Lippmann–Schwinger Spectrum, Composite Materials Eigenstates and Their Role in Computational Homogenization

Focusing on the homogenization of periodic composite materials, this study investigates computational methods based on volume integral equations. Such formulations are revisited from the standpoint of the preconditioning of the original cell problem by the introduction of a comparison material. This allows for to recovery of simple convergence criteria for iterative steepest-descent and fixed-point schemes for composites with general non-linear behaviour. In the case of linear materials, the preconditioned volume integral formulation coincides with the well-known Lippmann–Schwinger equation. The spectral properties of the featured linear integral operator, which is bounded and self-adjoint, are investigated to shed light on the behaviour of conventional computational homogenization methods. The so-called Lippmann–Schwinger spectrum is analyzed, with its bounds governing the convergence rate of iterative solution methods. The associated eigenvectors, which constitute the eigenstates of the composite material considered, are also described in detail to understand their role in constructing the solution to the cell problem and ultimately in computing the effective properties. Formulated in the continuous setting, this analysis is followed by the investigation of a discrete representation of the integral operator considered. A number of examples on synthetic microstructures are finally considered in the conductivity setting to illustrate the obtained theoretical results and highlight the role of the spectral properties in the operation of computational homogenization methods. This paves the way for the development of reduced models and more efficient computations.