The backward substitution method is a recently proposed semi-analytical meshless collocation method. The main idea of the back substitution method is to obtain the boundary approximation from the boundary data, and this approximation does not need to satisfy the governing equation. Then, the traditional linear combination of basis functions is used to construct a correcting approximation that satisfies the homogeneous boundary conditions, the numerical solution is given by a linear combination system of boundary approximation and modified approximation, which satisfies the governing equation. In this way, the backward substitution method can be easily used to solve general problems without fundamental solutions or particular solutions. This paper makes the first attempt to extend the backward substitution method to the simulation of inverse Cauchy problems in piezoelectric structures. For the inverse Cauchy problems, the absence of data on some boundaries often leads to extreme instability of numerical solutions. A simple method is applied in this paper. For the points on the boundary without boundary information, the governing equation is enforced. Numerical results demonstrate that it will greatly increase the stability of the solution process.

中文翻译：

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电弹性压电结构中柯西反问题的无网格后向代入法


后向替代法是最近提出的一种半解析无网格配置方法。回代法的主要思想是从边界数据中得到边界近似，并且该近似不需要满足控制方程。然后，利用传统的基函数线性组合构造满足齐次边界条件的修正近似，通过边界近似和修正近似的线性组合系统给出数值解，满足控制方程。这样，可以很容易地用后向代入法来解决一般问题，而无需根本解或特殊解。本文首次尝试将后向代入法推广到压电结构中柯西反问题的模拟。对于反柯西问题，某些边界上缺乏数据通常会导致数值解的极度不稳定。本文采用了一种简单的方法。对于边界上没有边界信息的点，强制执行控制方程。数值结果表明，它将大大提高求解过程的稳定性。