Isogeometric Analysis for Compressible Flows with Application in Turbomachinery

Computer-aided design (CAD) and finite element analysis (FEA) tools are extensively used in industrial design processes. The idea of isogeometric analysis (IGA) is to bring those worlds closer together. Combining them significantly increases the efficiency of the design and development processes. The general idea of the IGA approach is to use non-uniform rational B-splines (NURBS) that are used to represent the geometry in the CAD tool also as a basis for numerical analysis of the partial differential equations (PDE) via FEA. The simplest setting is to use the space spanned by NURBS as a search/test space in the standard Galerkin method. Nowadays, increasing the efficiency of turbo-engines and therefore decreasing the emissions of mainly CO_2 and NO_x is a field of very active research. Therefore, design optimization of turbomachines, largely based on numerical simulations, is extensively performed in industry. An additional motivation for solving the underlying flow problems by IGA is its ability to exactly represent the complex shapes of domains, which is an important feature for flow solvers that need to accurately resolve boundary layers. The topic of this thesis project: application of isogeometric analysis to compressible flow problems is very broad and it is beyond the scope of this thesis to answer all questions arising during the pilot implementation of an IGA solver for compressible flows. Therefore, it was decided to set the implementation of a B-spline-based IGA solver for the compressible Euler equations as the main goal. To achieve this goal several intermediate milestones were set. The first was the implementation of a simple B-spline basis generator and evaluator. The next step was to solve the Poisson problem with IGA. The next step consisted in implementing the solver for the convection-diffusion equation in stationary and time-dependent variants. Those steps led to the final milestone - the implementation of the compressible inviscid flow solver based on the IGA approach. It is well known that the standard Galerkin finite element method as well as its isogeometric counterpart suffer from infamous oscillatory behaviour for convection-dominated problems and problems including discontinuities or steep gradients in the domain. Therefore, the algebraic flux correction (AFC) stabilization was implemented to avoid oscillations and non-physical values in the solution. The main novelty of this thesis is to combine AFC with IGA for the compressible Euler equations. All solvers implemented during the thesis work were successfully validated using common benchmark problems. This work sets a base for further research and development that will hopefully lead to a productive implementation of IGA-based optimisation in industrial turbomachinery design.

Subject

IGA
isogeometric analysis
AFC
algebraic flux correction
B-splines
compressible Euler equations

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master thesis

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