Conservative polytopal mimetic discretization of the incompressible Navier–Stokes equations

Beltman, René; Anthonissen, M.J.H.; Koren, Barry

doi:10.1016/j.cam.2018.02.007

Cited by 5 publications

(7 citation statements)

References 39 publications

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“…For the convection term we use the central approximation introduced in [7], which conserves both momentum and kinetic energy. This discretization of the convection term fits well in the framework presented above [6].…”

Section: Bṽ (0) Bsupporting

confidence: 70%

“…The method conserves mass, momentum and energy, even in cut cells, and determines a vorticity corresponding to a physically correct global vorticity. For proofs and numerical verifications of these properties see [6].…”

Section: Numerical Results: the Flow Around A Cylindermentioning

confidence: 99%

“…We choose the orientation of the dual cells in relation to the orientation of their corresponding primal cells. From this it follows [6] that the incidence matrices on the dual meshG, i.e.,D (1,2) :C (2) →C (1) andD (0,1) :C (1) →C (0) are given bỹ D (1,2) = D (2,1) T andD (0,1) = D (1,0) T . For Eq.…”

Section: The Incidence Matricesmentioning

confidence: 92%

“…We use the mimetic scalar product matrices [5], which can also be interpreted as interpolation operators [6]. These operators are constructed locally for each 2-cell and the global operator is constructed by an assembly process as in finite element methods.…”

Section: The Discrete Hodge Operatorsmentioning

confidence: 99%

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Conservative Mimetic Cut-Cell Method for Incompressible Navier-Stokes Equations

Beltman

Anthonissen

Koren

2019

Lecture Notes in Computational Science and Engineering

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We introduce a mimetic Cartesian cut-cell method for incompressible viscous flow that conserves mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. In particular we discuss how the no-slip boundary conditions should be applied in a conservative way on objects immersed in the Cartesian mesh. We use the method to compute the flow around a cylinder. We find a good comparison between our results and benchmark results for both a steady and an unsteady test case. To compute fluid flow in complicated geometries often curvilinear or unstructured meshes are used. The generation of these meshes is difficult and time-consuming. When the geometry depends on time, the mesh has to be updated after every time step and the cost of mesh generation will take a significant part of the total computing time. Immersed boundary methods form an increasingly popular alternative. Immersed boundary methods are methods in which one Cartesian mesh is used for the complete flow domain, with the boundaries of objects immersed in this Cartesian mesh. Near the immersed boundary the Cartesian method is adapted for the no-slip boundary condition. Within the class of immersed boundary methods essentially two approaches for modeling the boundary conditions exist. In the first approach the influence of the boundary on the fluid is modeled by an extra force term in the Navier-Stokes equations. In the second approach the sharp interface of the boundary is maintained and the boundary condition is taken into account by adjusting the discretized Navier-Stokes equations.

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Section: Bṽ (0) Bsupporting

confidence: 70%

Section: Numerical Results: the Flow Around A Cylindermentioning

confidence: 99%

Section: The Incidence Matricesmentioning

confidence: 92%

Section: The Discrete Hodge Operatorsmentioning

confidence: 99%

See 2 more Smart Citations

Conservative Mimetic Cut-Cell Method for Incompressible Navier-Stokes Equations

Beltman

Anthonissen

Koren

2019

Lecture Notes in Computational Science and Engineering

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“…In this work however, we consider only (12) as the natural complex to obtain conservation properties.…”

Section: The Data-driven Exterior Calculusmentioning

confidence: 99%

Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs

Trask¹,

Huang²,

Hu³

2020

Preprint

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As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly topological objects lacking the metric information fundamental to PDE discretization. We present an approach where this missing metric information may be learned from data, using graphs as coarse-grained mesh surrogates that inherit desirable conservation and exact sequence structure from the combinatorial Hodge theory. The resulting datadriven exterior calculus (DDEC) may be used to extract structure-preserving surrogate models with mathematical guarantees of well-posedness. The approach admits a PDE-constrained optimization training strategy which guarantees machine-learned models enforce physics to machine precision, even for poorly trained models or small data regimes. We provide analysis of the method for a class of models designed to reproduce nonlinear perturbations of elliptic problems and provide examples of learning H(div)/H(curl) systems representative of subsurface flows and electromagnetics.

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