The Streamline Upwind/Petrov-Galerkin (SU/PG) method is applied to higher-order finite-element discretizations of the Euler equations in one dimension and the Navier-Stokes equations in two dimensions. The unknown flow quantities are discretized on meshes of triangular elements using triangular Bezier patches. The nonlinear residual equations are solved using an approximate Newton method with a pseudotime term. The resulting linear system is solved using the Generalized Minimum Residual algorithm with block diagonal preconditioning. The exact solutions of Ringleb flow and Couette flow are used to quantitatively establish the spatial convergence rate of each discretization. Examples of inviscid flows including subsonic flow past a parabolic bump on a wall and subsonic and transonic flows past a NACA 0012 airfoil and laminar flows including flow past a a flat plate and flow past a NACA 0012 airfoil are included to qualitatively evaluate the accuracy of the discretizations. The scheme achieves higher order accuracy without modification. Based on the test cases presented, significant improvement of the solution can be expected using the higher-order schemes with little or no increase in computational requirements. The nonlinear system also converges at a higher rate as the order of accuracy is increased for the same number of degrees of freedom; however, the linear system becomes more difficult to solve. Several avenues of future research based on the results of the study are identified, including improvement of the SU/PG formulation, development of more general grid generation strategies for higher order elements, the addition of a turbulence model to extend the method to high Reynolds number flows, and extension of the method to three-dimensional flows. An appendix is included in which the method is applied to inviscid flows in three dimensions. The three-dimensional results are preliminary but consistent with the findings based on the two-dimensional scheme. / Ph. D.

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Ce travail porte sur le développement d’une nouvelle méthode numérique intitulée « Meshless local Petrov Galerkin (MLPG) combinée à la méthode des volumes finis (MVF) » appliquée au calcul de structures. Elle est basée sur la résolution de la forme faible des équations aux dérivées partielles par une méthode de Petrov Galerkin comme en éléments finis, mais par contre l’approximation du champ de déplacement introduite dans la forme faible ne nécessite pas de maillage. Seul un ensemble de nœuds est réparti dans le domaine et l’approximation du champ de déplacement en un point ne dépend que de la distance de ce point par rapport aux nœuds qui l’entourent et non de l’appartenance à un certain élément fini. Les déformations et les déplacements sont déterminés aux différents nœuds par interpolation locale en utilisant les moindres carrés mobiles (MLS). Les valeurs des déformations aux nœuds sont exprimées en termes de valeurs nodales interpolées indépendamment des déplacements, en imposant simplement la relation déformation déplacement directement par collocation aux points nodaux. La procédure de calcul pour cette méthode est implémentée dans un programme de calcul développé sous MATLAB. Le code obtenu a été validé sur un certain nombre de cas tests par comparaison avec des solutions analytiques de référence et des calculs éléments finis comme ABAQUS. L’ensemble de ces tests a montré un bon comportement de la méthode (environs 0.0001% d’erreurs par rapport à la solution exacte). L’approche est étendue pour l’étude des poutres minces et pour l’analyse dynamique et stabilité. / This work concerns the development of a new numerical method entitled “Meshless Local Petrov- Galerkin (MLPG) combined with the Finite Volumes Method (FVM)” applied to the structural analysis. It is based on the resolution of the weak form of the partial differential equations by a method of Petrov Galerkin as in finite elements, but the approximation of the field of displacement introduced into the weak form does not require grid. The displacements and strains are given with the various nodes by local interpolation by using moving least squares (MLS). The values of the nodal strains are expressed in terms of interpolated nodal values independently of displacements, by simply imposing the strain displacement relationship directly by collocation at the nodal points. The procedure of calculation for this method is implemented in a computer code developed in MATLAB. The developed code was validated on a certain number of test cases by comparison with analytical solutions and finite elements results like ABAQUS. The whole of these tests showed a good behaviour of the method (about 0.0001% of errors in compared to the exact solution). The approach is also extended for the study of the thin beams and the dynamic analysis and stability.

O método SUPG (\"Streamline Upwind Petrov-Galerkin\") é analisado quanto a sua capacidade de estabilizar oscilações numéricas decorrentes de escoamentos convectivo-difusivos, e de manter a consistência nos resultados. Para esta finalidade, é elaborado um programa computacional como uma implementação algorítmica do método, e simulado o escoamento sobre um cilindro fixo a diferentes números de Reynolds. Ao final, é feita uma revelação sobre a solidez do método. Palavras-chave: escoamento, simulação numérica, método Petrov- Galerkin. / The \"Streamline Upwind Petrov-Galerkin\" method (SUPG) is analyzed with regard to its capability to stabilize numerical oscillations caused by convective-diffusive flows, and to maintain consistency in the results. To this aim, a computational program is elaborated as an algorithmic implementation of the method, and simulated the flow around a fixed cylinder at different Reynolds numbers. At the end, a revelation is made on the method\'s robustness. Keywords: flow, numerical simulation, Petrov-Galerkin method.

