用于任意网格上稳健有限元分析的非连续分片多项式广义移动最小二乘法方案

IF 8.7 2区 工程技术 Q1 Mathematics Engineering with Computers Pub Date : 2024-08-16 DOI:10.1007/s00366-024-02036-5
Paul Kuberry, Pavel Bochev, Jacob Koester, Nathaniel Trask
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引用次数: 0

摘要

本研究开发了一种无网格离散化的变分法,可对网格质量较差的偏微分方程进行精确、稳健的数值模拟。传统的有限元方法使用网格来离散几何域和定义有限元形状函数。后者在网格质量和有限元基础属性之间产生了依赖关系,可能会对离散化问题的精度产生不利影响。我们提出了一种定义有限元形状函数的新方法,它打破了这种依赖关系,将网格质量与离散化质量分离开来,我们称之为非连续片断多项式广义移动最小二乘法(DPP-GMLS)。该方法的核心是形状函数的无网格定义,它将网格的目的限制在表示几何域和积分基函数上,而不影响其近似质量。由此产生的非顺应空间可以在标准的非连续 Galerkin 框架内使用,为在低质量网格上求解偏微分方程提供了严格的基础。我们展示了一系列数值实验,证明我们的方法适用于广泛的环境:强胁迫椭圆问题、可压缩状态下的线性弹性问题和静态斯托克斯问题。我们证明了所有问题的收敛性和元素对的稳定性,而这些问题通常需要符合方法的 inf-sup 兼容性,我们还提到了通过对称内部惩罚 Galerkin 框架进行的微小修改,以稳定传统上不稳定的元素对。与连续 Galerkin 有限元方法相比,我们提供了一个例子,说明我们的方法在精度上提供了大于 5(\times)的改进,并允许对高度变形的网格采取 8(\times)更大的稳定时间步。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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A discontinuous piecewise polynomial generalized moving least squares scheme for robust finite element analysis on arbitrary grids

A variational approach is developed with a meshless discretization to enable accurate and robust numerical simulation of partial differential equations for meshes that are of poor quality. Traditional finite element methods use the mesh to both discretize the geometric domain and to define the finite element shape functions. The latter creates a dependence between the quality of the mesh and the properties of the finite element basis that may adversely affect the accuracy of the discretized problem. We propose a new approach for defining finite element shape functions that breaks this dependence and separates mesh quality from the discretization quality, which we call discontinuous piecewise polynomial generalized moving least squares (DPP-GMLS). At the core of the approach is a meshless definition of the shape functions, which limits the purpose of the mesh to representing the geometric domain and integrating the basis functions without having any role in their approximation quality. The resulting non-conforming space can be utilized within a standard discontinuous Galerkin framework, providing a rigorous foundation for solving partial differential equations on low-quality meshes. We present a collection of numerical experiments demonstrating our approach in a wide range of settings: strongly coercive elliptic problems, linear elasticity in the compressible regime, and the stationary Stokes problem. We demonstrate convergence for all problems and stability for element pairs for problems which usually require inf-sup compatibility for conforming methods, also referring to a minor modification possible through the symmetric interior penalty Galerkin framework for stabilizing element pairs that would otherwise be traditionally unstable. Mesh robustness is particularly critical for elasticity, and we provide an example that our approach provides a greater than 5\(\times\) improvement in accuracy and allows for taking an 8\(\times\) larger stable timestep for a highly deformed mesh, compared to the continuous Galerkin finite element method.

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来源期刊
Engineering with Computers
Engineering with Computers 工程技术-工程:机械
CiteScore
16.50
自引率
2.30%
发文量
203
审稿时长
9 months
期刊介绍: Engineering with Computers is an international journal dedicated to simulation-based engineering. It features original papers and comprehensive reviews on technologies supporting simulation-based engineering, along with demonstrations of operational simulation-based engineering systems. The journal covers various technical areas such as adaptive simulation techniques, engineering databases, CAD geometry integration, mesh generation, parallel simulation methods, simulation frameworks, user interface technologies, and visualization techniques. It also encompasses a wide range of application areas where engineering technologies are applied, spanning from automotive industry applications to medical device design.
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