Code Generation for Generally Mapped Finite Elements

Kirby, Robert C.; Mitchell, Lawrence

doi:10.1145/3361745

Cited by 20 publications

(17 citation statements)

References 56 publications

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“…A range of finite elements: in all examples considered herein, we utilised a continuous Q2-Q1 element pair for velocity and pressure with a Q2 discretisation for temperature (with the exception of one set of examples in Section 5.1, where we demonstrated the use of a Q1 temperature discretisation). Accordingly, we have not demonstrated Firedrake's support for a wide-range of finite elements, including continuous, discontinuous, H(div) and H(curl) discretisations, and elements with continuous derivatives such as the Argyris and Bell elements (see Kirby and Mitchell, 2019, for an overview). Some of these could offer major advantages for geodynamical simulation.…”

Section: Discussionmentioning

confidence: 81%

Automating Finite Element Methods for Geodynamics via Firedrake

Davies

Kramer

Ghelichkhan

et al. 2022

Preprint

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Abstract. Firedrake is an automated system for solving partial differential equations using the finite element method. By applying sophisticated performance optimisations through automatic code-generation techniques, it provides a means to create accurate, efficient, flexible, easily extensible, scalable, transparent and reproducible research software, that is ideally suited to simulating a wide-range of problems in geophysical fluid dynamics. Here, we demonstrate the applicability of Firedrake for geodynamical simulation, with a focus on mantle dynamics. The accuracy and efficiency of the approach is confirmed via comparisons against a suite of analytical and benchmark cases of systematically increasing complexity, whilst parallel scalability is demonstrated up to 12288 compute cores, where the problem size and the number of processing cores are simultaneously increased. In addition, Firedrake's flexibility is highlighted via straightforward application to different physical (e.g. complex nonlinear rheologies, compressibility) and geometrical (2-D and 3-D Cartesian and spherical domains) scenarios. Finally, a representative simulation of global mantle convection is examined, which incorporates 230 Myr of plate motion history as a kinematic surface boundary condition, confirming its suitability for addressing research problems at the frontiers of global mantle dynamics research.

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Section: Discussionmentioning

confidence: 81%

Automating Finite Element Methods for Geodynamics via Firedrake

Davies

Kramer

Ghelichkhan

et al. 2022

Preprint

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“…The system (52) was discretized using H 2 (0, L)-conforming cubic Hermite finite elements [40,41] using Firedrake [58]. The arising linear systems were solved using the sparse LU factorization of PETSc [5].…”

Section: Zeidler (1988)mentioning

confidence: 99%

Deflation for semismooth equations

Farrell

Croci

Surowiec

2019

Optimization Methods and Software

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Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite-and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics. KEYWORDSDeflation, semismooth Newton, variational inequalities, complementarity problems. AMS CLASSIFICATION65K15 Numerical methods for variational inequalities and related problems, 65P30 Bifurcation problems, 65H10 Systems of equations, 35M86 Nonlinear unilateral problems and nonlinear variational inequalities of mixed type, 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions).

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“…Moreover, the elements are typically not affine equivalent; i.e. the basis functions cannot be mapped to each element using a reference element in the standard way, and more complicated approaches are needed [24,36,38]. In order to avoid the use of such C 1 elements, other types of finite elements can be used, leading to nonconforming methods in which the finite-element space is not a subspace of H 2 (Ω), such as Morley and cubic Hermite elements [21,24,54].…”

mentioning

confidence: 99%

“…Unlike conforming methods, our approach works effectively in both two and three dimensions. Strongly imposing essential boundary conditions with some finite-element basis functions is difficult [36]. In addition, it can, sometimes, negatively affect properties of the finite-element method, such as its stability and accuracy [33,35].…”

mentioning

confidence: 99%

“…Applications of Nitsche's method to second-order PDEs can be found in [30,33,35]. Moreover, Nitsche-type penalty methods have been used to impose essential boundary conditions for some discretizations of the biharmonic and other fourth-order problems [16,26,36]. While we are able to impose a variety of boundary conditions directly in our variational formulation, we utilize Nitsche-type penalty methods for a particular case where strong enforcement of the boundary conditions leads to problems establishing inf-sup stability of the discretization.…”

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confidence: 99%

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A new mixed finite-element method for $H^2$ elliptic problems

Farrell¹,

Abdalaziz²,

MacLachlan³

2021

Preprint

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Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finiteelement discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

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Code Generation for Generally Mapped Finite Elements

Cited by 20 publications

References 56 publications

Automating Finite Element Methods for Geodynamics via Firedrake

Automating Finite Element Methods for Geodynamics via Firedrake

Deflation for semismooth equations

A new mixed finite-element method for $H^2$ elliptic problems

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