O método SUPG (\"Streamline Upwind Petrov-Galerkin\") é analisado quanto a sua capacidade de estabilizar oscilações numéricas decorrentes de escoamentos convectivo-difusivos, e de manter a consistência nos resultados. Para esta finalidade, é elaborado um programa computacional como uma implementação algorítmica do método, e simulado o escoamento sobre um cilindro fixo a diferentes números de Reynolds. Ao final, é feita uma revelação sobre a solidez do método. Palavras-chave: escoamento, simulação numérica, método Petrov- Galerkin. / The \"Streamline Upwind Petrov-Galerkin\" method (SUPG) is analyzed with regard to its capability to stabilize numerical oscillations caused by convective-diffusive flows, and to maintain consistency in the results. To this aim, a computational program is elaborated as an algorithmic implementation of the method, and simulated the flow around a fixed cylinder at different Reynolds numbers. At the end, a revelation is made on the method\'s robustness. Keywords: flow, numerical simulation, Petrov-Galerkin method.

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 .

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

Engineering problems frequently require solving a sequence of dual linear systems. This paper introduces recycling BiCG, that recycles the Krylov subspace from one pair of linear systems to the next pair. Augmented bi-Lanczos algorithm and modified two-term recurrence are developed for using the recycle space. Recycle space is built from the approximate invariant subspace corresponding to eigenvalues close to the origin. Recycling approach is extended to the CGS and the BiCGSTAB algorithms. Experiments on a convection-diffusion problem give promising results. / Master of Science

Dans cette these, nous etudions les schemas boite. Ils ont ete introduits par H.B. Keller en 1971. Dans un premier temps, on s'est interesse a des problemes elliptiques de type Poisson. Plusieurs schemas boite pour des domaines de $\mathbb(R)^2$ mailles par des triangles ou des rectangles ont ete introduits. Dans ce cas, la discretisation s'effectue sur la forme mixte du probleme en prenant la moyenne des deux equations (conservation et flux) sur les cellules du maillage. La methode peut etre qualifiee de ``methode volumes finis mixte de type Petrov-Galerkin ``. Une des difficultes du design de cette famille de schemas reside dans le choix des differents espaces de fonctions (approximation et test) qui doivent satisfaire des conditions de compatibilite de type Babuska-Brezzi. En revanche, cette methode de discretisation ne necessite qu'un seul maillage (le maillage du domaine). De plus, on montre dans la plupart des cas que le schema obtenu est equivalent a un probleme découplé : la résolution d'un probleme variationnel pour l'inconnue principale et une formule locale pour le gradient (le flux). Cette formulation facilite le calcul des inconnues discretes. Des resultats de stabilite et les calculs d'erreurs reposant sur la theorie des elements finis ont ete etablis. Une etude numérique valide ces resultats pour quelques cas tests. Dans le cadre du Groupement de Recherche MoMaS pour le stockage des dechets nucleaires dans la Meuse, j'ai ensuite etudie des problemes de convection-diffusion instationnaires. Un schéma boite permettant d'approcher ces equations dans le cas monodimensionnel a ete introduit. Des coefficients de decentrement propres a chaque maille permettent de controler le schema (precision, stabilite). Afin de generaliser rapidement ce schema au cas bidimensionnel, je me suis concentree sur une extension du schema boite monodimensionnel par la methode ADI (Alternating Direction Implicit).

Coercivity is a characteristic property of the bilinear term in a weak form of a partial differential equation in both infinite space and the corresponding finite space utilized by a numerical scheme. This concept implies \textit{stability} and \textit{well-posedness} of the weak form in both the exact solution and the numerical solution. In fact, the loss of this property especially in finite dimension cases leads to instability of the numerical scheme. This phenomenon occurs in three major families of problems consisting of advection-diffusion equation with dominant advection term, elastic analysis of very thin beams, and associated plasticity and non-associated plasticity problems. There are two main paths to overcome the loss of coercivity, first manipulating and stabilizing a weak form to ensure that the discrete weak form is coercive, second using an automatically stable method to estimate the solution space such as the Discontinuous Petrov Galerkin (DPG) method in which the optimal test space is attained during the design of the method in such a way that the scheme keeps the coercivity inherently. In this dissertation, A stable numerical method for the aforementioned problems is proposed. A stabilized finite element method for the problem of migration risk problem which belongs to the family of the advection-diffusion problems is designed and thoroughly analyzed. Moreover, DPG method is exploited for a wide range of valuing option problems under the black-Scholes model including vanilla options, American options, Asian options, double knock barrier options where they all belong to family of advection-diffusion problem, and elastic analysis of Timoshenko beam theory. Besides, The problem of American option pricing, migration risk, and plasticity problems can be categorized as a free boundary value problem which has their extra complexity, and optimization theory and variational inequality are the main tools to study these families of the problems. Thus, an overview of the classic definition of variational inequalities and different tools and methods to study analytically and numerically this family of problems is provided and a novel adjoint sensitivity analysis of variational inequalities is proposed.