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In this thesis, evaluation of H(div)-conforming finite elements is implemented in a high-performance setting and used to solve the incompressible Navier-Stokes equation, obtaining an exactly point-wise divergence-free velocity field. In particular, the anisotropic Raviart-Thomas tensor-product polynomial space is considered, where the finite element operators are evaluated with quadrature in a matrix-free fashion using sum-factorization on tensor-product meshes. The implementation includes evaluation over elements and faces in two- and three-dimensional space, supporting non-conforming meshes with hanging nodes, and using the contravariant Piola transformation to preserve normal components on element boundaries. In terms of throughput, the implementation achieves up to an order of magnitude faster evaluation of finite element operators compared to a matrix-based evaluation. Correctness is demonstrated with optimal convergence rates for various polynomial degrees, as well as exactly divergence-free solutions for the velocity field.

In summary, this thesis focuses on developing an a priori theory for geometric finite element discretizations of a Cosserat rod model, which is derived from incompatible elasticity. This theory will be supported by corresponding numerical experiments to validate the convergence behavior of the proposed method. The main result describes the qualitative behavior of intrinsic H1-errors and L2-errors in terms of the mesh diameter 0 < h ≪ 1 of the approximation scheme. Geometric Finite Element functions uh with its subclasses Geodesic Finite Elements and Projection- based Finite Elements as conforming path-independent and objective discretizations of Cosserat rod configurations were used. Existence, regularity, variational bounds and vector field transport estimates of the Cosserat rod model were derived to ob- tain an intrinsic a-priori theory. In the second part, this thesis concerns the derivation of the Cosserat rod from 3D elasticity featuring prestress together with numerical experiments for microheteroge- neous prestressed materials.

[pt] Nos últimos anos na cidade de Lima as escavações profundas alcançaram uma presença maior, devido à recente demanda pela construção de edifícios altos e centros comerciais com vários níveis de subsolos, onde utilizam-se preferencialmente as cortinas atirantadas para a contenção dos solos. No entanto, a técnica de solo grampeado mostra-se como uma alternativa interessante, dado que também permite fornecer suporte lateral e limitar os movimentos do solo. A técnica consiste em reforçar o bloco do solo atrás da parede através das inclusões passivas, criando uma estrutura de gravidade consistente que permite aumentar a resistência geral ao cisalhamento e restringir os deslocamentos. Este estudo de caso procura avaliar os deslocamentos horizontais em uma escavação de 15m de altura em um entorno urbano, onde o solo é predominantemente de pedregulho denso mal graduado, parte do depósito aluvial do Conglomerado de Lima. A estimativa dos parâmetros de resistência baseou-se em ensaios de cisalhamento direto in situ em grande escala e ensaios de arrancamento encontrados na literatura. O acompanhamento dos deslocamentos ao longo do processo construtivo, foi realizado a través de inclinômetros instalados no perímetro, em locais estratégicos para avaliar os deslocamentos mais críticos na escavação. A avaliação de erros sistemáticos em inclinômetros é também abordada no estudo. Complementarmente, realizaram-se análises numéricas com programas de equilíbrio limite e elementos finitos, que permitiram avaliar os fatores de segurança e os deslocamentos com o objetivo de comparar os dados de campo com os resultados calculados. Através de análise paramétrica e calibração do modelo foi possível conhecer a influência dos parâmetros geotécnicos e geométricos de escavações profundas grampeadas. / [en] In recent years in the city of Lima, the execution of deep excavations growth, due to the recent demand for the construction of high-rise buildings and shopping malls having various levels of basements, where anchorage walls systems are preferably used to soil retention. However, the soil nailing technique is an interesting alternative, by reason to provide lateral support and limit movements. The soil nailing technique consists of reinforcing the soil block behind the wall through passive inclusions, producing a consistent gravity structure that allows the increas of the shear strength and restricting displacements. This case study aims to evaluate the horizontal displacements generated in a 15m high deep excavation in an urban environment, where the existing soil is predominantly of poorly graded dense gravel, component of the alluvial deposit of the Lima Conglomerate. The estimate of strength parameters for this type of soil was based on large-scale in situ direct shear tests and pullout tests found in the literature. The monitoring of displacements throughout the construction process was carried out using vertical inclinometers installed on the perimeter, in strategic locations to assess the most critical displacements that the excavation could experience. The evaluation of systematic errors in inclinometers is also approached in the study. In addition, numerical analyzes were carried out with limit equilibrium and finite element programs, which allowed evaluating the safety factors and displacements in order to compare the field data with the calculated results. Through parametric analysis and model calibration, it was possible to know the influence of geotechnical and geometric parameters of nailed deep excavations.

