Mathematical Sciences: Finite Element Superconvergence in Computational Mechanics

数学科学：计算力学中的有限元超收敛

• 批准号: 9626193
• 负责人: Zhimin Zhang
• 金额: $ 3.4万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626193&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Finite; Element; Superconvergence

9626193 Zhang The investigator studies: (1) Superconvergence points for some typical equations in computational mechanics on different mesh patterns and for various finite element bases; (2) Theoretical justification for superconvergence phenomena for newly developed, highly effective patch recovery techniques in the engineering community; (3) Superconvergence and its recovery for equations with small parameters that model some typical mechanical problems, such as beams, plates, shells, and the Stokes problem; (4) Superconvergence recovery technique for equations with singular solutions, and interior or local estimates for recovery techniques in the presence of boundary singularities. Extensive numerical tests are performed to evaluate the effectness of the proposed recovery techniques for the Poisson equation, the nearly incompressible elasticity equation, the Reissner-Mindlin plate model, and the Stokes equation for different finite element bases. The computational experimentation consider various mesh geometries and some domains with reentrant corners, such as the L-shaped domain and the cracked domain. The primary objective of this project is the study of derivative superconvergence and its recovery in finite element computations. Such study is of fundamental importance in computational mechanics, because engineers are interested in more effective ways to estimate the stress, strain energy, etc. At present, most of the commercial codes used by the engineering community are required to have adaptive capabilities, which allow computations to be carried out in the most efficient way. A knowledge of superconvergence is essential for this adaptivity design through a posteriori error estimates. This project provides a firm theoretical basis for developing reliable and computationally inexpensive superconvergent recovery techniques. The results of this research affect large-scale scientific computing on problems of practical interest arising in structural me chanics.

9626193张研究员研究了：(1)计算力学中一些典型方程在不同网格模式和各种有限元基上的超收敛点；(2)工程界新发展的高效面片恢复技术的超收敛现象的理论证明；(3)具有小参数的方程的超收敛及其恢复，例如梁、板、壳和Stokes问题；(4)具有奇异解的方程的超收敛恢复技术，以及在存在边界奇异性的情况下恢复技术的内部或局部估计。通过大量的数值试验，对泊松方程、近不可压弹性方程、Reissner-Mindlin板模型和Stokes方程的恢复技术在不同有限元基上的有效性进行了评估。计算实验考虑了不同的网格几何形状和一些具有凹角的区域，如L形域和裂纹域。本项目的主要目标是研究有限元计算中的导数超收敛及其恢复。这种研究在计算力学中具有基本的重要性，因为工程师们感兴趣的是更有效的方法来估计应力、应变能等。目前，工程界使用的大多数商业代码都要求具有自适应能力，使计算能够以最有效的方式进行。通过后验误差估计，超收敛的知识对于这种自适应设计是必不可少的。该项目为开发可靠、计算成本低的超收敛采油技术提供了坚实的理论基础。这项研究的结果影响了在结构力学中产生的实际感兴趣的问题的大规模科学计算。