What Is a Good Linear Finite Element? Interpolation, Conditioning, Anisotropy, and Quality Measures
What Is a Good Linear Finite Element? Interpolation, Conditioning, Anisotropy, and Quality Measures
复制标题
DOI：
--
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
J. Shewchuk
中科院分区：
文献类型：
--
作者：
J. Shewchuk
当使用单纯元素（三角形或四面体）的网格形成函数的分段线性近似时，近似的精度取决于元素的大小和形状。在有限元方法中，刚度矩阵的调节还取决于单元的尺寸和形状。本文解释了网格几何、插值误差、离散化误差和刚度矩阵条件之间的数学联系。这些关系通过误差界限和元素质量度量来表达，这些度量确定三角形或四面体对于插值或实现低条件数的适合度。不幸的是，用于这些目的的质量测量并不完全一致；例如，小角度不利于矩阵调节，但不利于插值或离散化。这里给出的插值误差和单元刚度矩阵条件的上限和下限比文献中通常看到的更严格，因此质量度量可能是单元适应性的异常精确的指标。边界包含各向异性情况，其中长而薄的单元比等边单元的性能更好。令人惊讶的是，在某些情况下，不同纵横比或方向的元素可以最好地满足插值、调节和离散化误差的要求。部分由国家科学基金会 ACI-9875170、CMS-9980063、CCR-0204377 和 EIA9802069 奖项支持，部分由大川基金会捐赠。本文件中的观点和结论是作者的观点和结论。它们并未得到大川基金会或美国政府的认可，也不一定反映其立场或政策。
When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations. Supported in part by the National Science Foundation under Awards ACI-9875170, CMS-9980063, CCR-0204377, and EIA9802069, and in part by a gift from the Okawa Foundation. The views and conclusions in this document are those of the author. They are not endorsed by, and do not necessarily reflect the position or policies of, the Okawa Foundation or the U. S. Government.