极坐标系下弹性问题的重心插值配点法

Barycentric interpolation collocation method for solving elastic problems

针对岩土工程中的孔洞及曲梁问题,提出一种在极坐标系下求解二维弹性问题的重心插值配点法。该方法分别在r和θ方向分别布置m和n个节点,生成求解区域上的节点。以一维重心Lagrange插值的张量积插值形式近似二维弹性问题的位移函数,代入位移表达的平衡方程和边界条件,平衡方程和边界条件分别在所有的计算节点和边界节点上精确成立,得到极坐标下弹性力学平衡方程和边界条件的离散代数表达式。利用一维重心Lagrange插值微分矩阵,将离散的平衡方程和边界条件表达为矩阵形式。利用置换法施加边界条件,求得在计算节点处的位移,进而通过微分矩阵直接求得计算节点处的应力。数值算例表明:极坐标下重心插值配点法具有计算格式简单、程序实施容易和计算精度高的特点。

Aiming at the hole and curved beam problem of geotechnical engineering, barycentric interpolation collocation method, a high accuracy meshless method for solving elastic problems in polar coordinate system was presented. The computational nodes were generated by tensor product from m and n nodes located in r and 0 directions, respectively. The functions of displacement were approximated by tensor product of barycentric Lagrange interpolation in one dimension （1D）. The collocation method was used to obtain the systems of algebraic equations of governing equations and boundary conditions. The simple matrix form of the systems of algebraic equations was given using the notation of differential matrices from barycentric Lagrange interpolation in 1D. The boundary conditions were imposed by replacement method. The numerical results of displacements on nodes were obtained by solving the systems of algebraic equations of governing equations. The stresses values on nodes were directly computed using differential matrices. The numerical examples demonstrate that the proposed method has advantages of simple computational formulations, being easy to program and high precision.