Browsing by Subject "Computational Mechanics"

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    Model reduction framework in space and time for the Generalized single step single solve family of algorithms
    (2017-11) Deokar, Rohit
    This thesis presents the developments in the field of model order reduction framework in space and time for the Generalized single step single solve (GSSSS) family of algorithms. The GSSSS framework has been developed in the past two decades as a unified theory encompassing all the computationally competitive time integration schemes for first and second order systems over the past 50 years. Using the underlying versatility of the GSSSS framework, a novel model order reduction procedure in space is proposed to eliminate spurious high frequency participation in dynamical systems. Spurious high frequency participation are vestiges of numerical discretization and often pose serious numerical issues degrading solution accuracy. Numerically dissipative schemes which were originally proposed to deal with these high frequency participation lose energy over time and damp out the physics in the system. The proposed method for elimination of high frequency participation deals with this very problem by combining the advantages of the energy conserving and numerically dissipative algorithms through projection techniques. The DAE (iIntegration) framework which was recently proposed, extends the GSSSS family of algorithms to constrained mechanical systems (DAEs) while preserving the optimal properties that are desired from time integration schemes. This thesis extends the proposed model reduction methodology in space to the GSSSS DAE framework thereby reducing the computational complexity which can otherwise be daunting for constrained subdomain systems with subcycling. In addition, the so called "Finite element in time" framework for the GSSSS family of algorithms is developed using the weighted residual methodology. Based on the finite element in time methodology, a novel general purpose a posteriori error estimator for first and second order systems under the umbrella of GSSSS family of algorithms is proposed to foster adaptive time stepping. The applicability of the proposed estimator to several existing time integration algorithms including the well known schemes like the Newmark method, HHT-Alpha, Classical midpoint rule, Crank Nicolson and in addition, new algorithms and designs as well is demonstrated with single and multi-degree of freedom, linear and nonlinear dynamical problems. Lastly, model reduction in space and time through the so called staggered space-time MOR procedure is proposed which aims at refining the discretizations in space while employing a reduced dimension in time. Conversely, a reduced dimension in space is used to improve the discretization in time and the process is performed in an iterative fashion
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    Novel Design and Development of Isochronous Time Integration Architectures for Ordinary Differential Equations and Differential-Algebraic Equations: Computational Science and Engineering Applications
    (2014-12) Shimada, Masao
    Recently, the novel designs and developments encompassing isochronous integrators [iIntegrators] for systems of ordinary differential equations (ODE-iIntegrators) have been invented that entail most of the research to-date developed over the past 50 years or so including new and novel optimal schemes for both second-order and first-order transient systems. This present thesis next takes upon the daunting challenges for the extensions of the ODE iIntegrators to systems of differential-algebraic equations (DAEs). The iIntegrators for DAEs (DAE-iIntegrators) is an extremely powerful time integration toolkit with new and contemporary schemes that are novel and suitable to DAEs of any index which can be applied both for second- and first-order systems; and it includes most single step single solve implicit/semi-explicit schemes which preserve second-order time accuracies in all the variables (this is the novelty and it is not trivial and is not readily achievable with current state of the art for the differential and algebraic quantities to-date due to lack of fundamental understanding, poor or improper designs and implementation). Sub-cases include the classical algorithms in second-order systems such as Newmark, HHT-alpha, WBZ methods and many others, including mechanical integrators, and more new and optimal algorithms and designs for second-order systems; and this very same computational framework (hence, the name isochronous integration) readily adapts to the simulation of first-order systems as well as an added bonus and includes most of the classical developments such as Crank-Nicholson method, Gear's method, MacCormack's method and so on including more new and optimal designs encompassing both implicit and explicit schemes for first-order systems as well under the umbrella of a single unified toolkit. The new and novel DAE-iIntegration architecture is envisioned as the next generation toolkit, and can also be widely used, for example, as an added bonus for applicability to multi-physics problems such as fluid-structure, thermal-structure interaction problems. Additional studies on the multiple subdomain DAE simulations and model order reduction by the proper orthogonal decomposition (POD) for ODE systems are also investigated.