\r\n\t \r\n\tComputer graphics are not entirely an original topic, because it defines and solves problems using some already established techniques such as geometry, algebra, optics, and psychology. The geometry provides a framework for describing 2D and 3D space, while the algebraic methods are used for defining and evaluating equality related to the specific space. The science of optics enables the application of the model for the description of the behavior of light, while psychology provides models for visualization and color perception. \r\n\t \r\n\t3D computer graphics (or 3D graphics, three-dimensional computer graphics, three-dimensional graphics) is a term describing the different methods of creating and displaying three-dimensional objects by using computer graphics. \r\n\tThe first types of graphic interpretations were put in the plane (two-dimensional 2D). Requirements for a universal interpretation led to a three-dimensional (3D) interpretation content. From these creations have arisen applied mathematics and information disciplines of graphic interpretation of content - computer graphics. It relies on the principles of Mathematics, Descriptive Geometry, Computer Science and Applied Electronics. \r\n\t \r\n\t3D computer graphics or three-dimensional computer graphics use a three-dimensional representation of geometric data (often in terms of the Cartesian coordinate system) that is stored on a computer for the purpose of doing the calculation and creating 2D images. The images that are made can be stored for later use (probably as animation) or can be displayed in real-time. \r\n\t \r\n\tObjects within the 3D computer graphics are often called 3D models. Unlike rendered (generated) images, data that are ""tied"" to the model are inside graphic files. The 3D model is a mathematical representation of a random three-dimensional object. The model can be displayed visually as a two-dimensional image through a process called 3D rendering or can be used in non-graphical computer simulations and calculations. With 3D printing, models can be presented in real physical form. \r\n\t \r\n\tComputer graphics have remained one of the most interesting areas of modern technology, and it is the area that progresses the fastest. It has become an integral part of both application software, and computer systems in general. Computer graphics is routinely applied in the design of many products, simulators for training, production of music videos and television commercials, in movies, in data analysis, in scientific studies, in medical procedures, and in many other fields.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"de29c8802680e89528bdbecf055dffd1",bookSignature:"Dr. Dragan Mladen Cvetković",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8770.jpg",keywords:"Vector Graphics, Graphic design, 3D model, Computer-Aided Design (CAD), Computer-Aided Architectural Design (CAAD), 3D Rendering, Virtual engineering, 3D Mapping, 3D projection on 2D planes, Video games, 3D Printing, 3D Computer Graphics in Science",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 28th 2019",dateEndSecondStepPublish:"March 6th 2020",dateEndThirdStepPublish:"May 5th 2020",dateEndFourthStepPublish:"July 24th 2020",dateEndFifthStepPublish:"September 22nd 2020",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"101330",title:"Dr.",name:"Dragan",middleName:"Mladen",surname:"Cvetković",slug:"dragan-cvetkovic",fullName:"Dragan Cvetković",profilePictureURL:"https://mts.intechopen.com/storage/users/101330/images/system/101330.jpg",biography:"Dragan Cvetković graduated in Aeronautics from the Faculty of Mechanical Engineering, University of Belgrade, in 1988. He defended his doctoral dissertation in December 1997.\n\nHe has published 64 books, scripts and practicums about computers and computer programs, aviation weapons and flight mechanics. He has also published a large number of scientific papers, both nationally and internationally.\n\nIn 2014, he became a full professor in the field of Informatics and Computing at Singidunum University, Belgrade. Previously he had served as an assistant professor. 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From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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1. Introduction
In this section, the advanced Oxidation Processes concepts will be related to use of the bulk and the nano- catalysts as vital materials for easily generating a highly oxidizing species and reactive oxygen species (ROSs) such as in aqueous or alcoholic solution [1]. ROSs are contains three primary kinds: superoxide anion (O2•−), hydrogen peroxide (H2O2) and the hydroxyl radical (HO•) [2], which produced from reaction of adsorbed oxygen molecule on catalyst’s surface with one electron in conductive band under illumination by light as UV, or visible or solar light, this mechanism is useful to reduce the recombination process and increased the life time of hole in valance band [3, 4]. As explained in Figure 1.
Figure 1.
Essential mechanism for generating the ROSs under illumination of photo-catalyst particles [1].
The ROSs are having the electron configurations as tabled in Table 1 [5, 6, 7, 8].
Table 1.
Electronic configurations and chemical formulas for the ROSs types.
2. Advance oxidation process applications
In the last few years, several researches have predominated in many universities and research centers on the scientific ventures to mainly treat the contaminations that produced by textile factories [9, 10, 11], reduced the degradation of food’s dye [12], decolorization of colored organometallic complexes [13], degradation of toxic cyclic compounds [14] and produced a hydrogen from alcohol as renewable energy [15]. The effective materials for all above mention research are generated the hydroxyl radical in aqueous solution with maximum oxidation power equals to 2.8 V [1]. Based on to the AOPs, the common sources for creation of.OH in AOPs are illustrated in Figure 2, which regards as power to star the dark or photo reactions [1, 16, 17, 18, 19].
Figure 2.
Schematic diagram of common sources of.OH in advanced oxidation processes.
Fortunately, the benefits of AOPs are more than those of drawbacks. The benefits of AOPs are summarized up as [1, 20] follows to:
Create a large number of free radicals species.
Have the appropriate potential to depress the hazardous organic pollutants by complete their mineralization and producing CO2 and H2O.
Reduce the time of dark or photoreaction.
Have low economic cost.
Whereas, the drawbacks of AOPs [1, 21] are quenching the reaction rate with increasing the scavenger contains (mostly peroxide ion) and may be generated the undesirable hazardous products that prevented the complete of mineralization process, hence, the altered of pH or using further cost steps may be essentially to treat their problems.
3. Bulk and nano-catalysts
In general, the catalysts may be metal or alloy or semiconductor. Semiconductor is wide used as catalyst and can be element or compound as amorphous or crystalline or rock salt crystal. Because of semiconductors have intermediate properties between metal and insulator, which has given them rescannable electronic and structural properties, hence, semiconductor is classified as a better-known kinds, as mentioned in Figure 3 [22, 23, 24].
