We propose in this paper an effective meshless and integration-free method for the numerical solution of multidimensional inverse heat conduction problems. Due to the use of fundamental solutions as basis functions, the method leads to a global approximation scheme in both the spatial and time domains. To tackle the ill-conditioning problem of the resultant linear system of equations, we apply the Tikhonov regularization method based on the generalized cross-validation criterion for choosing the regularization parameter to obtain a stable approximation to the solution. The effectiveness of the algorithm is illustrated by several numerical two and three-dimensional examples.

The method of fundamental solution for solving multidimensional inverse heat conduction problems

Solution of inverse heat conduction problems using control volume approach

The meshless local Petrov–Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2-D and 3-D axisymmetric bodies. A 3-D axisymmetric body is generated by rotating a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduce the original 3-D boundary value problem to a 2-D problem. The analyzed domain is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIEs) on the boundaries of the chosen subdomains. The time integration schemes are formulated based on the Laplace transform technique and the time difference approach, respectively. The local integral equations are non-singular and take a very simple form. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. Singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

Inverse heat conduction problems by meshless local Petrov–Galerkin method

Estimation of directional-dependent thermal properties in a carbon-carbon composite

Evaluation of heat transfer coefficient during heat treatment by inverse analysis

This paper presents a finite element algorithm for two-dimensional nonlinear inverse heat conduction analysis. The proposed method is capable of handling both unknown surface heat flux and unknown surface temperature of solids using temperature histories measured at a few discrete points. The proposed algorithms were used in the study of the thermofracture behavior of leaking pipelines with experimental verifications.

N. S. Sun

Papers

Finite Element Formulation for Two-Dimensional Inverse Heat Conduction Analysis