Multiple Grid Multiple Time-Scale (MGMT) Simulations in Linear Structural Dynamics Open Access
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The work presented in this dissertation describes a general algorithm and its Finite Element (FE) implementation for performing concurrent multiple sub-domain simulations in linear structural dynamics. Using this approach one can solve problems in which the domain under analysis can be selectively discretized spatially and temporally, hence allowing the user to obtain a desired level of accuracy in critical regions whilst improving computational efficiency globally. The mathematical background for this approach is largely derived from the fundamental principles of Domain Decomposition Methods (DDM) and Lagrange Multipliers, used to obtain coupled equations of motion for distinct regions of a continuous domain. These methods when combined together systematically yield constraint forces that not only ensure conservation of energy, but also enforce continuity of field quantities across sub-domain interfaces. Multiple Grid (MG) coupling between conforming or non-conforming sub-domains is achieved in the form of linear multi-point constraints that are modeled using Mortar Finite Element Method (M-FEM); whereas coupled Multiple Time-scale (MT) equations are derived for the classical Newmark integration scheme and its constituent algorithms. A rigorous proof of stability is provided using Energy Method and necessary conditions for enforcing energy balance are discussed in reference with field variables that are selected to enforce sub-domain interface continuity. Fully discretized equations of motion for component sub-domains, augmented with an interface continuity condition are then solved using block elimination method and Crout factorization. A step-by-step solution approach, utilizing recursive black box sub-routines, is modeled in order to allow efficient implementation within existing finite element frameworks. Proposed MGMT Method and corresponding solution algorithm is systematically implemented by using the finite element approach and programming in FORTRAN 90. Resulting in-house code - FEAPI (Finite Element Analysis Programming Interface) is capable of solving linear structural dynamics problems that are modeled using independently discretized sub-domains. Auxiliary sub-routines for defining pre simulation parameters and for viewing global/component sub-domain results are built into FEAPI and work in conjugation with GiD; a universal, adaptive and user-friendly pre and post-processor. Overall stability, numerical accuracy and computational efficiency of MGMT Method is evaluated and verified using a series of benchmark examples. Verification matrices take into consideration performance evaluation factors such as energy balance (at global and component-sub-domain levels), interface continuity, evolution/distribution of kinematic quantities and propagation of structural waves across connecting sub-domains. Assessment of computational efficiency is derived by comparing the size of respective FE problems (nodes, elements, number of equations, skyline storage requirements) and the required computation times (CPU solution time). Discussed examples highlight the greatest advantage of MGMT Method; which is significant gain in simulation speedups (at the cost of reasonably small errors).
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