Nitsche methods for constrained problems in mechanics

Abstract

We present guidelines for deriving new Nitsche Finite Element Methods to enforce equality and inequality constraints that act on the value of the unknown mechanical quality. We first formulate the problem as a stabilized finite element method for the saddle point formulation where a Lagrange multiplier enforces the underlying constraint. The Nitsche method is then presented in a general minimization form, suitable for nonlinear finite element methods and allowing straightforward computational implementation with automatic differentation. To validate these ideas, we present Nitsche formulations for a range of problems in solid mechanics and give numerical evidence of the convergence rates of the Nitsche method.

Figures (10)

• Figure 1: Numerical solution for two membranes in contact using the Nitsche method.
• Figure 2: Two membrane contact problem convergence rate follows the theoretical linear convergence in the $H^1$ norm for linear finite element basis.
• Figure 3: Condition numbers of the Jacobian in the Newton iterations \ref{['eq:newtonsystem']} for the penalty and Nitsche variants of the two membrane contact problem with quadratic elements. The blue lines depict condition numbers of the penalty method and red lines of the Nitsche method, respectively. The markers distinguish problems of different sizes. Compared to the Nitsche method's $\gamma = \alpha h^2$, the penalty method requires $\gamma = \alpha h^3$ to retain optimal convergence gustafsson2017finite and has thus consistently larger conditioning numbers with slower convergence.
• Figure 4: Numerical solution for the membrane against elastic solid problem using the Nitsche method.
• Figure 5: Convergence rate in the $H^1$ norm for the membrane against elastic solid problem.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3