In this thesis, a parallel solver was developed for the modelling of blood flow through a number of patient-specific geometries. A locally conservative Galerkin (LCG) spatial discretisation was applied along with an artificial compressibility and characteristic based split (CBS) scheme to solve the 3D incompressible Navier-Stokes equations. The Spalart-Allmaras one equation turbulence model was also optionally employed. The solver was constructed using FORTRAN and the Message Passing Interface (MPI). Parallel testing demonstrated linear or better than linear speedup on hybrid patient-specific meshes. These meshes were unstructured with structured boundary layers. From the parallel testing it is clear that the significance of inter-processor communication is negligible in a three dimensional case. Preliminary tests on a short patient-specific carotid geometry demonstrated the need for ten or more boundary layer meshes in order to sufficiently resolve the peak wall shear stress (WSS) along with the peak time-averaged WSS. A time sensitivity study was also undertaken along with the assessment of the order of the real time step term. Three backward difference formulae (BDF) were tested and no significant difference between them was detected. Significant speedup was possible as the order of time discretisation increased however, making the choice of BDF important in producing a timely solution. Followed by the preliminary investigation, four more carotid geometries were investigated in detail. A total of six haemodynamic wall parameters have been brought together to analyse the regions of possible atherogenesis within each carotid. The investigations revealed that geometry plays an overriding influence on the wall parameter distribution. Each carotid artery displayed high time-averaged WSS at the apex, although the value increased significantly with a proximal stenosis. Two out of four meshes contained a region of low time-averaged WSS distal to the flow divider and within the largest connecting artery (internal or external carotid artery), indicating a potential region of atherosclerosis plaque formation. The remaining two meshes already had a stenosis in the corresponding region. This is in excellent agreement with other established works. From the investigations, it is apparent that a classification system of stenosis severity may be possible with potential application as a clinical diagnosis aid. Finally, the flow within a thoracic aortic aneurysm was investigated in order to assess the influence of a proximal folded neck. The folded neck had a significant effect on the wall shear stress, increasing by up to 250% over an artificially smoothed neck. High wall shear stresses may be linked to aneurysm rupture. Being proximal to the aneurysm, this indicated that local geometry should be taken into account when assessing the rupture potential of an aneurysm.

abstract: This dissertation describes a process for interface capturing via an arbitrary-order, nearly quadrature free, discontinuous Galerkin (DG) scheme for the conservative level set method (Olsson et al., 2005, 2008). The DG numerical method is utilized to solve both advection and reinitialization, and executed on a refined level set grid (Herrmann, 2008) for effective use of processing power. Computation is executed in parallel utilizing both CPU and GPU architectures to make the method feasible at high order. Finally, a sparse data structure is implemented to take full advantage of parallelism on the GPU, where performance relies on well-managed memory operations. With solution variables projected into a kth order polynomial basis, a k+1 order convergence rate is found for both advection and reinitialization tests using the method of manufactured solutions. Other standard test cases, such as Zalesak's disk and deformation of columns and spheres in periodic vortices are also performed, showing several orders of magnitude improvement over traditional WENO level set methods. These tests also show the impact of reinitialization, which often increases shape and volume errors as a result of level set scalar trapping by normal vectors calculated from the local level set field. Accelerating advection via GPU hardware is found to provide a 30x speedup factor comparing a 2.0GHz Intel Xeon E5-2620 CPU in serial vs. a Nvidia Tesla K20 GPU, with speedup factors increasing with polynomial degree until shared memory is filled. A similar algorithm is implemented for reinitialization, which relies on heavier use of shared and global memory and as a result fills them more quickly and produces smaller speedups of 18x. / Dissertation/Thesis / Doctoral Dissertation Aerospace Engineering 2015

Orientador: Jose Luiz Boldrini / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-07-22T03:31:13Z (GMT). No. of bitstreams: 1 PobleteCantelano_MarianoEdgardo_M.pdf: 2237950 bytes, checksum: 29fe3f66b8ac123f3886b8e7779345a6 (MD5) Previous issue date: 1997 / Resumo: Não informado / Abstract: Not informed / Mestrado / Mestre em Matemática Aplicada

