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Deducting Conserved Quantities for Numerical Schemes using Parametric Groebner SystemsMajrashi, Bashayer 05 1900 (has links)
In partial differential equations (PDEs), conserved quantities like mass and momentum are fundamental to understanding the behavior of the described physical
systems. The preservation of conserved quantities is essential when using numerical
schemes to approximate solutions of corresponding PDEs. If the discrete solutions
obtained through these schemes fail to preserve the conserved quantities, they may
be physically meaningless and unreliable.
Previous approaches focused on checking conservation in PDEs and numerical
schemes, but they did not give adequate attention to systematically handling parameters. This is a crucial aspect because many PDEs and numerical schemes have parameters that need to be dealt with systematically. Here, we investigate if the discrete
analog of a conserved quantity is preserved under the solution induced by a parametric finite difference method. In this thesis, we modify and enhance a pre-existing
algorithm to effectively and reliably deduce conserved quantities in the context of
parametric schemes, using the concept of comprehensive Groebner systems.
The main contribution of this work is the development of a versatile algorithm
capable of handling various parametric explicit and implicit schemes, higher-order
derivatives, and multiple spatial dimensions. The algorithm’s effectiveness and efficiency are demonstrated through examples and applications. In particular, we illustrate the process of selecting an appropriate numerical scheme among a family of
potential discretization for a given PDE.
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A Fractional Step Zonal Model and Unstructured Mesh Generation Frame-work for Simulating Cabin FlowsTarroc Gil, Sergi January 2021 (has links)
The simulation of physical systems in the early stages of conceptual designs has shown to be a key factor for adequate decision making and avoiding big and expensive issues downstream in engineering projects. In the case of aircraft cabin design, taking into account the thermal comfort of the passengers as well as the proper air circulation and renovation can make this difference. However, current numerical fluid simulations (CFD) are too computationally expensive for integrating them in early design stages where extensive comparative studies have to be performed. Instead, Zonal Models (ZM) appear to be a fast-computation approach that can provide coarse simulations for aircraft cabin flows. In this thesis, a Zonal Model solver is developed as well as a geometry-definition and meshing framework, both in Matlab®, for performing coarse, flexible and computationally cheap flow simulations of user-defined cabin designs. On one hand, this solver consists of a Fractional Step approach for coarse unstructured bi-dimensional meshes. On the other, the cabin geometry can be introduced by hand for simple shapes, but also with Computational Aided Design tools (CAD) for more complex designs. Additionally, it can be chosen to generate the meshes from scratch or morph them from previously generated ones. / <p>The presentation was online</p>
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