一种求解平面热传导反问题的新型无网格方法

A New Boundary Meshless Method for the Heat Conduction Inverse Problem

平均源边界点法(average source boundary node method,ASBNM)是一种新型无网格方法。采用该方法可避免边界元方法中的强弱奇异积分计算,克服了基本解法中的虚假边界问题。首次采用平均源边界点法与截断奇异值分解(TSVD)和Tikhonov正则化技术相结合模拟平面热传导Cauchy反问题,通过广义交叉校验准则(GCV)来确定正则化参数。提出的无网格方法基于一种完全规则化边界积分方程,通过加减去奇异和平均积分的思想,消除了基本解的源点奇异性,具有无网格、无积分、仅需边界离散、半解析的特性。3个典型数值算例的结果表明:该方法在求解平面热传导反问题时具有简单、精确、稳定的优势,即使边界数据噪音水平达到5%,仍可获得高精度的数值解,对平面热传导反问题的研究具有参考意义,并拓展了平均源边界点法的应用领域。

Average source boundary node method（ ASBNM）,a recently novel boundary meshfree technique,can efficiently avoid the calculations of the weakly and strongly singular integrals in the boundary element methods,and overcome the fictitious boundary issue in the method of fundamental solutions. This paper firstly attempts to deal with the two-dimensional inverse heat conduction problems by using the ASBNM. In the process,the truncated singular value decomposition and Tikhonov regularization techniques combining with the generalized cross validation criterion are employed to solve the resulting matrix equation which is highly ill-conditioned. The proposed method is based on a completely regularized boundary integral equation with direct variables,removes the singularity computation via the sum of the off-diagonal elements and an average source technique,andtherefore is simple and easy-to-implement due to its boundary-only discretization and semi-analytical nature. Three benchmark numerical examples indicate that the proposed method is computationally efficient and numerically stable for the solution of the two-dimensional inverse heat conduction problems. Even if the level of noise in input data reach to 5%,the method still can obtain a very highly accurate numerical solution. It provides a new path for dealing with such problems.Meanwhile,it extends the application field of the ASBNM.