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11	Deducting Conserved Quantities for Numerical Schemes using Parametric Groebner Systems Majrashi, 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.
12	A Fractional Step Zonal Model and Unstructured Mesh Generation Frame-work for Simulating Cabin Flows Tarroc 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> Zonal Model Thermal Comfort Fluid Simulation Aircraft Cabin Design NURBS Morphing Radial Basis Functions Meshing Co-located Mesh Unstructured Mesh Mesh Quality Mesh Smoothing Enlarged Orientation Theorem Fractional Step Method Boussinesq Adam Bashforth Helmholtz-Hodge Theorem Rhie and Chow Difference Schemes Gradient Evaluation Differential Heated Cavity Driven Cavity Draught Rate Aerospace Engineering Rymd- och flygteknik Fluid Mechanics and Acoustics Strömningsmekanik och akustik

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Deducting Conserved Quantities for Numerical Schemes using Parametric Groebner Systems

A Fractional Step Zonal Model and Unstructured Mesh Generation Frame-work for Simulating Cabin Flows