Elsevier

Computers & Fluids

Volume 205, 15 June 2020, 104580
Computers & Fluids

Locally adaptive artificial viscosity strategies for Lagrangian hydrodynamics

https://doi.org/10.1016/j.compfluid.2020.104580Get rights and content

Highlights

  • New method for constructing an adaptive, artificial viscosity in the context of one-dimensional, staggered-grid Lagrangian hydrodynamics.

  • The adaptive, artificial viscosity depends locally on density in a neighborhood around the shock.

  • The algorithm to compute the adaptive coefficient is automated using a data-driven approach based on a set of optimal reference values for isolated shocks.

  • This methodology produces more accurate results for a variety of tests with propagating shock waves.

Abstract

To accurately model inviscid flow with shock waves using staggered-grid Lagrangian hydrodynamics, artificial viscosity is introduced to convert kinetic energy into internal energy, thereby providing a mechanism to generate the required entropy increase across shocks. In this paper, we propose a new method for constructing an adaptive, artificial viscosity in the context of one-dimensional, staggered-grid Lagrangian hydrodynamics. Our adaptive, artificial viscosity is defined in terms of two parameters that depend on density locally in a neighborhood around a shock, and hence, vary cell-by-cell. Our methodology is based on building a reference set of pre-computed optimal, globally constant artificial viscosity coefficients for a family of isolated shock test problems. For arbitrary flows, the evaluation of the unknown coefficients is automated by first estimating shock intensity locally, and second computing the corresponding adaptive parameter value via interpolation over the data from the pre-computed reference set of optimal, globally constant coefficient values. To illustrate the performance of our new approach, we compare our results against two existing methods. The first method is a limiter-based approach, which relies on estimating velocity gradients of the flow, and the second method is utilized in several commercial codes. We demonstrate that our new adaptive methodology produces more accurate results for a variety of tests with propagating shock waves, as well as for the aforementioned family of isolated shock problems.

Section snippets

Background and rationale

Algorithms based on staggered-grid Lagrangian hydrodynamics [1], [2], [3] provide a robust computational framework for modeling high-speed, multimaterial flows. In the setting of staggered-grid Lagrangian hydrodynamics, the discretization of material position and velocity are defined at grid points (or nodes) of the computational mesh, while all other state variables, such as density and specific internal energy, are defined inside cells (or zones) [1], [3]. One of the staggered-grid

Lagrangian staggered-grid discretization in 1D

In this section, we describe a standard, staggered-grid discretization for Lagrangian hydrodynamics in 1D (for more details see [1], [3]).

At each time step tn, the computational mesh is defined by a set of distributed nodes (points), which are enumerated by an integer index i, and have coordinates xin. Here and below subscripts i denote a spatial index, and superscripts n denote a time step index. Since the mesh moves with the fluid in the Lagrangian frame its coordinates also depend on time.

A Family of exact shock solutions

In this section, we describe a set of exact solutions for a family of isolated shocks parametrized by their shock strength. In the next section, we use this family to find reference values for the“optimal”, globally constant coefficient C for a given shock intensity η.

We consider a specific family of isolated shocks problems where the jump in velocity across the shock is fixed, so thatuL=0,uR=1.We specify the right density, ρR=1, and define the left density in terms of the intensity η, as

Globally constant, optimal artificial viscosity coefficient values

In this section, we explain how we initialize the discrete version of the isolated shock problem used later for computing numerical solutions to the family of isolated shocks described in the previous section. We also describe several important numerical artifacts of the approximate solution resulting from the staggered-grid scheme.

The density in each cell obtained from the staggered-grid discretization is interpreted as the cell-volume averaged density. Therefore, when computing the error with

Adaptive local coefficients of artificial viscosity

In this section, we describe how to define a locally adaptive coefficient of artificial viscosity for each cell. The general idea is to find a local estimate of shock strength based on the flow state in neighboring cells and then use the value of the coefficient obtained from an approximation to the curve through the optimal artificial viscosity coefficients as a function of shock strength described in the previous section.

First, we recall that artificial viscosity is only added to cells under

Numerical results

In this section, we illustrate the performance of our adaptive algorithm using several test problems. We also compare the results we obtained for our algorithm against several alternatives. In particular, we offer a comparison against an approach based on limiters [3] and a method applied in the commercial codes DYNA and ABAQUS [21]. Note that the latter method employs separate coefficients for the linear and quadratic artificial viscosity terms (see below). In our comparison, we will apply

Conclusions and future work

We have developed a new automated method to determine an adaptive artificial viscosity coefficient in the context of one-dimensional, staggered-grid Lagrangian hydrodynamics. Our new technique is based on first creating a set of reference values for a constant in space (and time) optimal artificial viscosity coefficient from a family of isolated shock test problems parameterized by shock intensity, measured in terms of the ratio in density across the shock. We have chosen this parametrization

In memoriam

This paper is dedicated to the memory of Dr. Douglas Nelson Woods (*January 11th, 1985–†September 11th, 2019), promising young scientist and post-doctoral research fellow at Los Alamos National Laboratory. Our thoughts and wishes go to his wife Jessica, to his parents Susan and Tom, to his sister Rebecca and to his brother Chris, whom he left behind.

CRediT authorship contribution statement

Jason Albright: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Visualization, Investigation. Mikhail Shashkov: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Visualization, Investigation.

Acknowledgements

The authors would like to thank L. Margolin, W. Rider, S. Runnels, M. Owen, D. Miller, R. Rieben, R. Loubere, P.-H. Maire, K. Lipnikov, B. Wendroff, P. Vachal, M. Kucharik, A. Barlow, S. Ramsey, and D. Bruney for many stimulating discussions.

This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The authors gratefully acknowledge the support of the US

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