Figure 3.
Better-known kinds of semiconductors.
The usages of the bulk and nano catalysts are increment with increasing the development of life activations. The catalysts were known for the long time to increase the rate of reaction with decreasing the time of reaction and the activation energy in dark reaction or photoreaction. In order to use the catalyst in photoreaction as photo catalyst, must have a band gap with raged about 1.1 eV to 5.0 eV [1, 24]. Referring to Figure 4, several band gap energy positions of some common photo catalysts can be displayed [1, 25, 26, 27].
Figure 4.
Band gap energy positions of different photo-semiconductor at pH = 1.
The mainly problem in bulk and nano catalyst is recombination process, which results in diminishing the efficiency of used photocatalyst by returning the photoelectron from conductive band to valance band and reacting with photohole immediately. The recombination includes four kinds can be followed in Table 2 and Figure 5 [1, 28, 29, 30].
Kinds
Other name
Info
Type of photocatalyst
Direct recombination
Band-to- band recombination
In this kind, the transition occurrs as a radiative transition in direct band gap semiconductor. It is created when the Free photo electron in CB drops directly into free photo hole (an unoccupied state) in the VB and associated together. Note Figure 5(A).
ZnO have a direct band gap.
Volume recombination
Centers recombination or Trap-assisted recombination
This case obtains, when defect of semiconductor by impurities that given a new levels (as traps of photoelectron and photohole). It leads to liberate heat as phonon in indirect band gap semiconductor. Note Figure 5(B).
Pure TiO2 and defect of TiO2 by metal, which had given an indirect band gap.
Surface recombination
Recombination of an exciton
This case occurs at low temperature, when the traps at or near the surface or interface of the semiconductor, capture the photo electron- hole as exciton. That attitude to dangling bonds caused by the sudden discontinuation of the semi-conductor crystal with energy just below the band gap value. Note Figure 5(C).
It happed in solar cells and light emitting diode (LED) containing shallow levels.
Auger recombination
—
This recombination involves three carriers: Free photo electron, free photo whole recombine, and the emitting the energy as heat or as a photon (non-radiative process). The transition of energy deals with as intra-band transitions, which resulting when either electron elevates in higher levels of conduction band or hole deeper push into the valence band. Note Figure 5(D).
This case can be obtained wit short lifetime when heavy doping defects (like Ag) in direct-gap semiconductors under present sunlight.
Table 2.
The most common recombination types concepts.
Figure 5.
The schematic diagram of the most common recombination kinds.
In order to improve the activity of photocatalysts must depress the recombination with modify their surfaces with three main methods: surface sensitization, metalized photocatalyst surface and coupled for two or more photocatalysts as Composite. The details of these modification methods are mention in Table 3 and Figure 6 [40].
Table 3.
The description of the methods for modifying photocatalysts [31, 32, 33, 34, 35, 36, 37, 38, 39].
Figure 6.
Schematic diagram for modification of photocatalyst surface [40].
4. Used of bulk or nano catalyst in AOPs
There are many common application of AOPs in environment fields by using the white photocatalyst or its modified such as ZnO, TiO2 ZrO2, ZnS, WO3, CdS and Mn3O4. The efficiencies with used these photocatalysts are altered with using AOPs methods. The efficiency of the photoreaction depends mostly on the concentration of colored material, initial pH which affected on the surface of photocatalyst and the temperature. As shown in Table 4.
Some applications of bulk and nano photocatalydts in AOPs, with environment chemistry and green chemistry.
5. Conclusions
This chapter focuses on the source of hydroxyl radical which produces via the advance oxidation process. Indeed, this process interests in the forming the different species, which in the final step generates a hydroxyl radical. The photocatalyst enhances the generating of hydroxyl radicals (2.8 V) in aqueous solution under Uv- light or visible or solar. The photoexitation of photocatalyst leads to jump of electon to conductive band then return to valance band and liberates a hot this process called recombination. It is depressed the efficiency of photoreaction. However, some procedures used to modify the photocatalyst surface.
Acknowledgments
The author wants to thank his family for helping him in carrying out this work.
\n',keywords:"nanocatalysts, bulk catalyst, advanced oxidation processes, wastewater treatment, photocatalysis, Fenton reaction, photo-Fenton",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/73905.pdf",chapterXML:"https://mts.intechopen.com/source/xml/73905.xml",downloadPdfUrl:"/chapter/pdf-download/73905",previewPdfUrl:"/chapter/pdf-preview/73905",totalDownloads:94,totalViews:0,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"May 21st 2020",dateReviewed:"September 29th 2020",datePrePublished:"November 3rd 2020",datePublished:"February 17th 2021",dateFinished:"November 3rd 2020",readingETA:"0",abstract:"Advanced oxidation processes (AOPs) are considered to be vital methods for treating the contaminations produced mainly by the human activations. In present-day, UV light or solar light, bulk and nano- photocatalysts are often used to enhance this technology by creating the highly reactive species such as the hydroxyl radicals. Extreme hydroxyl radical is considered as a key to start the photoreaction. Photoreaction is widely used in treatment of Lab and industrial contaminations, preparation of compounds and produced the renewable energy, so it’s classified as green technique. In order to improve the efficiency of this reaction with fabrication the surface of the used photocatalyst such as metal doped, sensitized and produced a composite as bulk catalyst or nano catalyst.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/73905",risUrl:"/chapter/ris/73905",book:{slug:"oxidoreductase"},signatures:"Luma Majeed Ahmed",authors:[{id:"321806",title:"Prof.",name:"Luma",middleName:"Majeed",surname:"Majeed Ahmed",fullName:"Luma Majeed Ahmed",slug:"luma-majeed-ahmed",email:"lumamajeed2013@gmail.