基本解方法求解一个三维线弹性力学反问题

Method of fundamental solutions for a three-dimensional inverse problem in linear elasticity

将用于求解椭圆型偏微分方程边值问题的基本解方法应用于求解一个三维线弹性反问题,即Navier方程组的Cauchy问题.基本解方法离散方程所得的线性方程组是高度病态的,常见的求解方法如最小二乘法等无法得到合理的解.文中应用Tikhonov正则化和截断奇异值分解这两种正则化方法求解线性方程组,所需正则化参数则根据L-曲线确定,克服了问题的病态性.数值算例表明,本文方法能有效地求解三维线弹性力学反问题,而且这两种正则化方法所得到的结果精度相当.

The application of the method of fundamental solutions is considered for the numerical solution to a threedimensional inverse problem in linear elasticity, i. c. , the Cauchy problem associated with the Navier system. The coefficient matrix arising from the method of fundamental solutions is highly ill-posed, and standard methods for solving matrix equations fail to give an acceptable solution. Regularization methods, i. e. , the Tikhonov regularization method and truncated singular value decomposition are employed to solve the resulting matrix equations, with the regularization parameter determined by the L-curve method. Numerical experiments indicate that the method proposed can yield stable, accurate solutions to the inverse problem, and the solution is convergent with respect to decreasing amount of data noise. It＇s found that the two regularization methods lead to solutions of comparable accuracy.