自由表面问题和分片检验函数的数学方法研究
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摘要
力学的发展不断提出数学问题,力学与数学的结合促进了它们自身的发展和新领域的诞生。如何利用现代数学知识解决力学问题已经成为力学和数学工作者共同的目标。另外利用现代数学知识解决历史遗留的数学问题也为人们所关注。本文本着利用新知识解决问题的思想,根据现代数学知识的发展,从三个方面分别讨论了以下力学问题:
一求解渗流自由面:有自由面的渗流问题是工程中经常遇到的问题,自由表面要同时满足水头函数和压力条件而其位置不能预先给定,它的互补和非线性性质给问题的求解带来很大的困难,而且表现为非光滑的形式。本文根据近些年来求解可动边界的经验和非光滑分析的发展,建立了求解渗流自由面的数学模型,提出了求解渗流自由面的有限元混合不动点法和非光滑牛顿法。
基于有自由面渗流问题的高斯点法,建立了求解渗流问题的非光滑非线性方程组模型和求解此类问题的混合不动点法,此类方法属固定网格法,只需划分一次网格,不需要对数据做任何近似处理,完全利用计算机数值计算确定渗流自由面。本文讨论了非光滑方程组解的存在性和不动点算法的收敛性。通过结点压强插值绘制出渗流自由面。算例结果表明,该方法简单且收敛速度快。本文对不动点法的收敛性分析为迭代法的收敛提供了理论依据。
在前面建立的非光滑方程组数学模型和固定网格法基础上,利用广义导数的概念给出了求解渗流自由面的一种新方法-----非光滑阻尼牛顿法,该法是对非光滑方程组求导,适当的处理广义导数矩阵使其非奇异,利用非光滑牛顿法求解。
二变系数KdV方程解析解:浅水波问题原本属于带自由表面波的传播问题,原则上可以按上面的方法建立非光滑非线性方程组的数学模型。浅水波理论经过长期研究,按摄动展开已经建立了一套浅水波的非线性偏微分方程理论,本文旨在深入研究非线性偏微分方程表示的浅水波方程,首次开展空间变系数非线性偏微分方程解析解研究,在求解变水深KdV方程时,引入非线性变换,求出了一类变水深KdV方程的解析解。
三有限元增强型分片检验:Mindlin板和圆柱薄壳有限元法一直没有完整的分片检验提法。本文首次提出并建立了Mindlin板和圆柱薄壳有限元增强型分片检验的检验函数。证明了常规轴对称有限元检验函数不含常剪应变项,常应力分片检验的剪应变必为零。轴对称C~1偶应力有限元检验函数为二次函数,本文证明二次检验函数也不含常剪应变项。
With the development of Mechanics, many mathematic problems were presented. Combination mathematics with mechanics propels them forward and new fields arise to meet requirements. How to solve Mechanical problems by using mathematical knowledge is a common object for both Mechanical and mathematical researchers. Moreover, using modern mathematical approaches to settle historical mathematical problems is also a noticeable subject. Based on the new knowledge and the development of mathematics, the thesis discusses the following Mechanical problems:
Solving the seepage free surface problem: Seepage problems with free surface are usually met in engineering. The free surface satisfies the water head function as well as the pore pressure conditions, and its location is unknown beforehand. It is very difficult to identify seepage free Surface due to the strong nonlinearity. Based on the experience in action boundary and the development of nonsmooth analysis, the thesis establishes the nonsmooth equations model for seepage problem .At the same time, the mix fixed point method and the nonsmooth Newton method for finite element method are also presented
Based on the Gauss point method for seepage problem with free surface, the non-smooth equations model for seepage problem is proposed as well as the mix fixed point method which belongs to the fixed mesh method. The seepage free surface is determined by using computer numerical computation with only once mesh plotting and no other approximate processing for the data. This thesis also discusses the existence of the solution of non-smooth equations and the convergence of the proposed fixed point method. Moreover, the free surface of seepage is plotted through interpolation of pressure intensity on the nodes. Numerical simulation shows that the new method is simple and possesses rapid convergence rate. It also offers a theoretical base for convergence analysis of iterative method.
For the previous nonsmooth equations model and fixed mesh method, a new nonsmooth damped Newton method is given based on the definition of the generalized derivative. The derivatives of the nonsmooth equations are computed to make the generalized derivative matrix nonsingular. Then the nonsmmoth damped Newton method is used to solve them.
Analytic solutions of the KdV equation with variable coefficients: Shallow water wave problems belong to problems of free surface wave, and can be molded by the nonsmooth equations model given above in principle. After a long research of shallow water wave, a set of nonlinear partial differential equation theory has been established base on perturbation expanding. Aiming at the analytic solution of partial differential equation of shallow water wave, this thesis firstly studies the analytic solution of the KdV equation with space variable. The nonlinear transformation is introduced during solving the KdV equation with variable depth, and a class of analytic solution of KdV equations for variable depth is solved.
Enhanced patch test of finite element methods: There is no complete patch test proposed for the finite element analysis on Mindlin plate and thin cylindrical shell before. In this thesis, an enhanced patch test function for Mindlin plate and thin cylindrical shell elements is proposed. It also proves that patch test function for axisymmetric element can not contain constant shear. Shear of constant stess patch test must be zero. The patch test function for C~1 axisymmetric couple stress is quadratic function. And it proves that the quadratic patch test function does not contain constant shear strain term either.
