Garth N. Wells

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

DOLFIN: Automated finite element computing

We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled, and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This article discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code.

From continuous to discontinuous failure in a gradient-enhanced continuum damage model

Abstract A computational framework for the description of the combined continuous–discontinuous failure in a regularised strain-softening continuum is proposed. The continuum is regularised through the introduction of gradient terms into the constitutive equations. At the transition to discrete failure, the problem fields are enhanced through a discontinuous interpolation based on the partition of unity paradigm of finite-element shape functions. The inclusion of internal discontinuity surfaces, where boundary conditions are applied without modifications of the original finite-element mesh, avoids the unrealistic damage growth typical of this class of regularised continuum models. Combined models allow for the analysis of the entire failure process, from diffuse microcracking to localised macrocracks. The discretisation procedure is described in detail and numerical examples illustrate the performance of the combined continuous–discontinuous approach.

A discontinuous Galerkin method for the Cahn-Hilliard equation

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn-Hilliard equation. The Cahn-Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite element methods, coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth-order spatial derivatives. The variational formulation of the discontinuous Galerkin method, its implementation and numerical examples are presented. In this communication, it is also shown under what conditions the method is stable, and an error estimate in an energy-type norm is presented. The method is evaluated by comparison with a standard finite element treatment in which the Cahn-Hilliard equation is decomposed into two coupled partial differential equations.

On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture

In this paper, strong discontinuities embedded in finite elements are used to model discrete cracking in quasi-brittle materials. Special attention is paid to (i) the constitutive models used to describe the localized behaviour of the discontinuities, (ii) the enforcement of the continuity of the crack path and (iii) mixed-mode crack propagation. Different constitutive relations are adopted to describe the localized behaviour of the discontinuities, namely two damage laws and one plasticity law. A numerical algorithm is introduced to enforce the continuity of the crack path. In the examples studied, an objective dissipation of energy with respect to the mesh is found. Examples of mode-I and mixed-mode crack propagation are presented, namely a double notch tensile test and a single-edge notched beam subjected to shear. In the former case different crack patterns are obtained depending on the notch offset; in the latter case special emphasis is given to the effect of shear on the global structural response. In particular, both the peak load and the softening response of the structure are related to the amount of shear tractions allowed to develop between crack faces. The results obtained are compared to experimental results. As a general conclusion, it is found that crack path continuity allows for the development of crack patterns similar to those found in experiments, even when reasonably coarse meshes are used.

Three-dimensional embedded discontinuity model for brittle fracture

Abstract Three-dimensional elements with discontinuous shape functions have been developed to model brittle fracture in unstructured meshes. Through the incorporation of discontinuous shape functions, highly localised deformations can be captured in a coarse finite element mesh. By carefully examining the variational formulation, traction continuity is assured in a weak sense across discontinuities and a discrete constitutive (traction–separation) model is applied. Using discrete constitutive models avoids the need for numerical parameters to achieve mesh objective results with respect to energy dissipation. Further, kinematic enhancements allow objective analysis in unstructured meshes. Examples of three-dimensional analysis are shown to illustrate the performance of the model.

Unified form language: A domain-specific language for weak formulations of partial differential equations

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.

Sensitivity of the scale partition for variational multiscale large-eddy simulation of channel flow

The variational multiscale method has been shown to perform well for large-eddy simulation (LES) of turbulent flows. The method relies upon a partition of the resolved velocity field into large- and small-scale components. The subgrid model then acts only on the small scales of motion, unlike conventional LES models which act on all scales of motion. For homogeneous isotropic turbulence and turbulent channel flows, the multiscale model can outperform conventional LES formulations. An issue in the multiscale method for LES is choice of scale partition and sensitivity of the computed results to it. This is the topic of this investigation. The multiscale formulation for channel flows is briefly reviewed. Then, through the definition of an error measure relative to direct numerical simulation (DNS) results, the sensitivity of the method to the partition between large- and small-scale motions is examined. The error in channel flow simulations, relative to DNS results, is computed for various partitions between large- and small-scale spaces, and conclusions drawn from the results.

A discontinuous Galerkin formulation for a strain gradient-dependent damage model

Abstract The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand C 1 continuity of the shape functions. Here, recent developments in discontinuous Galerkin methods are explored and exploited for the solution of a prototype non-linear strain gradient-dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard C 0 shape functions. The formulation is tested in one dimension for the simplest possible finite element formulation: continuous piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed formulation.

Some observations on embedded discontinuity models

The formulation of finite elements with incompatible discontinuous modes is examined rigorously. Both weak and strong discontinuities are considered. Starting from a careful elaboration of the kinematics for both types of discontinuities a comprehensive finite element formulation is derived based on a three‐field variational statement. Similarities and differences are highlighted between the various formulations which ensue.