value, Δpi is the interval radius. Then, the IFEA equation is accordingly rewritten as follows:


where the superscript “I” hereinafter represents an interval input. The exact solution set of this interval equation can be expressed as:


It is noted that interdependencies among entries of the response vector are introduced due to sharing the common input vector and a nonconvex polyhedron is always defined [7], which makes it extremely difficult to obtain the exact solution [5]. However, only individual ranges of some components in the response vector are of interest for real‐life problems. Therefore, by neglecting the aforementioned interdependencies, the smallest hypercube approximation denoted as UI around the exact solution set is an alternative object for current IFEA. The kth component of UI is expressed as follows:


where superscripts “L” and “U” represent the lower and upper bounds of an interval variable, respectively; N is the total number of response components of interest. Accordingly, the smallest hypercube solution of IFEA equation is expressed as:


where “T” is a transposition operator.

2. Core algorithms

From published literatures, four types of algorithms for IFEA have been well established. Most of the current schemes are formulated based on these core algorithms.

2.1. Arithmetic approach

The key point of arithmetic approach is to translate the complete deterministic numerical FE procedure to an interval procedure using the arithmetic operations. Each substep of the interval algorithm calculates the range of the intermediate subfunction instead of the deterministic result. Based on this principle, the interval bounds of the output can be obtained. The original solution procedure for IFEA is the interval arithmetic approaches [710], in which all basic deterministic algebraic operations are replaced by their interval arithmetic counterparts.

The major advantage of the arithmetic approach is its simplicity. However, the major drawback of this method is its repeated vulnerability to conservatism. It is shown that these methods suffer considerably from the overestimation effect, also referred to as the dependency problem, and for the real‐life problems, the resulting overestimation may render the final result totally useless [5]. A simple example is shown as follows. Consider the function

applied on the interval x=[0,1]. Applying arithmetic approach, both terms are assumed independently. This leads to the interval solution f(x)=[0,2]. However, the exact range of the function equals f(x)=[34,1]. That is to say, an arithmetic interval operation introduces conservatism in its result if neglecting the correlation that exists between the operands. Besides, the integration of interval arithmetic approaches with software for FEA is also a challenge in real applications.

2.2. Perturbation approach

The perturbation approach has been widely applied in structural response analyses and other applications. Compared to arithmetic approaches, perturbation methods are more popular due to its simplicity and efficiency in IFEA and can be available in the original, improved, and modified versions.

2.2.1. Original version

The first‐order Taylor expansions of the interval stiffness matrix and load vector at the nominal (mid‐) values of interval parameters were firstly obtained as:


where pc is the nominal (mid‐) value of the interval input vector and ΔpiI=[Δpi,Δpi] is the interval radius of the ith interval parameter, i.e.,


And the interval radiuses of the stiffness matrix and load vector in Eq. (7) are expressed as follows, respectively.


The FEA model for the perturbed system can be rewritten as follows:


By expanding Eq. (10) and neglecting the second‐order perturbed term, the following equations can be obtained.


Substituting Eq. (10) into Eq. (11) yields the interval radius of the response vector as:


And the radius vector of the response vector is estimated in the original interval perturbation method [11] as follows:


The smallest hypercube solution can thus be determined as:


The major drawback of this method is that a significant overestimation is introduced by the original interval perturbation method, indicating that it applies to the interval analysis of problems with “small” interval parameters.

2.2.2. Improved version

The most typical improved interval perturbation method was proposed in Ref. [12], in which the radius vector of the response vector was calculated as follows:


Accordingly, the smallest hypercube solution of IFEA can also be determined by Eq. (14). Although with better accuracy compared to the original one, an interval translation effect, i.e., the translation of the resulting interval w.r.t. the accurate one, is always introduced by the improved interval perturbation method.

