Browsing by Subject "Time Integration"

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    A Novel Computational Framework Integrating Different Space Discretization and Time Discretization Methods with Multiple Subdomains and Reduced Order Modeling
    (2020-09) Tae, David
    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.
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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.