Model reduction framework in space and time for the Generalized single step single solve family of algorithms

Abstract

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