Unified framework for finite element assembly

Abstract

Over the last fifty years, the finite element method has emerged as a successful methodology for solving a wide range of partial differential equations. At the heart of any finite element simulation is the assembly of matrices and vectors from finite element variational forms. In this paper, we present a general and unified framework for finite element assembly. Based on this framework, we propose a specific software interface called Unified Form-assembly Code (UFC) between problem-specific and general-purpose components of finite element programs. The interface is general in the sense that it applies to a wide range of finite element problems (including mixed finite elements and discontinuous Galerkin methods) and may be used with libraries that differ widely in their design. The interface consists of a minimal set of abstract C++ classes and data transfer is via plain C arrays.We discuss how one may use the UFC interface to build a plug-and-play system for finite element simulation where basic components such as computational meshes, linear algebra and, in particular, variational form evaluation may come from different libraries and be used interchangeably. We further discuss how the UFC interface is used to glue together components from the FEniCS suite of software to provide an integrated highlevel environment where variational forms may be entered as expressions directly in Python and assembled efficiently into sparse matrices.A central design goal for the interface is to minimize dependency on external libraries for the problem-specific code used in applications. Thus, the UFC interface consists of a single C++ header file and does not rely on external libraries for its operation. In particular, the UFC interface does not depend on any other FEniCS components. As a result, finite element code developers may use the interface to detach equationspecific details from general-purpose library code, allowing very flexible connections to alternative libraries. We encourage developers of finite element libraries to incorporate the interface in their libraries. The UFC interface is released into the public domain.

Optimized code generation for finite element local assembly using symbolic manipulation

“…(4) Apply common sub-expression elimination to the symbolic expressions of the matrix computed in the previous step to find an efficient evaluation strategy (Section 5). (5) Generate a UFC [Alnaes et al 2009] compliant cell integration function. We note that EXCAFÉ was not designed as a form compiler, and we generate UFC compliant code primarily to facilitate comparisons against FFC.…”

“…We note that in a strict mathematical derivation, it is possible for the local-to-global scaling factor to become negative if the cell orientation has been reversed. The UFC [Alnaes et al 2009] specification defines cell-local vertex numbering in terms of global vertex numbering, which does not easily permit the preservation of cell orientation. Therefore, we take the modulus of the scaling factor instead, which allows cell orientation to be reversed without negating the value of the cell integral.…”

“…FFC takes as input the Unified Form Language (UFL), a domain specific language that describes the variational form to be integrated. The output is a C++ implementation conforming to the Unified Form-assembly Code (UFC) specification [Alnaes et al 2009], an interface defined by the FEniCS project that specifies how different components of a finite element assembly implementation interact.…”

Automated Modelling of Evolving Discontinuities

“…The form code can be entered into a text file, and FFC can be called from the command line to generate C++ code from this input. The generated code conforms to the Unified Form-assembly Code (UFC) specification [14,15] and serves as input to any assembly library which supports the UFC interface, such as DOLFIN [7,16]. Alternatively, the above input to the form compiler code can be included in the Python interface of DOLFIN which will then use just-in-time complication to generate and compile C++ code on demand.…”

“…FFC generates C++ code which conforms to the Unified Formassembly Code (UFC) specification [14,15] (the generated code conforms to the upcoming version 1.2 of UFC). The UFC specification is a C++ interface for the assembly of variational forms and provides a specification against which automated code generators can work.…”