Weak imposition of Dirichlet boundary conditions for analyses using Powell–Sabin B‐splines

Powell–Sabin B-splines are enjoying an increased use in the analysis of solids and fluids, including fracture propagation. However, the Powell–Sabin B-spline interpolation does not hold the Kronecker delta property and, therefore, the imposition of Dirichlet boundary conditions is not as straightforward as for the standard finite elements. Herein, we discuss the applicability of various approaches developed to date for the weak imposition of Dirichlet boundary conditions in analyses which employ Powell–Sabin B-splines. We take elasticity and fracture propagation using phase-field modeling as a benchmark problem. We first succinctly recapitulate the phase-field model for propagation of brittle fracture, which encapsulates linear elasticity, and its discretization using Powell–Sabin B-splines. As baseline solution we impose Dirichlet boundary conditions in a strong sense, and use this to benchmark the Lagrange multiplier, penalty, and Nitsche's methods, as well as methods based on the Hellinger-Reissner principle, and the linked Lagrange multiplier method and its modified version.