Optimization of Artificial Viscosity in Production Codes Based on Gaussian Regression Surrogate Models

Abstract

To accurately model flows with shock waves using staggered-grid Lagrangian hydrodynamics, the artificial viscosity has to be introduced to convert kinetic energy into internal energy, thereby increasing the entropy across shocks. Determining the appropriate strength of the artificial viscosity is an art and strongly depends on the particular problem and experience of the researcher. The objective of this study is to pose the problem of finding the appropriate strength of the artificial viscosity as an optimization problem and solve this problem using machine learning (ML) tools, specifically using surrogate models based on Gaussian Process regression (GPR) and Bayesian analysis. We describe the optimization method and discuss various practical details of its implementation. The shock-containing problems for which we apply this method all have been implemented in the LANL code FLAG (Burton in Connectivity structures and differencing techniques for staggered-grid free-Lagrange hydrodynamics, Tech. Rep. UCRL-JC-110555, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1992, in Consistent finite-volume discretization of hydrodynamic conservation laws for unstructured grids, Tech. Rep. CRL-JC-118788, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Tech. Rep. UCRL-JC-118306, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, in FLAG, a multi-dimensional, multiple mesh, adaptive free-Lagrange, hydrodynamics code. In: NECDC, 1992). First, we apply ML to find optimal values to isolated shock problems of different strengths. Second, we apply ML to optimize the viscosity for a one-dimensional (1D) propagating detonation problem based on Zel’dovich-von Neumann-Doring (ZND) (Fickett and Davis in Detonation: theory and experiment. Dover books on physics. Dover Publications, Mineola, 2000) detonation theory using a reactive burn model. We compare results for default (currently used values in FLAG) and optimized values of the artificial viscosity for these problems demonstrating the potential for significant improvement in the accuracy of computations.

Keywords

Optimization / Artificial viscosity / Gaussian regression surrigate model

References

[1.] Aslam T. Shock temperature dependent rate law for plastic bonded explosives. J. Appl. Phys., 2018, 123,

[2.] Barlow A, Shashkov MJ, Maire P-H, Rieben R, Rider W. Arbitrary Lagrangian-Eulerian methods for modeling high-speed compressible multimaterial flows. J. Comput. Phys., 2016, 322: 603-665,

[3.] Bishop CM. . Pattern Recognition and Machine Learning, 2006 Berlin Springer

[4.] Burton, D.: Connectivity structures and differencing techniques for staggered-grid free Lagrange hydrodynamics. Tech. Rep. UCRL-JC-110555, Lawrence Livermore National Laboratory, Livermore, CA, 1992 (1992)

[5.] Burton, D.: Consistent finite-volume discretization of hydrodynamic conservation laws for unstructured grids. Tech. Rep. CRL-JC-118788, Lawrence Livermore National Laboratory, Livermore, CA, 1992 (1994)

[6.] Burton, D.: Multidimensional discretization of conservation laws for unstructured polyhedral grids. Tech. Rep. UCRL-JC-118306, Lawrence Livermore National Laboratory, Livermore, CA, 1992 (1994)

[7.] Burton, D.E.: FLAG: a multi-dimensional, multiple mesh, adaptive free-Lagrange, hydrodynamics code. In: Nuclear Explosives Code Development Conference (1992)

[8.] Campbell J, Shashkov M. A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys., 2001, 172: 739-765,

[9.] Caramana E, Burton D, Shashkov M, Whalen P. The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. J. Comput. Phys., 1998, 146: 227-262,

[10.] Fickett W, Davis W. . Detonation: Theory and Experiment, Dover Books on Physics, 2000 Minoela Dover Publications

[11.] Gramacy RB. . Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences, 2020 Boca Raton Chapman Hall/CRC,

[12.] Kenamond M, Bement M, Shashkov M. Compatible, total energy conserving and symmetry preserving arbitrary Lagrangian-Eulerian hydrodynamics in 2D rz-cylindrical coordinates. J. Comput. Phys., 2014, 268: 154-185,

[13.] Kennedy MC, O’Hagan A. Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B (Statistical Methodology), 2001, 63: 425-464,

[14.] Landshoff, R.: A numerical method for treating fluid flow in the presence of shocks. Tech. Rep. LA-1930, Los Alamos National Laboratory, Los Alamos, NM (1955)

[15.] Lipnikov K, Shashkov M. A framework for developing a mimetic tensor artificial viscosity for Lagrangian hydrocodes on arbitrary polygonal meshes. J. Comput. Phys., 2010, 229: 7911-7941,

[16.] Mockus J, Tiesis V, Zilinskas A. The application of Bayesian methods for seeking the extremum. Towards Glob. Optim., 1978, 2: 2

[17.] Price, A.: ZND verification tests for reactive burn models in FLAG. Tech. Rep. LA-UR-20-21911, Los Alamos National Laboratory (1990)

[18.] Prunty S. . Introduction to Simple Shock Waves in Air. With Numerical Solutions Using Artificial Viscosity, 2019 Berlin Springer,

[19.] Rasmussen CE, Williams CKI. . Gaussian Processes for Machine Learning, 2006 Cambridge Massachusetts Institute of Technology

[20.] Schonlau M, Welch WJ, Jones DR. Global versus local search in constrained optimization of computer models. Lect. Notes-Monogr. Ser., 1998, 34: 11-25

[21.] Toro E. . Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 2009 Berlin Springer,

[22.] Neumann J, Richtmyer R. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 1950, 21: 232-237,

[23.] Wescott D, Stewart BL, Davis W. Equation of state and reaction rate for condensed-phase explosives. J. Appl. Phys., 2005, 98: 053514,

[24.] Wilkins ML. Use of artificial viscosity in multidimensional fluid dynamic calculations. J. Comput. Phys., 1980, 36: 281-303,

Funding

Office of Defense Nuclear Security(89233218CNA000001)