Immersed normalized B-spline finite elements – A convergence study for 2D problems

Sanches et al. (2011) [21] introduced a novel B-spline based immersed finite element method (FEM), called I-spline FEM, for the computation of geometrically and topologically complex problems. Away from the domain boundaries, the standard B-spline basis functions are used for the finite element interpolation, however, close to domain boundaries, the B-spline basis functions are modified, so that they locally interpolate the Dirichlet boundary conditions. Although the new technique is demonstrated to be robust and efficient, in the first tests it appeared to present very small convergence rates considering the B-spline order. In this work we propose a quadrature procedure for boundary cells based on triangles subdivision and linear mapping and make use of 2D elasticity and Poisson problems to study the I-spline finite element method convergence. Cubic B-splines are considered and all changes due to the B-spline modifications are studied regarding its influence in convergence rate. Finally, we show that good convergence rates, cubic for displacements and quadratic for stresses, may be achieved.