Nitsche methods for constrained problems in mechanics

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 differentiation. 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. • We derive a novel interpretation of the Nitsche Finite Element Method to solve constrained problems in solid mechanics. • We use the new formulation to solve several contact problems and validate the convergence rates numerically. • The computational implementation utilizes a general minimization form and automatic differentiation handles the variational derivatives.