2.2.3. Modified versions

Compared with the original version of the perturbation approach where only first‐order terms are considered, the main aspect of the following two modified interval perturbation methods [13, 14] is that the interval bounds are calculated by retaining part of higher order terms in Neumann series. Therefore, the modified methods can obtain more accurate response bounds. The key expressions are summarized as follows:






Different estimations of the radius vector of the response vector were, respectively, obtained as follows:


It should be pointed out that significant unpredicted estimation is always introduced by Eqs. (19) and (20). A more reasonable estimation of the radius vector of the response vector is simultaneously determined herein as follows:


And a slight conservatism is alternatively resulted in by Eq. (21). The smallest hypercube solution for the IFEA is finally determined as Eq. (14). It is worth mentioning that the spectral radius of (Kc)1ΔK increases with the increase in ΔKI. (Kc+ΔK)1 can be expanded with a Neumann series if and only if (Kc)1ΔK is less than 1 based on the criteria of convergence for a Neumann series. Therefore, these methods applies to the interval analysis of nonlinear problems with “small” interval parameters and the accuracy for those with “large” interval inputs can be improved by the subinterval technique in Section 3.1. Furthermore, the integration of all interval perturbation methods with current FEA software for the system simulation remains a great challenge.

2.3. Sampling approach

2.3.1. Vertex method

The vertex method was originally developed in Ref. [15], which can be viewed as a sampling technique with vertices being input samples of the FEA model. This method has been popular for the implementation of IFEA [1621] due to its main aspect of simple formulation and the black‐box property. If the behavior of the target response w.r.t. uncertain parameters can be guaranteed to be monotonic, the vertex method firstly proposed in Ref. [15] yields the exact solution. It should be pointed out that the concept of monotonicity in this section means monotonic along all principal directions where only one parameter is changing at a time. However, it is very hard—if not impossible —to prove the property of monotonicity in a general way, e.g., in the application of structural dynamics [22]. The number of FEA runs necessary for the vertex method is given as:

where n is the number of interval parameters. It is noted that the computational cost for the vertex method exponentially increases w.r.t. the number of interval parameters, which results in a dimensionality curse.

2.3.2. Transformation method

To promote the accuracy of the vertex method for nonmonotonic problems, transformation methods for the epistemic uncertainty propagation were developed. Its original version was firstly proposed in literature [23]. This method is based on the αlevel strategy and on each αlevel the interval problem is defined. The interval solution strategy then consists of a dedicated sampling strategy in the space spanned by αcut of fuzzy parameters. This method is available in a general, a reduced, and an extended form, with the most appropriate form to be selected depending on the type of model to be evaluated [23, 24]. If the behavior of the target response w.r.t. uncertain parameters can be guaranteed to be monotonic, the reduced transformation method yields the exact solution. If it shows nonmonotonic behavior, instead, the extended transformation method can be applied, in which more observation points were added in a well‐directed way to the search domain after rating the monotonicity of the response w.r.t. different uncertain parameters on the basis of a classification criterion [24].

The computational cost of the transformation method is governed by the number of FEA runs N to be performed. In the case of the general transformation method, this number is given as:

where m is the number of discrete αlevels and n is the number of fuzzy parameters. It is noted that the number of FEA runs grows exponentially w.r.t. the number of fuzzy inputs, which makes the general transformation method computational tedious for high‐dimensional problems. The main aspect of the transformation method, its characteristic property of reducing fuzzy arithmetic to multiple crisp‐number operations entails that this method can be implemented without major problems into an existing software environment for system simulation. Expensive rewriting of the program codes is not required [25]. Some of the most recent applications can be found in Refs. [2532]. Besides, a program named as FAMOUS (fuzzy arithmetical modeling of uncertain systems) has been developed [25], which provides an interface to commercial software environments. Primarily developed in Matlab environment, FAMOUS actually works as a standalone application on both Windows and Linux platforms.

