Weak impositions of Dirichlet boundary conditions in solid mechanics: a critique of current approaches and extension to partially prescribed boundaries

Abstract

In this article we first review various approaches developed to date for the weak imposition of Dirichlet boundary conditions in fictitious domain analysis for elasticity problems. The Hellinger-Reissner (H-R) principle, the linked Lagrange multiplier (LLM) method, the implicit boundary method and the fat boundary method are discussed along with the well-known Lagrange multiplier, penalty and Nitsche’s methods. We state these approaches in a common form starting with energy functionals and weak forms, and discretise using the fictitious domain finite element method. Previous formulations of these methods were in general developed for full prescription along the Dirichlet boundary, which generally implies no local effect of boundary inclination. However, partially prescribed conditions (such as the structural roller boundary condition) with inclination have wide practical applications in engineering. Here we provide techniques of imposing such boundary conditions in these methods in detail. For those methods that contain algorithmic parameters, such as the penalty and Nitsche’s methods, extra computation or empirical estimation is necessary to decide values of the parameters, and hence we discuss parametric and convergence behaviours through numerical examples to provide guidance on the choice of parameters.