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure correction schemes used for the incompressible Navier Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. Exploring the influence of boundary conditions on CGP, the minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions. We discuss the CGP method as a guide for partial mesh refinement of incompressible flow computations and show its application for simulations of flow over a backward facing step and flow past a cylinder. / Master of Science / Coarse Grid Projection (CGP) methodology is a new multigrid technique applicable to pressure projection methods for solving the incompressible Navier-Stokes equations. In the CGP approach, the nonlinear momentum equation is evolved on a fine grid, and the linear pressure Poisson equation is solved on a corresponding coarsened grid. Mapping operators transfer the data between the grids. Hence, one can save a considerable amount of CPU time due to reducing the resolution of the pressure filed while maintaining excellent to reasonable accuracy, depending on the level of coarsening.

[pt] O concreto de ultra alto desempenho (UHPC) reforçado com fibras é um material que foi desenvolvido nas últimas décadas para atender à necessidade de estruturas modernas por um material mais resistente e durável. Suas características altamente não lineares tanto na tração quanto na compressão levam a um comportamento complexo. Além disso, a distribuição não homogênea das fibras e a alta resistência à tração, quando comparada ao concreto convencional, resultam em menor ductilidade para vigas de UHPC. A análise de elementos finitos mostra ser uma ferramenta adequada para representar a resposta de elementos estruturais de UHPC, mas a calibragem do modelo deve ser aplicada corretamente e técnicas de modelagem coerentes devem ser usadas para representar corretamente os tramos pós-pico de curvas força-deslocamento para vigas de UHPC submetidas a testes de flexão de quatro pontos. Foi realizada uma extensa caracterização do material tanto em tração quanto em compressão. Testes axiais monotônicos foram conduzidos para obter curvas tensão-deformação na compressão e tensão-abertura de fissura na tração. Testes cíclicos foram realizados para determinar a evolução do dano experimental em compressão e na tração. Esses dados serviram como referência para calibrar modelos uniaxiais e modelos de evolução de dano de acordo com expressões analíticas disponíveis na literatura. Modelos heterogêneos simulando a dispersão do material nas propriedades mecânicas do UHPC ao longo do volume das vigas foram utilizados para obter uma seção transversal que apresentasse resistência otimizada, mantendo a ductilidade desejada. Finalmente, cinco vigas foram testadas, com diferentes formas e porcentagens de reforço, e estratégias de modelagem foram comparadas aos dados experimentais das vigas. / [en] Ultra-high performance concrete is a material which has been developed in the last decades to fulfill modern structures need for a more resistant and durable material. Its highly nonlinear characteristics in both tension and compression leads to a complex behavior. In addition to that, the inhomogeneous distribution of the fibers and the high tensile strength when compared to conventional concrete result in reduced ductility for UHPC beams. Finite element analysis is shown to be an adequate tool to represent UHPC structural element s response but the model calibration must be correctly applied and coherent modeling techniques must be used to correctly model the post-peak branches of load-displacement curves for UHPC beams subjected to four-point load bending tests. An extensive material characterization in both tension and compression was conducted. Monotonic axial tests were conducted to obtain stress-strain curves in compression and stress-crack opening in tension and cyclic tests were made to determine the experimental damage evolution in compression and in tension. These data served as input to calibrate uniaxial models and damage evolution models according to analytical expressions available in the literature. Heterogeneous models simulating the material dispersion of the mechanical properties of the UHPC over structural beams were used to obtain a cross-section that presented optimized resistance while maintaining target ductility. Finally, five beams were tested, with different shapes and reinforcement ratios and the modeling strategies were benchmarked to the beams experimental data.

L’objectif de ce travail de thèse est de proposer la Méthodes des éléments finis de Galerkin discontinus (DGFEM) à la discrétisation des équations compressibles de Navier-Stokes. Plusieurs challenges font l’objet de ce travail. Le premier aspect a consisté à montrer l’ordre de convergence optimal de la méthode DGFEM en utilisant les polynômes d’interpolation d’ordre élevé. Le deuxième aspect concerne l’implémentation de méthodes de ‘‘shock-catpuring’’ comme les limiteurs de pentes et les méthodes de viscosité artificielle pour supprimer les oscillations numériques engendrées par l’ordre élevé (lorsque des polynômes d’interpolation de degré p>0 sont utilisés) dans les écoulements transsoniques et supersoniques. Ensuite nous avons implémenté des estimateurs d’erreur a posteriori et des procédures d ’adaptation de maillages qui permettent d’augmenter la précision de la solution et la vitesse de convergence afin d’obtenir un gain de temps considérable. Finalement, nous avons montré la capacité de la méthode DG à donner des résultats corrects à faibles nombres de Mach. Lorsque le nombre de Mach est petit pour les écoulements compressibles à la limite de l’incompressible, la solution souffre généralement de convergence et de précision. Pour pallier ce problème généralement on procède au préconditionnement qui modifie les équations d’Euler. Dans notre cas, les équations ne sont pas modifiées. Dans ce travail, nous montrons la précision et la robustesse de méthode DG proposée avec un schéma en temps implicite de second ordre et des conditions de bords adéquats. / The aim of this thesis is to deal with compressible Navier-Stokes flows discretized by Discontinuous Galerkin Finite Elements Methods. Several aspects has been considered. One is to show the optimal convergence of the DGFEM method when using high order polynomial. Second is to design shock-capturing methods such as slope limiters and artificial viscosity to suppress numerical oscillation occurring when p>0 schemes are used. Third aspect is to design an a posteriori error estimator for adaptive mesh refinement in order to optimize the mesh in the computational domain. And finally, we want to show the accuracy and the robustness of the DG method implemented when we reach very low mach numbers. Usually when simulating compressible flows at very low mach numbers at the limit of incompressible flows, there occurs many kind of problems such as accuracy and convergence of the solution. To be able to run low Mach number problems, there exists solution like preconditioning. This method usually modifies the Euler. Here the Euler equations are not modified and with a robust time scheme and good boundary conditions imposed one can have efficient and accurate results.

Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

The response of submerged structures subject to loading by underwater blast waves is governed by complex interactions between the moving or deforming structure and the surrounding fluid and these phenomena need to be thoroughly understood in order to design structural components against underwater blast. This thesis has addressed the response of simple structural systems to blast loading in shallow or deep water environment. Analytical models have been developed to examine the one-dimensional response of both water-backed and air-backed submerged rigid plates, supported by linear springs and loaded by underwater shock waves. Cavitation phenomena as well as the effect of initial static fluid pressure are explicitly included in the models and their predictions were found in excellent agreement with detailed FE simulations. Then, a novel experimental apparatus has been developed, to reproduce controlled blast loading in initially pressurised liquids. It consists of a transparent water shock tube and allows observing the structural response as well as the propagation of cavitation fronts initiated by fluid-structure interaction in a blast event. This experimental technique was then employed to explore the one-dimensional response of monolithic plates, sandwich panels and double-walled structures subject to loading by underwater shock waves. The performed experiments provide great visual insight into the cavitation process and the experimental measurements were found to be in good agreement with analytical predictions and dynamic FE results. Finally, underwater blast loading of circular elastic plates has been investigated by theoretically modelling the main phenomena of dynamic plate deformation and fluid-structure interaction. In addition, underwater shock experiments have been performed on circular composite plates and the obtained measurements were found in good correlation with the corresponding analytical predictions. The validated analytical models were then used to determine the optimal designs of circular elastic plates which maximise the resistance to underwater blast.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: In this work the Method of Manufactured Solutions (MMS) is introduced for the code veri cation of full-wave frequency dependent electromagnetic computational software. At rst the method is sketched in the context of the veri cation and validation process and the need for proper code veri cation is highlighted. Subsequently, the MMS is investigated in its natural context: the Finite Element Method, speci cally for the E- eld Vector Wave Equation. The usefulness of the method to detect error in a computational code is demonstrated. The selection of Manufactured Solutions is discussed and it is demonstrated how it can be used to nd the probable cause of bugs. Mutation testing is introduced and used to show the ability to detect errors present in code. The MMS is nally applied in a novel manner to a Method of Moments (MoM) code. The challenges of numerical integration associated with the application of the operator is discussed and correct integration is successfully demonstrated. Subsequently the MMS is demonstrated to be successfully applied to the MoM and mutation testing is used to demonstrate the practical e cacy of the method. The application of the MMS to the MoM is the main contribution of this work. / AFRIKAANSE OPSOMMING: Die Metode van Vervaardigde Oplossings (MVO) word hier bekend gestel vir die veri kasie van numeriese volgolf frekwensie-afhanklike elektromagnetise kode. Die metode word eerstens in die bre e konteks van algemene veri kasie en validasie geplaas en gevolglik word die noodsaaklikheid van kode veri kasie beklemtoon. Daarna, word die toets-metode in die konteks van die Eindige Element Metode vir die E-veld vektorgolf vergelyking bestudeer. Die MVO is oorspronklik ontwikkel in die di erentiaalvergelyking omgewing. Die bruikbaarheid van die metode vir elektromagnetiese simulasies word prakties gedemonstreer deur die opsporing van werklike foute. Die metode word ook verder ondersoek vir die oorsprong van foute. Mutasietoetsing word bekendgestel en word gebruik om die metode verder prakties te veri eer. Die MVO word laastens in 'n nuwe manier gebruik om 'n Moment Metode kode te veri eer. Die praktiese probleme betrokke by numeriese integrasie word ondersoek en die korrekte toepassing van die integraal operator word prakties gedemonstreer. Daarna, word die MVO in hierdie konteks gedemonstreer deur verskeie voorbeelde te ondersoek. Mutasietoetsing word weereens gebruik om na die e ektiewiteit van die MVO te kyk om 'n Moment Metode kode te toets. Die toepassing van die MVO op 'n Moment Metode kode is die hoof bydrae van hierdie werk.