固体中短波传播单位分解有限元法的解析积分

A special integration scheme for partition of unity finite element method for short wave propagation in solids

针对固体中短波传播数值模拟的单位分解有限元法中单元矩阵积分的被积函数的强烈振荡特性,应用直角坐标系下标准有限元形函数和单元内的波动方向知识提出了一种单元矩阵的解析积分方案。它对于平面三,六,四,八和九节点的直边单位分解有限单元是完全解析的,对于与这些单元相应的曲边单元则是半解析的。数值结果显示所提出的积分方案在计算效率上比高斯-勒让德积分有大幅度提高。

A special integration scheme is developed for computing element matrices of partition of unity finite elements for short wave propagation in solids. The finite element spaces are constructed by multiplying the standard finite element shape functions referred to orthogonal coordinates, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which approximately reproduce the highly oscillatory nature of the solution and include a priori knowledge of the wave directions in each element. It is analytical for the 3, 4, 6, 8 and 9-noded plane elements with straight edges and semi-analytical for the elements with curved edges. The numerical results exhibit a great saving in the computational efforts for the proposed integration scheme as compared with the Gauss-Legendre one.