The accuracy of a fluid flow simulation depends not only on the numerical method used for discretizing the governing equations, but also on the distribution and topology of the mesh elements. Mesh adaptation is a technique for automatically modifying the mesh in order to improve the simulation accuracy in an attempt to reduce the manual work required for mesh generation. The conventional approach to mesh adaptation is based on a feature-based criterion that identifies the distinctive features in the flow field such as shock waves and boundary layers. Although this approach has proved to be simple and effective in many CFD applications, its implementation may require a lot of trial and error for determining the appropriate criterion in certain applications. An alternative approach to mesh adaptation is the residual-based approach in which the discretization error of the fluid flow quantities across the mesh faces is used to construct an adaptation criterion. Although this approach provides a general framework for developing robust mesh adaptation criteria, its incorporation leads to significant computational overhead. The main objective of the thesis is to present a methodology for developing an appropriate mesh adaptation criterion for fluid flow problems that offers the simplicity of a feature-based criterion and the robustness of a residual-based criterion. This methodology is demonstrated in the context of a second-order accurate cell-centred finite volume method for simulating laminar steady incompressible flows of constant property fluids. In this methodology, the error of mass and momentum flows across the faces of each control volume are estimated with a Taylor series analysis. Then these face flow errors are used to construct the desired adaptation criteria for triangular isotropic meshes and quadrilateral anisotropic meshes. The adaptation results for the lid-driven cavity flow show that the solution error on the resulting adapted meshes is 80 to 90 percent lower than that of a uniform mesh with the same number of control volumes. The advantage of the proposed mesh adaptation method is the capability to produce meshes that lead to more accurate solutions compared to those of the conventional methods with approximately the same amount of computational effort.

The accuracy of a fluid flow simulation depends not only on the numerical method used for discretizing the governing equations, but also on the distribution and topology of the mesh elements. Mesh adaptation is a technique for automatically modifying the mesh in order to improve the simulation accuracy in an attempt to reduce the manual work required for mesh generation. The conventional approach to mesh adaptation is based on a feature-based criterion that identifies the distinctive features in the flow field such as shock waves and boundary layers. Although this approach has proved to be simple and effective in many CFD applications, its implementation may require a lot of trial and error for determining the appropriate criterion in certain applications. An alternative approach to mesh adaptation is the residual-based approach in which the discretization error of the fluid flow quantities across the mesh faces is used to construct an adaptation criterion. Although this approach provides a general framework for developing robust mesh adaptation criteria, its incorporation leads to significant computational overhead. The main objective of the thesis is to present a methodology for developing an appropriate mesh adaptation criterion for fluid flow problems that offers the simplicity of a feature-based criterion and the robustness of a residual-based criterion. This methodology is demonstrated in the context of a second-order accurate cell-centred finite volume method for simulating laminar steady incompressible flows of constant property fluids. In this methodology, the error of mass and momentum flows across the faces of each control volume are estimated with a Taylor series analysis. Then these face flow errors are used to construct the desired adaptation criteria for triangular isotropic meshes and quadrilateral anisotropic meshes. The adaptation results for the lid-driven cavity flow show that the solution error on the resulting adapted meshes is 80 to 90 percent lower than that of a uniform mesh with the same number of control volumes. The advantage of the proposed mesh adaptation method is the capability to produce meshes that lead to more accurate solutions compared to those of the conventional methods with approximately the same amount of computational effort.

