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Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions of Second-Order Elliptic Problems

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Communications in Computational Physics 2008年第4期3卷 852-877页

作者： Richard Liska mikhail shashkov Czech Technical University in Prague Faculty of Nuclear Sciences and Physical EngineeringBrehova 711519 Prague 1Czech Republic. Theoretical Division Group T-7MS-B284Los Alamos National LaboratoryLos AlamosNM 87545USA.

The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value *** preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key *** is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in *** this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic *** approach is based on repair technique,which is a posteriori correction of the discrete *** method is based on constrained *** tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.

关键词： Second-order elliptic problems linear finite element solutions discrete maximum principle constrained optimization.

THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES

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Acta Mathematica Scientia 2011年第6期31卷 2195-2202页

作者： Richard Liska mikhail shashkov Burton Wendroff Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Group XCP-4 MS-F644Los Alamos National Laboratry Group T-5 MS-B284Los Alamos National Laboratry

We give a brief discussion of some of the contributions of Peter Lax to Com- putational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We de- velop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax- Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh＇s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.

关键词： Lax-Friedrichs Lax-Wendroff conservation laws Lagrangian coordinates,triangular grid

The Error-Minimization-Based Strategy for Moving Mesh Methods

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Communications in Computational Physics 2006年第1期1卷 53-80页

作者： Konstantin Lipnikov mikhail shashkov Los Alamos National Laboratory MS B284Los AlamosNM87545USA

The typical elements in a numerical simulation of fluid flow using moving meshes are a time integration scheme,a rezone method in which a new mesh is defined,and a remapping(conservative interpolation)in which a solution is transferred to the new *** objective of the rezone method is to move the computational mesh to improve the robustness,accuracy and eventually efficiency of the *** this paper,we consider the onedimensional viscous Burgers’equation and describe a new rezone strategy which minimizes the L2 norm of error and maintains mesh *** efficiency of the proposed method is demonstrated with numerical examples.

关键词： Moving meshes Burgers’equation error estimates

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

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Communications on Applied Mathematics and Computation 2024年第3期6卷 1521-1550页

作者： Vitaliy Gyrya Evan Lieberman Mark Kenamond mikhail shashkov Applied Mathematics and Plasma Physics Theoretical DivisionMail Stop B284Los Alamos National LaboratoryLos AlamosNM 87545USA Continuum Models and Numerical Methods Group X-Computational Physics DivisionMail Stop F644Los Alamos National LaboratoryLos AlamosNM 87545USA Lagrangian Codes Group X-Computational Physics DivisionMail Stop T086Los Alamos National LaboratoryLos AlamosNM 87545USA

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 re

关键词： Optimization Artificial viscosity Gaussian regression surrigate model

A Pressure Relaxation Closure Model for One-Dimensional, Two-Material Lagrangian Hydrodynamics Based on the Riemann Problem

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Communications in Computational Physics 2010年第5期7卷 927-976页

作者： James R.Kamm mikhail J.shashkov Applied Physics Division Los Alamos National LaboratoryLos AlamosNM 87545USA Theoretical Division Los Alamos National LaboratoryLos AlamosNM 87545USA

Despite decades of development, Lagrangian hydrodynamics of strengthfree materials presents numerous open issues, even in one dimension. We focus on theproblem of closing a system of equations for a two-material cell under the assumptionof a single velocity model. There are several existing models and approaches, eachpossessing different levels of fidelity to the underlying physics and each exhibitingunique features in the computed solutions. We consider the case in which the changein heat in the constituent materials in the mixed cell is assumed equal. An instantaneous pressure equilibration model for a mixed cell can be cast as four equations infour unknowns, comprised of the updated values of the specific internal energy andthe specific volume for each of the two materials in the mixed cell. The unique contribution of our approach is a physics-inspired, geometry-based model in which theupdated values of the sub-cell, relaxing-toward-equilibrium constituent pressures arerelated to a local Riemann problem through an optimization principle. This approachcouples the modeling problem of assigning sub-cell pressures to the physics associated with the local, dynamic evolution. We package our approach in the frameworkof a standard predictor-corrector time integration scheme. We evaluate our model using idealized, two material problems using either ideal-gas or stiffened-gas equationsof state and compare these results to those computed with the method of Tipton andwith corresponding pure-material calculations.

关键词： Lagrangian hydrodynamics compressible flow multi-material flow pressure relaxation

Machine Learning Approaches for the Solution of the Riemann Problem in Fluid Dynamics:a Case Study

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Communications on Applied Mathematics and Computation 2024年第3期6卷 1832-1859页

作者： Vitaly Gyrya mikhail shashkov Alexei Skurikhin Svetlana Tokareva Los Alamos National Laboratory Applied Mathematics and Plasma Physics(T-5)Los AlamosUSA Los Alamos National Laboratory Continuum Models and Numerical Methods(XCP-4)Los AlamosUSA Los Alamos National Laboratory Space Data Science and Systems(ISR-3)Los AlamosUSA

We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid *** Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock *** solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin ***,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial *** exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear *** to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their *** compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)*** these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.

