Multiscale temporal integration

Abstract We study the application of the variational multiscale method to the problem of temporal integration, with the final goal of designing integration schemes that go beyond the classical notion of upwinding. We develop a formulation based on a mixed hybrid finite element method, where the fine scale mode problems automatically decouple at the element level without the need to resort to a localization assumption. We give general orthogonality conditions for the trial and test spaces that allow to construct hierarchical p methods. We test some simple ideas for the modeling of the unresolved scales. The resulting algorithms are analyzed using classical analytical measures.

[1]  T. Belytschko,et al.  Computational Methods for Transient Analysis , 1985 .

[2]  Thomas J. R. Hughes,et al.  Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics , 1976 .

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

[4]  Franco Brezzi,et al.  Recent results in the treatment of subgrid scales , 2002 .

[5]  C. Bottasso.  A new look at finite elements in time: a variational interpretation of Runge-Kutta methods , 1997 .

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

[7]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[8]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[9]  Thomas J. R. Hughes,et al.  Space-time finite element methods for second-order hyperbolic equations , 1990 .

[10]  Arieh Iserles,et al.  Geometric integration: numerical solution of differential equations on manifolds , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Olivier A. Bauchau,et al.  Robust integration schemes for flexible multibody systems , 2003 .

[12]  M. Borri,et al.  A general framework for interpreting time finite element formulations , 1993 .

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