Orientador: Jose Luiz Boldrini / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-07-23T10:39:30Z (GMT). No. of bitstreams: 1 RojasMedar_MariaDrina_D.pdf: 1133068 bytes, checksum: d84ec877405eed4dc5426546e18a1355 (MD5) Previous issue date: 1998 / Resumo: Não informado / Abstract: Not informed / Doutorado / Doutor em Matemática

Submitted by Marcele Costal de Castro (costalcastro@gmail.com) on 2017-10-04T17:13:38Z No. of bitstreams: 1 PAULO AUGUSTO BERQUÓ DE SAMPAIO D.pdf: 6576641 bytes, checksum: 71355f6eedcf668b2236d4c10f1a2551 (MD5) / Made available in DSpace on 2017-10-04T17:13:38Z (GMT). No. of bitstreams: 1 PAULO AUGUSTO BERQUÓ DE SAMPAIO D.pdf: 6576641 bytes, checksum: 71355f6eedcf668b2236d4c10f1a2551 (MD5) Previous issue date: 1991-09 / The basic difficulties associated with the numerical solution of the incompressible Navier-Stokes equations in primitive variables are identified and analysed. These difficulties, namely the lack of self-adjointness of the flow equations and the requirement of choosing compatible interpolations for velocity and pressure, are addressed with the development of consistent Petrov-Galerkin formulations. In particular, the solution of incompressible viscous flow problems using simple equal order interpolation for all variables becomes possible .

This paper presents a wavelet Galerkin scheme for the fast solution of boundary integral equations. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators which yields quasi-sparse system matrices. These matrices can be compressed such that the complexity for solving a boundary integral equation scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. Based on the wavelet Galerkin scheme we present also an adaptive algorithm. By numerical experiments we provide results which demonstrate the performance of our algorithm.

The current work concerns the study and the implementation of a modern algorithm for a posteriori error estimation in Computational Fluid Dynamics (CFD) simulations based on partial differential equations (PDEs). The estimate involves the use of duality argument and proper consistent discretisation of primal and dual problem.A key element is the construction of the adjoint form of the primal differential operators where the data term is a quantity of interest depending on the application. In engineering, this is typically a physical functional of the solution. So, by solving this adjoint problem, it is possible to obtain important information about local sensitivity of the error with respect to the current target quantity and thereby, we are able to perform an a posteriori error representation based on adjoint data. Through this, we provide local error indicators which can drive an adaptive meshing algorithm in order to optimally reduce the target error. Therefore, we first derive and solve the discrete primal problem in agreementwith the chosen numerical method. According to consistency and compatibility conditions, we can use the same discretisation for solving the adjoint problem, simply by swapping the position of the unknowns and the test functions in the linearised variational operator. Remembering that the corresponding adjoint problem always remains linear, the computational cost for obtaining these data is limited compared to the effort needed to solve the primal nonlinear problem.This procedure, fully developed for Discontinuous Galerkin (DG) and Finite Volume (FV) methods, is here for the first time applied in a fully consistent way for Petrov-Galerkin (PG) discretisations. Differently from the latter, the biggest issue for the PG method becomes the need to handle two different functional spaces in the discretisation, one of which is often not even continuous. Stabilized finite element schemes such as Streamline Upwind (SUPG), bubble stabilized (BUBBLE) Petrov-Galerkin and stabilized Residual Distribution (RD) have been selected for implementation and testing. Indeed, based on local advection information, these schemes are naturally more suitable for solving hyperbolic problems and therefore, interesting alternatives for fluid dynamics applications.A scalar linear advection equation is used as a model problem for convergence rate of both primal and adjoint solutions and target quantity. In addition, it is also applied in order to verify the accuracy of the adjoint-based a posteriori error estimate. Next, we apply the methods to a complete collection of numerical examples, starting from scalar Burgers’ problem till 2D compressible Euler equations. Through suited quantities of interest, we illustrate aspects of the adjoint mesh refinement by comparing its efficiency with respect to the standard a posteriori error estimation. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

