Browsing by Author "Shimada, Masao"

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    iIntegration framework and algorithms by design: implicit and explicit families of generalized single step single solve algorithms by design in two- and single-field forms
    (2013-08) Shimada, Masao
    Time dependent problems are of interest here, and the designs of time integration algorithms for linear and nonlinear dynamical systems have been widely studied for the past 50 years or so and continue to be of much interest. Numerous issues need to be still resolved for transient problems so as to capture as much physics as possible and this thesis addresses these features. This thesis shows the detailed developments towards the frameworks of the generalized single step single solve (GSSSS/GS4) family of algorithms, and leading to the general frameworks comprising of families of implicit and explicit time integration schemes in the two- and single-field forms. The basic idea of designing the time integration schemes is based upon and emanates from the time weighted residual methodology. The major developments include the following considerations: (i)All the resulting developments are strictly second-order time accurate which is an important design concern, (ii) all algorithms possess a consistent time level in the discretized equations which is not well understood to-date, (iii) Linear dynamics and algorithms and designs are first addressed, (iv) the design of implicit frameworks and the corresponding predictor-corrector explicit algorithms and designs then follow; (v) how to properly extend linear dynamics algorithms to nonlinear dynamics applications is then addressed using a novel normalized time weighted residual methodology and leading to those termed as symplectic-momentum conserving and energy momentum conserving designs, and (iv) lastly, a new and novel iIntegration framework that is applicable to both second order systems and first order systems is finally designed for applicability to general computational engineering and science problems. The various relations to scenarios emanating from other methods of development and typical of variational algorithms and exact energy-momentum conserving algorithms to the time integration framework presented in this thesis are also carefully discussed. Both N-body systems and continuum elastodynamics applications are illustrated and numerous numerical experiments of a wide variety of applications confirm the theoretical developments. Most of the designs of algorithms within the past 50 years or so and related to LMS methods are part of the present unified framework; and also new avenues and algorithm designs are an additional contribution including optimal designs of algorithms. Consequently, one has to simply implement the present technology which provides a wide variety of choices to the analyst in a simple setting whilst permitting to switch algorithms from one design to another based upon the problem at hand.
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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.