Solution of geometric inverse heat conduction problems by smoothed fixed grid finite element method

This paper presents a methodology for solving the nonlinear inverse geometry heat transfer problems where the observations are temperatures measured at points on the external boundary and the unknowns are the shape and the location of a cavity inside the problem domain. The algorithm is based on the minimization of the squared error between the measured temperatures and those calculated by the proposed smoothed fixed grid finite element method where a non-boundary-fitted mesh is used to solve the direct problem. This eliminates mesh adaptation and re-meshing processes as needed in the standard finite element method and reduces the analysis cost significantly. The proposed method uses the advantages of smoothed finite element method and fixed grid finite element method in which a fixed computational mesh is used to solve the variable domain problem and the numerical integration is performed through gradient smoothing technique. The present method is more accurate and facilitates integration over boundary intersecting elements. The shape gradients of the shape identification problem are obtained by direct differentiation method. The domain parameterization technique with cubic splines is adopted to manipulate the shape variation of the cavity. To stabilize the optimization process, a geometric regularization function is used. Some numerical examples are solved to evaluate the method. In these examples effects of initial guess, measurement errors, cavity shape and cavity size on the results are examined. It is shown that the results are in good agreement with exact solutions.