Collaborative Research: The Least-Squares Meshfree Particle Finite Element

虽然有限元方法在解决工程和科学中的各种问题上取得了惊人的成功，但它也有明显的缺点：网格生成和网格重新划分非常困难和耗时。无网格方法可以通过完全根据一组节点来构造近似函数来避免这些困难。大多数无网格法都是基于Galerkin原理，并使用移动最小二乘近似来构造形函数。虽然在构造移动最小二乘形函数时不需要显式网格，但需要单独的背景网格来积分弱形式，因此它们不是真正的无网格方法。由于移动最小二乘近似的非插值性，Galerkin公式中本质边界条件的实施相当困难。此外，移动最小二乘逼近的计算量比有限元插值法的计算量大。在所提出的研究中，我们将发展一种结合最小二乘有限元方法和无网格质点方法特点的最小二乘无网格质点有限元方法。基于一阶微分方程组残差的L2范数最小化的最小二乘有限元方法(LSFEM)是一种简单、高效和稳健的方法，几乎可以用相同的数学和计算公式来求解任何类型的偏微分方程组。由于最小二乘法没有利用分块积分将区域积分转化为边界积分，而无网格质点法采用了通常的基于粒子的有限元内插，所有困扰基于Garlerkin的无网格法的问题都消失了。最小二乘无网格质点有限元方法总是得到一个对称的正定线性代数方程组。无矩阵逐个粒子共轭梯度法可以在并行计算机上求解非常大的问题，且实现简单。该项目的目的是开发一种新的计算机方法，以更高的精度和效率模拟复杂的工程设计和复杂的多物理过程。该项目的成果将使数值模拟在许多涉及国家利益的重要应用中超越目前的能力，包括汽车碰撞安全分析、汽车降噪、充满电池的能效、半导体器件的降热等。

Although the finite element method has been astonishingly successful in solving various problems in engineering and science, it has significant drawbacks: mesh generation and remeshing are very difficult and time-consuming. Meshfree methods may avoid these difficulties by constructing approximation functions entirely in terms of a set of nodes. Most meshfree methods are based on the Galerkin principle and employ moving least-squares approximation for the construction of shape functions. Although there is no need for an explicit mesh in the construction of moving least-squares shape functions, a separate background mesh is required to integrate the weak form, so they are not truly meshfree methods. Due to the non-interpolative character of the moving least-squares approximation, the enforcement of essential boundary conditions in the Galerkin formulation is quite awkward. Moreover, the moving least-squares approximation is more expensive computationally than the finite element interpolation. In the proposed research, we will develop a least-squares meshfree particle finite element method which combines the features of the least-squares finite element method and the meshfree particle method. The least-squares finite element method (LSFEM), based on minimization of the L2 norm of the residuals of a first-order system of differential equations, is a simple, efficient and robust technique, and can solve almost any kind of partial differential equation with the same mathematical/computational formulation. Since the least-squares method doesn't make use of the integration by parts for converting domain integration into boundary integration, and the meshfree particle method employs the usual finite element interpolations based on particles, all troubles that plague the Garlerkin-based meshfree methods disappear. The least-squares meshfree particle finite element method always leads to a symmetric positive definite system of linear algebraic equations. The matrix-free particle-by-particle conjugate gradient method can be used to solve very large problems on parallel computers, and the implementation is straightforward.. The purpose of this project is to develop a new computer method to simulate complicated engineering designs and sophisticated multi-physical processes with much greater accuracy and efficiency. Achievements of this project would enable numerical simulations beyond current capabilities in many important applications of national interest, including car crash safety analysis, noise reduction of cars, energy efficiency in full cells, heat reduction in semiconductor devices, etc.