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A fast method for solving microstructural problems defined by digital images: a space Lippmann–Schwinger scheme
Author(s) -
Yvonnet Julien
Publication year - 2012
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4334
Subject(s) - fourier transform , mathematics , finite element method , discretization , computation , inverse problem , linear map , iterative method , algorithm , mathematical analysis , physics , pure mathematics , thermodynamics
SUMMARY A fast numerical method is proposed to solve thermomechanical problems over periodic microstructures whose geometries are provided by experimental techniques, such as X‐ray microtomography images. In such configuration, the phase properties are defined over regular grids of voxels. To overcome the limitations of calculations on such fine models, an iterative scheme is proposed, avoiding the construction and storage of finite element matrices. Equilibrium equations are written in the form of a Lippmann–Schwinger integral equation, which can be solved iteratively. Unlike previous algorithms based on the Fourier transform, the present scheme strictly operates in the real‐space domain and removes the numerical Fourier and inverse Fourier transforms at each iteration. For this purpose, the linear operator related to the Lippmann–Schwinger equation is constructed numerically by means of transformation tensors in the real‐space domain. The convergence and accuracy of the method are evaluated through examples in both steady‐state thermics and linear elasticity problems. Computational times are found to scale linearly with the number of degrees of freedom, and parallel computations can be carried out straightforwardly. The method is also illustrated on many examples involving complex microstructures, including a problem defined by a microtomography image. Copyright © 2012 John Wiley & Sons, Ltd.