Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods
Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods
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DOI：
10.1016/j.cma.2013.01.007
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发表时间：
2013-05
影响因子：
7.2
通讯作者：
Y. Sudhakar;W. Wall
中科院分区：
文献类型：
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作者：
Y. Sudhakar;W. Wall
基于矩拟合方程构造求积方案，以在任意凸/凹体积上积分多项式，这些体积尤其出现在丰富单位分割有限元方法（EPUM）中。该方案的构建模块涉及多变量微积分的发散定理，用于集成的基函数。提出了一种高效、鲁棒的点分布方法，并通过求解最小二乘问题获得相应点处的正交权值。该方法最初适用于集成给定的多项式函数在复杂的体积，并进一步模拟简单的三维流体动力学问题，涉及非常复杂的体积时，解决与EPUM。通过与现有精确/数值解的比较，证明了该方法的准确性，并通过与广泛使用的曲面细分方法的计算时间比较，证明了该方法的有效性。
Quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex/concave volumes that arise, among others, in Enriched Partition of Unity finite element Methods (EPUM). The building block of the scheme involves the divergence theorem of multivariable calculus, which is used to integrate the base functions. An efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least-squares problem. The method is applied initially to integrate given polynomial functions over complex volumes, and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with EPUM. Accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact/numerical solutions, and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method.