2.4. Optimization approach

In essence, calculating the solution set expressed in Eq. (3) is equivalent to performing a global optimization, aimed at the minimization and maximization of the components of the deterministic analysis results {U}. The lower and upper bounds of the output of a classical FEA model are determined by the optimization approach through a search algorithm within the domain spanned by the interval parameters. If the global minimum and maximum of the analysis result are found by the search algorithm, it returns the smallest hypercube solution around the exact one. The optimization is performed independently on each element of the response vector. Furthermore, as the behavior of the target response w.r.t. uncertain parameters is rather unpredictable, the computational cost of the optimization approach in general is strongly problem‐dependent. It is noted that the optimization approach is immune to the excessive conservatism for the interval arithmetic approaches because the optimization strategy approaches the smallest hypercube solution from its inside, which means that it does not guarantee conservatism until the actual bounds are captured. Additionally, the smooth behavior of the target response w.r.t. uncertain parameters facilitates the search for the global extrema over the space spanned by uncertain parameters. The directional search‐based algorithm [16, 33, 34], linear programming [35], and genetic algorithm [36] were utilized to formulate the procedure of IFEA or FFEA. More applications can be found in [3739]. It is worth mentioning that the optimization approach and Monte Carlo simulation can be adopted to verify the accuracy of other schemes for IFEA and FFEA.

3. Preprocessing procedures

Except for the aforementioned core algorithms for IFEA/FFEA, two types of preprocessing procedures are always adopted to improve either the accuracy or efficiency.

3.1. Subinterval technique

For the accuracy improvement, the subinterval technique w.r.t. interval inputs is developed [11, 40] and can be integrated with the interval arithmetic and perturbation approaches. The main aspect of the subinterval technique is the ability to relax requirements of “small” or “narrow” interval inputs for nonlinear responses. However, there remain two challenges as follows:

  1. Convergence validation. Similar to the prior determination of the sample size of MC in the probabilistic analysis, the subinterval number for each interval parameter should be first determined to guarantee the convergence of the analysis result.

  2. Efficiency sacrifice. An exponential increase of the computational cost is introduced as increasing the subinterval number to guarantee the convergence of the analysis result. For example, the computational cost increases by mn times where n is the number of parameters and m is the number of subintervals for each interval parameter. Thus, the most dominant advantage in efficiency for the interval arithmetic and perturbation approaches over other interval algorithms is significantly sacrificed.

3.2. Surrogate model

To enhance the efficiency of IFEA and FFEA, many surrogate models of the real numerical model are always adopted when dealing with engineering design problems often involving large‐scale FEA models. The main aspect of the surrogate model is to avoid the large amount of computational time. Apart from the conventional surrogate models always used in the optimization procedure of IFEA and FFEA, e.g. response surface models [41, 42], Kriging models [4345], radial basis function models [4648] and sparse grid meta‐models [4951], those for the sampling and optimization approaches including the high dimensional model representation (HDMR) and the component mode synthesis (CMS) are gaining momentum in recent years. CMS was originally introduced in Ref. [52], in which a Ritz‐type transformation to each individual component of a structure was adopted. The deformation of each component is approximated using a limited number of component modes. For each of these vectors, only a single degree of freedom (DOF) was retained in the reduced component model, yielding a large reduction in DOF for each component and the entire structure. Thus, the computational cost for the FEA is drastically reduced. From this viewpoint, CMS can also be seen as a special surrogate model of the expensive numerical FEA for the improvement in the computational efficiency. The repeated FEAs required in the context of IFEA can benefit from this computational time reduction obtained by CMS.

4. Hybrid algorithms

Numerous schemes for IFEA and FFEA have been developed based on the core algorithms and preprocessing procedure, which can be classified into the following three cases.

4.1. Subinterval‐based hybrid algorithms

Divide the large interval parameter piI(i=1,2,,n) into Ni subintervals and its rith subinterval can be expressed as follow:


The number of subintervals for each interval parameter may be different. Nsub combinations can be produced by taking a subinterval out of each interval parameter.

For each subinterval combination, the IFEA model can be rewritten as:


where pr1r2rnI stands for a subinterval combination and is composed of the r1th subinterval of the first interval parameter, the r2th subinterval of the second one and up to the rnth subinterval of the nth one. In a conclusion, Eq. (26) stands for Nsub subinterval IFEA equations. For each subinterval IFEA equation, the response vector can be obtained by using core algorithms in Section 2, e.g., interval arithmetic approaches, perturbation approaches, and vertex method. For two adjacent subinterval vector pr1rrrnI and pr1rr+1rnI, the following formulae hold true, i.e.,


where prrU and prrL are the upper bound of prrI and lower bound of prr+1L, respectively. Thus, the intersection of U(pr1rrrnI) and U(pr1rr+1rnI) does not equal to an empty set, i.e.,


It is shown from Eq. (29) that the interval response vectors for each subinterval combination are simply connected. Therefore, the interval response vector can be obtained as follows by using the interval union operation.


