Locally adaptive artificial viscosity strategies for Lagrangian hydrodynamics
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 . 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 thatWe specify the right density, 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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