Eigendecomposition-based convergence analysis of the Neumann series for laminated composites and discretization error estimation

In computational homogenization for periodic composites, the Lippmann-Schwinger integral equation constitutes a convenient formulation to devise numerical methods to compute local fields and their macroscopic responses. Among them, the iterative scheme based on the Neumann series is simple and efficient. For such schemes, a priori global error estimates on local fields and effective property are not available, and this is the concern of this article, which focuses on the simple, but illustrative, conductivity problem in laminated composites. The global error is split into an iteration error, associated with the Neumann series expansion, and a discretization error. The featured nonlocal Green's operator is expressed in terms of the averaging operator, which circumvents the use of the Fourier transform. The Neumann series is formulated in a discrete setting, and the eigendecomposition of the iterated matrix is performed. The ensuing analysis shows that the local fields are computed using a particular subset of eigenvectors, the iteration error being governed by the associated eigenvalues. Quadratic error bounds on the effective property are also discussed. The discretization error is shown to be related to the accuracy of the trapezoidal quadrature scheme. These results are illustrated numerically, and their extension to other configurations is discussed.