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.
University of Minnesota M.S. thesis. August 2013. Major: Mechanical Engineering. Advisor:Kumar K. Tamma. 1 computer file (PDF); xlii, 647 pages.
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.
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