瞬态分析中的比例边界有限元法递推公式  

Recursive formulation for scaled boundary finite element method in transient analysis

作　　者：李上明[1] 

机构地区：[1]中国工程物理研究院总体工程研究所,绵阳621900

出　　处：《应用力学学报》2013年第6期882-886,954,共5页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(10902060;11272299);中国博士后科学基金(201003123);中国工程物理研究院科学技术发展基金(2011B0201036)

摘　　要：为改进现有比例边界有限元法卷积积分公式的计算效率,推导了一种新的递推公式。本文求解了比例边界有限元法的动态刚度矩阵,并将其分解成高频和低频两部分;对低频部分利用无约束高斯-牛顿迭代算法建立其对应的有理式公式,确定递推系数;将高频和低频两部分通过傅里叶逆变换建立时域公式,其中高频部分对应阻尼矩阵,低频部分对应刚度矩阵与递推项。结果表明:当递推阶数m=15时,递推公式的解与解析解曲线形状相似,在最大值处的相对误差为0.8%;当分析步数为3183时,卷积积分公式的计算时间为递推公式的14倍。此外,在瞬态分析中该递推公式可用常规有限元法求解;通过具有解析解的坝库耦合算例,验证了该递推公式的正确性和有效性。In order to improve the efficiency of the convolution integral formulation in the scaled boundary finite element method(SBFEM), a recursive formulation is proposed. Firstly, the dynamic stiffness matrix of the SBFEM is evaluated and decomposed into high and low frequency parts. Secondly, an unconstrained Gauss-Newton iterative method is used to derive a rational formulation for low-frequency part, which determines coefficients of the recursive formulation. Finally, the recursive formulation is derived by applying the inverse Fourier transformation to the rational formulation and high-frequency part. The high-frequency part is corresponding to a damping matrix, while the rational formulation results in a stiffness matrix and recursive terms. The recursive formulation can be solved by the traditional finite element method techniques in a transient analysis. A dam-reservoir interaction example, which analytical solution can be available, is analyzed to validate its accuracy and efficiency. When the recursive order is 15, results from the recursive formulation and analytical solutions are almost the same. When the analysis step number is 3183, the time cost of convolution integral formulation is 14 times of that of recursive formulation.

关 键 词：比例边界有限元法 坝库系统 无限库水 地震激励 瞬态分析 

分 类 号：TV312[水利工程—水工结构工程]