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Generalized finite difference method for solving two-dimensional non-linear obstacle problems

机译：二维非线性障碍物问题的广义有限差分法

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摘要

In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The CFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using CFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer-Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems.
机译：在这项研究中，障碍问题，也称为非线性自由边界问题，通过广义有限差分法（GFDM）和虚拟时间积分法（FTIM）进行了分析。 CFDM是一种新开发的域类型无网格方法，在本研究中用于空间离散化。使用CFDM可以避免网格生成和数值积分的任务，并且可以保持数值结果的高精度。障碍问题是很难用任何数值方案解决的，因为两种不同类型的控制方程被施加在计算域上，并且这两个区域之间的界面是未知的。障碍问题将在数学上公式化为非线性互补问题（NCP），然后将使用GFDM和Fischer-Burmeister NCP函数形成非线性代数方程组（NAE）。然后，使用FTIM作为NAE的简单而强大的求解器来求解NAE的系统。 FTIM无需计算雅可比矩阵的逆矩阵。提供了三个数值示例，以验证所提出的用于处理二维障碍问题的无网格数值方案的简单性和准确性。

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