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Advance oxidation process applications",level:"1"},{id:"sec_3",title:"3. Bulk and nano-catalysts",level:"1"},{id:"sec_4",title:"4. Used of bulk or nano catalyst in AOPs",level:"1"},{id:"sec_5",title:"5. Conclusions",level:"1"},{id:"sec_6",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Ahmed L, and Hussein F. Roles of Photocatalytic Reactions of Platinized TiO2 Nanoparticales. 1st ed. LAP Lambert Academia Published; 2014. 103 P. ISBN-10: 3659538817'},{id:"B2",body:'Collin F. Chemical Basis of Reactive Oxygen Species Reactivity and Involvement in Neurodegenerative Diseases. Int. J. Mol. Sci. 2019; 20, 2407:1-17. DOI: 10.3390/ijms20102407'},{id:"B3",body:'Ahmed L, Ivanova I, Hussein F and Bahnemann D. Role of Platinum Deposited on TiO2 in Photocatalytic Methanol Oxidation and Dehydrogenation Reactions. International Journal of Photoenergy. 2014; 1:1-9. DOI: 10.1155/2014/503516'},{id:"B4",body:'Ahmed L, Hussen F, Mahdi A. Photocatalytic Dehydrogenation of Aqueous Methanol Solution by Naked and Platinized TiO2 Nanoparticles. 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Journal of the SID . 2012; 20(1): 63- 69S'},{id:"B29",body:'Pierret R. Advanced semiconductor Fundemantals. 1st ed. vol. VI, Modular series on solid state devices, New Jersey : Pearson Education, Inc. Upper Saddle River;2002. 07458 p. CH5'},{id:"B30",body:'Hayawi M. Preparation,Characterization of Spinel Mn3O4/ ZrO2 Composite and Application on Colored Material[MSC thesis]. Karbala: Kerbala University:2020'},{id:"B31",body:'Giwa A, Nkeonye P., Bello K., Kolawole G. and Campos A. Solar Photocatalytic Degradation of Reactive Yellow 81 and Reactive Violet 1 in Aqueous Solution Containing Semiconductor Oxides. International Journal of Applied Science and Technology. 2012; 2(4): 90-105'},{id:"B32",body:'Ahmed L. Photo-decolorization kinetics of acid red 87 dye in ZnO suspension under different types of UV-A light. Asian J. Chem. 2018; 30(9): 2134-2140'},{id:"B33",body:'Pare B, Singh P and Jonnalagadda S, Visible Light-drive Photocatalytic Degradation and Minieralization of neutral Red dye in a sulurry Photoreactor. Indian Journal Chemistry Technoogy.2010; 17: 391-395'},{id:"B34",body:'Jasim K, and Ahmed L. TiO2 Nanoparticles Sensitized by Safranine O Dye using UV-A Light System. In IOP Conference Series. Materials Science and Engineering. 2019; 571(012064):1-9'},{id:"B35",body:'Fadhil F, Ahmed L, and Mohammed A. Effect of silver doping on structural and photocatalytic circumstances of ZnO nanoparticles. Iraqi Journal of Nanotechnolog, Synthesis and Application. 2020; 1: 13-20'},{id:"B36",body:'Mahammed B, and Ahmed L. Enhanced Photocatalytic Properties of Pure and Cr-Modified ZnS Powders Synthesized by Precipitation Method. Journal of Geoscience and Environment Protection. 2017; 5: 101-111'},{id:"B37",body:'Mohammed B, and Ahmed L. Improvement the Photo Catalytic Properties of ZnS nanoparticle with Loaded Manganese and Chromium by Co-Precipitation Method. Journal of Global Pharma Technology. 2018; 10(7):129-138'},{id:"B38",body:'Jawad T. Synthesis and Characterization of Nano-Composite WO3/ TiO2 by Using Ultrasonic and it is Application on Photodecolorization of Eosin Yellow Dye [ MSC thesis]. Karbala: Kerbala University: 2020'},{id:"B39",body:'Fakhri F, and Ahmed L. Incorporation CdS with ZnS as nanocomposite and Using in Photo-Decolorization of Congo Red Dye. Indones. J. Chem.2019; 19(4):936-943'},{id:"B40",body:'Fakhri F. Preparation, Characterization of ZnS/CdS Composites nano particles and using in photocatalytic-decolorization of Congo red dye[ MSC thesis]. Karbala: Kerbala University: 2019'},{id:"B41",body:'Mashkour M, Al-Kaim A, Ahmed L, and Hussein F. Zinc Oxide Assisted Photocatalytic Decolourization of Reactive Red 2 Dye. Int. J. Chem. Sci. 2011; 9(3): 969-979'},{id:"B42",body:'Zuafuani S, and Ahmed L. Photocatalytic Decolourization of Direct Orange Dye by Zinc Oxide under UV Irradiation. Int. J. Chem. Sci.2015; 13(1):187-196'},{id:"B43",body:'Ahmed L, Tawfeeq F, Abed Al-Ameer F, Abed Al-Hussein K. and Athaab A. Photo-Degradation of Reactive Yellow 14 Dye (A Textile Dye) Employing ZnO as Photocatalyst. Journal of Geoscience and Environment Protection.2016; 4: 34-44'},{id:"B44",body:'Abbas S, Hassan Z, and Ahmed L. Influencing the Artificial UV-A light on decolorization of Chlorazol black BH Dye via using bulk ZnO Suspensions. In Journal of Physics: Conference Series. IOP Publishing. 052050. 2019; 1294(5):1-8'},{id:"B45",body:'Eesa M, Juda A, and Ahmed L. Kinetic and thermodynamic study of the photocatalytic decolourization of light green SF yellowish (acid green 5) dye using commercial bulk Titania and commercial Nanotitania. Int. J. Sci. Res.2016; 5(11): 1495-1500'},{id:"B46",body:'Hussein Z, Abbas S, and Ahmed, L. UV-A activated ZrO2 via photodecolorization of methyl green dye. In IOP Conference Series: Materials Science and Engineering. 012132, IOP Publishing. 2018; 454(1): 1-11'},{id:"B47",body:'Hayawi M, Kareem M, and Ahmed L. Synthesis of Spinel Mn3O4 and Spinel Mn3O4/ZrO2 Nanocomposites and Using Them in Photo-Catalytic Decolorization of Fe(II)-(4,5-Diazafluoren-9-One 11) Complex . Periódico Tchê Química. 2020; 17(34): 689-699'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Luma Majeed Ahmed",address:"luma.ahmed@uokerbala.edu.iq",affiliation:'
Department of Chemistry, College of Science, University of Kerbala, Kerbala, Iraq
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1. Introduction
The fractional calculus has recently been widely used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. This branch of mathematical analysis has emerged in recent years as an effective and powerful tool for the mathematical modeling of various engineering, industrial, and materials science applications [1, 2, 3]. The fractional-order operators are useful in describing the memory and hereditary properties of various materials and processes, due to their nonlocal nature. It clearly reflects from the related literature produced by leading fractional calculus journals that the primary focus of the investigation had shifted from classical integer-order models to fractional order models [4, 5]. Fractional calculus has important applications in hereditary solid mechanics, fluid dynamics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation, nanotechnology, etc. [6, 7, 8].
