Abstract
AbstractWe present a Lippmann‐Schwinger equation for the explicit jump discretization of thermal computational homogenization. Our solution scheme is based on the fast Fourier transform and thus fast and memory‐efficient. We reformulate the explicit jump discretization using harmonically averaged thermal conductivities and obtain a symmetric positive definite system. Thus, a Lippmann‐Schwinger formulation is possible. In contrast to Fourier and finite difference based discretization methods the explicit jump discretization does not exhibit ringing and checkerboarding artifacts.
Highlights
A conductivity tensor field A : Y → Sym(d) on a rectangular box Y ⊆ Rd, d = (2, 3) and a macroscopic temperature gradient ξare given
We present a Lippmann-Schwinger equation for the explicit jump discretization of thermal computational homogenization
Besides the continuous and symmetric formulation of the corrector equation as a partial differential equation presented in Eq (1) it is possible to formulate the problem of heat conduction into a continuous but non-symmetric boundary integral equation
Summary
A conductivity tensor field A : Y → Sym(d) on a rectangular box Y ⊆ Rd, d = (2, 3) and a macroscopic temperature gradient ξare given. We seek a periodic temperature field θ : Y → R solving the corrector equation div A(ξ + ∇θ) = 0. Besides the continuous and symmetric formulation of the corrector equation as a partial differential equation presented in Eq (1) it is possible to formulate the problem of heat conduction into a continuous but non-symmetric boundary integral equation. Wiegmann and Zemitis used a discrete version of this boundary integral formulation for the Explicit Jump Immersed Interface Method (EJIIM) [1]. Since the resulting equations of this formulation are non-symmetric they are solved using BiCGSTAB. We want to reformulate the explicit jump discretization into a symmetric and positive definite system which enables us to use CG while retaining to positive features of explicit jump
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