Tae, David2022-12-022022-12-022020-09https://hdl.handle.net/11299/250069University of Minnesota Ph.D. dissertation. September 2020. Major: Mechanical Engineering. Advisor: Kumar Tamma. 1 computer file (PDF); xxxiii, 375 pages.This thesis presents advances and developments in the field of spatial discretization and time integration. Along with the growth of the FEM, there has been a steady development of particle discretization methods such as Moving Particle Semi-implicit method or Smoothed Particle Hydrodynamics method. We propose a novel generalized approach to describe numerous existing particle methods by exploiting Taylor series expansion and the weighted residual method. The method is then validated through various problems in first and second order systems. The FEM and the particle methods have their own strengths and weaknesses. With the concept of subdomains and Differential Algebraic Equations (DAE) framework, we can divide a body and implement different methods in different regions of the body targeting an area with a specific method which can fully utilize its best features. We propose an implementation of multi-spatial method, multi-time scheme subdomain DAE framework allowing a mix of different space discretization methods and different time schemes on a single body analysis. This is not possible in the current state of the technology as it shows limitations in order of accuracy, and consistency. Various combinations of spatial methods and time schemes between subdomains are tested in linear and nonlinear problems for first and second order systems. Lastly, we introduced reduced order modeling via Proper Orthogonal Decomposition (POD) which decreases the size of the system based on its eigenvalues. The snapshot data are used to establish the reduced order basis. We additionally propose the integration of POD into the subdomain DAE framework. As the required amount of snapshot data are unknown and problem specific, we present an iterative process to ensure the snapshot data to accurately capture the physics of the system. In addition, the iteration approach is extended to include the convergence check in time on the solution for implicit time schemes. The proposed DAE POD framework is tested on numerous linear and nonlinear problems for first and second order systems. In all cases, we see time savings in computational effort.enDifferential Algebraic EquationFinite Element AnalysisParticle MethodProper Orthogonal DecompositionSubdomainTime IntegrationThesis or Dissertation