Higher order and continuity [formula omitted] projections with piece-wise constant test functions

Abstract

We focus on the isogeometric L2 projection problem employing higher-order C1 continuity basis functions that preserve the partition of unity. We show that the rows of the system of linear equations can be combined, and the test functions can be sum up to 1 using the partition of unity property at the quadrature points. Thus, the test functions in higher continuity IGA can be set to piece-wise constants. This formulation is equivalent to testing with piece-wise constant basis functions, with supports span over some parts of the domain. The resulting method is a Petrov–Galerkin formulation with piece-wise constant test functions. This observation has the following consequences. The numerical integration cost can be reduced because we do not need to evaluate the test functions since they are equal to 1. This observation is valid for any basis functions preserving the partition of unity property. It is independent of the problem dimension and geometry of the computational domain. The resulting method is equivalent to a linear combination of the collocations at points and with weights resulting from applied quadrature over the spans defined by supports of the piece-wise constant test functions. We show the algorithm for finding optimal supports of the piece-wise constant test functions using the inf–sup condition argument. We also discuss how the piece-wise constants’ replacement of the test functions can be utilized in modern integration methods, with weighted quadrature, sum factorization, and row by row assembly. We test our method on the isogeometric L2 projection, the explicit dynamics simulations in two and three dimensions, the heat transfer problem, and the three-dimensional elastic wave propagation simulations.