The above subinterval method is shown in Figure 2 with 50 subintervals when considering one uncertain parameter x.


Figure 2.

The diagram of subinterval method.

The interval arithmetic approach, subinterval technique and Taylor series expansion were integrated [40]. More applications can be found in [13, 53, 54].

4.2. Surrogate model‐based hybrid algorithms

Taylor series expansion was integrated with the interval arithmetic approach in [40] and a method named as Taylor expansion with extrema management was proposed by integrating the higher order Taylor series expansion and the optimization approach [55] to detect possible nonmonotonic influences.

The transformation method was integrated with HDMR in Ref. [25]. And a component mode transformation method was developed [56] by combing the CMS with the transformation method to provide a significant reduction of the computational cost for large mechanical problems with uncertain parameters. Besides, a hybrid method was proposed for the interval frequency response analysis by integrating the optimization and interval arithmetic approach in [57], which was further integrated with CMS in Ref. [22]. An acceptable computational cost and a limited amount of conservatism in the analysis result were achieved by these hybrid algorithms.

4.3. Hybrid core algorithms

The aforementioned core algorithms can be combined together to achieve a better tradeoff between the accuracy and efficiency, e.g., frameworks [22, 5760] formulated by the global optimization methods and interval arithmetic ones.

To improve the computational efficiency, any core algorithm in Section 2 can be integrated with reanalysis method [61], which is fundamentally an intrusive FEA. It is noted that the major computational cost of IFEA consists of repeated solutions of the deterministic FEA systems while the main goal of the re‐analysis method is to accelerate this conventional FEA solution process. It is shown that the application of the re‐analysis method in the context of IFEA can reduce the computational cost by one order of magnitude compared to those based on the conventional FEA strategy [5].

5. Conclusions

This chapter presents the state‐of‐the‐art and recent advances in nonprobabilistic finite element analyses. The main advantages and shortcomings of each nonprobabilistic finite element analysis method are discussed.

The arithmetic approach is the most straightforward strategy for nonprobabilistic finite element analyses. However, this chapter further shows that the interval arithmetic implementation of the finite element procedure is conservative. Therefore, the development of an adequate methodology for solving the uncertain parameter dependency problem is still the main challenge in the domain of arithmetic approach. The perturbation approach has been widely used in structural response analyses and other applications due to its simplicity and efficiency. The accuracy of the original perturbation methods can be improved by retaining part of higher order terms in Neumann series or Taylor series as shown in the improved and modified versions. The sampling approach like vertex method yields the exact solution under the condition that the behavior of the target response w.r.t. uncertain parameters can be guaranteed to be monotonic and has been popular for the implementation of IFEA due to its main aspect of simple formulation and the black‐box property. However, when tackling the nonmonotonic problems, the extended transformation methods should be applied by adding more observation points in a well‐directed way. The optimization approach is more and more acknowledged as standard procedure in an interval finite element context except for the high computational cost.

Moreover, in this context, two typical preprocessing procedures, e.g., subinterval technique and surrogate model to improve either the accuracy or efficiency are described in detail. Additionally, novel hybrid algorithms, including subinterval‐based hybrid algorithms, surrogate model‐based hybrid algorithms and hybrid core algorithms can be conceived by combining the aforementioned core algorithms and preprocessing procedures to achieve a better tradeoff between the accuracy and efficiency for the specific problems and the future work in this research field can be fostered.