Automation of finite element analysis based on a fully adaptive <i>p</i>-refinement procedure can reduce the magnitude of discretization error to the desired accuracy with minimum computational cost and computer resources. This study aims to 1) develop an efficient <i>p</i>-refinement procedure with a non-uniform <i>p</i> analysis capability for solving 2-D and 3-D elastostatic mechanics problems, and 2) introduce a stress error estimate. An element-by-element algorithm that decides the appropriate order for each element, where element orders can range from 1 to 8, is described. Global and element data structures that manage the complex data generated during the refinement process are introduced. These data structures are designed to match the concept of object-oriented programming where data and functions are managed and organized simultaneously. The stress error indicator introduced is found to be more reliable and to converge faster than the error indicator measured in an energy norm called the residual method. The use of the stress error indicator results in approximately 20% fewer degrees of freedom than the residual method. Agreement of the calculated stress error values and the stress error indicator values confirms the convergence of final stresses to the analyst. The error order of the stress error estimate is postulated to be one order higher than the error order of the error estimate using the residual method. The mapping of a curved boundary element in the working coordinate system from a square-shape element in the natural coordinate system results in a significant improvement in the accuracy of stress results. Numerical examples demonstrate that refinement using non-uniform <i>p</i> analysis is superior to uniform <i>p</i> analysis in the convergence rates of output stresses or related terms. Non-uniform <i>p</i> analysis uses approximately 50% to 80% less computational time than uniform <i>p</i> analysis in solving the selected stress concentration and stress intensity problems. More importantly, the non-uniform <i>p</i> refinement procedure scales the number of equations down by 1/2 to 3/4. Therefore, a small scale computer can be used to solve equation systems generated using high order <i>p</i>-elements. In the calculation of the stress intensity factor of a semi-elliptical surface crack in a finite-thickness plate, non-uniform <i>p</i> analysis used fewer degrees of freedom than a conventional <i>h</i>-type element analysis found in the literature. / Ph. D.

The vector finite element method has gained great attention since overcoming the deficiencies incurred by the scalar basis functions for the vector Helmholtz equation. Most implementations of vector FEM have been non-adaptive, where a mesh of the domain is generated entirely in advance and used with a constant degree polynomial basis to assign the degrees of freedom. To reduce the dependency on the users' expertise in analyzing problems with complicated boundary structures and material characteristics, and to speed up the FEM tool, the demand for adaptive FEM grows high. For efficient adaptive FEM, error estimators play an important role in assigning additional degrees of freedom. In this proposal study, hierarchical vector basis functions and four error estimators for p-refinement are investigated for electromagnetic applications.

Title: Adaptive choice of parameters in stabilization methods for convection- diffusion equations Author: Bc. Petr Lukáš (e-mail: luk.p@post.cz) Department: Department of Numerical Mathematics Supervisor: Doc. Mgr. Petr Knobloch, Dr. (e-mail: knobloch@karlin.mff.cuni.cz) Abstract: The aim of the work is to propose suitable approaches for adap- tive choice of parameters in stabilization methods for convection-difusion equations discretized by the finite element method. We introduce the L-SR1 method, compare it with other nonlinear methods of minimizing functions with large number of variables, and introduce and compare the adaptive methods based on minimizing of the error indicator. Keywords: Adaptive choice of parameters, finite element method, stabiliza- tion methods, convection-diffusion equation, L-SR1 method, error indicator

