A conservative finite element method for the incompressible Euler equations with variable density

Gawlik, Evan S.; Gay–Balmaz, François

doi:10.1016/j.jcp.2020.109439

Cited by 20 publications

(14 citation statements)

References 31 publications

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“…For example, the stream function / vorticity formulation, which proved particular effective in 2D, does not exist in the same simple form there. One alternative would be to use a structure preserving discretization of the Navier-Stokes equations, e.g., [20,21], as a basis. The built-in constraints, such as an inherently divergence-free velocity vector field, would then again aid the physicality of the learned solution and simplify the learning task.…”

Section: Discussionmentioning

confidence: 99%

Structure preservation for the Deep Neural Network Multigrid Solver

Margenberg

Lessig

Richter

2021

etna

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The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering, and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural properties such as divergence freedom can and should be respected. Our work is based on DNN-MG, a deep neural network multigrid technique, that we introduced recently and which uses a neural network to represent fine grid fluctuations not resolved by a geometric multigrid finite element solver. Although DNN-MG provides solutions with very good accuracy and is computationally highly efficient, we noticed that the neural network-based corrections substantially violate the divergence freedom of the velocity vector field. In this contribution, we discuss these findings and analyze three approaches to address the problem: a penalty term to encourage divergence freedom of the network output; a penalty term for the corrected velocity field; and a network that learns the stream function and which hence yields divergence-free corrections by construction. Our experimental results show that the third approach based on the stream function outperforms the other two and not only improves the divergence freedom but also the overall fidelity of the simulation.

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Section: Discussionmentioning

confidence: 99%

Structure preservation for the Deep Neural Network Multigrid Solver

Margenberg

Lessig

Richter

2021

etna

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“…As a consequence of the symmetry (12), one can associate to L the Lagrangian (u, ρ) in Eulerian form, as follows…”

Section: Compressible Flowsmentioning

confidence: 99%

“…In particular the discretization corresponds to a weak form of the compressible fluid equation that doesn't seem to have been used in the finite element literature. An incompressible version of this expression of the weak form has been used in [12] for the incompressible fluid with variable density. The setting that we develop applies in general to 2D and 3D fluid models that can be written in Euler-Poincaré form.…”

Section: Introductionmentioning

confidence: 99%

A Variational Finite Element Discretization of Compressible Flow

Gawlik¹,

Gay–Balmaz²

2019

Preprint

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We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart-Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation that doesn't seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations.

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“…The approach we develop in this paper is built on our earlier work on conservative methods for compressible fluids (Gawlik & Gay-Balmaz 2020b) and for incompressible MHD with variable density in Gawlik & Gay-Balmaz (2020a. Two notable differences that arise in the viscous, resistive compressible setting are the change in boundary conditions for the velocity and magnetic fields, and the fact that the magnetic field is not advected as a vector field when the fluid is compressible; that is, curl(B × u) does not coincide with the Lie derivative of the vector field B along u when divu = 0.…”

Section: Introductionmentioning

confidence: 99%

A structure-preserving finite element method for compressible ideal and resistive magnetohydrodynamics

Gawlik

Gay–Balmaz

2021

J. Plasma Phys.

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We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total energy and magnetic helicity, and preserves mass and the constraint $\text {div}B = 0$ to machine precision, both at the spatially and temporally discrete levels. In particular, conservation of energy and magnetic helicity hold at the discrete levels in the ideal case. It is observed that cross-helicity is well conserved in our simulation in the ideal case.

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A conservative finite element method for the incompressible Euler equations with variable density

Cited by 20 publications

References 31 publications

Structure preservation for the Deep Neural Network Multigrid Solver

Structure preservation for the Deep Neural Network Multigrid Solver

A Variational Finite Element Discretization of Compressible Flow

A structure-preserving finite element method for compressible ideal and resistive magnetohydrodynamics

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