Collaborative Research: The Least-Squares Meshfree Particle Finite Element Method

合作研究：最小二乘无网格粒子有限元法

• 批准号: 0310609
• 负责人: Bo-Nan Jiang
• 金额: $ 12万
• 依托单位: Oakland University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0310609&HistoricalAwards=false
• 关键词: Collaborative; Research; Least; Squares; Meshfree

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.

虽然有限元法在解决工程和科学中的各种问题方面取得了巨大的成功，但它也有明显的缺点：网格生成和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构造近似函数来避免这些困难。大多数无网格方法是基于伽辽金原理，并采用移动最小二乘近似的形状函数的建设。虽然在构造移动最小二乘形状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质，Galerkin公式中本质边界条件的施加是相当困难的。此外，移动最小二乘近似是更昂贵的计算比有限元插值。在本研究中，我们将结合最小二乘有限元法与无网格质点法的特性，发展一种最小二乘无网格质点有限元法。最小二乘有限元法（LSFEM）基于一阶微分方程组残差的L2范数最小化，是一种简单、高效和鲁棒的方法，可以用相同的数学/计算公式求解几乎任何类型的偏微分方程。 由于最小二乘法不采用分部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质点的有限元插值，因此消除了基于Garlerkin的无网格法的所有问题。最小二乘无网格质点有限元法总是导出一个对称正定的线性代数方程组。无矩阵逐粒子共轭梯度法可用于在并行计算机上解决非常大的问题，并且实现简单。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来模拟复杂的工程设计和复杂的多物理过程。这一项目的成果将使国家利益的许多重要应用，包括汽车碰撞安全分析、汽车降噪、全电池能效、半导体器件散热等方面的数值模拟超越目前的能力。