Vários modelos computacionais que implementam o transporte de soluto em meio poroso saturado surgem constantemente em publicações científicas devido à suma importância dada à compreensão e previsão do transporte de constituintes dissolvidos em água subterrânea. As soluções numéricas obtidas por esquemas computacionais não estão imunes aos erros de discretização. No entanto, a confiabilidade nos resultados obtidos das complexas operações provenientes da dinâmica de fluidos computacional pode ser aumentada através de estimadores de erro a posteriori que indicam a precisão da solução numérica de um modelo matemático que simula o fenômeno físico de interesse. Neste trabalho é apresentado um estimador residual para a equação parabólica que descreve os fenômenos de advecção-dispersão-reação (ADR) em meio poroso saturado, considerando o transporte em regime de pequena advecção. A solução numérica da equação ADR é obtida pelo método dos elementos finitos que emprega termos upwind para minimizar as inconvenientes oscilações espúrias. A implementação do código computacional para obter essa solução numérica e o seu correspondente erro a posteriori, é feita em linguagem JAVA na plataforma Eclipse seguindo o paradigma da Programação Orientada a Objetos (POO). A solução numérica da equação elíptica do fluxo subterrâneo e o seu estimador de erro com características de recuperação do gradiente, o estimador ZZ, também são implementados no código JAVA. Assim, a solução da equação do transporte é obtida em função da reusabilidade POO prevista na implementação da equação do fluxo. A comparação da solução numérica do modelo ADR 2D com a correspondente solução analítica disponível na literatura, demonstra que o estimador residual apresenta excelentes índices de eficiência. Os resultados numéricos obtidos mostraram que o estimador residual encontra-se limitado inferior e superiormente pelo erro real da solução em malha grosseira. O estimador ZZ mostrou-se inadequado para a análise do erro de aproximação das equações ADR. Os exemplos selecionados para verificação e aplicação do estimador residual abrangem, em diferentes escalas, modelos que descrevem reação de primeira ordem e modelos com fenômenos de sorção e retardamento na migração do contaminante em meio poroso saturado. Em conseqüência, o estimador residual proposto provou ser computável, eficiente e robusto no sentido de abranger uma grande variedade das aplicações dos fenômenos de transporte de contaminantes em meio poroso saturado e regime de pequena advecção. / Several computational models that implement the solute migration in saturated porous media constantly appear in scientific publications due to the great importance given to the understanding and forecast of the solute transport in groundwater. The numerical solutions obtained by computational schemes are not immune to errors related to the discretization process. However, the reliability of the results obtained by the complex operations of the computational fluids dynamics can be enhanced by a posteriori error estimates that indicate the accuracy of the numerical solution. In this work a residual error estimator is presented for the parabolic equation that describes the advection-dispersion-reaction phenomena (ADR) in saturated porous media, considering the transport in small advection regime. The numerical solution of the ADR equation is obtained by the finite element method using upwind terms to minimize the spurious oscillations. The computational code and the correspondent a posteriori error estimates are implemented in Java language following the Object Oriented Programming (OOP) paradigm in Eclipse platform. The numerical solution of the elliptic groundwater flow equation and the respective error estimates with gradient recovery characteristic, the ZZ-estimator, are also implemented in the JAVA code. The solution of the transport equation is obtained as a consequence of the OOP reusability intended in the implementation of the flow equation. The numerical solution of the ADR 2D simulation compared to the analytical solution available in the literature, demonstrate the excellent effectivity index presented by the residual error estimator. The obtained results indicate that the residual error estimator is lower and upper bounded by a solution in coarse mesh. The ZZ-estimator showed to be inadequate for the error analysis of the ADR equations. The examples selected for validation and application of the residual estimator include, in distinct scales, models that describe reaction of first order and models with sorption and retardation phenomena in the pollutant migration in saturated porous media. Therefore, the proposed residual error estimator proved to be computable, efficient and robust in the sense of solving a great variety of applications of transport phenomena in saturated porous media at small advection regime.

