Shape Aware Quadratures

• Published in: Journal of computational physics 2018-12, Vol.374, p.1239-1260
• Main Authors: Thiagarajan, Vaidyanathan, Shapiro, Vadim
• Format: Article
• Language: English
• Subjects: Approximation; Bivariate analysis; Computational physics; Divergence theorem; Domains; Integration; Mathematical analysis; Moment fitting equations; Numerical analysis; Numerical integration; Octree/quadtree; Polynomials; Quadratures; Sensitivity analysis; Shape sensitivity analysis; Theorems; Topological sensitivity analysis
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.05.024

With Shape Aware Quadratures (SAQ), 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 (Ω0) that is a reasonable approximation of the original domain (Ω) that are then corrected for the deviation of the shape of Ω0 from Ω via shape correction factors. This idea was already successfully utilized in the Adaptively Weighted Numerical Integration (AW) method [1–3], where moments were computed over simplified but homeomorphic domain and then corrected using first-order shape sensitivity, allowing efficient and accurate integration of integrable functions over arbitrary domains. With SAQ, more general shape correction factors can be derived based on a variety of sensitivity analysis techniques such as shape sensitivity (SSA) and topological sensitivity (TSA). Using the right kind/order of shape correction factors for moment approximation enables the resulting quadrature (from the moment fitting equations) to efficiently adapt to the shape of the original domain even in the presence of thousands of small features. We derive shape correction factors based on SSA and TSA of moment integrals. We also demonstrate the efficacy of SAQ in integrating bivariate/trivariate polynomials over several 2D/3D domains in the presence of small features. •Non-trivial generalization of our earlier work (AW method) to efficiently account for topological features in addition to geometric features.•Generalization of the notion of moment approximations and correction factors using a variety of sensitivity analysis tools.•Detailed derivation of first/second-order shape and topological sensitivity of moment integrals.•Demonstration of polynomial integration over 2D/3D domains with several small features using non-conforming meshes.