Numerous mathematicians have contributed to the history of fractional calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. Then, Laplace defined a fractional derivative by means of an integral in 1812.
Lacroix introduced the first fractional order derivative which appeared in a calculus in 1819, where he expressed the nth derivative of the function y=xm as follows:
dndxn=Γm+1Γm‐n+1xm‐nE1
Liouville assumed that dvdxveax=aveaxforv>0 to obtain the following fractional order derivative:
dvx‐adxv=‐1vΓa+vΓax‐a‐vE2
Laurent has been using the Cauchy’s integral formula for complex valued analytical functions to define the integration of arbitrary order v>0 as follows:
cDxvfx=cDxm‐ρfx=dmdxm1Γρ∫cxx−tρ−1ftdt,0<ρ≤1E3
where cDxv denotes differentiation of order v of the function f along the x‐axis.
Cauchy introduced the following fractional order derivative:
f+α=∫fτt‐τα‐1Γ‐αdτE4
Caputo introduced his fractional derivative of order α<0 to be defined as follows:
D∗αft=1Γm‐α∫0tfmτt‐τα+1‐mdτ,m−1<α<m,α>0E5
Recently, research on nonlinear generalized magneto-thermoelastic problems has received wide attention due to its practical applications in various fields such as geomechanics, geophysics, petroleum and mineral prospecting, earthquake engineering, astronautics, oceanology, aeronautics, materials science, fiber-optic communication, fluid mechanics, automobile industries, aircraft, space vehicles, plasma physics, nuclear reactors, and other industrial applications. Due to computational difficulties in solving nonlinear generalized magneto-thermoelastic problems in general analytically, many numerical techniques have been developed and implemented for solving such problems [9, 10, 11, 12, 13, 14, 15, 16, 17]. The boundary element method (BEM) [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] has been recognized as an attractive alternative numerical method to domain methods [32, 33, 34, 35, 36] like finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. The superior feature of BEM over domain methods is that only the boundary of the domain needs to be discretized, which often leads to fewer elements and easier to use. This advantage of BEM over other domain methods has significant importance for modeling and optimization of thermoelastic problems which can be implemented using BEM with little cost and less input data. Nowadays, the BEM has emerged as an accurate and efficient computational technique for solving complicated inhomogeneous and non-linear problems in physical and engineering applications [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69].
In the present chapter, we introduce a practical engineering application of fractal analysis in the field of thermoelasticity, where the thermal field is described by time fractional three-temperature radiative heat conduction equations. Fractional order derivative considered in the current chapter has high ability to remove the difficulty of our numerical modeling. A new boundary element method for modeling and optimization of 3T fractional order nonlinear generalized thermoelastic multi-material initially stressed multilayered functionally graded anisotropic (ISMFGA) structures subjected to moving heat source is investigated. Numerical results show that the fractional order parameter has a significant effect on the sensitivities of displacements, total three-temperature, and thermal stresses. Numerical examples show that the fractional order parameter has a significant effect on the final topology of ISMFGA structures. Numerical results of the proposed model confirm the validity and accuracy of the proposed technique, and numerical examples results demonstrate the validity of the BESO multi-material topology optimization method.
A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of fractional order problems and their applications. Section 2 describes the physical modeling of fractional order problems in three-temperature nonlinear generalized magneto-thermoelastic ISMFGA structures. Section 3 outlines the BEM implementation for modeling of 3T fractional nonlinear generalized magneto-thermoelastic problems of multi-material ISMFGA structures subjected to moving heat source. Section 4 introduces an illustration of the mechanisms of solving design sensitivities and optimization problem of the current chapter. Section 5 presents the new numerical results that describe the effects of fractional order parameter on the problem’s field variations and on the final topology of multi-material ISMFGA structures.
2. Formulation of the problem
Consider a multilayered structure with n functionally graded layers in the xy‐plane of a Cartesian coordinate. The x‐axis is the common normal to all layers as shown in Figure 1. The thickness of the layer is denoted by h. The considered multilayered structure has been placed in a primary magnetic field H0 acting in the direction of the y‐axis.
Figure 1.
Geometry of the considered problem.
According to the three-temperature theory, the governing equations of nonlinear generalized magneto-thermoelasticity in an initially stressed multilayered functionally graded anisotropic (ISMFGA) structure for the ith layer can be written in the following form:
σab,b+τab,b−Γab=ρix+1mu¨aiE6
σab=x+1mCabfgiuf,gi−βabiTαi−Tα0i+τ1Ṫα1E7
τab=μix+1mh˜aHb+h˜bHa−δbah˜fHfE8
Γab=Pix+1m∂uai∂xb−∂ubi∂xaE9
According to Fahmy [10], the time fractional order two-dimensional three-temperature (2D-3 T) radiative heat conduction equations in nondimensionless form can be expressed as follows:
where σab, τab, and uki are mechanical stress tensor, Maxwell’s electromagnetic stress tensor, and displacement vector in the ith layer, respectively, cα(α = c, I, p) are constant Tα0i, Tαi, Cabfgi, and βabi are, respectively, reference temperature, temperature, constant elastic moduli, and stress-temperature coefficients in the ith layer: μi, h˜, Pi, ρi, and csαi are, respectively, magnetic permeability, perturbed magnetic field, initial stress, density, isochore specific heat coefficients in the ith layer; τ is the time; τ0 and τ1 are the relaxation times; i=1,2,…,n represents the parameters in multilayered structure; and m is a functionally graded parameter. Also, we considered in the current study that the medium is subjected to a moving heat source of constant strength moving along x‐axis with a constant velocity v. This moving heat source is assumed to have the following form:
Qxτ=Q0δx−vτE15
where, Q0 is the heat source strength and δ is the delta function.
where inertia term, temperature gradient, and initial stress terms are treated as the body forces.
