Reduction of dissipation in Lagrange cell-centered hydrodynamics (CCH) through corner gradient reconstruction (CGR)

Abstract

This work presents an extension of a second order cell-centered hydrodynamics scheme on unstructured polyhedral cells [13] toward higher order. The goal is to reduce dissipation, especially for smooth flows. This is accomplished by multiple piecewise linear reconstructions of conserved quantities within the cell. The reconstruction is based upon gradients that are calculated at the nodes, a procedure that avoids the least-square solution of a large equation set for polynomial coefficients. Conservation and monotonicity are guaranteed by adjusting the gradients within each cell corner. Results are presented for a wide variety of test problems involving smooth and shock-dominated flows, fluids and solids, 2D and 3D configurations, as well as Lagrange, Eulerian, and ALE methods. stress and momentum are solved on offset control volumes such that the logical center of each lies on the boundary of the other. Cell-centered hydrodynamics schemes (CCH) in which all conservation equations are solved on a common control volume have been widely applied to Eulerian methods, but only recently widely applied to Lagrange. A cell-centered Lagrange method was first suggested by Godunov [49,48]. Early implementations such as that in the CAVEAT code [2,39] solved an approximate Riemann problem at cell faces rather than at the nodes. It was not until the seminal work of Després and Mazeran [33] that a solution at the nodes was found. Since then, interest in CCH methods has increased and at least three additional formulations have arisen [74,3,13] giving rise to extensions in many areas [25,32,58,75,76,71,77,70,46,45,73,72,21,14,81,15,80]. Although the discussion in this paper is centered about the CCH2 formulation of [13,15], it is relevant to all.We begin with an overview of the CCH2 method. Details are in the cited references that we summarize here in very general terms. Reference [13] describes the basic XY formulation while [15] describes an RZ formulation based on conservative fluxes that preserves symmetry on equiangular polar meshes. The cyclic ordering of the steps depends upon the particular implementation. For purposes of discussion, we begin with the finite volume integration.Finite volume integration. The extensive evolution or rate equations for momentum, deformation, and total energy are expressed as surface integralsUin which M z is the cell mass and {u z ,γ z ,τ z } are respectively specific cell average rates of change of velocity, deformation, and total energy. The fundamental challenge in CCH is the determination of the surface fluxes u p and σ p that are solutions to a Riemann problem. The decomposition of the total energy into kinetic and internal rates k z ,ė z will be discussed later in Section 2.1. A stress rateσ z is derived from the internal energy rate and deformation rate through a constitutive model such as that described in Appendix A. The rate equations are integrated to yield new values for velocity, stress, and total energy.Conservative reconstruction. The finite volume method provides no direct infor...

A 3D finite element ALE method using an approximate Riemann solution

“…[21]. The amount of rotation at the tip of the vortex is similar to the CCH method, CCH2, presented in [11], although not as much as the amount produced using their corner gradient reconstruction method. This test problem was calculated using a fixed Eulerian mesh.…”

“…The aforementioned Riemann jump relation modifies the one used in [10,11,13,30] by using Eq. The first relation is where u ´i s the Riemann velocity at the cell center, the subscript c.p/ is the cell corner between node p and cell´, is the shock impedance, and a is a unit vector in the approximate direction of the shock.…”

“…A preponderance of previous algorithmic research in ALE hydrodynamics has focused on the finite volume staggered grid hydrodynamic (SGH) method [3][4][5][6] and the cell-centered hydrodynamic (CCH) method [7][8][9][10][11][12][13]. The SGH method stores the solution variables at different locations.…”

“…The multidimensional Riemann-like problems were first proposed for Lagrangian CCH [10][11][12]26]; since then, researchers have extended these Riemann problems to finite volume SGH [27][28][29][30] and edge-based finite element PCH [16,21]. The version of this test problem commonly studied in the literature [11,40] contains three regions and two materials with the first high-pressure region and the low-pressure region having D 1:5, while the second high-pressure region has D 1:4. A finite element formulation is attractive because it offers the possibility of using higher-order elements for improved accuracy for situations involving the elastic-plastic deformation of solid materials [20].…”

“…The multidimensional Riemann-like problems were first proposed for Lagrangian CCH [10][11][12]26]; since then, researchers have extended these Riemann problems to finite volume SGH [27][28][29][30] and edge-based finite element PCH [16,21]. The multidimensional Riemann-like problems were first proposed for Lagrangian CCH [10][11][12]26]; since then, researchers have extended these Riemann problems to finite volume SGH [27][28][29][30] and edge-based finite element PCH [16,21].…”