1 - Ben‐Haim Y, Elishakoff I. Convex models of uncertainty in applied mechanics. Amsterdam: Elsevier Science Publishers; 1990.
2 - Elishakoff I. Essay on uncertainties in elastic and viscoelastic structures: From A. M. Freudenthal’s criticisms to modern convex modeling. Computers & Structures. 1995;56(6):871–895. doi:
3 - Elishakoff I. Possible limitations of probabilistic methods in engineering. Applied Mechanics Reviews. 2000;53(2):19–36. doi:10.1115/1.3097337
4 - Moens D, Vandepitte D. A survey of non‐probabilistic uncertainty treatment in finite element analysis. Computer Methods in Applied Mechanics and Engineering. 2005;194(12–16):1527–1555. doi:
5 - Moens D, Hanss M. Non‐probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances. Finite Elements in Analysis and Design. 2011;47(1):4–16. doi:
6 - Nguyen HT. A note on the extension principle for fuzzy sets. Journal of Mathematical Analysis and Applications. 1978;64(2):369–380. doi:‐247X(78)90045-8
7 - Moore RE, Kearfott RB, Cloud MJ. Introduction to Interval Analysis. Philadelphia, PA: Society for Industrial and Applied Mathematics; 2009.
8 - Degrauwe D, Lombaert G, De Roeck G. Improving interval analysis in finite element calculations by means of affine arithmetic. Computers & Structures. 2010;88(3–4):247–254. doi:
9 - Behera D, Chakraverty S. Fuzzy finite element analysis of imprecisely defined structures with fuzzy nodal force. Engineering Applications of Artificial Intelligence. 2013;26(10):2458–2466. doi:
10 - Serhat Erdogan Y, Gundes Bakir P. Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic. Engineering Applications of Artificial Intelligence. 2013;26(1):357–367. doi:10.1016/j.engappai.2012.10.003
11 - Qiu Z, Elishakoff I. Antioptimization of structures with large uncertain‐but‐non‐random parameters via interval analysis. Computer Methods in Applied Mechanics and Engineering. 1998;152(3):361–372. doi:
12 - McWilliam S. Anti‐optimisation of uncertain structures using interval analysis. Computers & Structures. 2001;79(4):421–430. doi:
13 - Xia B, Yu D, Liu J. Interval and subinterval perturbation methods for a structural‐acoustic system with interval parameters. Journal of Fluids and Structures. 2013;38:146–163. doi:
14 - Wang C, Qiu Z, Wang X, Wu D. Interval finite element analysis and reliability‐based optimization of coupled structural‐acoustic system with uncertain parameters. Finite Elements in Analysis and Design. 2014;91:108–114. doi:
15 - Dong W, Shah HC. Vertex method for computing functions of fuzzy variables. Fuzzy Sets and Systems. 1987;24(1):65–78. doi:
16 - Rao S, Sawyer JP. Fuzzy finite element approach for analysis of imprecisely defined systems. AIAA Journal. 1995;33(12):2364–2370. doi:10.2514/3.12910
17 - Chen L, Rao SS. Fuzzy finite‐element approach for the vibration analysis of imprecisely‐defined systems. Finite Elements in Analysis and Design. 1997;27(1):69–83. doi:
18 - Akpan UO, Koko TS, Orisamolu IR, Gallant BK. Practical fuzzy finite element analysis of structures. Finite Elements in Analysis and Design. 2001;38(2):93–111. doi:
19 - Qiu Z, Wang X, Chen J. Exact bounds for the static response set of structures with uncertain‐but‐bounded parameters. International Journal of Solids and Structures. 2006;43(21):6574–6593. doi:
20 - Qiu Z, Xia Y, Yang J. The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem. Computer Methods in Applied Mechanics and Engineering. 2007;196(49–52):4965–4984. doi:10.1016/j.cma.2007.06.022
21 - Xu M, Qiu Z, Wang X. Uncertainty propagation in SEA for structural–acoustic coupled systems with non‐deterministic parameters. Journal of Sound and Vibration. 2014;333(17):3949–3965. doi:
22 - De Gersem H, Moens D, Desmet W, Vandepitte D. Interval and fuzzy dynamic analysis of finite element models with superelements. Computers & Structures. 2007;85(5–6):304–319. doi:
