This thesis presents a preconditioned solenoidal basis method to solve the algebraic system arising from the linearization and discretization of primitive variable formulations of Navier-Stokes equations for incompressible fluid flows. The system is restricted to a discrete divergence-free space which is constructed from the incompressibility constraint. This research work extends an earlier work on the solenoidal basis method for two-dimensional flows and three-dimensional flows that involved the construction of the solenoidal basis P using circulating flows or vortices on a uniform mesh. A localized algebraic scheme for constructing P is detailed using mixed finite elements on an unstructured mesh. A preconditioner which is motivated by the analysis of the reduced system is also presented. Benchmark simulations are conducted to analyze the performance of the proposed approach.

We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.

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We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.

Past studies (primarily on steady state problems) that have compared the penalty and the velocity-pressure finite element formulations on a variety of problems have concluded that both methods yield solutions of comparable accuracy, and that the choice of one method over the other is dictated by which of the two is more efficient. In this work, we show that the penalty finite element method yields inaccurate solutions at large times on a class of transient problems, while the velocity-pressure formulation yields solutions that are in good agreement with the analytical solution. Numerical studies are conducted on various problems to compare these two formulations on the basis of rates of convergence, total number of equations to be solved and accuracy of results. We found that both formulations give almost the same rates of convergence in all problems, however the penalty formulation involves lesser number of equations than the velocity-pressure formulation due to implicit treatment of pressure field, and hence is more efficient. In some of the problems we have also compared a finite volume method with the penalty and velocity-pressure formulations on the basis of accuracy and computational cost.

Experimental data convincingly show that viscosity of a fluid may change significantly with pressure. This observation leads to various generalizations of well-known models, like Darcy's law, Stokes' law or the Navier-Stokes equations, among others. This thesis investigates three such models in a series of three published papers. Their unifying topic is development of existence theory and finding a weak solution to systems of partial differential equations stemming from the considered models.

This project presents the development and implementation of a GPU-accelerated meshless two-phase incompressible fluid flow solver. The solver uses a variant of the Generalized Finite Difference Meshless Method presented by Gerace et al. [1]. The Level Set Method [2] is used for capturing the fluid interface. The Compute Unified Device Architecture (CUDA) language for general-purpose computing on the graphics-processing-unit is used to implement the GPU-accelerated portions of the solver. CUDA allows the programmer to take advantage of the massive parallelism offered by the GPU at a cost that is significantly lower than other parallel computing options. Through the combined use of GPU-acceleration and a radial-basis function (RBF) collocation meshless method, this project seeks to address the issue of speed in computational fluid dynamics. Traditional mesh-based methods require a large amount of user input in the generation and verification of a computational mesh, which is quite time consuming. The RBF meshless method seeks to rectify this issue through the use of a grid of data centers that need not meet stringent geometric requirements like those required by finite-volume and finite-element methods. Further, the use of the GPU to accelerate the method has been shown to provide a 16-fold increase in speed for the solver subroutines that have been accelerated.

Tato diplomová práce prezentuje metodu spektrálních prvků. Tato metoda je použita k řešení stacionárního 2-D laminárního proudění Newtonovské nestlačitelné tekutiny. Proudění je popsáno stacionarní Navier-Stokesovou rovnicí. Dohromady s okrajovou pod- mínkou tvoří Navier-Stokesův problém. Na slabou formulaci této úlohy je aplikována metoda spektrálních prvků. Touto discretizací se získá soustava nelineárních rovnic. K obrdžení lineární soustavy je použita Newtonova iterační metoda. Podorobný algorit- mus tvoří jádro Navier-Stokeseva solveru, který je naprogramován v Matlabu. Na závěr jsou pomocí tohoto solveru řešeny dva příklady: proudění v kavitě a obtékání válce. Přík- lady jsou řešeny pro různé Reynoldsovy čísla. První od 1 do 1000 a druhý od 1 do 100.

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