In this section, we are interested in using a boundary element method for modeling the two-dimensional three-temperature radiation heat conduction equations coupled with electron, ion, and phonon temperatures.
According to finite difference scheme of Caputo at times f+1Δt and fΔτ, we obtain [1].
DτaTαif+1+DτaTαif≈∑j=0kWa,jTαif+1−jr−Tαif−jrE17
where
Wa,0=Δτ−aΓ2−a,Wa,j=Wa,0j+11−a−j−11−aE18
Based on Eq. (17), the fractional order heat Eq. (10) can be replaced by the following system:
Now, according to Fahmy [10], and applying the fundamental solution which satisfies (19), the boundary integral equations corresponding to (10) without heat sources can be expressed as
Tαiξ=∫STαiqi∗−Tαi∗qidC−∫RfabTαi∗dRE20
Thus, the governing equations can be written in operator form as follows:
Lgbufi=fgb,E21
LabTαi=fabE22
where the operators Lgb, fgb, Lab, and fab are as follows:
The differential Eq. (21) can be solved using the weighted residual method (WRM) to obtain the following integral equation:
∫RLgbufi−fgbudai∗dR=0E26
Now, the fundamental solution udfi∗ and traction vectors tdai∗ and tai can be written as follows:
Lgbudfi∗=−δadδxξE27
tdai∗=Cabfgudf,gi∗nbE28
tai=t¯aix+1m=Cabfguf,gi−βabiTαi+τ1TαinbE29
Using integration by parts and sifting property of the Dirac distribution for (26), then using Eqs. (27) and (29), we can write the following elastic integral representation formula:
In order to convert the domain integral in (42) into the boundary, we approximate the source vector SA by a series of known functions fAEq and unknown coefficients αEq as
SA≈∑q=1EfAEqαEqE51
Thus, the representation formula (42) can be written as follows:
By applying the point collocation procedure of Gaul et al. [43] to Eqs. (51) and (61), we obtain
Sˇ=Jα¯,Ui=J′γ,E65
Similarly, applying the same point collocation procedure to Eqs. (64), (46), (47), (48), and (49) yields
SˇTαi=BTγE66
SAu=ψUiE67
SˇTαι̇=Γ¯AFU̇iE68
SˇTαι¨=δ¯AFU¨iE69
Sˇu¨=Ⅎ¯U¨iE70
where ψ¯, Γ¯AF, δ¯AF, and Ⅎ¯ are assembled using the submatrices ψ, ΓAF, δAF, and Ⅎ, respectively.
Solving the system (65) for α¯ and γ yields
α¯=J−1Sˇ,γ=J′−1UiE71
Now, the coefficient α¯ can be written in terms of the unknown displacements Ui, velocities U̇i, and accelerations U¨i as
α¯=J−1Sˇ0+BTJ′−1+ψ¯Ui+Γ¯AFU̇i+Ⅎ¯+δ¯AFU¨iE72
An implicit-implicit staggered algorithm has been implemented for use with the BEM to solve the governing equations which can now be written in a suitable form after substitution of Eq. (72) into Eq. (60) as
M⏞U¨i+Γ⏞U̇i+K⏞Ui=Q⏞iE73
X⏞T¨αi+A⏞Ṫαi+B⏞Tαi=Z⏞U¨i+R⏞E74
where V=η℘ˇ−ζUˇJ−1,M⏞=VℲ¯+δ¯AF,Γ⏞=VΓ¯AF,K⏞=−ζˇ+VBTJ′−1+ψ¯, Q⏞i=−ηTˇ+VSˇ0,X⏞=−ρicsαiτ0,A⏞=−Kαi,B⏞=ξ∇Kαi∇,Z⏞=βabiTα0iτ0,R⏞=−Q0δx−vτ.
where U¨i,U̇i,Ui,Ti and Q⏞i are, respectively, acceleration, velocity, displacement, temperature, and external force vectors, and V,M⏞, Γ⏞, K⏞, A⏞, and B⏞ are, respectively, volume, mass, damping, stiffness, capacity, and conductivity matrices.
In many applications, the coupling term Z⏞U¨n+1i that appear in the heat conduction equation is negligible. Therefore, it is easier to predict the temperature than the displacement.
Hence Eqs. (73) and (74) lead to the following coupled system of differential-algebraic equations (DAEs):
M⏞U¨n+1i+Γ⏞U̇n+1i+K⏞Un+1i=Q⏞n+1ipE75
X⏞T¨αn+1i+A⏞Ṫαn+1i+B⏞Tαn+1i=Z⏞U¨n+1i+R⏞E76
where Q⏞n+1ip=ηTαn+1ip+VSˇ0 and Tαn+1ip is the predicted temperature.
Now, a displacement predicted staggered procedure for the solution of (80) and (85) is as follows:
The first step is to predict the propagation of the displacement wave field: Un+1ip=Uni. The second step is to substitute for U̇n+1i and U¨n+1i from Eqs. (77) and (75), respectively, in Eq. (85) and solve the resulted equation for the three-temperature fields. The third step is to correct the displacement using the computed three-temperature fields for the Eq. (80). The fourth step is to compute U̇n+1i, U¨n+1i, Ṫαn+1i, and T¨αn+1i from Eqs. (79), (81), (82), and (86), respectively.
The continuity conditions for temperature, heat flux, displacement, and traction that have been considered in the current chapter can be expressed as
Tαixzτx=hi=Tαi+1xzτx=hiE87
qixzτx=hi=qi+1xzτx=hiE88
ufixzτx=hi=ufi+1xzτx=hiE89
t¯aixzτx=hi=t¯ai+1xzτx=hiE90
where n is the total number of layers, t¯a are the tractions which is defined by t¯a=σabnb, and i=1,2,…,n−1.
