A general approach to transforming finite elements

Kirby, Robert C.

doi:10.5802/smai-jcm.33

Cited by 19 publications

(45 citation statements)

References 29 publications

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“…Kjolstad et al [29] present the Tensor Algebra Compiler (taco), which generates efficient code for primitive tensor operations, especially for various sparse array formats. While finite element local assembly mostly involves dense tensor operations, Kirby [26] describes a theory of reference transformation that requires multiplications with sparse matrices for finite elements with stronger than C 0 continuity.…”

Section: Related Workmentioning

confidence: 99%

TSFC: A Structure-Preserving Form Compiler

Homolya¹,

Mitchell²,

Luporini³

et al. 2018

SIAM J. Sci. Comput.

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A form compiler takes a high-level description of the weak form of partial differential equations and produces low-level code that carries out the finite element assembly. In this paper we present the Two-Stage Form Compiler (TSFC), a new form compiler with the main motivation to maintain the structure of the input expression as long as possible. This facilitates the application of optimizations at the highest possible level of abstraction. TSFC features a novel, structure-preserving method for separating the contributions of a form to the subblocks of the local tensor in discontinuous Galerkin problems. This enables us to preserve the tensor structure of expressions longer through the compilation process than other form compilers. This is also achieved in part by a two-stage approach that cleanly separates the lowering of finite element constructs to tensor algebra in the first stage, from the scheduling of those tensor operations in the second stage. TSFC also efficiently traverses complicated expressions, and experimental evaluation demonstrates good compile-time performance even for highly complex forms.Algorithm 1 Generated loop nests for the expression (4.2).1: for all q do 2:3: for all q, r do 4: 1) q,r * c r 5: for all q do 6:J (1,2) [q] ← 0 7: for all q, r do 8: 2) q,r * c r 9: for all q do 10: J (2,1) [q] ← 0 11: for all q, r do 12: J (2,1) [q] += C (2,1) q,r * c r 13: for all q do 14: J (2,2) [q] ← 0 15: for all q, r do 16: J (2,2) [q] += C (2,2) q,r * c r

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Section: Related Workmentioning

confidence: 99%

TSFC: A Structure-Preserving Form Compiler

Homolya¹,

Mitchell²,

Luporini³

et al. 2018

SIAM J. Sci. Comput.

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“…that can facilitate the construction of a wider class of finite elements, still based on a reference element. This mechanism has been the standard way of defining elements like Hermite, Argyris or Raviart-Thomas in GetFEM and has independently been proposed by other authors [Domínguez and Sayas 2008;Kirby 2018]. The use of Eq.…”

Section: Finite Element Descriptionmentioning

confidence: 99%

GetFEM

Renard

Poulios

2020

ACM Trans. Math. Softw.

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This article presents the major mathematical and implementation features of a weak form language (GWFL) for an automated finite-element (FE) solution of partial differential equation systems. The language is implemented in the GetFEM framework and strategic modeling and software architecture choices both for the language and the framework are presented in detail. Moreover, conceptual similarities and differences to existing high-level FE frameworks are discussed. Special attention is given to the concept of a generic transformation mechanism that contributes to the high expressive power of GWFL, allowing to interconnect multiple computational domains or parts of the same domain. Finally, the capabilities of the language for expressing strongly coupled multiphysics problems in a compact and readable form are shown by means of modeling examples.

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“…As we show in Kirby [2018], we can take a simple linear combination of the pullbacks of the Hermite basis functions onK to obtain the Hermite basis functions on K. Let {ψ 3i } 2 i=0 be the basis functions taking unit value at vertex v i ,…”

Section: )mentioning

confidence: 99%

“…Since there is only a single directional derivative on each edge, pairs of basis function corresponding to a gradient (in some coordinates) cannot be adjusted by the Jacobian like in for the Hermite case. In Kirby [2018], we develop a three-step mapping technique that generalizes the approach in Domínguez and Sayas [2008]. First, one extends the reference and physical nodal sets with additional nodes such that their spans do coincide.…”

Section: )mentioning

confidence: 99%

“…This works naturally when reference and physical elements are equivalent under affine or Piola pullback, but implementing elements (see Fig. 2) such as Hermite [Ciarlet and Raviart 1972], Morley [Morley 1971], Bell [Bell 1969], and Argyris [Argyris et al 1968] that require more complex transformations [Domínguez and Sayas 2008;Kirby 2018] easily leads to a proliferation of special cases. It also requires encoding basis tabulation and transformation in completely different code components.…”

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confidence: 99%

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Code Generation for Generally Mapped Finite Elements

Kirby

Mitchell

2019

ACM Trans. Math. Softw.

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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 C 1 discretizations.

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A general approach to transforming finite elements

Cited by 19 publications

References 29 publications

TSFC: A Structure-Preserving Form Compiler

TSFC: A Structure-Preserving Form Compiler

GetFEM

Code Generation for Generally Mapped Finite Elements

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