Shape reconstruction in transient heat conduction problems based on radial integration boundary element method

Abstract

In order to accurately identify the geometric boundary, the radial integration boundary element method (RIBEM) combined with the modified Levenberg-Marquardt (LM) algorithm is proposed for shape reconstruction in transient heat conduction problems. Compared with the finite element method (FEM), the boundary element method (BEM) only discretizes the boundary rather than the whole domain, so it has incomparable advantages in the shape reconstruction. Especially, the radial integration method still maintains the superiority of boundary discretization in transient heat conduction problems. For the iterative scheme of LM algorithm, the shape reconstruction greatly depends on whether the sensitivity matrix can be obtained accurately. Therefore, complex variable derivation method (CVDM) is firstly introduced to carry out shape reconstruction in the transient heat conduction problems, which accurately solves the sensitivity coefficients of node temperature to the reconstructed boundary, and the RIBEM is transformed from the real domain to the complex domain to solve the transient heat conduction problems. The modified LM algorithm effectively overcomes the shortcoming of the excessive dependence on the step size of finite difference scheme. It should be noted that compared with the finite difference scheme, the proposed method can greatly reduce the computational cost of the direct problem in the multi-variate optimization process, thereby greatly improving the efficiency of identifying the boundary shape. Finally, different reconstruction strategies are applied in numerical models with three types of variable boundary, and reconstruction results demonstrate the great accuracy and efficiency of the proposed method in 2D and 3D shape reconstruction, and it also shows good robustness in the presence of measurement errors.