The initial and boundary conditions of the present study are
ufixz0=u̇fixz0=0forxz∈R∪CE91
ufixzτ=Ψfxzτforxz∈C3E92
t¯aixzτ=Φfxzτforxz∈C4,τ>0E93
Tαixz0=Tαixz0=0forxz∈R∪CE94
Tαixzτ=f¯xzτforxz∈C1,τ>0E95
qixzτ=h¯xzτforxz∈C2,τ>0E96
where Ψf, Φf, f, and h¯ are prescribed functions, C=C1∪C2=C3∪C4, and C1∩C2=C3∩C4=0.
4. Design sensitivity and optimization
According to Fahmy [58, 60], the design sensitivities of displacements components and total 3T can be performed by implicit differentiation of (75) and (76), respectively, which describe the structural response with respect to the design variables, and then we can compute thermal stresses sensitivities.
The bi-directional evolutionary structural optimization (BESO) is the evolutionary topology optimization method that allows modification of the structure by either adding or removing material to or from the structure design. This addition or removal depends on the sensitivity analysis. Sensitivity analysis is the estimation of the response of the structure to the modification of design variables and is dependent on the calculation of derivatives [70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80].
The homogenized vector of thermal expansion coefficients αH can be written in terms of the homogenized elasticity matrix DH and the homogenized vector of stress-temperature coefficients βH as follows:
αH=DH−1βHE97
For the material design, the derivative of the homogenized vector of thermal expansion coefficients can be written as
∂αH∂Xklm=DH−1∂βH∂Xklm−∂DH∂XklmαHE98
where ∂DH∂Xklm and ∂βH∂Xklm for any lth material phase can be calculated using the adjoint variable method [73] as
In order to show the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic thermoelastic material which has the following physical constants [57].
Mass density ρ=7820kg/m3 and heat capacity c = 461 J/kg K.
The proposed technique that has been utilized in the present chapter can be applicable to a wide range of three-temperature nonlinear generalized thermoelastic problems of ISMFGA structures. The main aim of this chapter was to assess the impact of fractional order parameter on the sensitivities of total three-temperature, displacement components, and thermal stress components.
Figure 2 shows the variation of the total temperature sensitivity along the x‐axis. It was shown from this figure that the fraction order parameter has great effects on the total three-temperature sensitivity.
Figure 2.
Variation of the total 3T sensitivity along x-axis.
Figures 3 and 4 show the variation of the displacement components u1 and u2 along the x‐axis for different values of fractional order parameter. It was noticed from these figures that the fractional order parameter has great effects on the displacement sensitivities.
Figure 3.
Variation of the displacement u1 sensitivity along x-axis.
Figure 4.
Variation of the displacement u2 sensitivity along x-axis.
Figures 5–7 show the variation of the thermal stress components σ11, σ12, and σ22, respectively, along the x‐axis for different values of fractional order parameter. It was noted from these figures that the fractional order parameter has great influences on the thermal stress sensitivities.
Figure 5.
Variation of the thermal stress σ11 sensitivity along x-axis.
Figure 6.
Variation of the thermal stress σ12 sensitivity along x-axis.
Figure 7.
Variation of the thermal stress σ22 sensitivity along x-axis.
Since there are no available results for the three-temperature thermoelastic problems, except for Fahmy’s research [10, 11, 12, 13, 14]. For comparison purposes with the special cases of other methods treated by other authors, we only considered one-dimensional numerical results of the considered problem. In the special case under consideration, the displacement u1 and thermal stress σ11 results are plotted in Figures 8 and 9. The validity and accuracy of our proposed BEM technique were demonstrated by comparing our BEM results with the FEM results of Xiong and Tian [81], it can be noticed that the BEM results are found to agree very well with the FEM results.
Figure 8.
Variation of the displacement u1 sensitivity along x-axis.
Figure 9.
Variation of the thermal stress σ11 waves along x-axis.
Example 1. Short cantilever beam.
The mean compliance has been minimized, to obtain the maximum stiffness, when the structure is subjected to moving heat source. In this example, we consider a short cantilever beam shown in Figure 10, where the BESO final topology of considered short cantilever beam shown in Figure 11a for α=0.5 and shown in Figure 11b for α=1.0. It is noticed from this figure that the fractional order parameter has a significant effect on the final topology of the multi-material ISMFGA structure.
Figure 10.
Design domain of a short cantilever beam.
Figure 11.
The final topology of a short cantilever beam: (a) α = 0.5 and (b) α = 1.0.
Example 2. MBB beam.
It is known that extraordinary thermo-mechanical properties can be accomplished by combining more than two materials phases with conventional materials [75]. For this reason, it is essential that the topology optimization strategy permits more than two materials phases within the design domain. In this example, we consider a MBB beam shown in Figure 12, where the BESO final topology of MBB beam has been shown in Figure 13a for α=0.5 and shown in Figure 13b for α=1.0 to show the effect of fractional order parameter on the final topology of the multi-material ISMFGA structure.
Figure 12.
Design domain of a MBB beam.
Figure 13.
The final topology of MBB beam: (a) α = 0.5 and (b) α = 1.0.
Example 3. Roller-supported beam.
In this example, we consider a roller-supported beam shown in Figure 14, where the BESO final topology of a roller-supported beam shown in Figure 15a for α=0.5 and shown in Figure 15b for α=1.0.
Figure 14.
Design domain of a roller-supported beam.
Figure 15.
The final topology of a multi-material roller-supported beam: (a) α = 0.5 and (b) α = 1.0.
Example 4. Cantilever beam (validation example).
In order to demonstrate the validity of our implemented BESO topology optimization technique, we consider isotropic case of a cantilever beam shown in Figure 16 as a special case of our anisotropic study to interpolate the elasticity matrix and the stress-temperature coefficients using the design variables XM, then we compare our BESO final topology shown in Figure 17a with the material interpolation scheme of the solid isotropic material with penalization (SIMP) shown in Figure 17b.
