Abstract In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.

中文翻译：

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求解复合结构热传导问题的局部最小二乘元微分法

摘要 本文提出了一种全新的数值方法，称为局部最小二乘元微分法（LSEDM），用于求解二阶偏微分方程控制的一般工程问题。该方法是一种强形式有限元方法。在该方法中，等参单元相对于全局坐标的一组微分公式被用来配置所考虑问题的控制微分方程和诺依曼边界条件，以生成配置单元的内部节点和边界节点的方程组. 对于每个外边界或单元界面，使用 Neumann 边界条件生成一个方程，因此可以为与多个单元界面关联的每个节点生成多个方程。最小二乘法用于通过在最小二乘法公式中优化局部物理变量，将这些界面方程转换为一个方程。因此，求解系统具有与当前热传导问题的节点总数一样多的方程。所提出的 LSEDM 最终可以保证跨单元表面的热通量的保守性，并且可以有效地提高单元微分法在解决材料特性差异很大的问题时的解稳定性，这是无网格方法中的一个具有挑战性的问题。给出了二维和三维热传导问题的数值例子，以证明所提出方法的稳定性和效率。