Reconstructing displacements from the solution to the periodic Lippmann-Schwinger equation discretized on a uniform grid

Abstract : Uniform grid solvers of the periodic Lippmann–Schwinger equation have been introduced by Moulinec and Suquet for the numerical homogenization of heterogeneous materials. Based on the fast Fourier transform, these methods use the strain as main unknown and usually do not produce displacement fields. While this is generally not perceived as a restriction for homogenization purposes, some tasks might require kinematically admissible displacement fields. In this paper, we show how the numerical solution to the periodic Lippmann–Schwinger equation can be post-processed to reconstruct a displacement field. Our procedure is general: it applies to any variant of the Moulinec–Suquet solver. The reconstruction is formulated as an auxiliary elastic equilibrium problem of a homogeneous material, which is solved with displacement-based finite elements. Taking advantage of the periodicity, the uniformity of the grid and the homogeneity of the material, the resulting linear system is formulated and solved efficiently in Fourier space. The cost of our procedure is lower than that of one iteration of the Lippmann–Schwinger solver. An application of this post-processing procedure is proposed, in which the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus of some model microstructure. This is the pre-peer reviewed version of the following article: Reconstructing displacements from the solution to the periodic Lippmann–Schwinger equation discretized on a uniform grid, which has been published in final form at 10.1002/nme.5263. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Sections 1 (Introduction) and 5 (Numerical results) have been significantly reworked in the published (final) version of this article. The remainder of this article (including Sections 3 and 4 where the method itself is presented) is essentially unchanged.
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International Journal for Numerical Methods in Engineering, Wiley, 2017, 109 (4), pp.459-486. 〈10.1002/nme.5263〉
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Sébastien Brisard. Reconstructing displacements from the solution to the periodic Lippmann-Schwinger equation discretized on a uniform grid. International Journal for Numerical Methods in Engineering, Wiley, 2017, 109 (4), pp.459-486. 〈10.1002/nme.5263〉. 〈hal-01304603〉

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