Figure 16.
Design domain of a cantilever beam.
Figure 17.
The final topology of a cantilever beam: (a) MMA and (b) BESO.
The BESO topology optimization problem implemented in Examples 1 and 4, to find the distribution of the M material phases, with the volume constraint can be stated as
Find XM
That minimize CM=12PMTuM=12fM,ter+fM,mecTuM
Subject to VM,∗−∑i=1NViMXiM=0
KMuM=PM
XiM=xminV1
where XM is the design variable; CM is the mean compliance; P is the total load on the structure, which is the sum of mechanical and thermal loads; uM is the displacement vector; VM,∗ is the volume of the solid material; N is the total number of elements; KM is the global stiffness matrix; xmin is a small value (e.g., 0.0001), which it guarantee that none of the elements will be removed completely from design domain; fM,mec is the mechanical load vector; and fM,ter is the thermal load vector. Also, the BESO parameters considered in Examples 1–4 can be seen in Tables 1–4, respectively.
Variable name
Variable description
Variable value
VfM
Final volume fraction
0.5
ERM
Evolutionary ratio
1%
ARmaxM
Volume addition ratio
5%
rminM
Filter ratio
3 mm
τ
Convergence tolerance
0.1%
N
Convergence parameter
5
Table 1.
BESO parameters for minimization of a short cantilever beam.
Variable name
Variable description
Variable value
Vf1M
Final volume fraction of the material 1 for both interpolations
0.10
Vf2M
Final volume fraction of the material 2 for both interpolations
0.20
ERM
Evolutionary ratio for interpolation 1
2%
ERM
Evolutionary ratio for interpolation 2
3%
ARmaxM
Volume addition ratio for interpolation 1
3%
ARmaxM
Volume addition ratio for interpolation 2
2%
rminM
Filter ratio for interpolation 1
4 mm
rminM
Filter ratio for interpolation 2
3 mm
τ
Convergence tolerance for both interpolations
0.01%
N
Convergence parameter for both interpolations
5
Table 2.
Multi-material BESO parameters for minimization of a MBB beam.
Variable name
Variable description
Variable value
Vf1M
Final volume fraction of the material 1 for both interpolations
0.25
Vf1M
Final volume fraction of the material 2 for both interpolations
0.25
ERM
Evolutionary ratio for interpolation 1
3%
ERM
Evolutionary ratio for interpolation 2
3%
ARmaxM
Volume addition ratio for interpolation 1
1%
ARmaxM
Volume addition ratio for interpolation 2
1%
rminM
Filter ratio for interpolation 1
4 mm
rminM
Filter ratio for interpolation 2
4 mm
τ
Convergence tolerance for both interpolations
0.5 %
N
Convergence parameter for both interpolations
5
Table 3.
Multi-material BESO parameters for minimization of a roller-supported beam.
Variable name
Variable description
Variable value
VfM
Final volume fraction
0.4
ERM
Evolutionary ratio
1.2%
ARmaxM
Volume addition ratio
3%
rminM
Filter ratio
0.19 mm
τ
Convergence tolerance
0.1%
N
Convergence parameter
5
Table 4.
BESO parameters for minimization of a cantilever beam.
The BESO topology optimization problem implemented in Examples 2 and 3, to find the distribution of the two materials in the design domain, which minimize the compliance of the structure, subject to a volume constraint in both phases can be stated as.
Find XM
That minimize CM=12PMTuM=12fM,ter+fM,mecTuM
Subject to VjM,∗−∑i=1NViMXijM−∑i=1j−1ViM,∗=0;j=1,2
KMuM=PM
XiM=xminV1;j=1,2
where VjM,∗ is the volume of jth material phase and i and j denote the element ith which is made of jth material.
6. Conclusion
The main purpose of this chapter is to describe a new boundary element formulation for modeling and optimization of 3T time fractional order nonlinear generalized thermoelastic multi-material ISMFGA structures subjected to moving heat source, where we used the three-temperature nonlinear radiative heat conduction equations combined with electron, ion, and phonon temperatures.
Numerical results show the influence of fractional order parameter on the sensitivities of the study’s fields. The validity of the present method is examined and demonstrated by comparing the obtained outcomes with those known in the literature. Because there are no available data to confirm the validity and accuracy of our proposed technique, we replace the three-temperature radiative heat conduction with one-temperature heat conduction as a special case from our current general study of three-temperature nonlinear generalized thermoelasticity. In the considered special case of 3T time fractional order nonlinear generalized thermoelastic multi-material ISMFGA structures, the BEM results have been compared graphically with the FEM results; it can be noticed that the BEM results are in excellent agreement with the FEM results. These results thus demonstrate the validity and accuracy of our proposed technique. Numerical examples are solved using the multi-material topology optimization algorithm based on the bi-evolutionary structural optimization method (BESO). Numerical results of these examples show that the fractional order parameter affects the final result of optimization. The implemented optimization algorithm has proven to be an appropriate computational tool for material design.
Nowadays, the knowledge of 3T fractional order optimization of multi-material ISMFGA structures, can be utilized by mechanical engineers for designing heat exchangers, semiconductor nano materials, thermoelastic actuators, shape memory actuators, bimetallic valves and boiler tubes. As well as for chemists to observe the chemical processes such as bond breaking and bond forming.
