New approaches for adjoint-based error estimation and mesh adaptation in stabilized finite element methods with an emphasis on solid mechanics applications

Abstract

In this thesis, we present an approach to automate the process of adjoint-based error estimation and mesh adaptation to lower the barrier of entry for solid mechanics practitioners. This approach has been developed to be applicable to both Galerkin and stabilized finite element methods, but we mainly emphasize stabilized finite elements. In particular, we demonstrate the effectiveness of this approach for two and three dimensional problems in incompressible elasticity and elastoplasticity. Further, we demonstrate the ability of this approach to execute effectively on parallel machines.

In a finite element simulation, not all of the computed data is of equal importance. Rather, the goal of an engineering practitioner is often to accurately assess only a small number of critical outputs, such as the displacement at a point or the von-Mises stress over a domain. When these outputs can be expressed as functionals, a strategy known as adjoint-based error estimation can be employed to accurately assess output errors. Using this error information, mesh adaptation can then be utilized to reduce and control output errors. The use of adjoint-based error estimation and mesh adaptation is much more prevalent in computational fluid dynamics applications when compared to computational solid mechanics. This can in part be explained by the high level of expertise required to derive and implement adjoint-based error estimation routines in computational solid mechanics.

The variational multiscale (VMS) method is a particular methodology that allows one to develop a stabilized finite element method. As a further research endeavor, we develop and investigate a novel approach for adjoint-based error estimation and mesh adaptation for VMS methods. In particular, we develop an approach for adjoint enrichment based on VMS techniques.