A fast method for solving microstructural problems defined by digital images: a space Lippmann–Schwinger scheme

Abstract

SUMMARYA fast numerical method is proposed to solve thermomechanical problems over periodic microstructures whose geometries are provided by experimental techniques, like X-ray micro tomography 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 micro tomography image.

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