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.

中文翻译：

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使用 Powell-Sabin B 样条分析的 Dirichlet 边界条件的弱强加

Powell-Sabin B 样条在固体和流体分析中的应用越来越广泛，包括裂缝扩展。然而，Powell-Sabin B 样条插值不具备 Kronecker delta 属性，因此，Dirichlet 边界条件的施加并不像标准有限元那样直接。在此，我们讨论了迄今为止为在采用 Powell-Sabin B 样条的分析中弱强加 Dirichlet 边界条件而开发的各种方法的适用性。我们将使用相场建模的弹性和裂缝扩展作为基准问题。我们首先简明扼要地概括脆性断裂传播的相场模型，该模型封装了线性弹性，并使用 Powell-Sabin B 样条对其进行了离散化。