FFTbased method for homogenization of periodic media: Theory and applications
FFTbased method for homogenization of periodic media: Theory and applications
dc.contributor.advisor  Zeman Jan  
dc.contributor.author  Vondřejc Jaroslav  
dc.date.accessioned  20130130T01:01:38Z  
dc.date.available  20130130T01:01:38Z  
dc.date.issued  20130130  
dc.date.submitted  20130130 02:00:04.0  
dc.identifier  KOS185379145005  
dc.identifier.uri  http://hdl.handle.net/10467/14623  
dc.description.abstract  This dissertation is devoted to an FFTbased 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, FFTbased homogenization, that was proposed by Moulinec and Suquet, is a numerical algorithm derived from LippmannSchwinger equation. Its equivalence to a corresponding weak formulation is shown; it eliminates a reference homogeneous material, a parameter of LippmannSchwinger equation. Next, Galerkin approximation with numerical integration is introduced to produce MoulinecSuquet 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 nonsymmetric 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 FFTbased homogenization. A general technique is proposed to allow for efficient calculation by FFT algorithm and to maintain the upperlower 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 FFTbased homogenization to realworld 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 twostep homogenization for a highly porous aluminium foam.  
dc.description.abstract  This dissertation is devoted to an FFTbased 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, FFTbased homogenization, that was proposed by Moulinec and Suquet, is a numerical algorithm derived from LippmannSchwinger equation. Its equivalence to a corresponding weak formulation is shown; it eliminates a reference homogeneous material, a parameter of LippmannSchwinger equation. Next, Galerkin approximation with numerical integration is introduced to produce MoulinecSuquet 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 nonsymmetric 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 FFTbased homogenization. A general technique is proposed to allow for efficient calculation by FFT algorithm and to maintain the upperlower 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 FFTbased homogenization to realworld 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 twostep 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/autorskepravo/013982006.pdf and the citation ethics http://www.cvut.cz/sites/default/files/content/d1dc93cd58944521b799c7e715d3c59e/cs/20160901metodickypokync12009ododrzovanietickychprincipupripripravevysokoskolskych.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/autorskepravo/013982006.pdf a citační etikou http://www.cvut.cz/sites/default/files/content/d1dc93cd58944521b799c7e715d3c59e/cs/20160901metodickypokync12009ododrzovanietickychprincipupripripravevysokoskolskych.pdf  cze 
dc.subject  homogenization, Fourier transform, FFT, discretization, finite element method, convergence, guaranteed bounds  cze 
dc.title  FFTbased method for homogenization of periodic media: Theory and applications  
dc.title  FFTbased method for homogenization of periodic media: Theory and applications  eng 
dc.type  disertační práce  cze 
dc.date.updated  20130130T01:01:38Z  
dc.date.accepted  20130129 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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Disertační práce  11000 [498]