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

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

##### Type of document

disertační práce##### Author

Vondřejc Jaroslav

##### Supervisor

Zeman Jan

##### Opponent

Kruis Jaroslav

##### Field of study

Matematika ve stavebním inženýrství##### Study program

Stavební inženýrství##### Institutions assigning rank

Fakulta stavební##### Defended

2013-01-29 00:00:00.0##### 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.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.

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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. 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.

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