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

Abstract

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. Copyright © 2011 John Wiley & Sons, Ltd.Keywords: sensitivity analysis; shape optimization; speed method, level set method IntroductionThe internal thermal properties of an object, the presence of cracks or voids, or the shape of some unknown portion of the boundary can be determined by a technique called thermal imaging. This technique is widely utilized in non-destructive testing and evaluation. A heat source is used on an object, and the resulting temperature response is observed near the object's surface. Thermal imaging has been significantly investigated as a method for detecting damage or corrosion in industrial machines, vehicles, or aircrafts. Industrial non-destructive testing uses this technique for broad range of materials ranging from composite materials to electronics [6,14]. We mention the reconstruction of small inclusions from boundary measurements of temperature [1] and study of conductivity interface problems by layer potential techniques [3]. These authors have studied also other types of thermal imaging.We elaborate a specific problem of crack, voids, and impurities identification inside an object with nonlinear thermal conductivity. We attempt to identify the inhomogeneities from the measurements of the heat equilibrium. The model equation is thus steady-state heat equation with unknown u and with coefficients nonlinearly dependent on u. or by f x . We frequently use the restriction of a function. Therefore to simplify the notation, we use the expression f 2 L 2 . / even for f : D ! R instead of a longer notation f j 2 L 2 . /. Mathematical modelD represents the object under consideration inside that there are some inhomogeneities. The domain represents these inhomogeneities. Note that can consist of several disjoint parts. Their number, position, and shape are to be determined. Consider a function u : D ! R representing a temperature distribution and assume that the function b : D R ! R is defined piecewise by where b 1 , b 2 are smooth nonlinear functions. The material occupying D n has nonlinear thermal conductivity represented by a nonlinear function b 2 , satisfying some properties listed later. In the case of crack or void identification, the domain representing the voids is filled with air, water, or some other liquid or gas. Forward model for steady-state temperature distributi...

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