Showing papers by "Garth N. Wells published in 2012"

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

Abstract: 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.

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

Abstract: 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 multi-field 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.

DOLFIN: a C++/Python Finite Element Library

Abstract: DOLFIN is a C++/Python library that functions as the main user interface of FEniCS. In this 4806 chapter, we review the functionality of DOLFIN. We also discuss the implementation of some key 4807 features of DOLFIN in detail.

FFC: the FEniCS Form Compiler

Abstract: One of the key features of FEniCS is automated code generation for the general and efficient 7018 solution of finite element variational problems. This automated code generation relies on a form 7019 compiler for offline or just-in-time compilation of code for individual forms. Two different form 7020 compilers are available as part of FEniCS.

Energy Stable and Momentum Conserving Hybrid Finite Element Method for the Incompressible Navier-Stokes Equations

Abstract: A hybrid method for the incompressible Navier-Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier-Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, and global energy stability are proved for the time-continuous case and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations.

Finite Element Assembly

Abstract: The finite element method may be viewed as a method for forming a discrete linear system 4064 AU = b or nonlinear system b(U) = 0 corresponding to the discretization of the variational form of a 4065 differential equation.A central part of the implementation of finite element methods is therefore the 4066 computation of matrices and vectors from variational forms.In this chapter, we describe the standard 4067 algorithm for computing the discrete operator (tensor) A.

Applications in solid mechanics

Abstract: Problems in solid mechanics constitute perhaps the largest field of application of finite element methods. The vast majority of solid mechanics problems involve the standard momentum balance equation, posed in a Lagrangian setting, with different models distinguished by the choice of nonlinear or linearized kinematics, and the constitutive model for determining the stress. For some common models, the constitutive relationships are rather complex. This chapter addresses a number of canonical solid mechanics models in the context of automated modeling, and focuses on some pertinent issues that arise due to the nature of the constitutive models. The solution of equations with second-order time derivatives, which characterizes dynamic problems, is also considered.

Energy stable and momentum conserving hybrid finite element method for the incompressible

Abstract: A hybrid method for the incompressible Navier-Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the ve- locity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier-Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appro- priate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, and global energy stability are proved for the time-continuous case and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations.

Quadrature representation of finite element variational forms

Abstract: This chapter addresses the conventional run-time quadrature approach for the numerical integration of local element tensors associated with finite element variational forms, and in particular automated optimizations that can be performed to reduce the number of floating point operations. An alternative to the run-time quadrature approach is the tensor representation presented in Chapter 8. Both the quadrature and tensor approaches are implemented in FFC (see Chapter 11). In this chapter we discuss four strategies for optimizing the quadrature representation for run-time performance of the generated code and show that optimization strategies lead to a dramatic improvement in run-time performance over a naive implementation. We also examine performance aspects of the quadrature and tensor approaches for different equations, and this will motivate the desirability of being able to choose between the two representations.

Finite elements for incompressible fluids

Abstract: The structure of the finite element method offers a user a range of choices. This is especially true for solving incompressible fluid problems, where theory points to a number of stable finite element formulations. Using automation tools, we implement and examine various stable formulations for the steady-state Stokes equations. It is demonstrated that the expressiveness of the FEniCS Project components allows solvers for the Stokes problem that use various element formulations to be created with ease.

Modeling evolving discontinuities

Abstract: We present a framework for solving partial differential equations with discontinuities in the solution across evolving surfaces. The partition-of-unity/extended finite element approach is adopted, 1 and it is demonstrated that such methods can be used in combination with a form compiler to generate equation-specific parts of a program. The automated generation of code makes it straightforward to incorporate discontinuities in formulations involving multiple fields, using both Lagrange and non-Lagrange basis functions. The approach is illustrated through some salient code extracts.

Solving electromagnetic wave propagation problems using fenics

Abstract: The FEniCS Project [1] is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the automated solution of differential equations by finite element methods. The open source project started in 2003 and has reached sufficient maturity to be used in new and full scale applications. The project does not only support classical Lagrange finite elements, but also a.o. discontinuous and Whitney elements. Many features and aspect of PDE simulations are continuously being improved to suit the needs of the users. The main appeal of this software is the high level interfaces reconciled with performance in parallelisation. These allow fast implementation of finite element solvers for many PDEs due to the large library of supported elements. The FEniCS software can easily be used to solve the Maxwell wave equation, which will be illustrated. With the help of the dolfin-adjoint library [2], also backward problems, such as inverse electromagnetic scattering, can be solved using this software.