Cross-Loop Optimization of Arithmetic Intensity for Finite Element Local Assembly

Luporini, Fabio; Vărbănescu, Ana Lucia; Rathgeber, Florian; Bercea, Gheorghe-Teodor; Ramanujam, J.; Ham, David A.; Kelly, Paul H. J.

doi:10.1145/2687415

Cited by 56 publications

(68 citation statements)

References 23 publications

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“…The various discretizations given above were implemented using the Themis software framework ( [73]), which shares a front-end with the Firedrake [74] project consisting of UFL (Unified Form Language, [75]), TSFC (Two-Stage Form Compiler, [76,77]), FInAT (FInAT/FInAT: a smarter library of finite elements, [78]), COFFEE (COmpiler For Fast Expression Evaluation, [79,80]) and FIAT (FInite element Automatic Tabulator, [81]). In fact, as currently implemented the model code runs using both Themis and Firedrake without changes.…”

Section: Methodsmentioning

confidence: 99%

A quasi-Hamiltonian discretization of the thermal shallow water equations

Eldred

Dubos

Kritsikis

2019

Journal of Computational Physics

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The rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model.

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Section: Methodsmentioning

confidence: 99%

A quasi-Hamiltonian discretization of the thermal shallow water equations

Eldred

Dubos

Kritsikis

2019

Journal of Computational Physics

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“…In order to apply the FEM, the previously introduced CAD model is subdivided into tetrahedral elements using the meshing tool Trelis [41]. The finite element tool Firedrake [42,43,44,45,46,47,48,49] assembles the matrices by approximating the weak form using a weighted residual method. The time and space dependency of the weak form is separated, such that the time dependency can be factorized [38].…”

Section: The Digital Realizationmentioning

confidence: 99%

A comparison of second-order model order reduction methods for an artificial fishtail

Saak

Siebelts

Werner

2019

At - Automatisierungstechnik

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In order to apply control theory in small autonomous vehicles, mathematical models with small numbers of states are required for using the limited computational power in embedded programming. In this paper, we consider an artificial fishtail as an example for a complex mechanical system with a second-order large-scale model, which is derived by using the finite element method. To meet the above limitations, the several hundreds of thousands of degrees of freedom need to be reduced to merely a handful of surrogate degrees of freedom. We seek to achieve this task by various second-order model order reduction methods. All methods are applied on the fishtail’s matrices and their results are evaluated and compared in the frequency domain as well as in the time domain.

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“…UFL Unified Form Language [4,2], a domain-specific language for the specification of finite element variational forms. Firedrake also relies on PyOP2 [37] and COFFEE [30].…”

Section: Product Finite Elements Within Finite Element Exterior Calcumentioning

confidence: 99%

Automated Generation and Symbolic Manipulation of Tensor Product Finite Elements

McRae¹,

Bercea²,

Mitchell³

et al. 2016

SIAM J. Sci. Comput.

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Abstract. We describe and implement a symbolic algebra for scalar and vector-valued finite elements, enabling the computer generation of elements with tensor product structure on quadrilateral, hexahedral, and triangular prismatic cells. The algebra is implemented as an extension to the domain-specific language UFL, the Unified Form Language. This allows users to construct many finite element spaces beyond those supported by existing software packages. We have made corresponding extensions to FIAT, the FInite element Automatic Tabulator, to enable numerical tabulation of such spaces. This tabulation is consequently used during the automatic generation of low-level code that carries out local assembly operations, within the wider context of solving finite element problems posed over such function spaces. We have done this work within the code-generation pipeline of the software package Firedrake; we make use of the full Firedrake package to present numerical examples.

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Cross-Loop Optimization of Arithmetic Intensity for Finite Element Local Assembly

Cited by 56 publications

References 23 publications

A quasi-Hamiltonian discretization of the thermal shallow water equations

A quasi-Hamiltonian discretization of the thermal shallow water equations

A comparison of second-order model order reduction methods for an artificial fishtail

Automated Generation and Symbolic Manipulation of Tensor Product Finite Elements

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