基于几何指标的边界元法单元积分精度评估和修正  

Accuracy evaluation and correction of the element integration based on a geometric index in the boundary element method

作　　者：黄君豪 陈永强[1] HUANG Junhao;CHEN Yongqiang(Department of Mechanics and Engineering Science,College of Engineering,Peking University,Beijing 100871,China)

机构地区：[1]北京大学工学院力学与工程科学系

出　　处：《清华大学学报（自然科学版）》2019年第11期953-960,共8页Journal of Tsinghua University(Science and Technology)

基　　金：国家重点研发计划项目（2018YFC0809700,2017YFC0803300）

摘　　要：该文提出基于几何指标来评估边界元法单元积分精度并对其进行修正来改善求解精度的方法。奇异性几何指标定义为源点到被积单元的最短距离与单元长度的比值。对于离散后的边界元法网格,利用求积误差上界公式得到各单元奇异性几何指标与积分精度的关系,通过该网格节点几何信息估算出全部单元的积分精度。通过矩阵的误差传递公式估算代数方程组求解结果的最大相对误差,将其作为全局精度指标。若该指标大于指定值,说明存在局部单元积分精度不足而影响结果精度的情况,必须对单元积分精度进行修正和提高。提出采用sinh变换法对精度不满足要求的单元积分进行修正。数值结果表明:该方法可以仅基于网格的几何信息,在不改变原计算流程和几乎不增加计算量的条件下,保证边界元法整体刚度矩阵系数精度和最终求解精度,且易于数值实施。This paper presents a scheme to evaluate the element integration accuracy for the boundary element method based on a singular geometric index to improve the accuracy of the final solution.The singular geometric index is defined as the ratio of the shortest distance from the source point to the integral element to the element length.For a discretized boundary mesh,the upper bound of the quadrature error is used to obtain the relationship between the singular geometric index and the element integration accuracy with the integrated accuracies of all the elements estimated from the node geometries.Furthermore,the maximum relative error of the solution which is a global precision index is estimated from the matrix error transfer formula.If the index is larger than a threshold,some of the element integrals are not accurate and should be corrected to improve the solution accuracy.This method uses the sinh transformation method to correct the element integrals of low accuracy elements.Numerical results show that this method can guarantee the accuracy of the overall stiffness matrix coefficient of the boundary element method and the final solution accuracy using only the geometric node information.This scheme does not need to change the original calculation, requires little additional computational cost,and can be easily implemented.

关 键 词：边界元法 单元积分精度 几何指标 精度修正 sinh变换 

分 类 号：O241[理学—计算数学]