Computational Solution of an Inverse Boundary-Value Problem for Heat Transfer in a Composite Material

In the numerical simulation of composite material models, it is often necessary to recover boundary values of the solution to parabolic problems from integral constraints. In this work, we consider an inverse problem of determining a Dirichlet boundary condition for a heat equation with multiple interfaces and integral overspecification on a part of the spatial domain. After establishing the well-posedness of the direct problem, we propose an efficient numerical method for identifying an unknown Dirichlet boundary condition. The method decomposes the global inverse problem into a sequence of local subproblems, solved independently within each layer, including the accurate reconstruction of the solution at the interfaces. The approach relies solely on explicit schemes, employing an unconditionally stable Saulyev-type discretization and a novel interface treatment that avoids matrix inversion. Results from numerical experiments are presented and discussed.