The inverse methods based on S-FEMs with an adaptive SVD regularization technique for solving Cauchy inverse heat transfer problems

Abstract

This paper presents three inverse models based on S-FEMs with adaptive singular value decomposition (SVD) regularization technique for solving inverse heat transfer problems of Cauchy type with noisy input data “measured” on the domain boundary. The smoothed finite element methods (S-FEMs) are used as forward solvers, including edge-based S-FEM (ES-FEM), node-based S-FEM (NS-FEM) and cell-based S-FEM (CS-FEM that is the same as the standard FEM using T3 elements). First, the full-size S-FEM system equation is created based on the S-FEM theory. A set of local nodal temperature equations is then extracted from the full-size equations through a matrix partitioning operation, considering the nodes related to the heat flux on the Cauchy boundary. The Fourier heat convection theory is used to establish a relationship between the local nodal temperatures and the known heat fluxes on the Cauchy boundary. This procedure effectively converts a Cauchy type inverse problem to a forward-like problem with a set of ill-posed equations for inverse analyses. In order to mitigate the ill-posedness and to achieve a reliable and accurate inverse solution, we propose an adaptive SVD technique to regularize the solution, through minimizing the error between the approximate heat fluxes of the inverse model and the given (“measured”) flux values on the Neumann boundaries. The proposed adaptive procedure effectively determines the number of small singular values to be deleted, and hence the noises in the input data cannot be magnified. The present techniques are tested using a number of Cauchy type heat transfer problems, and it is concluded that our inverse procedure is much more effective compared to the widely used Tikhonov regularization technique, and it is a systematic procedure for accurate, reliable and stable inverse solutions with noisy input data.