Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.

Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.

The goal of the present doctoral research is to create a theoretical framework and develop a numerical implementation for a shell finite element that can potentially achieve higher performance (i.e. combination of speed and accuracy) than current Continuum-based (CB) shell finite elements (FE), in particular in applications related to soft biological tissue dynamics. Specifically, this means complex and irregular geometries, large distortions and large bending deformations, and anisotropic incompressible hyperelastic material properties. The critical review of the underlying theories, formulations, and capabilities of the existing CB shell FE revealed that a general nonlinear CB shell FE with the abovementioned capabilities needs to be developed. Herein, we propose the theoretical framework of a new such CB shell FE for dynamic analysis using the total and the incremental updated Lagrangian (UL) formulations and explicit time integration. Specifically, we introduce the geometry and the kinematics of the proposed CB shell FE, as well as the matrices and constitutive relations which need to be evaluated for the total and the incremental UL formulations of the dynamic equilibrium equation. To verify the accuracy and efficiency of the proposed CB shell element, its large bending and distortion capabilities, as well as the accuracy of three different techniques presented for large strain analysis, we implemented the element in Matlab and tested its application in various geometries, with different material properties and loading conditions. The new high performance and accuracy element is shown to be insensitive to shear and membrane locking, and to initially irregular elements.

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

La Simulation aux Grandes Echelles (SGE) est de plus en plus utilisée dans les processus de développement et la conception des réacteurs aéronautiques industriels. L'une des raisons pour ce besoin résulte dans la capacité de la SGE à fournir des informations instantanées d'un écoulement turbulent augmentant la quantité des prédictions de la composition des gaz d'échappement. Ce manuscrit de thèse aborde deux sujets récurrents de la SGE. D'une part, les schémas numériques pour la SGE nécessitent certaines propriétés, notamment une précision élevée avec une diffusivité faible pour ne pas nuire aux modèles de turbulence. Afin de répondre à ce pré requis, une famille de schémas d'intégration temporelle d'ordre élevée est proposée, permettant de modifier la diffusion numérique du schéma. D'autre part, la SGE étant intrinsèquement instationnaire, elle est très consommatrice en temps CPU. De plus, une géométrie complexe prend beaucoup de temps de simulation même avec les super calculateurs d'aujourd'hui. Dans le cas particulier d'intérêt et souvent rencontré dans les applications industrielles, l'approche bas-Mach est constitue une alternative intéressante permettant de réduire le coût et le temps de retour d'une simulation LES. L'impact et la comparaison des formalismes compressible et incompressible sont toutefois rarement quantifiés, ce qui est proposé dans ce travail pour une configuration représentative d'un brûleur swirlé industriel mesuré au CORIA / Large-Eddy Simulation (LES) becomes a more and more demanded tool to improve the design of aero-engines. The main reason for this request stems from the constraints imposed on the next generation low-emission engines at the industrial development level and the ability for LES to provide information on the instantaneous turbulent flow field which greatly contributes to improving the prediction of mixing and combustion thereby offering an improved prediction of the exhaust emission. The work presented in this thesis discusses two recurring issues of LES. For one, numerical schemes for LES require certain properties, i.e. low-diffusion schemes of high order of accuracy so as not to interfere with the turbulence models. To meet this purpose in the context of fully unstructured solvers, a new family of high-order time-integration schemes is proposed. With this class of schemes, the diffusion implied by the numerical scheme become adjustable and built-in. Second, since fully unsteady by nature, LES is very consuming in terms of CPU time. Even with today's supercomputers complex problems require long simulation times. Due to the low flow velocities often occurring in industrial applications, the use of a low-Mach number solver seems suitable and can lead to large reductions in CPU time if comparable to fully compressible solvers. The impact of the incompressibility assumption and the different nature of the numerical algorithms are rarely discussed. To partly answer the question, detailed comparisons are proposed for an experimental swirled configuration representative of a real burner that is simulated by LES using a fully explicit compressible solver and an incompressible solution developed at CORIA

