FFT-based method for homogenization of periodic media: Theory and applications

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 dc.contributor.advisor Zeman Jan dc.contributor.author Vondřejc Jaroslav dc.date.accessioned 2013-01-30T01:01:38Z dc.date.available 2013-01-30T01:01:38Z dc.date.issued 2013-01-30 dc.date.submitted 2013-01-30 02:00:04.0 dc.identifier KOS-185379145005 dc.identifier.uri http://hdl.handle.net/10467/14623 dc.description.abstract This dissertation is devoted to an FFT-based homogenization scheme, a numerical method for the evaluation of the effective (homogenized) matrix of periodic linear heterogeneous materials. A problem is explained and demonstrated on a scalar problem modeling electric conduction, heat conduction, or diffusion. Originally, FFT-based homogenization, that was proposed by Moulinec and Suquet, is a numerical algorithm derived from Lippmann-Schwinger equation. Its equivalence to a corresponding weak formulation is shown; it eliminates a reference homogeneous material, a parameter of Lippmann-Schwinger equation. Next, Galerkin approximation with numerical integration is introduced to produce Moulinec-Suquet algorithm; trigonometric polynomials are taken as the trial space. Convergence of approximate solutions to the solution of weak formulation is provided using a standard finite element approach together with approximation properties of trigonometric polynomials. Then, the solution of assembled non-symmetric linear system by Conjugate gradients, proposed by Zeman et al., is clarified. Next, we study arbitrary accurate guaranteed bounds of homogenized matrix introduced by Dvořák for a scalar problem and later independently by Wieckowski for linear elasticity. This approach is also applicable for FFT-based homogenization. A general technique is proposed to allow for efficient calculation by FFT algorithm and to maintain the upper-lower bound structure. Dual formulation is employed to obtain lower bounds - for odd number of discretization points, the solution of dual formulation can be avoided. A general number of discretization points leads to a more complicated theory in both discretization and numerical treatment. Finally, applications of FFT-based homogenization to real-world problems is demonstrated. The method is used to calculate homogenized matrix for cement paste, gypsum and aluminum alloy with local data obtained from nanoindentation. Next, it is employed as a part of two-step homogenization for a highly porous aluminium foam. dc.description.abstract This dissertation is devoted to an FFT-based homogenization scheme, a numerical method for the evaluation of the effective (homogenized) matrix of periodic linear heterogeneous materials. A problem is explained and demonstrated on a scalar problem modeling electric conduction, heat conduction, or diffusion. Originally, FFT-based homogenization, that was proposed by Moulinec and Suquet, is a numerical algorithm derived from Lippmann-Schwinger equation. Its equivalence to a corresponding weak formulation is shown; it eliminates a reference homogeneous material, a parameter of Lippmann-Schwinger equation. Next, Galerkin approximation with numerical integration is introduced to produce Moulinec-Suquet algorithm; trigonometric polynomials are taken as the trial space. Convergence of approximate solutions to the solution of weak formulation is provided using a standard finite element approach together with approximation properties of trigonometric polynomials. Then, the solution of assembled non-symmetric linear system by Conjugate gradients, proposed by Zeman et al., is clarified. Next, we study arbitrary accurate guaranteed bounds of homogenized matrix introduced by Dvořák for a scalar problem and later independently by Wieckowski for linear elasticity. This approach is also applicable for FFT-based homogenization. A general technique is proposed to allow for efficient calculation by FFT algorithm and to maintain the upper-lower bound structure. Dual formulation is employed to obtain lower bounds - for odd number of discretization points, the solution of dual formulation can be avoided. A general number of discretization points leads to a more complicated theory in both discretization and numerical treatment. Finally, applications of FFT-based homogenization to real-world problems is demonstrated. The method is used to calculate homogenized matrix for cement paste, gypsum and aluminum alloy with local data obtained from nanoindentation. Next, it is employed as a part of two-step homogenization for a highly porous aluminium foam. eng dc.language.iso eng dc.publisher České vysoké učení technické v Praze. Vypočetní a informační centrum. cze dc.rights A university thesis is a work protected by the Copyright Act. Extracts, copies and transcripts of the thesis are allowed for personal use only and at one’s own expense. The use of thesis should be in compliance with the Copyright Act http://www.mkcr.cz/assets/autorske-pravo/01-3982006.pdf and the citation ethics http://www.cvut.cz/sites/default/files/content/d1dc93cd-5894-4521-b799-c7e715d3c59e/cs/20160901-metodicky-pokyn-c-12009-o-dodrzovani-etickych-principu-pri-priprave-vysokoskolskych.pdf eng dc.rights Vysokoškolská závěrečná práce je dílo chráněné autorským zákonem. Je možné pořizovat z něj na své náklady a pro svoji osobní potřebu výpisy, opisy a rozmnoženiny. Jeho využití musí být v souladu s autorským zákonem http://www.mkcr.cz/assets/autorske-pravo/01-3982006.pdf a citační etikou http://www.cvut.cz/sites/default/files/content/d1dc93cd-5894-4521-b799-c7e715d3c59e/cs/20160901-metodicky-pokyn-c-12009-o-dodrzovani-etickych-principu-pri-priprave-vysokoskolskych.pdf cze dc.subject homogenization, Fourier transform, FFT, discretization, finite element method, convergence, guaranteed bounds cze dc.title FFT-based method for homogenization of periodic media: Theory and applications dc.title FFT-based method for homogenization of periodic media: Theory and applications eng dc.type disertační práce cze dc.date.updated 2013-01-30T01:01:38Z dc.date.accepted 2013-01-29 00:00:00.0 dc.contributor.referee Kruis Jaroslav dc.description.department katedra mechaniky cze theses.degree.name Ph.D. cze theses.degree.discipline Matematika ve stavebním inženýrství cze theses.degree.grantor Fakulta stavební cze theses.degree.programme Stavební inženýrství cze evskp.contact ČVUT cze
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