一求解渗流自由面:有自由面的渗流问题是工程中经常遇到的问题,自由表面要同时满足水头函数和压力条件而其位置不能预先给定,它的互补和非线性性质给问题的求解带来很大的困难,而且表现为非光滑的形式。本文根据近些年来求解可动边界的经验和非光滑分析的发展,建立了求解渗流自由面的数学模型,提出了求解渗流自由面的有限元混合不动点法和非光滑牛顿法。
基于有自由面渗流问题的高斯点法,建立了求解渗流问题的非光滑非线性方程组模型和求解此类问题的混合不动点法,此类方法属固定网格法,只需划分一次网格,不需要对数据做任何近似处理,完全利用计算机数值计算确定渗流自由面。本文讨论了非光滑方程组解的存在性和不动点算法的收敛性。通过结点压强插值绘制出渗流自由面。算例结果表明,该方法简单且收敛速度快。本文对不动点法的收敛性分析为迭代法的收敛提供了理论依据。
在前面建立的非光滑方程组数学模型和固定网格法基础上,利用广义导数的概念给出了求解渗流自由面的一种新方法-----非光滑阻尼牛顿法,该法是对非光滑方程组求导,适当的处理广义导数矩阵使其非奇异,利用非光滑牛顿法求解。
二变系数KdV方程解析解:浅水波问题原本属于带自由表面波的传播问题,原则上可以按上面的方法建立非光滑非线性方程组的数学模型。浅水波理论经过长期研究,按摄动展开已经建立了一套浅水波的非线性偏微分方程理论,本文旨在深入研究非线性偏微分方程表示的浅水波方程,首次开展空间变系数非线性偏微分方程解析解研究,在求解变水深KdV方程时,引入非线性变换,求出了一类变水深KdV方程的解析解。
三有限元增强型分片检验:Mindlin板和圆柱薄壳有限元法一直没有完整的分片检验提法。本文首次提出并建立了Mindlin板和圆柱薄壳有限元增强型分片检验的检验函数。证明了常规轴对称有限元检验函数不含常剪应变项,常应力分片检验的剪应变必为零。轴对称C~1偶应力有限元检验函数为二次函数,本文证明二次检验函数也不含常剪应变项。
With the development of Mechanics, many mathematic problems were presented. Combination mathematics with mechanics propels them forward and new fields arise to meet requirements. How to solve Mechanical problems by using mathematical knowledge is a common object for both Mechanical and mathematical researchers. Moreover, using modern mathematical approaches to settle historical mathematical problems is also a noticeable subject. Based on the new knowledge and the development of mathematics, the thesis discusses the following Mechanical problems:
Solving the seepage free surface problem: Seepage problems with free surface are usually met in engineering. The free surface satisfies the water head function as well as the pore pressure conditions, and its location is unknown beforehand. It is very difficult to identify seepage free Surface due to the strong nonlinearity. Based on the experience in action boundary and the development of nonsmooth analysis, the thesis establishes the nonsmooth equations model for seepage problem .At the same time, the mix fixed point method and the nonsmooth Newton method for finite element method are also presented
Based on the Gauss point method for seepage problem with free surface, the non-smooth equations model for seepage problem is proposed as well as the mix fixed point method which belongs to the fixed mesh method. The seepage free surface is determined by using computer numerical computation with only once mesh plotting and no other approximate processing for the data. This thesis also discusses the existence of the solution of non-smooth equations and the convergence of the proposed fixed point method. Moreover, the free surface of seepage is plotted through interpolation of pressure intensity on the nodes. Numerical simulation shows that the new method is simple and possesses rapid convergence rate. It also offers a theoretical base for convergence analysis of iterative method.
For the previous nonsmooth equations model and fixed mesh method, a new nonsmooth damped Newton method is given based on the definition of the generalized derivative. The derivatives of the nonsmooth equations are computed to make the generalized derivative matrix nonsingular. Then the nonsmmoth damped Newton method is used to solve them.
Analytic solutions of the KdV equation with variable coefficients: Shallow water wave problems belong to problems of free surface wave, and can be molded by the nonsmooth equations model given above in principle. After a long research of shallow water wave, a set of nonlinear partial differential equation theory has been established base on perturbation expanding. Aiming at the analytic solution of partial differential equation of shallow water wave, this thesis firstly studies the analytic solution of the KdV equation with space variable. The nonlinear transformation is introduced during solving the KdV equation with variable depth, and a class of analytic solution of KdV equations for variable depth is solved.
Enhanced patch test of finite element methods: There is no complete patch test proposed for the finite element analysis on Mindlin plate and thin cylindrical shell before. In this thesis, an enhanced patch test function for Mindlin plate and thin cylindrical shell elements is proposed. It also proves that patch test function for axisymmetric element can not contain constant shear. Shear of constant stess patch test must be zero. The patch test function for C~1 axisymmetric couple stress is quadratic function. And it proves that the quadratic patch test function does not contain constant shear strain term either.
引文
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