Nitsche methods for constrained problems in mechanics

TL;DR

This paper develops generalized Nitsche finite element methods for enforcing equality and inequality constraints in mechanics problems, providing a flexible approach for nonlinear and constrained finite element analysis.

Contribution

It introduces a unified Nitsche formulation for various constraints in mechanics, extending beyond classical boundary conditions with straightforward implementation.

Findings

Numerical experiments demonstrate optimal convergence rates.

The method effectively enforces constraints in nonlinear mechanics problems.

Formulations are validated across multiple solid mechanics applications.

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 quantity. 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 adding constraints to nonlinear finite element methods and allowing straightforward computational implementation with automatic differentation. This extends the method beyond classical boundary condition enforcement. 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.