Journal Article

The Augmented Lagrangian Method as a Framework for Stabilised Methods in Computational Mechanics

Hansbo, Peter, Burman, Erik, Larson, Mats G.

2023

Overview

In this paper we will present a review of recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalising the constraint equations, and is well known as the basis for iterative algorithms for constrained optimisation problems. Its use as a stabilisation methods in computational mechanics has, however, only recently been appreciated. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

• Publisher: Springer Netherlands,Springer Nature B.V
• Subject: Approximation / Augmented lagrange multiplier methods / Augmented Lagrangian methods / Computational mechanics / Constrained optimi-zation problems / Constrained optimization / Constraint equation / Constraints / Engineering / Finite element analysis / Galerkin Least Squares / Inequality / Iterative algorithm / Iterative algorithms / Iterative methods / Lagrange multiplier / Lagrange multiplier method / Lagrange multipliers / Mathematical and Computational Engineering / Mechanics / Methods / Optimization / Partial differential equations / Penalty term / Review Article / Stabilization methods / Stabilized method