Stabilized shock hydrodynamics: I. A Lagrangian method

Abstract A new SUPG-stabilized formulation for Lagrangian hydrodynamics of materials satisfying Mie–Gruneisen equation of state is proposed. It allows the use of simplex-type (triangular/tetrahedral) meshes as well as the more commonly used brick-type (quadrilateral/hexahedral) meshes. The proposed method yields a globally conservative formulation, in which equal-order interpolation (P1 or Q1 isoparametric finite elements) is applied to velocities, displacements, and pressure. As a direct consequence, and in contrast to traditional cell-centered multidimensional hydrocode implementations, the proposed formulation allows a natural representation of the pressure gradient on element interiors. The SUPG stabilization involves additional design requirements, specific to the Lagrangian formulation. A discontinuity capturing operator in the form of a Noh-type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions. A set of challenging shock hydrodynamics benchmark tests for the Euler equations of gas dynamics in one and two space dimensions is presented. In the two-dimensional case, computations performed on quadrilateral and triangular grids are analyzed and compared. These results indicate that the new formulation is a promising technology for hydrocode applications.

[1]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[2]  Thomas J. R. Hughes,et al.  Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations , 1984 .

[3]  D. Estep,et al.  Global error control for the continuous Galerkin finite element method for ordinary differential equations , 1994 .

[4]  Thomas J. R. Hughes,et al.  The Continuous Galerkin Method Is Locally Conservative , 2000 .

[5]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[6]  Tayfun E. Tezduyar,et al.  Stabilization and shock-capturing parameters in SUPG formulation of compressible flows , 2004 .

[7]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[8]  Peter Monk,et al.  Continuous finite elements in space and time for the heat equation , 1989 .

[9]  Pierre Jamet.  Stability and Convergence of a Generalized Crank-Nicolson Scheme on a Variable Mesh for the Heat Equation , 1980 .

[10]  N. N. I︠A︡nenko.  Difference methods for solutions of problems of mathematical physics, 1 , 1967 .

[11]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations , 1991 .

[12]  Antoinette M. Maniatty,et al.  Stabilized finite element method for viscoplastic flow: formulation with state variable evolution , 2003 .

[13]  Thomas J. R. Hughes,et al.  Encyclopedia of computational mechanics , 2004 .

[14]  Guillermo Hauke,et al.  Simple stabilizing matrices for the computation of compressible flows in primitive variables , 2001 .

[15]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[16]  B. Hulme.  Discrete Galerkin and related one-step methods for ordinary differential equations , 1972 .

[17]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[18]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[19]  Yong Liu,et al.  Stabilized finite element method for viscoplastic flow: formulation and a simple progressive solution strategy , 2001 .

[20]  Mikhail Shashkov,et al.  Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations , 1998 .

[21]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[22]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[23]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[24]  Donald A. French,et al.  Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems , 1994 .

[25]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[26]  Claes Johnson,et al.  On the convergence of a finite element method for a nonlinear hyperbolic conservation law , 1987 .

[27]  Tayfun E. Tezduyar,et al.  Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces , 2004 .

[28]  L. Sedov.  Similarity and Dimensional Methods in Mechanics , 1960 .

[29]  Mark L. Wilkins,et al.  Use of artificial viscosity in multidimensional fluid dynamic calculations , 1980 .

[30]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[31]  David J. Benson,et al.  A new two-dimensional flux-limited shock viscosity for impact calculations , 1991 .

[32]  Yong Liu,et al.  Higher order stabilized finite element method for hyperelastic finite deformation , 2002 .

[33]  Donald A. French,et al.  A space-time finite element method for the wave equation* , 1993 .

[34]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[35]  Guglielmo Scovazzi.  Multiscale methods in science and engineering , 2004 .

[36]  T. Tezduyar.  Computation of moving boundaries and interfaces and stabilization parameters , 2003 .

[37]  Donald A. French,et al.  A continuous space-time finite element method for the wave equation , 1996, Math. Comput..

[38]  D. Benson.  Computational methods in Lagrangian and Eulerian hydrocodes , 1992 .

[39]  M. Shephard,et al.  A stabilized mixed finite element method for finite elasticity.: Formulation for linear displacement and pressure interpolation , 1999 .

[40]  A. E. Fincham,et al.  Parallel Computation , 1999, Algorithms and Theory of Computation Handbook.

[41]  Anders Szepessy,et al.  Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions , 1989 .

[42]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[43]  Guglielmo Scovazzi,et al.  Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations☆ , 2007 .

[44]  Guillermo Hauke,et al.  a Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation of the K - Model. , 1994 .

[45]  Pavel B. Bochev,et al.  Stability of the SUPG finite element method for transient advection-diffusion problems , 2004 .

[46]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[47]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[48]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms , 1991 .

[49]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[50]  Peter Hansbo,et al.  On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws , 1990 .

[51]  Thomas J. R. Hughes,et al.  A comparative study of different sets of variables for solving compressible and incompressible flows , 1998 .

[52]  Wulf G. Dettmer,et al.  An analysis of the time integration algorithms for the finite element solutions of incompressible Navier-Stokes equations based on a stabilised formulation , 2003 .