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Title: Compatible, energy conserving, bounds preserving remap of hydrodynamic fields for an extended ALE scheme

Abstract

From the very origins of numerical hydrodynamics in the Lagrangian work of von Neumann and Richtmyer [83], the issue of total energy conservation as well as entropy production has been problematic. Because of well known problems with mesh deformation, Lagrangian schemes have evolved into Arbitrary Lagrangian–Eulerian (ALE) methods [39] that combine the best properties of Lagrangian and Eulerian methods. Energy issues have persisted for this class of methods. We believe that fundamental issues of energy conservation and entropy production in ALE require further examination. The context of the paper is an ALE scheme that is extended in the sense that it permits cyclic or periodic remap of data between grids of the same or differing connectivity. The principal design goals for a remap method then consist of total energy conservation, bounded internal energy, and compatibility of kinetic energy and momentum. We also have secondary objectives of limiting velocity and stress in a non-directional manner, keeping primitive variables monotone, and providing a higher than second order reconstruction of remapped variables. Particularly, the new contributions fall into three categories associated with: energy conservation and entropy production, reconstruction and bounds preservation of scalar and tensor fields, and conservative remap of nonlinear fields. Ourmore » paper presents a derivation of the methods, details of implementation, and numerical results for a number of test problems. The methods requires volume integration of polynomial functions in polytopal cells with planar facets, and the requisite expressions are derived for arbitrary order.« less

Authors:
ORCiD logo [1]; ORCiD logo [1];  [1];  [1]; ORCiD logo [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1411348
Report Number(s):
LA-UR-17-21696
Journal ID: ISSN 0021-9991; TRN: US1800221
Grant/Contract Number:
AC52-06NA25396
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 355; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Lagrangian; hydrodynamics; remap; mimetic; cell-centered; ALE; exactintersection; KEfixup; finite-volume

Citation Formats

Burton, Donald E., Morgan, Nathaniel Ray, Charest, Marc Robert Joseph, Kenamond, Mark Andrew, and Fung, Jimmy. Compatible, energy conserving, bounds preserving remap of hydrodynamic fields for an extended ALE scheme. United States: N. p., 2017. Web. doi:10.1016/j.jcp.2017.11.017.
Burton, Donald E., Morgan, Nathaniel Ray, Charest, Marc Robert Joseph, Kenamond, Mark Andrew, & Fung, Jimmy. Compatible, energy conserving, bounds preserving remap of hydrodynamic fields for an extended ALE scheme. United States. doi:10.1016/j.jcp.2017.11.017.
Burton, Donald E., Morgan, Nathaniel Ray, Charest, Marc Robert Joseph, Kenamond, Mark Andrew, and Fung, Jimmy. 2017. "Compatible, energy conserving, bounds preserving remap of hydrodynamic fields for an extended ALE scheme". United States. doi:10.1016/j.jcp.2017.11.017.
@article{osti_1411348,
title = {Compatible, energy conserving, bounds preserving remap of hydrodynamic fields for an extended ALE scheme},
author = {Burton, Donald E. and Morgan, Nathaniel Ray and Charest, Marc Robert Joseph and Kenamond, Mark Andrew and Fung, Jimmy},
abstractNote = {From the very origins of numerical hydrodynamics in the Lagrangian work of von Neumann and Richtmyer [83], the issue of total energy conservation as well as entropy production has been problematic. Because of well known problems with mesh deformation, Lagrangian schemes have evolved into Arbitrary Lagrangian–Eulerian (ALE) methods [39] that combine the best properties of Lagrangian and Eulerian methods. Energy issues have persisted for this class of methods. We believe that fundamental issues of energy conservation and entropy production in ALE require further examination. The context of the paper is an ALE scheme that is extended in the sense that it permits cyclic or periodic remap of data between grids of the same or differing connectivity. The principal design goals for a remap method then consist of total energy conservation, bounded internal energy, and compatibility of kinetic energy and momentum. We also have secondary objectives of limiting velocity and stress in a non-directional manner, keeping primitive variables monotone, and providing a higher than second order reconstruction of remapped variables. Particularly, the new contributions fall into three categories associated with: energy conservation and entropy production, reconstruction and bounds preservation of scalar and tensor fields, and conservative remap of nonlinear fields. Our paper presents a derivation of the methods, details of implementation, and numerical results for a number of test problems. The methods requires volume integration of polynomial functions in polytopal cells with planar facets, and the requisite expressions are derived for arbitrary order.},
doi = {10.1016/j.jcp.2017.11.017},
journal = {Journal of Computational Physics},
number = C,
volume = 355,
place = {United States},
year = 2017,
month =
}

Journal Article:
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  • We present a Lagrangian time integration scheme and compatible discretization for total energy conservation in multicomponent mixing simulations. Mixing behavior results from relative motion between species. Species velocities are determined by solving species momentum equations in a Lagrangian manner. Included in the species momentum equations are species artificial viscosity (since each species can undergo compression) and inter-species momentum exchange. Thermal energy for each species is also solved, including compression work and thermal dissipation caused by momentum exchange. The present procedure is applicable to mixing of an arbitrary number of species that may not be in pressure or temperature equilibrium. Amore » traditional staggered stencil has been adopted to describe motion of each species. The computational mesh for the mixture is constructed in a Lagrangian manner using the mass-averaged mixture velocity. Species momentum equations are solved at the vertices of the mesh, and temporary species meshes are constructed and advanced in time using the resulting species velocities. Following the Lagrangian step, species quantities are advected (mapped) from the species meshes to the mixture mesh. Momentum exchange between species introduces work that must be included in an energy-conserving discretization scheme. This work has to be transformed to dissipation in order to effect a net change in species thermal energy. The dissipation between interacting species pairs is obtained by combining the momentum exchange work. The dissipation is then distributed to the species involved using a distribution factor based on species specific heats. The resulting compatible discretization scheme provides total energy conservation of the whole mixture. In addition, the numerical scheme includes conservative local energy exchange between species in mixture. Due to the relatively large species interaction coefficients, both the species momenta and energies are calculated implicitly. Sample calculations have yielded excellent results, including conservation of total energy in Lagrangian steps, symmetry preservation, and correct steady-state behavior.« less
  • Cited by 7
  • This work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions. Cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations. Written in Lagrangian form, these equations are discretized utilizing compatible, control volume differencing with a staggered-grid placement of the spatial variables. The concept of compatibility means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations togethermore » define the total energy as a quantity that is exactly conserved in time in discrete form. Multiple pressures are utilized in each zone; they produce forces that resist spurious vorticity generation. This difficulty can severely limit the utility of the Lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions. An edge-centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks. The particular difficulty of exactly preserving one-dimensional spherical symmetry in three-dimensional geometry is solved. This problem has both practical and pedagogical significance. The algorithm is suitable for both structured and unstructured grids. Limitations that symmetry preservation imposes on the latter type of grids are delineated.« less