Isogeometric analysis with piece-wise constant test functions

TL;DR

This paper introduces a Petrov-Galerkin formulation in isogeometric analysis where test functions are simplified to piece-wise constants, reducing computational costs while maintaining accuracy for higher-order C1 basis functions.

Contribution

The paper demonstrates that higher-order C1 continuity basis functions in IGA can use piece-wise constant test functions, simplifying computations and reducing integration costs.

Findings

Reduced numerical integration cost due to constant test functions

Applicable to any partition of unity preserving basis functions

Valid for problems in any dimension and geometry

Abstract

We focus on the finite element method computations with 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. It also can be used in time-dependent problems, e.g., in the explicit dynamics computations, where we can reduce the cost of generation of the right-hand side. This summation of test functions can be performed for an arbitrary linear differential operator resulting from the Galerkin method applied to a PDE where we discretize with C1 continuity basis functions. 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.