Acceleration of FFT-based homogenisation by low-rank tensors approximation

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21110%2F19%3A00334198" target="_blank" >RIV/68407700:21110/19:00334198 - isvavai.cz</a>

Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Acceleration of FFT-based homogenisation by low-rank tensors approximation

Popis výsledku v původním jazyce

The main task of the homogenization is to homogeneously de- scribe material properties of periodically heterogeneous materials, based on knowledge of microstructure geometry and properties of its phases. Evalu- ation of the homogenised properties with high accuracy requires a detailed knowledge of materials’ microstructure. Unfortunately, this knowledge comes hand in hand with high memory and time requirements of approximate so- lution to homogenisation problem. To find a solution we use method based on the Fast Fourier Transform (FFT), which has been introduced in 1994 by Moulinec and Suquet [1] and lately explained as Fourier-Galerkin method by Vondˇrejc, Zeman and Marek in [2]. FFT based methods has turned out to be an effective computa- tional approach for numerical homogenisation of periodic media. Its com- putational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. We accelerated this method by low-rank tensor approximation techniques for a solution field. The idea of low-rank tensors, or low-rank approxima- tions is to express large multidimensional tensors by fewer parameters. This compression can lead to a huge reduction of the mentioned computational requirements. Hackbusch explained whole concept on variety of low-rank tensor formats in [3]. We have tested canonical, Tucker and tensor train formats. The last one was introduced by Oseledets and Tyrtyshnikov in [4]. The advantages of this approach against those using full tensors will be demonstrated using numerical examples for the model homogenisation prob- lem that consists of a scalar linear elliptic variational problem defined in two and three dimensional setting with continuous and discontinuous heteroge- neous material coefficients. This approach has the potential for an efficient reduced order modelling of large scale engineering problems with heteroge- neous material.

Název v anglickém jazyce

Acceleration of FFT-based homogenisation by low-rank tensors approximation

Popis výsledku anglicky

The main task of the homogenization is to homogeneously de- scribe material properties of periodically heterogeneous materials, based on knowledge of microstructure geometry and properties of its phases. Evalu- ation of the homogenised properties with high accuracy requires a detailed knowledge of materials’ microstructure. Unfortunately, this knowledge comes hand in hand with high memory and time requirements of approximate so- lution to homogenisation problem. To find a solution we use method based on the Fast Fourier Transform (FFT), which has been introduced in 1994 by Moulinec and Suquet [1] and lately explained as Fourier-Galerkin method by Vondˇrejc, Zeman and Marek in [2]. FFT based methods has turned out to be an effective computa- tional approach for numerical homogenisation of periodic media. Its com- putational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. We accelerated this method by low-rank tensor approximation techniques for a solution field. The idea of low-rank tensors, or low-rank approxima- tions is to express large multidimensional tensors by fewer parameters. This compression can lead to a huge reduction of the mentioned computational requirements. Hackbusch explained whole concept on variety of low-rank tensor formats in [3]. We have tested canonical, Tucker and tensor train formats. The last one was introduced by Oseledets and Tyrtyshnikov in [4]. The advantages of this approach against those using full tensors will be demonstrated using numerical examples for the model homogenisation prob- lem that consists of a scalar linear elliptic variational problem defined in two and three dimensional setting with continuous and discontinuous heteroge- neous material coefficients. This approach has the potential for an efficient reduced order modelling of large scale engineering problems with heteroge- neous material.

Druh

O - Ostatní výsledky

CEP obor

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OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GC17-04150J" target="_blank" >GC17-04150J: Robustní dvojúrovňové simulace založené na Fourierově metodě a metodě konečných prvků: Odhady chyb, redukované modely a stochastika</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů