Nitsche methods for constrained problems in mechanics
For researchers in computational mechanics, this offers a systematic framework for implementing constraints in nonlinear finite element methods, though it is an incremental extension of existing Nitsche methods.
The paper provides guidelines for deriving Nitsche Finite Element Methods to enforce equality and inequality constraints in mechanics, extending the method beyond classical boundary conditions. Numerical evidence demonstrates convergence rates for various solid mechanics problems.
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.