关键词： Machine learning(ML) Neural network(NN) Gaussian process(GP) Riemann problem Numerical fluxes Finite-volume method

A Multi-Material CCALE-MOF Approach in Cylindrical Geometry

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Communications in Computational Physics 2014年第2期15卷 330-364页

作者： Marie Billaud Friess Jerome Breil Pierre-Henri Maire mikhail shashkov LUNAM Universite GeMUMR CNRS 6183Ecole Centrale NantesUniversite de Nantes1 rue de la NoeBP 9210144321 Nantes Cedex 3France Univ.Bordeaux CEACNRSCELIAUMR5107F-33400 TalenceFrance CEA CESTA 15 Avenue des Sabli`eresCS 6000133116 Le Barp CedexFrance Los Alamos National Laboratory XCP-4Los AlamosNM 87545USA.

In this paper we present recent developments concerning a Cell-Centered Arbitrary Lagrangian Eulerian(CCALE)strategy using the Moment Of Fluid(MOF)interface reconstruction for the numerical simulation of multi-material compressible fluid flows on unstructured grids in cylindrical ***,our attention is focused here on the following ***,we propose a new formulation of the scheme used during the Lagrangian phase in the particular case of axisymmetric ***,the MOF method is considered for multi-interface reconstruction in cylindrical ***,a method devoted to the rezoning of polar meshes is ***,a generalization of the hybrid remapping to cylindrical geometries is *** explorations are validated by mean of several test cases using unstructured grid that clearly illustrate the robustness and accuracy of the new method.

关键词： Cell-centered scheme Lagrangian hydrodynamics ALE MOF interface reconstruction Rezoning algorithm polar meshes hybrid remapping axisymmetric geometries.

Machine learning-based method to adjust electron anomalous conductivity profile to experimentally measured operating parameters of Hall thruster

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Plasma Science and Technology 2022年第6期24卷 148-156页

作者： Andrey shashkov mikhail TYUSHEV Alexander LOVTSOV Dmitry TOMILIN Dmitrii KRAVCHENKO JSC'Keldysh Research Center' 8 Onezhskaya St.Moscow 125438Russia

The problem of determining the electron anomalous conductivity profile in a Hall thruster,when its operating parameters are known from the experiment,is *** solve the problem,we propose varying the parametrically set anomalous conductivity profile until the calculated operating parameters match the experimentally measured ones in the best *** axial 1D3V hybrid model was used to calculate the operating parameters with parametrically set *** of the conductivity profile was performed using Bayesian optimization with a Gaussian process(machine learning method),which can resolve all local minima,even for noisy *** calculated solution corresponding to the measured operating parameters of a Hall thruster in the best way proved to be unique for the studied operating modes of *** local plasma parameters were calculated and compared to the measured ones for four different operating *** results show the qualitative *** agreement between calculated and measured local parameters can be improved with a more accurate model of plasma-wall interaction.

关键词： Hall thruster anomalous conductivity machine learning Bayesian optimization Gaussian process electric propulsion

Numerical study of viscosity and heat flux role in heavy species dynamics in Hall thruster discharge

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Plasma Science and Technology 2023年第1期25卷 165-180页

作者： Andrey shashkov Alexander LOVTSOV Dmitri TOMILIN Dmitrii KRAVCHENKO JSC'Keldysh Research Center' 8 Onezhskaya st.Moscow 125438Russia

A two-and three-dimensional velocity space axisymmetric hybrid-PIC model of Hall thruster discharge called Hybrid2D has been *** particle-in-cell(PIC) method was used for neutrals and ions(heavy species),and fluid dynamics on a magnetic field-aligned(MFA) mesh was used for electrons.A time-saving method for heavy species moment interpolation on a MFA mesh was *** method comprises using regular rectangle and irregular triangle meshes,connected to each other on a pre-processing *** electron fluid model takes into account neither inertia terms nor viscous terms and includes an electron temperature equation with a heat flux *** developed model was used to calculate all heavy species moments up to the third one in a stationary *** analysis of the viscosity and the heat flux impact on the force and energy balance has shown that for the calculated geometry of the Hall thruster,the viscosity and the heat flux terms have the same magnitude as the other terms and could not be ***,it was shown that the heat flux is not proportional to the temperature gradient and,consequently,the highest moments should be calculated to close the neutral fluid equation *** the same time,ions can only be modeled as a cold non-viscous fluid when the sole aim of modeling is the calculation of the operating parameters or distribution of the local parameters along the centerline of the discharge *** is because the magnitude of the viscosity and the temperature gradient terms are negligible at the ***,when a simulation’s focus is either on the radial divergence of the plume or on magnetic pole erosion,three components of the ion temperature should be taken into *** non-diagonal terms of ion pressure tensor have a lower impact than the diagonal *** to the study,a zero heat flux condition could be used to close the ion equation system in calculated geometry.

关键词： Hall thruster hybrid-PIC model moments of distribution function magnetic-field-aligned mesh