900 research outputs found
Efficient Compilation of a Class of Variational Forms
We investigate the compilation of general multilinear variational forms over
affines simplices and prove a representation theorem for the representation of
the element tensor (element stiffness matrix) as the contraction of a constant
reference tensor and a geometry tensor that accounts for geometry and variable
coefficients. Based on this representation theorem, we design an algorithm for
efficient pretabulation of the reference tensor. The new algorithm has been
implemented in the FEniCS Form Compiler (FFC) and improves on a previous
loop-based implementation by several orders of magnitude, thus shortening
compile-times and development cycles for users of FFC.Comment: ACM Transactions on Mathematical Software 33(3), 20 pages (2007
A general approach to transforming finite elements
The use of a reference element on which a finite element basis is constructed
once and mapped to each cell in a mesh greatly expedites the structure and
efficiency of finite element codes. However, many famous finite elements such
as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence
needed to work with a reference element in the standard way. This paper gives a
generalizated approach to mapping bases for such finite elements by means of
studying relationships between the finite element nodes under push-forward.Comment: 28 page
Code generation for generally mapped finite elements
Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations
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