Reconstructing displacements from the solution to the periodic Lippmann–Schwinger equation discretized on a uniform grid
Reconstructing displacements from the solution to the periodic Lippmann–Schwinger equation discretized on a uniform grid
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DOI：
10.1002/nme.5263
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发表时间：
2017-01
影响因子：
2.9
通讯作者：
S. Brisard
中科院分区：
文献类型：
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作者：
S. Brisard
周期Lippmann-Schwinger方程的均匀网格求解器已由Moulinec和Suquite引入，用于非均质材料的数值均匀化。这些方法基于快速傅里叶变换，以应变为主要未知量，通常不产生位移场。虽然这通常不被视为均化目的的限制，但某些任务可能需要运动学上允许的位移场。在这篇文章中，我们展示了如何对周期Lippmann-Schwinger方程的数值解进行后处理以重建位移场。我们的程序适用于Moulinec-Suquite求解器的任何变体。将重构表示为均匀材料的辅助弹性平衡问题，并用基于位移的有限元进行求解。利用网格的周期性、网格的均匀性和材料的均匀性，在傅立叶空间中建立线性方程组，并进行高效求解。我们的程序比Lippmann-Schwinger求解器的一次迭代的成本要低。提出了两种在二维和三维方面的应用。在第一个应用中，重建的位移场被用来计算有效剪切模数的严格上限。在第二个应用中，对重建的质量进行了定量评估。版权所有©2016 John Wiley&Sons，Ltd.
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 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 periodicity, uniformity of the grid and 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. Two applications are proposed, in two and three dimensions. In the first application, the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus. In the second application, the quality of the reconstruction is assessed quantitatively. Copyright © 2016 John Wiley & Sons, Ltd.