Nitsche’s method for linear Kirchhoff–Love shells: Formulation, error analysis, and verification

Abstract

Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff–Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both displacement and normal rotation boundary conditions must be applied. A popular alternative is to employ Nitsche’s method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verification. In fact, existing formulations are variationally inconsistent and yield sub-optimal convergence rates when used with common boundary condition specifications. In this paper, we present a novel Nitsche-based formulation for the linear Kirchhoff–Love shell that is provably stable and optimally convergent for general sets of admissible boundary conditions. To arrive at our formulation, we first present a framework for constructing Nitsche’s method for any abstract variationally constrained minimization problem. We then apply this framework to the linear Kirchhoff–Love shell and, for the particular case of NURBS-based isogeometric analysis, we prove that the resulting formulation yields optimal convergence rates in both the shell energy norm and the standard L2-norm. To arrive at this formulation, we derive the Euler–Lagrange equations for general sets of admissible boundary conditions and show that the Euler–Lagrange boundary conditions typically presented in the literature are incorrect. We verify our formulation by manufacturing solutions for a new shell obstacle course that encompasses flat, parabolic, hyperbolic, and elliptic geometric configurations with a variety of common boundary condition specifications. These manufactured solutions allow us to robustly measure the error across the entire shell in contrast with current best practices where displacement and stress errors are only measured at specific locations. We use NURBS discretizations to represent the shell geometry and show optimal convergence rates in both the shell energy norm and the standard L2-norm with varying polynomial degrees for all of the problems in the obstacle course.