This thesis studies the numerical solution of non-linear convection-diffusion problems using the space- time discontinuous Galerkin method, which perfectly suits the space as well as time local adaptation. We aim to develop a posteriori error estimates reflecting the spatial, temporal, and algebraic errors. These estimates are based on the measurement of the residuals in dual norms. We derive these estimates and numerically verify their properties. Finally, we derive an adaptive algorithm and apply it to the numerical simulation of non-stationary viscous compressible flows. Powered by TCPDF (www.tcpdf.org)

Orientadores: Jorge Stolfi, Anamaria Gomide / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-04T01:31:55Z (GMT). No. of bitstreams: 1 Cardoso_ClaudioGuidoSilva_M.pdf: 3745382 bytes, checksum: 3bab54915892693628aea66a49348c48 (MD5) Previous issue date: 2004 / Resumo: Nesta dissertação, estudamos as aplicações de splines construídos sobre grades diádicas na aproximação de funções e solução de equações diferenciais. Nosso foco principal é a simulação de escoamento de petróleo em reservatórios naturais. Uma grade diádica é construída através de divisões sucessivas de uma célula raiz retangular, sempre pela metade, alternando a direção do corte ciclicamente pelos eixos de coordenadas. Desta forma, os splines diádicos podem ser refinados adaptativamente de acordo com as funções a serem aproximadas, até que satisfaçam o grau de precisão exigido. As grades diádicas também possuem uma estrutura muito simples e podem ser representadas e manipuladas computacionalmente com grande eficiência e facilidade. Trabalhamos em particular com uma família de splines diádicos multilineares e contínuos para uma grade arbitrária G, que denominamos por [1 [G]. Desenvolvemos algoritmos que geram duas bases padrões de elementos finitos (tendas diádicas) Bmax e Bmin para estes espaços. Para testarmos as potencialidades dos espaços [l[G], realizamos experimentos numéricos utilizando a técnica de aproximação por mínimos quadrados, e resolvemos alguns casos de equações diferenciais parciais lineares. Nestes experimentos, estudamos o uso de malhas diádicas de resolução variável no espaço e no tempo. Concluímos que grades adaptativas podem produzir os mesmos resultados que uma grade uniforme fina, a uma fração do custo. A última parte deste trabalho descreve em detalhes a construção e teste de um simulador para escoamento bifásico óleo/água (Simóleo), baseado em grades diádicas adaptativas. Revisamos, em linhas gerais, as equações que regem este processo, e sua discretização pelo método de elementos finitos (Galerkin). Desta forma, obtemos um sistema de equações diferenciais não linear, dependente do espaço e do tempo, que resolvemos de maneira iterativa. O desempenho do Simóleo foi avaliado utilizando um modelo popular de reservatório (5 poços) / Abstract: ln this dissertation we study the application of splines built over dyadic grids for function approximation and integration of differential equations. Our main focus is the simulation of oil flow in natural reservoirs. A dyadic grid is built by recursively splitting a rectangular root cell into equal halves, alternating the orientation of the cut cyclically over the coordinate axis. Therefore, dyadic splines may be refined adaptatively to fit the functions they are approximating, until a certain precision leveI is reached. Dyadic grids also have a simple structure which may be easily and efficiently represented and manipulated by computers. We worked in particular with a family of multilinear and continuous dyadic splines for a certain grid G, denoted by [1[G]. Algorithms were developed to build two standard basis of finite el¿ments (dyadic tents), Bmax and Bmin, for such function spaces. In order to test the potentialities of the spaces [1 [G], we performed some numerical experiments of least squares approximations and integration of linear partial differential equations. In these experiments, we studied the use of adaptive dyadic grids, whose resolution varies in time and space. We conclude that adaptive grids can produce the same results as a fine uniform grid, at a fraction of the cost. The last part of this work describes in detail the development and tests of an oiljwater flow simulator (Simóleo), based on adaptive dyadic grids. We review the oil flow equations, and their discretization by the finite element (Galerkin) method. Thus, we obtain a non linear differential equation system, depending on space and time, which we solve iteratively. The program's performance is measured with experiments using a popular reservoir model (5-spot) / Mestrado / Ciência da Computação / Mestre em Computação

