Weighted Quadrature on Unstructured Splines
For researchers in isogeometric analysis, this provides a fast assembly method for unstructured spline surfaces, addressing a known bottleneck in extending WQ to unstructured meshes.
This work extends weighted quadrature (WQ) to unstructured splines (ASUTS) by defining quadrature in the physical domain, using function-wise quadrature points, and employing truncated SVD to handle ill-conditioning. Numerical tests on Poisson, biharmonic, and nonlinear heat transfer problems show accuracy and efficiency gains.
This work presents a weighted quadrature (WQ) method to fast assemble Galerkin matrices based on unstructured spline surfaces. The method is developed upon a particular variant of unstructured splines, namely the bicubic analysis-suitable unstructured T-splines (ASUTS). While existing WQ approaches have significant speedup for structured splines (e.g., B-splines), their extension to unstructured splines faces several challenges: (1) lack of a global parametric domain for defining quadrature points, (2) a varying number of basis functions across elements that complicates the determination of the optimal number of quadrature points, and (3) ill-conditioned underdetermined linear systems that must be solved to find the quadrature weights. To solve these issues, we first define the WQ rule directly in the physical domain. Second, we specify the number of quadrature points function-wise (rather than element-wise), which naturally satisfies the well-posedness condition, namely the number of unknown weights no less than that of exactness constraints. Third, we employ the truncated Singular Value Decomposition to improve the conditioning of the underdetermined systems by discarding extremely small singular values, which are caused by the splines around extraordinary points. Several different model problems are studied, such as Poisson's problem, the biharmonic problem, and the nonlinear heat transfer problem. In the end, a variety of numerical tests are performed to demonstrate the accuracy and efficiency of the proposed method.