Inverse thermal imaging in materials with nonlinear conductivity by material and shape derivative method

摘要

The material and shape derivative method is used for an inverse problem in thermal imaging. The goal is to identify the boundary of unknown inclusions inside an object by applying a heat source and measuring the induced temperature near the boundary of the sample. The problem is studied in the framework of quasilinear elliptic equations. The explicit form is derived of the equations that are satisfied by material and shape derivatives. The existence of weak material derivative is proved. These general findings are demonstrated on the steepest descent optimization procedure. Simulations involving the level set method for tracing the interface are performed for several materials with nonlinear heat conductivity.