平面弹性问题位移—应力混合重心插值配点法

发布时间：2019-02-26 18:03

【摘要】：弹性力学问题可归结为二阶耦合椭圆形偏微分方程边值问题。工程中遇到的大部分问题都难以得到其解析解。为求解弹性力学方程,工程实际中广泛采用数值求解技术。本文提出数值分析平面弹性问题的位移-应力混合重心插值配点法。将弹性力学控制方程表达为位移和应力的耦合偏微分方程组,采用重心插值近似未知量,利用重心插值微分矩阵得到平面问题控制方程的矩阵形式离散表达式。使用重心插值离散位移和应力边界条件,采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,应用最小二乘法求解过约束方程组,得到平面弹性问题位移和应力数值解。对于不规则区域的弹性力学问题,采用重心Lagrange插值正则区域法,将不规则区域嵌入规则区域,在规则区域上采用重心Lagrange插值近似未知函数。利用配点法强迫微分方程在离散节点处精确成立,得到规则区域位移-应力混合方程组。在不规则区域的边界上取若干节点,由规则区域内的重心插值插值节点的未知函数,得到一个边界条件的约束代数方程。将位移-应力混合方程的离散方程和边界条件的约束方程组合成一个新的过约束代数方程组,应用最小二乘法求解过约束方程组,得到平面弹性问题位移和应力数值解。本文提供的5个规则区域的数值算例和4个不规则区域的数值算例结果表明:重心Lagrange插值配点法和重心插值正则区域法的运用,可以有效的解决规则区域和不规则区域的平面弹性问题。重心Lagrange插值配点法不仅计算公式简单、节点适应性好、程序通用性强、而且计算精度非常高。
[Abstract]:The elastic mechanics problem can be reduced to the boundary value problem of the second order coupled elliptic partial differential equation. Most of the problems encountered in the engineering are difficult to obtain its analytical solution. In order to solve elastic equation, numerical solution technology is widely used in engineering practice. In this paper, a displacement-stress mixed center of gravity interpolation method for numerical analysis of plane elastic problems is presented. The governing equations of elasticity are expressed as coupled partial differential equations of displacement and stress. The approximate unknown value of barycentric interpolation is used to obtain the matrix form discrete expression of governing equations of plane problems by using the differential matrix of barycentric interpolation. The boundary conditions of discrete displacement and stress are interpolated by the center of gravity, and the boundary conditions are imposed by the additional method. The overconstrained linear algebraic equations for solving the plane elastic problems are obtained, and the overconstrained equations are solved by the least square method. The displacement and stress numerical solutions of the plane elastic problem are obtained. For the elasticity problem of irregular regions, the barycentric Lagrange interpolation regular region method is used to embed irregular regions into regular regions, and the barycentric Lagrange interpolation is used to approximate unknown functions in regular regions. The collocation method is used to force the differential equation to be accurately established at the discrete nodes, and the displacement-stress mixed equations in the regular region are obtained. By taking some nodes on the boundary of irregular regions and from the unknown functions of barycentric interpolation nodes in regular regions, a constrained algebraic equation with boundary conditions is obtained. The discrete equation of the displacement-stress mixed equation and the constraint equation of the boundary conditions are combined into a new algebraic system of over-constraint. The numerical solution of displacement and stress of the plane elastic problem is obtained by using the least square method to solve the over-constrained equations. The numerical examples of five regular regions and four irregular regions are presented in this paper. The results show that the barycentric Lagrange interpolation method and the barycentric interpolation regular region method are used. It can effectively solve the plane elasticity problem of regular region and irregular region. The barycentric Lagrange interpolation collocation method not only has the advantages of simple calculation formula, good adaptability of nodes, strong generality of program, but also very high calculation precision.
【学位授予单位】：山东建筑大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82;O343

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