Orientadores : Philippe R. Bernard Devloo, Sonia Maria Gomes / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-07-31T18:02:06Z (GMT). No. of bitstreams: 1 DiazCalle_JorgeLizardo_D.pdf: 3677542 bytes, checksum: 11e118bb87aca506bf568e98a68c5293 (MD5) Previous issue date: 2002 / Resumo: O presente trabalho apresenta técnicas inovadoras para a aproximação numérica de leis de conservação sobre malhas não estruturadas. Implementa se um algoritmo h-adaptativo que utiliza um esquema numérico baseado em espaços de aproximação de funções polinomiais descontínuas. A escolha adaptável do refinamento h é feita mediante uma análise da regularidade da solução utilizando-se técnicas de análise wavelet. Esta análise permite determinar sub-domínios ou regiões de suavidade nos quais os elementos finitos são levados a níveis menos refinados, ou regiões de singularidade nas quais os elementos são refinados. Para evitar possíveis oscilações numéricas, um termo difusivo é aplicado no interior dos elementos finitos de uma região de singularidade ou próximo a ela. A análise wavelet também é utilizada para estabelecer a magnitude do termo difusivo. O esquema proposto aproveita idéias do método Runge-Kutta Galerkin descontínuo [16] e o método streamline difusion [33]. Como resultado, o esquema, na sua forma mais simples, é o método de volumes finitos h-adaptativo, e no caso de usar ordem de interpolação p = 1, é o método Galerkin descontínuo h-adaptativo com esquema Euler no tempo e dispensando o uso de limitadores. é apresentado um estudo para estabelecer uma relação adequada entre o valor do número CFL (condição de estabilidade - Courant Priedrichs Lewi) e o coeficiente máximo do termo difusivo interno de tal forma a garantir a estabilidade do esquema e obter precisão numérica ótima. Quando o termo difusivo d? 8, o esquema tem comportamento similar ao esquema de volumes finitos. O esquema foi implementado em um ambiente computacional na filosofia de programação orientada para objetos [21]. Neste contexto, foi elaborada uma biblioteca de classes que permite implementar espaços aproximantes descontínuos e realizar análise de regularidade local utilizando wavelets / Abstract: In this work an inovative technique is presented for the numerical approximation of conservation laws on unstructured meshes. The numerical scheme uses a discontinuous piecewise polynomial approximation with hadaptivity. The self-adaptive h-refinement strategy uses a regularity assessment of the solution based on wavelet techniques. This strategy determines the sub-domains or regions where the solution is smooth and where elements can be coarsened and/or regions of singularity where the elements are refined. Numerical oscillations within the discontinuous element are controlled by adding a difusive term. The wavelet analysis is also used to determine the magnitude of the diffusive term. The resulting scheme is based both on the Runga-Kutta Discontinuous Galerkin method [16] and the streamline diffusion method [33] : when the order of interpolation within the elements is zero it is a cell centered finite volume method, and for interpolation order p _ 1 , it is an h-adaptive discontinuous Galerkin method, using Euler time-stepping. Internal oscillations are entirely controlled by the added streamline diffusion operator. An optimal relationship between the time step (in terms of the CFL condition) and the size of the diffusion coeficient is analysed for numerical precision. When the diffusive term d?8 , the presented scheme reduces to the finite volume method. The scheme is implemented using the object oriented programming philosophy based on the environment described in [21]. Within this context, a library of classes was developed which implements piecewise polynomial discontinuous approximations and which analyses the regularity of the approximate solution using the wavelet technique / Doutorado / Doutor em Matemática Aplicada