Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods

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.

► A new robust and efficient quadrature rule is proposed.
► Method can handle arbitrary convex and concave volumes.
► Number of necessary quadrature points is one order of magnitude lower than with tessellation.
► Method is very robust and very simple to implement.