Thursday, 03 December 2020

Speaker: Prof. Erik Burman

University College of London

Augmented Lagrangian methods for finite element methods in computational mechanics

Abstract

The augmented Lagrangian method is a classical method in constrained optimisation, typically introduced in the context of iterative methods. In this talk we will revisit the Augmented Lagrangian method, but from the point of view of a tool for the derivation of stable and accurate discretisation methods for constrained problems. The target application are problems with nonlinearities or poor (physical) stability properties that make the design of computational methods challenging. We will take the imposition of Dirichlet boundary conditions as starting point and derive a classical methods due to Nitsche. We will then show how the same ideas can be extended to contact problems, drawing on ideas by Chouly, HIld and Renard, and in a second step to inverse boundary value problems, or data assimilation problems. The key message is that the use of Augmented Lagrangian methods allows for the systematic design of discretisation methods with good stability and accuracy properties also for problems where such results have been elusive.

Biography

Erik Burman defended his thesis on adaptive finite element methods for compressible two-phase flows at Chalmers University of Technology in 1998. The thesis was written under the supervision of Prof. C. Johnson at Chalmers and in collaboration with Prof. P. A Raviart and Dr. L. Sainsaulieu at CMAP, Ecole Polytechnique, Palaiseau. Dr. Burman then spent two years at CMAP working with Prof. V. Giovangigli and A. Ern on adaptive finite element methods for combustion. He then moved to Ecole Polytechnique Federale de Lausanne, where he first was a post doc in the group of Prof. J. Rappaz on a project on adaptive methdos for solidification problems. He then got a permanent position as researcher in the group of Alfio Quarteroni. In 2007 he was appointed professor at the University of Sussex and in 2013 Chair of Computational Mathematics at the University College London. His current research focuses on the integration of geometrical and and measured data in numerical methods for computational mechanics.

Notes

by Jessica Manganotti and Vincenzo Zarra.

The augmented Lagrangian method is developed from its classical version by adding a quadratic penalisation on its H1 norm

This method leads to a robust and accurate discretization and it is adapted to evaluate the variational inequality with the FEM.

It allows to apply the Newton method because it leads to non-linear equations.

Here are shown applications to:

Problem of the membrane with obstacle;

Frictionless contact problem for curved membranes;

Problem of the plates with Signorini conditions;