Algorithmic properties of the midpoint predictor-corrector time integrator.

Algorithmic properties of the midpoint predictor-corrector time integration algorithm are examined. In the case of a finite number of iterations, the errors in angular momentum conservation and incremental objectivity are controlled by the number of iterations performed. Exact angular momentum conservation and exact incremental objectivity are achieved in the limit of an infinite number of iterations. A complete stability and dispersion analysis of the linearized algorithm is detailed. The main observation is that stability depends critically on the number of iterations performed.

[1]  D. Flanagan,et al.  PRONTO 3D: A three-dimensional transient solid dynamics program , 1989 .

[2]  Mikhail Shashkov,et al.  Multi-Scale Lagrangian Shock Hydrodynamics on Q1/P0 Finite Elements: Theoretical Framework and Two-dimensional Computations. , 2008 .

[3]  C. L. Rousculp,et al.  A Compatible, Energy and Symmetry Preserving Lagrangian Hydrodynamics Algorithm in Three-Dimensional Cartesian Geometry , 2000 .

[4]  S. Krenk.  Dispersion-corrected explicit integration of the wave equation , 2001 .

[5]  T. J.R. Hughes,et al.  ANALYSIS OF TRANSIENT ALGORITHMS WITH PARTICULAR REFERENCE TO STABILITY BEHAVIOR. , 1983 .

[6]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[7]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[8]  A. Curnier.  Computational methods in solid mechanics , 1994 .

[9]  Brian H. Hahn,et al.  Essential Matlab For Engineers And Scientists , 2008 .

[10]  K. Park.  Practical aspects of numerical time integration , 1977 .

[11]  T. Belytschko,et al.  Dispersion analysis of finite element semidiscretizations of the two‐dimensional wave equation , 1982 .

[12]  J. P. Woodruff.  KOVEC user's manual , 1976 .

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

[14]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[15]  Samuel W. Key,et al.  Transient shell response by numerical time integration , 1973 .

[16]  Jintai Chung,et al.  Explicit time integration algorithms for structural dynamics with optimal numerical dissipation , 1996 .

[17]  L. P. Bindeman,et al.  Assumed strain stabilization of the eight node hexahedral element , 1993 .

[18]  Amos Gilat,et al.  Matlab, An Introduction With Applications , 2003 .

[19]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[20]  T. Belytschko,et al.  Stability of explicit‐implicit mesh partitions in time integration , 1978 .

[21]  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 .

[22]  Raphaël Loubère,et al.  A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods , 2005 .

[23]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[24]  R. B. Pember,et al.  A Comparison of Staggered-Mesh Lagrange Plus Remap and Cell-Centered Direct Eulerian Godunov Schemes for Rulerian Shock Hydrodynamics , 2000 .

[25]  James P. Braselton,et al.  Mathematica by Example , 1992 .

[26]  Sandia Report,et al.  Lagrangian Continuum Dynamics in ALEGRA , 2007 .

[27]  R. Menikoff,et al.  The Riemann problem for fluid flow of real materials , 1989 .

[28]  W. Cabot,et al.  A high-wavenumber viscosity for high-resolution numerical methods , 2004 .

[29]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[30]  D. Serre.  Matrices: Theory and Applications , 2002 .

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

[32]  M. Shashkov,et al.  A Multi-Scale Q1/P0 Approach to Lagrangian Shock Hydrodynamics , 2007 .

[33]  Ted Belytschko,et al.  Eigenvalues and Stable Time Steps for the Uniform Strain Hexahedron and Quadrilateral , 1984 .

[34]  Stephen H. Friedberg,et al.  Linear Algebra , 2018, Computational Mathematics with SageMath.

[35]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[36]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .