Transient three‐dimensional domain decomposition problems: Frame‐indifferent mortar constraints and conserving integration

The present work deals with transient large-deformation domain decomposition problems. The tying of dissimilar meshed grids is performed by applying the mortar method. In this connection, a reformulation of the original linear mortar constraints is proposed, which retains frame-indifference for arbitrary discretizations of the interface. Furthermore, a specific coordinate augmentation technique is proposed to make possible the design of an energy–momentum scheme. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy–momentum scheme for three-dimensional problems. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  S. Antman Nonlinear problems of elasticity , 1994 .

[2]  Peter Betsch,et al.  Energy-Momentum Conserving Schemes for Frictionless Dynamic Contact Problems , 2007 .

[3]  Michael A. Puso,et al.  A 3D mortar method for solid mechanics , 2004 .

[4]  F. Armero,et al.  Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems , 1998 .

[5]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

[6]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics , 2001 .

[7]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods , 2001 .

[8]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[9]  Clark R. Dohrmann,et al.  Methods for connecting dissimilar three-dimensional finite element meshes , 2000 .

[10]  T. Laursen,et al.  Improved implicit integrators for transient impact problems—geometric admissibility within the conserving framework , 2002, International Journal for Numerical Methods in Engineering.

[11]  Peter Betsch,et al.  Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints , 2002 .

[12]  Peter Betsch,et al.  Energy-momentum conserving integration of multibody dynamics , 2007 .

[13]  O. Gonzalez Mechanical systems subject to holonomic constraints: differential—algebraic formulations and conservative integration , 1999 .

[14]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[15]  Peter Betsch,et al.  Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics , 2001 .

[16]  David J. Benson,et al.  Sliding interfaces with contact-impact in large-scale Lagrangian computations , 1985 .

[17]  Tod A. Laursen,et al.  A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations , 2007 .

[18]  Oscar Gonzalez,et al.  Exact energy and momentum conserving algorithms for general models in nonlinear elasticity , 2000 .

[19]  T. Laursen,et al.  DESIGN OF ENERGY CONSERVING ALGORITHMS FOR FRICTIONLESS DYNAMIC CONTACT PROBLEMS , 1997 .

[20]  J. Z. Zhu,et al.  The finite element method , 1977 .

[21]  O. Gonzalez Time integration and discrete Hamiltonian systems , 1996 .

[22]  C. Truesdell,et al.  The Non-Linear Field Theories of Mechanics , 1965 .

[23]  K. Bathe,et al.  Stability and patch test performance of contact discretizations and a new solution algorithm , 2001 .

[24]  Peter Betsch,et al.  A comparison of computational methods for large deformation contact problems of flexible bodies , 2006 .

[25]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[26]  P. Tallec,et al.  Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact , 2006 .

[27]  Keith Hjelmstad,et al.  A finite element formulation of non-smooth contact based on oriented volumes for quadrilateral and hexahedral elements , 2007 .

[28]  Martin W. Heinstein,et al.  A three dimensional surface‐to‐surface projection algorithm for non‐coincident domains , 2003 .

[29]  J. Périaux,et al.  Domain Decomposition Methods in Science and Engineering , 1994 .

[30]  Tod A. Laursen,et al.  A segment-to-segment mortar contact method for quadratic elements and large deformations , 2008 .

[31]  R. Krause,et al.  Nonconforming decomposition methods: Techniques for linear elasticity , 2000 .

[32]  Anthony T. Patera,et al.  Domain Decomposition by the Mortar Element Method , 1993 .

[33]  Peter Betsch,et al.  A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems , 2009 .

[34]  J. C. Simo,et al.  A perturbed Lagrangian formulation for the finite element solution of contact problems , 1985 .

[35]  T. Laursen Computational Contact and Impact Mechanics , 2003 .

[36]  Tod A. Laursen,et al.  Two dimensional mortar contact methods for large deformation frictional sliding , 2005 .

[37]  P. Tallec,et al.  A discontinuous stabilized mortar method for general 3D elastic problems , 2007 .

[38]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.