Vários modelos computacionais que implementam o transporte de soluto em meio poroso saturado surgem constantemente em publicações científicas devido à suma importância dada à compreensão e previsão do transporte de constituintes dissolvidos em água subterrânea. As soluções numéricas obtidas por esquemas computacionais não estão imunes aos erros de discretização. No entanto, a confiabilidade nos resultados obtidos das complexas operações provenientes da dinâmica de fluidos computacional pode ser aumentada através de estimadores de erro a posteriori que indicam a precisão da solução numérica de um modelo matemático que simula o fenômeno físico de interesse. Neste trabalho é apresentado um estimador residual para a equação parabólica que descreve os fenômenos de advecção-dispersão-reação (ADR) em meio poroso saturado, considerando o transporte em regime de pequena advecção. A solução numérica da equação ADR é obtida pelo método dos elementos finitos que emprega termos upwind para minimizar as inconvenientes oscilações espúrias. A implementação do código computacional para obter essa solução numérica e o seu correspondente erro a posteriori, é feita em linguagem JAVA na plataforma Eclipse seguindo o paradigma da Programação Orientada a Objetos (POO). A solução numérica da equação elíptica do fluxo subterrâneo e o seu estimador de erro com características de recuperação do gradiente, o estimador ZZ, também são implementados no código JAVA. Assim, a solução da equação do transporte é obtida em função da reusabilidade POO prevista na implementação da equação do fluxo. A comparação da solução numérica do modelo ADR 2D com a correspondente solução analítica disponível na literatura, demonstra que o estimador residual apresenta excelentes índices de eficiência. Os resultados numéricos obtidos mostraram que o estimador residual encontra-se limitado inferior e superiormente pelo erro real da solução em malha grosseira. O estimador ZZ mostrou-se inadequado para a análise do erro de aproximação das equações ADR. Os exemplos selecionados para verificação e aplicação do estimador residual abrangem, em diferentes escalas, modelos que descrevem reação de primeira ordem e modelos com fenômenos de sorção e retardamento na migração do contaminante em meio poroso saturado. Em conseqüência, o estimador residual proposto provou ser computável, eficiente e robusto no sentido de abranger uma grande variedade das aplicações dos fenômenos de transporte de contaminantes em meio poroso saturado e regime de pequena advecção. / Several computational models that implement the solute migration in saturated porous media constantly appear in scientific publications due to the great importance given to the understanding and forecast of the solute transport in groundwater. The numerical solutions obtained by computational schemes are not immune to errors related to the discretization process. However, the reliability of the results obtained by the complex operations of the computational fluids dynamics can be enhanced by a posteriori error estimates that indicate the accuracy of the numerical solution. In this work a residual error estimator is presented for the parabolic equation that describes the advection-dispersion-reaction phenomena (ADR) in saturated porous media, considering the transport in small advection regime. The numerical solution of the ADR equation is obtained by the finite element method using upwind terms to minimize the spurious oscillations. The computational code and the correspondent a posteriori error estimates are implemented in Java language following the Object Oriented Programming (OOP) paradigm in Eclipse platform. The numerical solution of the elliptic groundwater flow equation and the respective error estimates with gradient recovery characteristic, the ZZ-estimator, are also implemented in the JAVA code. The solution of the transport equation is obtained as a consequence of the OOP reusability intended in the implementation of the flow equation. The numerical solution of the ADR 2D simulation compared to the analytical solution available in the literature, demonstrate the excellent effectivity index presented by the residual error estimator. The obtained results indicate that the residual error estimator is lower and upper bounded by a solution in coarse mesh. The ZZ-estimator showed to be inadequate for the error analysis of the ADR equations. The examples selected for validation and application of the residual estimator include, in distinct scales, models that describe reaction of first order and models with sorption and retardation phenomena in the pollutant migration in saturated porous media. Therefore, the proposed residual error estimator proved to be computable, efficient and robust in the sense of solving a great variety of applications of transport phenomena in saturated porous media at small advection regime.

This study proposes novel optimization methodologies for the optimization of problems that reveal non-physical step discontinuities. More specifically, it is proposed to use gradient-only techniques that do not use any zeroth order information at all for step discontinuous problems. A step discontinuous problem of note is the shape optimization problem in the presence of remeshing strategies, since changes in mesh topologies may - and normally do - introduce non-physical step discontinuities. These discontinuities may in turn manifest themselves as non-physical local minima in which optimization algorithms may become trapped. Conventional optimization approaches for step discontinuous problems include evolutionary strategies, and design of experiment (DoE) techniques. These conventional approaches typically rely on the exclusive use of zeroth order information to overcome the discontinuities, but are characterized by two important shortcomings: Firstly, the computational demands of zero order methods may be very high, since many function values are in general required. Secondly, the use of zero order information only does not necessarily guarantee that the algorithms will not terminate in highly unfit local minima. In contrast, the methodologies proposed herein use only first order information, rather than only zeroth order information. The motivation for this approach is that associated gradient information in the presence of remeshing remains accurately and uniquely computable, notwithstanding the presence of discontinuities. From a computational effort point of view, a gradient-only approach is of course comparable to conventional gradient based techniques. In addition, the step discontinuities do not manifest themselves as local minima. / Thesis (PhD)--University of Pretoria, 2010. / Mechanical and Aeronautical Engineering / unrestricted