Multiple Grid and Multiple Time-Scale (MGMT) Simulations in Continuum Mechanics

Abstract

The work presented in this paper describes a general formulation for implementation of Multiple Grid and Multiple Time-scale (MGMT) simulations in continuum mechanics. Using this method one can solve problems in structural dynamics in which the domain under consideration can be selectively discretized (spatially and temporally) in critical and remote regions, hence allowing the user to obtain a desired level of accuracy and save computational time. The formulation is based upon the fundamental principles of Domain Decomposition Methods (DDM) used to obtain the semi-discrete equation of motion for coupled sub-domains augmented with interface energy. Lagrange Multipliers, based on Schur’s dual formulation, are used to enforce interface conditions since they not only ensure energy balance but also enforce continuity of kinematic quantities across the interface. The Finite Element Tearing and Interconnecting (FETI) based Multi Time-step (MTS) coupling algorithm proposed by Prakash and Hjelmstad [1] is then used to obtain the evolution of unknown quantities in respective sub-domains using different time-steps and/or different variants of the Newmark Implicit Method. Our work is in the direction of coupling this MTS algorithm with multiple grid discretizations in respective subdomains. We propose using coarse grid discretization to define the mortar space between non-conforming sub-domains and show that this particular choice when combined with the implicit integration scheme yields a stable algorithm for MGMT simulations. The formulation is implemented, comprehensively, using Finite Element Methods and programming in FORTRAN 90. Several scenarios with different mesh densities and time-steps are evaluated to analyze the efficiency of MGMT simulations. The purpose of this paper is to study and evaluate its accuracy and stability by looking at evolution and distribution of quantities across the connecting interface. Results show that the interface coupling for non-conforming sub-domains with distinct integration time-steps can be efficiently modeled using this approach.