23 - Hanss M. The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems. 2002;130(3):277–289. doi:
24 - Hanss M. The extended transformation method for the simulation and analysis of fuzzy‐parameterized models. International Journal of Uncertainty, Fuzziness and Knowledge‐Based Systems. 2003;11(06):711–727. doi:doi:10.1142/S0218488503002491
25 - Hanss M, Turrin S. A fuzzy‐based approach to comprehensive modeling and analysis of systems with epistemic uncertainties. Structural Safety. 2010;32(6):433–441. doi:
26 - Turrin S, Hanss M, Gaul L. Fuzzy arithmetical vibration analysis of a windshield with uncertain parameters. In: Proceedings of the Ninth International Conference on Recent Advances in Structural Dynamics ‐ RASD, Southampton 2006
27 - Hanss M, Becker J, Maess M, Gaul L. Fuzzy arithmetical analysis of smart structures with uncertainties. In: Proceedings of the First International Conference on Uncertainty in Structural Dynamics, Sheffield; 2007
28 - Junge M, Brunner D, Becker J, Maess M, Roseira J, Hanss M. Combination of fuzzy arithmetic and a fast boundary element method for acoustic simulation with uncertainties. Journal of Computational Acoustics 2009;17(01): 45–69. doi:doi:10.1142/S0218396X09003811
29 - Hanss M, Klimke A. On the reliability of the influence measure in the transformation method of fuzzy arithmetic. Fuzzy Sets and Systems. 2004;143(3):371–390. doi:
30 - Allahviranloo T, Kiani NA, Motamedi N. Solving fuzzy differential equations by differential transformation method. Information Sciences. 2009;179(7):956–966. doi:
31 - Klimke A. An efficient implementation of the transformation method of fuzzy arithmetic. In: Fuzzy Information Processing Society, 2003. NAFIPS 2003. International Conference of the North American. New York: IEEE Xplore, 2003:468–473
32 - Gauger U, Turrin S, Hanss M, Gaul L. A new uncertainty analysis for the transformation method. Fuzzy Sets and Systems. 2008;159(11):1273–1291. doi:
33 - Rao SS, Berke L. Analysis of uncertain structural systems using interval analysis. AIAA Journal. 1997;35(4):727–735. doi:10.2514/2.164
34 - Rao SS, Chen L. Numerical solution of fuzzy linear equations in engineering analysis. International Journal for Numerical Methods in Engineering. 1998;43(3):391–408. doi:10.1002/(sici)1097‐0207(19981015)43:3<391::aid‐nme417>;2‐j
35 - Köylüog lu HUu, Elishakoff I. A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties. Computers & Structures. 1998;67(1–3):91–98. doi:
36 - Möller B, Graf W, Beer M. Fuzzy structural analysis using α‐level optimization. Computational Mechanics. 2000;26(6):547–565. doi:10.1007/s004660000204
37 - Moens D, Vandepitte D. Fuzzy finite element method for frequency response function analysis of uncertain structures. AIAA Journal. 2002;40(1):126–136. doi:10.2514/2.1621
38 - Farkas L, Moens D, Vandepitte D, Desmet W. Application of fuzzy numerical techniques for product performance analysis in the conceptual and preliminary design stage. Computers & Structures. 2008;86(10):1061–1079. doi:10.1016/j.compstruc.2007.07.012
39 - Farkas L, Moens D, Vandepitte D, Desmet W. Fuzzy finite element analysis based on reanalysis technique. Structural Safety. 2010;32(6):442–448. doi:10.1016/j.strusafe.2010.04.004
40 - Zhou YT, Jiang C, Han X. Interval and subinterval analysis methods of the structural analysis and their error estimations. International Journal of Computational Methods. 2006;3(2):229–244. doi:10.1142/S0219876206000771
41 - de Boor C, Ron A. On multivariate polynomial interpolation. Constructive Approximation. 1990;6(3):287–302. doi:10.1007/bf01890412
42 - Myers RH, Montgomery DC, Anderson‐Cook CM. Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Hoboken, NJ: John Wiley & Sons; 2011.
