Direct and inverse approximation for finite element solutions of the p and h-p versions and applications to three-dimensional propblem, nonlinear problems and Kirchhoff plate problem.

p 和 h-p 版本的有限元解的直接和逆近似以及在三维问题、非线性问题和基尔霍夫板问题中的应用。

• 批准号: RGPIN-2014-04642
• 负责人: Guo, Benqi
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611158
• 关键词: Direct; inverse; approximation; finite; element

Numerical Simulation plays an extremely important role for solving practical problems arising from various fields of engineering, science, finance, and health science as computer technology has rapidly developed in the recent decades. Numerical simulation involves numerical methods, error analysis, assessment of computed solutions, algorithms, and applications. The finite element method (FEM) is today's most adopted numerical method for numerical simulation, and the high-order (p and h-p) FEM provides reliable solutions with high accuracy and lower computational cost. My proposed research will focus on the p and h-p FEM in three dimensions and their applications to nonlinear problems and plate problems. The real world we are living is three-dimensional, and three dimensional models reflect precisely the nature of physical problems in the real world. Therefore Numerical simulation is of greater significance, and is also much more difficult for three-dimensional models than those simplified to one and two dimensions due to the complexity of geometry of polyhedral domains and anisotropic singularities of solutions. The optimal error estimation of the p and h-p version for problems on polyhedral domains is extremely difficult, and has been a challenging issue for decades. The inverse approximation of FE solution of p-version has been an untouched issue so far for three dimensional problems, which could indicate regularity of the solution based on the computed date to improve the efficiency of adaptive algorithms of p and h-p FEM. My proposed research will have important impact on the approximation theory of FEM and practical scientific/engineering computations in three dimensions. Corrosion is an electrochemical reaction with its environment, which causes severe damage to engineering structures such as ships, bridges, and metal containers. A commonly used technique in corrosion engineering is a so-called catholic protection system, which is modeled by a linear elliptical equation with a nonlinear Neumann boundary condition. In this system, impressed current is applied to control the electrical potential on the portion of surface of the engineering structure to be protected. The accurate numerical solution of the electrical potential is essential for the catholic protection. Since the Neumann condition is discontinuous at the changing point where the Neumann condition changes from a linear type to a nonlinear polarization functions, the singularity occurs there, and severely affects the global regularity and convergence of the FE solutions. My proposed research will investigate singularity near the changing point in a countably weighted Sobolev spaces and design a highly accurate h-p finite element method associated with geometric meshes. Obviously, the research will be of great impact on corrosion engineering as well as to the nonlinear analysis generally. Kirchhoff plate is a typical model of the plate problem, which is characterized by a biharmonic equation with various boundary conditions. It is known that the solutions have severe singularity of r^s-type with s >1 at vertices of the polygonal plate. The convergence of the p and h-p FE solutions with C0 and C1 continuity on a general quasi-uniform mesh will be analyzed. There are several key issues to be dealt with:(1) Optimal bound of approximation error in p and h for the singular function measured in the Sobolev H^2-norm on individual element;(2) Construction of a globally C1-continuous and piecewise polynomial without compromising the optimal error bound in the global H^2-norm;(3) Nonconforming p and h-p FEM with C0-elements for singular solution. The progress of the my proposed research will add new knowledge to numerical simulation for the plate problems.

数值模拟在解决工程，科学，金融和健康科学领域引起的实践问题方面起着极为重要的作用，因为近几十年来，计算机技术已经迅速发展。数值模拟涉及数值方法，错误分析，计算解决方案的评估，算法和应用。有限元方法（FEM）是当今用于数值模拟的数值方法，高阶（P和H-P）FEM提供具有高精度和较低计算成本的可靠解决方案。我提出的研究将在三个维度上的P和H-P FEM介绍，及其在非线性问题和板块问题上的应用。 我们所生活的现实世界是三维的，三维模型准确反映了现实世界中物理问题的本质。因此，由于多面体结构域的几何形状和解决方案的各向异性奇异性的复杂性，三维模型的数值模拟具有更大的重要性，对于三维模型而言，也更加困难。 P和H-P版本在多面体域上的问题的最佳误差估计非常困难，几十年来一直是一个艰巨的问题。到目前为止，对于三维问题，P-version的Fe解决方案的逆近似是一个未触及的问题，这可能表明基于计算日期的解决方案的规律性，以提高P和H-P FEM的自适应算法的效率。我提出的研究将对在三个维度上的FEM和实践科学/工程计算的近似理论产生重要影响。 腐蚀是对环境的电化学反应，它对工程结构（例如船舶，桥梁和金属容器）造成严重破坏。腐蚀工程中常用的技术是一种所谓的天主教保护系统，该系统由具有非线性Neumann边界条件的线性椭圆方程进行建模。在该系统中，印象深刻的电流用于控制要保护的工程结构表面部分的电势。电势的准确数值解决方案对于天主教保护至关重要。由于Neumann条件是不连续的，在变化点，Neumann条件从线性类型变为非线性极化函数，因此在那里发生奇异性，并严重影响FE溶液的全局规则性和收敛性。我提出的研究将在变化的Sobolev空间中调查变化点附近的奇异性，并设计一种与几何网格相关的高度精确的H-P有限元方法。显然，这项研究将对腐蚀工程以及一般非线性分析产生重大影响。 Kirchhoff板是板问题的典型模型，其特征是具有各种边界条件的Biharmonic方程。众所周知，在多边形板的顶点，溶液具有r^s型的严重奇异性。将分析P和H-P FE溶液与C0和C1连续性在一般准均匀网格上的收敛性。 There are several key issues to be dealt with:(1) Optimal bound of approximation error in p and h for the singular function measured in the Sobolev H^2-norm on individual element;(2) Construction of a globally C1-continuous and piecewise polynomial without compromising the optimal error bound in the global H^2-norm;(3) Nonconforming p and h-p FEM with C0-elements for singular solution.我提出的研究的进展将为板问题的数值模拟增加新知识。