Les structures composites sont souvent exposées à des ambiances dynamiques plus oumoins sévères. Ces vibrations peuvent développer différentes formes d’endommagement(fracture des fibres, délamination, fissuration de la matrice. . . ). Les défauts locaux sepropagent et affectent les propriétés mécaniques de la structure modifiant ainsi soncomportement dynamique global. Ces changements peuvent induire une dégradationrapide de la structure et une réduction de sa durée de vie. La thèse a pour objectif lamise en oeuvre de modèles de comportement pour le dimensionnement de structurescomplexes intégrant des sous-structures composites susceptibles d’être endommagées.La méthode des éléments finis est utilisée pour modéliser le comportement vibratoirelinéaire et non-linéaire de ces structures et l’endommagement est introduit via un modèlebilatéral, dans un premier temps. Durant le processus de résolution, une des difficultésrencontrées est le coût de calcul très élevé. Ainsi, un méta-modèle a été développé basésur les réseaux de neurones artificiels couplé avec la méthode de condensation par sousstructurationde Craig-Bampton. Les réseaux de neurones artificiels permettent d’estimer,à moindre cout numérique, le niveau d’endommagement sans avoir recours au calculexact. Le modèle d’endommagement bilatéral n’est pas adapté au cas de chargementsalternés ou périodiques. Par conséquent, la deuxième partie de la thèse est orientée versle développement d’un modèle d’endommagement unilatéral qui donne une meilleuredescription du comportement mécanique lorsque les micro-fissures sont fermées. De plus,dans plusieurs applications industrielles, les structures composites utilisées sont de faibleépaisseur. Par conséquent, elles peuvent avoir naturellement un comportement vibratoirenon-linéaire de type grands déplacements. Le modèle de comportement dynamique engrands déplacements et en présence de la non-linéarité matérielle d’endommagement estdéveloppé et validé. A l’issue de ces travaux de thèse, un outil numérique implémentésur MATLAB® a été développé intégrant deux modèles d’endommagement, bilatéralet unilatéral et une méta-modélisation permettant la localisation et l’estimation del’endommagement ainsi que la prédiction de la réponse dynamique des structures composites, totalement ou localement, endommagées. Le méta-modèle proposé permet deréduire significativement le coût de calcul tout en assurant une bonne précision en termesde localisation et d’estimation du niveau d’endommagement. Cet outil peut s’avérer utilepour diverses applications dans le domaine de surveillance de l’état de santé des structurescomposites. / Composite structures are often exposed to more or less severe dynamic perturbations.These vibrations can develop different forms of damage (fiber fracture, delamination,cracking of the matrix, etc.). Local defects propagate and affect the mechanical propertiesof the structure resulting to modify its global dynamic behavior. These changes can leadto the degradation of the structure and the reduction in its lifetime. This thesis focuseson the implementation of behavior models for the dimensioning of complex structuresintegrating damaged composite sub-structures. The finite element method is used tomodel the linear and nonlinear vibration behavior of these structures where the damageis introduced, initially, via a bilateral model. Since the high computational costs duringthe solving process, a meta-model was developed based on artificial neural networkscoupled with the condensation method of Craig-Bampton. Artificial neural networkspermit to estimate the damage severity at a lower numerical cost without resorting toexact calculation. The bilateral damage model is not adapted to the case of periodic loads.Consequently, the second part of the thesis is oriented towards the development of aunilateral damage model which gives a better description of the mechanical behaviorwhen the micro-cracks are closed. Moreover, in several industrial applications, the usedcomposite structures have small thickness. Therefore, they can naturally have a geometricnon-linear dynamic behavior. The model of dynamic behavior in large displacements andin the presence of material non-linearity of damage is developed and validated. At theend of this thesis, a numerical tool implemented on MATLAB® software was developedintegrating two models of damage, bilateral and unilateral, and a meta-modeling allowingthe localization and the estimation of the damage as well as the prediction of the linear andnon-linear dynamic responses of composite structures, totally or locally, damaged. Theproposed meta-model reduces significantly the computational cost and ensuring a goodaccuracy in terms of localization and estimation of the damage severity. Thereby, thistool can be useful in life-time estimation and monitoring strategies of composite structures.Thèse de

The present work deals with the numerical solution of dynamic contact and fracture problems. The contact problem is a Signorini problem with or without Coulomb friction. The fracture problem uses a cohesive zone model with a prescribed crack path. These problems are characterized by a non-regular boundary condition and can be formulated with evolutionary variational inequations or differential inclusions. For the numerical solution, we combine, as usual in solid dynamics, a finite element discretization in space and time-integration schemes. For the contact problem, we begin by comparing the main methods proposed in the literature. We then focus on the so-called modified mass method recently introduced by H. Khenous, P. Laborde et Y. Renard, for which we propose a semi-explicit variant. In addition, we prove a convergence result of the space semi-discrete solutions to a continuous solution in the frictionless viscoelastic case. We also analyze the space semi-discrete and fully discrete problems in the friction Coulomb case. For the dynamic fracture problem, using a fully explicit scheme is impossible or not robust enough. Therefore, we propose time-integration schemes where the boundary condition is treated in an implicit way. Finally, we present and analyze augmented Lagrangian methods for static fracture problems

