Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

Anderson, Robert; Dobrev, Veselin; Kolev, Tzanio V.; Rieben, Robert N.

doi:10.1002/fld.3965

Cited by 44 publications

(43 citation statements)

References 18 publications

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“…In cases where a geometric boundary condition is unknown, the fixed point projection (45) could be used to disable motion of the discrete geometry assuming that any attached interfaces are represented exactly. For the more general case of a priori unknown attached interfaces, such as the case considered in Section 5.4, some specification of a geometric boundary condition will be required in order to allow the interface attachment point to move along the surface.…”

Section: Geometric Boundary Conditionsmentioning

confidence: 99%

“…Only the point initially located at (x = 0, t = 1) has a nonzero derivative; however, a planar boundary condition (44) is applied, so that the derivative of the residual with respect to the t-component vanishes, thus constraining the point to the t = 1 plane. Points lying on the inflow plane are constrained to their initial position, according to (45), so that their derivative vanishes. In particular, the interface location at t = 0, ie,…”

Section: Figurementioning

confidence: 99%

“…Lagrangian or arbitrary Lagrangian‐Eulerian (ALE) methods, represented for higher‐order finite element methods by works such as those of Dobrev et al, Anderson et al, Persson et al, and Fidkowski, are another class of interface‐fitting techniques. These methods have numerous advantages, particularly regarding highly accurate and robust treatment of dynamic material interfaces or moving bodies.…”

Section: Introductionmentioning

confidence: 99%

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A moving discontinuous Galerkin finite element method for flows with interfaces

Corrigan

Kercher

Kessler

2018

Numerical Methods in Fluids

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Summary A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high‐order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space‐time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three‐dimensional flows that provide an initial assessment of the stability and accuracy of the method.

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Section: Geometric Boundary Conditionsmentioning

confidence: 99%

Section: Figurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A moving discontinuous Galerkin finite element method for flows with interfaces

Corrigan

Kercher

Kessler

2018

Numerical Methods in Fluids

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“…The conversion of kinetic energy into internal energy through shock waves, consistent with the second law of thermodynamics, is ensured by adding an artificial viscosity term. The staggered grid schemes employed in most hydro-codes have been remarkably successful over the past decades in solving complex multi-dimensional compressible fluid flows, refer for instance to [45,88,17,18,32,55,33,34,4]. However, they clearly have some theoretical and practical deficiencies such as mesh imprinting and symmetry breaking.…”

Section: Introductionmentioning

confidence: 99%

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

Vilar

Shu

Maire

2016

Journal of Computational Physics

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International audienceOne of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84]

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“…Moreover, the locality is lost if one wants to be independent of the order in which the re-distribution is performed, even if a few cells are concerned in principle [21]. It is finally worth mentioning that high-order staggered remap [23]-naturally motivated by high-order versions of Lagrangian SGH schemes-also represent a good way to decrease the kinetic energy error.…”

Section: Conservation and Entropy Control In Lagrange-plus-remap Algorithmsmentioning

confidence: 99%

Conservative and entropy controlled remap for multi-material ALE simulations with space-staggered schemes

Marbœuf

Claisse

Tallec

2019

Journal of Computational Physics

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The remapping strategy is crucial in any Arbitrary Lagrangian-Eulerian (ALE) algorithm based on a Lagrange-plusremap paradigm. This step is particularly challenging for space-staggered schemes since inconsistencies may appear between cell centered and node centered fields after remap if no special care is taken [1][2][3]. We propose here a space-staggered remapping strategy focusing on conservation properties and entropy control. The proposed algorithm conserves mass, total energy and respects the Second Law of Thermodynamics (for robustness) up to round-off errors. This is achieved at a low computational cost by introducing a consistent, explicit and local post processing of the linear momentum after remap. This new method is then analyzed showing that the strict momentum conservation is sacrificed. It is now conserved to the scheme's order, such as entropy. Other classical properties such that the "DeBar consistency" [4], the continuity with the Lagrangian step and the monotonicity are also discussed. This work is developed in the context of the intersection-based (or overlay-based) remap. Therefore, the rezoned mesh does not have to be close to the Lagrangian one and, even if it is not considered here, our study can be easily extended to rezoning strategies which modify the mesh connectivity.

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Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

Cited by 44 publications

References 18 publications

A moving discontinuous Galerkin finite element method for flows with interfaces

A moving discontinuous Galerkin finite element method for flows with interfaces

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

Conservative and entropy controlled remap for multi-material ALE simulations with space-staggered schemes

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