43 - Jones DR, Schonlau M, Welch WJ. Efficient global optimization of expensive black‐box functions. Journal of Global Optimization. 1998;13(4):455–492. doi:10.1023/a:1008306431147
44 - Martin JD, Simpson TW. Use of Kriging models to approximate deterministic computer models. AIAA Journal. 2005;43(4):853–863. doi:10.2514/1.8650
45 - Kleijnen JPC. Kriging metamodeling in simulation: A review. European Journal of Operational Research. 2009;192(3):707–716. doi:
46 - Park J, Sandberg IW. Universal approximation using radial‐basis‐function networks. Neural Computation. 1991;3(2):246–257. doi:10.1162/neco.1991.3.2.246
47 - Chen S, Cowan CFN, Grant PM. Orthogonal least squares learning algorithm for radial basis function networks. IEEE Transactions on Neural Networks. 1991;2(2):302–309. doi:10.1109/72.80341
48 - T Sev, Shin YC. Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems. IEEE Transactions on Neural Networks. 1994;5(4):594–603. doi:10.1109/72.298229
49 - Klimke A, Nunes RF, Wohlmuth BI. Fuzzy arithmetic based on dimension‐adaptive sparse grids: A case study of a large‐scale finite element model under uncertain parameters. International Journal of Uncertainty, Fuzziness and Knowledge‐Based Systems. 2006;14(5):561–577. doi:10.1142/S0218488506004199
50 - Klimke A, Willner K, Wohlmuth BI. Uncertainty modeling using fuzzy arithmetic based on sparse grids: Applications to dynamic systems. International Journal of Uncertainty, Fuzziness and Knowledge‐Based Systems. 2004;12(6):745–759. doi:10.1142/S0218488504003181
51 - Nunes RF, Klimke A, Arruda JRF. On estimating frequency response function envelopes using the spectral element method and fuzzy sets. Journal of Sound and Vibration. 2006;291(3–5):986–1003. doi:
52 - Hurty WC. Dynamic analysis of structural systems using component modes. AIAA Journal. 1965;3(4):678–685. doi:10.2514/3.2947
53 - Xia B, Yu D. Modified sub‐interval perturbation finite element method for 2D acoustic field prediction with large uncertain‐but‐bounded parameters. Journal of Sound and Vibration. 2012;331(16):3774–3790. doi:
54 - Wang C, Qiu Z, Li Y. Hybrid uncertainty propagation of coupled structural–acoustic system with large fuzzy and interval parameters. Applied Acoustics. 2016;102:62–70. doi:
55 - Massa F, Tison T, Lallemand B. A fuzzy procedure for the static design of imprecise structures. Computer Methods in Applied Mechanics and Engineering. 2006;195(9–12):925–941. doi:
56 - Giannini O, Hanss M. The component mode transformation method: A fast implementation of fuzzy arithmetic for uncertainty management in structural dynamics. Journal of Sound and Vibration. 2008;311(3–5):1340–1357. doi:
57 - Moens D, Vandepitte D. An interval finite element approach for the calculation of envelope frequency response functions. International Journal for Numerical Methods in Engineering. 2004;61(14):2480–2507. doi:10.1002/nme.1159
58 - De Gersem H, Moens D, Desmet W, Vandepitte D. A fuzzy finite element procedure for the calculation of uncertain frequency response functions of damped structures: Part 2—Numerical case studies. Journal of Sound and Vibration. 2005;288(3):463–486. doi:10.1016/j.jsv.2005.07.002
59 - Moens D, Vandepitte D. A fuzzy finite element procedure for the calculation of uncertain frequency‐response functions of damped structures: Part 1—Procedure. Journal of Sound and Vibration. 2005;288(3):431–462. doi:10.1016/j.jsv.2005.07.001
60 - De Munck M, Moens D, Desmet W, Vandepitte D. A response surface based optimisation algorithm for the calculation of fuzzy envelope FRFs of models with uncertain properties. Computers & Structures. 2008; 86(10):1080–1092. doi:10.1016/j.compstruc.2007.07.006
61 - Laszlo F, David M, Gersem HD, Dirk V. Efficient FE reanalysis method for fuzzy uncertainty analysis of a composite wing. In: 10th AIAA Non‐Deterministic Approaches Conference, Schaumburg 2008