Automated Computational Modelling for Complicated Partial Differential Equations

In engineering, physical phenomena are often described mathematically by partial differential equations (PDEs), and a commonly used method to solve these equations is the finite element method (FEM). Implementing a solver based on this method for a given PDE in a computer program written in source code can be tedious, time consuming and error prone. Recently, compilers that automatically generate source code from the mathematical representation of a given PDE expressed in a form language have been introduced. This approach to automated mathematical modelling, which is key in the FEniCS Project (\url{http://fenicsproject.org}), has reduced the burden of application developers working with the FEM when it comes to implementing solvers for new models.

In this thesis, the automated modelling framework of the FEniCS Project is extended such that discontinuous Galerkin methods can be handled; rapid prototyping of advanced models and applications is possible; and efficiency is maintained also for complex problems in general.

The extensions are implemented in various components of the FEniCS framework. For instance, the Unified Form Language (UFL) is extended by adding new abstractions that allow operators pertinent to discontinuous Galerkin methods to be represented in a straightforward fashion. The FEniCS Form Compiler (FFC) is also extended such that code can be generated from expressions that contain the discontinuous Galerkin operators introduced in UFL. In order to maintain computational efficiency for complex problems, various optimisation strategies for computing the local finite element tensor are implemented in the FFC. The central philosophy of the optimisation strategies is to manipulate the representation in such a way that the number of operations to compute the local element tensor decreases.

As an example, to demonstrate the extensions to the FEniCS framework developed in this work, a strain gradient plasticity model which includes a lifting-type discontinuous Galerkin formulation for the plastic multiplier is presented. It is demonstrated that the model is not suitable for softening problems. On the other hand, the model is able to capture size effects for a hardening problem in a micro-indentation simulation in three dimensions.