Lippmann-Schwinger solvers for the explicit jump discretization for thermal computational homogenization problems

We present a variational formulation and a Lippmann-Schwinger equation for the explicit jump discretization of thermal computational homogenization problems, together with fast and memory-efficient matrix-free solvers based on the fast Fourier transform (FFT). Wiegmann and Zemitis introduced the explicit jump discretization for volumetric image-based computational homogenization of thermal conduction. In contrast to Fourier and finite difference-based discretization methods classically used in FFT-based homogenization, the explicit jump discretization is devoid of ringing and checkerboarding artifacts. Originally, the explicit jump discretization was formulated as the discrete equivalent of a boundary integral equation for the jump in the temperature gradient. The resulting equations are not symmetric positive definite, and thus solved by the BiCGSTAB method. Still, the numerical scheme exhibits stable convergence behavior, also in the presence of pores. In this work, we exploit a reformulation of the explicit jump system in terms of harmonically averaged conductivities. The resulting system is intrinsically symmetric positive definite and admits a Lippmann-Schwinger formulation. A seamless integration into existing FFT-based software packages is ensured. We demonstrate our improvements by numerical experiments.