\n',keywords:"boundary element method, modeling and optimization, time fractional order, three-temperature, nonlinear generalized thermoelasticity, initially stressed multilayered functionally graded anisotropic structures, moving heat source",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/72883.pdf",chapterXML:"https://mts.intechopen.com/source/xml/72883.xml",downloadPdfUrl:"/chapter/pdf-download/72883",previewPdfUrl:"/chapter/pdf-preview/72883",totalDownloads:133,totalViews:0,totalCrossrefCites:0,dateSubmitted:"April 11th 2020",dateReviewed:"May 18th 2020",datePrePublished:"July 23rd 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"The main purpose of this chapter, which represents one of the chapters of a fractal analysis book, is to propose a new boundary element method (BEM) formulation based on time fractional order theory of thermoelasticity for modeling and optimization of three temperature (3T) multi-material initially stressed multilayered functionally graded anisotropic (ISMFGA) structures subjected to moving heat source. Fractional order derivative considered in the current chapter has been found to be an accurate mathematical tool for solving the difficulty of our physical and numerical modeling. Furthermore, this chapter shed light on the practical application aspects of boundary element method analysis and topology optimization of fractional order thermoelastic ISMFGA structures. Numerical examples based on the multi-material topology optimization algorithm and bi-evolutionary structural optimization method (BESO) are presented to study the effects of fractional order parameter on the optimal design of thermoelastic ISMFGA structures. The numerical results are depicted graphically to show the effects of fractional order parameter on the sensitivities of total temperature, displacement components and thermal stress components. The numerical results also show the effects of fractional order parameter on the final topology of the ISMFGA structures and demonstrate the validity and accuracy of our proposed technique.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/72883",risUrl:"/chapter/ris/72883",signatures:"Mohamed Abdelsabour Fahmy",book:{id:"9886",title:"Fractal Analysis",subtitle:"Selected Examples",fullTitle:"Fractal Analysis - Selected Examples",slug:"fractal-analysis-selected-examples",publishedDate:"September 9th 2020",bookSignature:"Robert Koprowski",coverURL:"https://cdn.intechopen.com/books/images_new/9886.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"50150",title:"Dr.",name:"Robert",middleName:null,surname:"Koprowski",slug:"robert-koprowski",fullName:"Robert Koprowski"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"233766",title:"Prof.",name:"Mohamed Abdelsabour",middleName:"Abdelsabour",surname:"Fahmy",fullName:"Mohamed Abdelsabour Fahmy",slug:"mohamed-abdelsabour-fahmy",email:"mohamed_fahmy@ci.suez.edu.eg",position:null,institution:{name:"Suez Canal University",institutionURL:null,country:{name:"Egypt"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Formulation of the problem",level:"1"},{id:"sec_3",title:"3. BEM numerical implementation",level:"1"},{id:"sec_4",title:"4. Design sensitivity and optimization",level:"1"},{id:"sec_5",title:"5. Numerical examples, results, and discussion",level:"1"},{id:"sec_6",title:"6. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Cattaneo C. Sur une forme de i’equation de la chaleur elinant le paradox d’une propagation instantanc. Comptes Rendus de l’Académie des Sciences. 1958;247:431-433'},{id:"B2",body:'Oldham KB, Spanier J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Mineola: Dover Publication; 2006'},{id:"B3",body:'Podlubny I. Fractional Differential Equations. San Diego, California, USA: Academic Press; 1999'},{id:"B4",body:'Ezzat MA, El Karamany AS, Fayik MA. Fractional order theory in thermoelastic solid with three-phase lag heat transfer. 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Computer Methods in Applied Mechanics and Engineering. 1987;65:147-164'},{id:"B40",body:'Partridge PW, Wrobel LC. The dual reciprocity boundary element method for spontaneous ignition. International Journal for Numerical Methods in Engineering. 1990;30:953-963'},{id:"B41",body:'Partridge PW, Brebbia CA. Computer implementation of the BEM dual reciprocity method for the solution of general field equations. Communications in Applied Numerical Methods. 1990;6:83-92'},{id:"B42",body:'Partridge PW, Brebbia CA, Wrobel LC. The Dual Reciprocity Boundary Element Method. Southampton: Computational Mechanics Publications; 1992'},{id:"B43",body:'Gaul L, Kögl M, Wagner M. Boundary Element Methods for Engineers and Scientists. Berlin: Springer-Verlag; 2003'},{id:"B44",body:'Fahmy MA. Implicit-explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic functionally graded solids. Engineering Analysis with Boundary Elements. 2013;37:107-115'},{id:"B45",body:'Fahmy MA. Generalized magneto-thermo-viscoelastic problems of rotating functionally graded anisotropic plates by the dual reciprocity boundary element method. Journal of Thermal Stresses. 2013;36:1-20'},{id:"B46",body:'Fahmy MA. A three-dimensional generalized magneto-thermo-viscoelastic problem of a rotating functionally graded anisotropic solids with and without energy dissipation. Numerical Heat Transfer, Part A: Applications. 2013;63:713-733'},{id:"B47",body:'Fahmy MA. A 2-D DRBEM for generalized magneto-thermo-viscoelastic transient response of rotating functionally graded anisotropic thick strip. International Journal of Engineering and Technology Innovation. 2013;3:70-85'},{id:"B48",body:'Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized Thermoelastic responses of functionally graded anisotropic rotating plates with one relaxation time. 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Design of multiphysics actuators using topology optimization—Part i: One-material structures. Computer Methods in Applied Mechanics and Engineering. 2001;190(49):6577-6604'},{id:"B75",body:'Sigmund O, Torquato S. Composites with extremal thermal expansion coefficients. Applied Physics Letters. 1996;69(21):3203-3205'},{id:"B76",body:'Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids. 1997;45(6):1037-1067'},{id:"B77",body:'Sigmund O, Torquato S. Design of smart composite materials using topology optimization. Smart Materials and Structures. 1999;8:365-379'},{id:"B78",body:'Yang XY, Xei YM, Steven GP, Querin OM. Bidirectional evolutionary method for stiffness optimization. AIAA Journal. 1999;37(11):1483-1488'},{id:"B79",body:'Wang Y, Luo Z, Zhang N, Wu T. Topological design for mechanical metamaterials using a multiphase level set method. Structural and Multidisciplinary Optimization. 2016b;54:937-954'},{id:"B80",body:'Xu B, Huang X, Zhou S, XIE Y. Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness. Composite Structures. 2016;150:84-102'},{id:"B81",body:'Xiong QL, Tian XG. Generalized magneto-thermo-microstretch response during thermal shock. Latin American Journal of Solids and Structures. 2015;12:2562-2580'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Mohamed Abdelsabour Fahmy",address:"mohamed_fahmy@ci.suez.edu.eg",affiliation:'
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