Adaptively weighted numerical integration over arbitrary domains

Abstract

In adaptively weighted numerical integration, for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain, and then are corrected for the deviation from the original domain using shape sensitivity analysis. Using divergence theorem, the moments reduce to integrals over the boundary of the simplified domain.The proposed approach supports accurate and efficient computation of quadrature weights for integration of a priori unknown functions over arbitrary 2D and 3D solid domains. Experimental results (2D) indicate that adaptively weighted integration compares favorably with more traditional approaches. Because the adaptively weighted integration avoids excessive domain subdivision, it is useful in many applications and meshfree analysis in particular.