梁方程降阶计算的重心插值配点法  

Barycentric interpolation collocation method based on depression of order for solving beam equations

作　　者：徐子康[1] 王兆清[1] 孙浩森[2] 李金[3] 

机构地区：[1]山东建筑大学工程力学研究所,山东济南250101 [2]山东建筑大学学报编辑部,山东济南250101 [3]山东建筑大学理学院,山东济南250101

出　　处：《山东建筑大学学报》2017年第3期245-250,共6页Journal of Shandong Jianzhu University

基　　金：国家自然科学基金面上项目(51379113);国家自然科学基金项目(11471195);山东省自然科学基金重点项目(ZR2016JL006)

摘　　要：采用重心插值配点法求解梁方程时,随着计算节点数量的持续增加,其计算精度将逐步下降。通过对降阶计算重心插值配点法的研究,可为数值求解梁方程提供一种数值稳定性好、计算精度高的新方法。文章基于重心Lagrange插值及其微分矩阵,推导了梁方程降阶计算重心插值配点法的公式,并通过数值算例验证其有效性。结果表明:随着计算节点数量的持续增加,降阶法的计算精度仍保持在10-10~10-12范围内;求解两端简支的梁方程时,两步降阶法的计算精度高于一步降阶法;直接法计算矩阵条件数与节点数的7次方是同阶的,而一步降阶法计算矩阵条件数与节点数的4次方是同阶的,降阶法可以有效地降低计算矩阵的条件数,提高计算精度;重心插值配点法采用矩阵—向量形式的计算公式,便于程序的编写,提高了计算效率。When the beam equations are solved by computational accuracy will decline gradually as the barycentric interpolation collocation method, number of nodes increases. The research on barycentrie interpolation collocation method based on depression of order, can provide the new method that has a good numerical stability and high computational accuracy for beam equations. Based on barycentric Lagrange interpolation and its differential matrices, the formula of barycentric interpolation collocation method based on depression of order is derived. Numerical examples are given to verify the effectiveness of the proposed method. The result shows that the computational accuracy of depression of order method remains the range of 10 -10 _ 10 -12 as the number of nodes increases. When the beam equations with simply supported ends are solved, the computational accuracy of the two-step depression of order method is higher than the one-step depression of order method. The condition number of the direct method is close to 7th power of the number of nodes, and the condition number of one-step depression of order method is close to 4th power of the number of nodes. The depression of order method can effectively reduce the condition number of computing matrix such that improves the computational accuracy. By applying the computational formula with matrix-vector form, the program is easy to write and the computational efficiency of barycentric interpolation collocation method can be improved remarkably.

关 键 词：梁方程 降阶法 重心Lagrange插值 配点法 

分 类 号：O302[理学—力学]