Development of a high-order cut-cell method for cfd-based design optimization

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Aeronautical engineering
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Computational fluid dynamics (CFD)-based aerodynamic design optimization requires automation, particularly when dealing with complex geometries that require boundary conforming meshes. To address this issue and help automate the design optimization process, non-boundary conforming meshes can be used, leading to cut-cell methods. However, cut-cell methods introduce their own challenges, one of which is a potentially ill-conditioned discretization due to some cut-cells being orders of magnitude smaller than regular shaped cells. Various techniques have been developed to tackle this issue, most of which require special treatment for dealing with small cut-cells. This thesis introduces a cut-cell method based on the discontinuous Galerkin difference discretization (cut-DGD) that does not require any special treatment of cut-cells to eliminate ill-conditioning. The first part of this thesis presents the cut-DGD discretization method, which uses an underlying discontinuous Galerkin (DG) discretization. The role of DGD basis functions and the stencil construction in maintaining a well-conditioned cut-DGD discretization is discussed, and numerical experiments are conducted to show that the condition numbers of the cut-DGD mass and stiffness matrices remain bounded as the size of the cut cell approaches zero. Furthermore, the accuracy of the cut-DGD discretization in solving the one-dimensional steady-state linear advection equation and the two-dimensional Poisson partial differential equation (PDE) is demonstrated in the presence of small cut-cells. The second part of the thesis focuses on performing numerical integration of arbitrarily shaped cut-cells. A level-set formulation is developed to approximate the geometries of general shapes, such as airfoils, which are of interest in aerodynamic applications. The impact of increasing the number of boundary points and applying high-order corrections to improve the accuracy of the level-set approximation is discussed. The level-set definition of a given geometry is necessary to apply the quadrature algorithms used in this work. Two different quadrature algorithms are explored: one that requires user-defined bounds on the level-set function, which can be challenging for geometries with discontinuities (e.g., trailing edges of airfoils), and another that only requires polynomial-type level-set functions without such bounds. Fluid-flow accuracy studies are conducted to demonstrate the high-order accuracy of cut-DGD, using both exact and approximate level-set functions, by solving two-dimensional Euler equations. The final part of the thesis presents contributions towards gradient-based design optimization. Specifically, the sensitivity analysis of cut-DGD is derived, and the semi-automatic differentiation of a cut-cell quadrature algorithm is performed. These derivatives are then verified against a finite-difference approximation. A discrete adjoint approach is used to perform sensitivity analysis and to compute adjoint variables necessary for solving functional sensitivity derivatives, which are further required for a gradient-based optimization algorithm. The residual sensitivity derivative is verified against finite-difference approximations for a model flow problem.
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Rensselaer Polytechnic Institute, Troy, NY
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