Dans cette thèse, nous étudions et développons différentes familles de schémas d’intégration en temps pour les EDO linéaires. Dans la première partie, après avoir introduit les définitions et propriétés utilisées pour construire les schémas en temps, nous présentons deux méthodes de discrétisation en espace et une revue des schémas de Runge-Kutta (RK) qui sont couramment utilisés dans la littérature. Dans la seconde partie on présente une méthodologie pour construire deux familles de schémas A-stable pour un ordre quelcomque. Puis on fournit des schémas explicites, construits en maximisant leur nombre CFL pour un profil de spectre donné. Ces schémas explicites sont ensuite combinés aux schémas implicites A-stable, pour construire des schémas localement implicites que nous décrivons. En plus des tests de validations des schémas pour des problèmes en dimension un et deux de l’espace, nous présentons des résultats numériques obtenus en résolvant des problèmes de propagation d’ondes acoustiques et électromagnétiques en dimensions trois dans la troisième partie. / In this thesis, we study and develop different families of time integration schemes for linear ODEs. After presenting the space discretisation methods and a review of classical Runge-Kutta schemes in the first part, we construct high-order A-stable time integration schemes for an arbitrary order with low-dissipation and low-dispersion effects in the second part. Then we develop explicit schemes with an optimal CFL number for a typical profile of spectrum. The obtained CFL number and the efficiency on the typical profile for each explicit scheme are given. Pursuing our aim, we propose a methodology to construct locally implicit methods of arbitrary order. We present the locally implicit methods obtained from the combination of the A-stable implicit schemes we have developed and explicit schemes with optimal CFL number. We use them to solve the acoustic wave equation and provide convergence curves demonstrating the performance of the obtained schemes. In addition of the different 1D and 2D validation tests performed while solving the acoustic wave equation, we present numerical simulation results for 3D acoustic wave and the Maxwell’s equations in the last part.

Doctor Scientiae - DSc / Robust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods.

<div>Uncertainty quantification techniques based on the spectral approach have been studied extensively in the literature to characterize and quantify, at low computational cost, the impact that uncertainties may have on large-scale engineering problems. One such technique is the <i>generalized polynomial chaos</i> (gPC) which utilizes a time-independent orthogonal basis to expand a stochastic process in the space of random functions. The method uses a specific Askey-chaos system that is concordant with the measure defined in the probability space in order to ensure exponential convergence to the solution. For nearly two decades, this technique has been used widely by several researchers in the area of uncertainty quantification to solve stochastic problems using the spectral approach. However, a major drawback of the gPC method is that it cannot be used in the resolution of problems that feature strong nonlinear dependencies over the probability space as time progresses. Such downside arises due to the time-independent nature of the random basis, which has the undesirable property to lose unavoidably its optimality as soon as the probability distribution of the system's state starts to evolve dynamically in time.</div><div><br></div><div>Another technique is the <i>time-dependent generalized polynomial chaos</i> (TD-gPC) which utilizes a time-dependent orthogonal basis to better represent the stochastic part of the solution space (aka random function space or RFS) in time. The development of this technique was motivated by the fact that the probability distribution of the solution changes with time, which in turn requires that the random basis is frequently updated during the simulation to ensure that the mean-square error is kept orthogonal to the discretized RFS. Though this technique works well for problems that feature strong nonlinear dependencies over the probability space, the TD-gPC method possesses a serious issue: it suffers from the curse of dimensionality at the RFS level. This is because in all gPC-based methods the RFS is constructed using a tensor product of vector spaces with each of these representing a single RFS over one of the dimensions of the probability space. As a result, the higher the dimensionality of the probability space, the more vector spaces needed in the construction of a suitable RFS. To reduce the dimensionality of the RFS (and thus, its associated computational cost), gPC-based methods require the use of versatile sparse tensor products within their numerical schemes to alleviate to some extent the curse of dimensionality at the RFS level. Therefore, this curse of dimensionality in the TD-gPC method alludes to the need of developing a more compelling spectral method that can quantify uncertainties in long-time response of dynamical systems at much lower computational cost.</div><div><br></div><div>In this work, a novel numerical method based on the spectral approach is proposed to resolve the curse-of-dimensionality issue mentioned above. The method has been called the <i>flow-driven spectral chaos</i> (FSC) because it uses a novel concept called <i>enriched stochastic flow maps</i> to track the evolution of a finite-dimensional RFS efficiently in time. The enriched stochastic flow map does not only push the system's state forward in time (as would a traditional stochastic flow map) but also its first few time derivatives. The push is performed this way to allow the random basis to be constructed using the system's enriched state as a germ during the simulation and so as to guarantee exponential convergence to the solution. It is worth noting that this exponential convergence is achieved in the FSC method by using only a few number of random basis vectors, even when the dimensionality of the probability space is considerably high. This is for two reasons: (1) the cardinality of the random basis does not depend upon the dimensionality of the probability space, and (2) the cardinality is bounded from above by <i>M+n+1</i>, where <i>M</i> is the order of the stochastic flow map and <i>n</i> is the order of the governing stochastic ODE. The boundedness of the random basis from above is what makes the FSC method be curse-of-dimensionality free at the RFS level. For instance, for a dynamical system that is governed by a second-order stochastic ODE (<i>n=2</i>) and driven by a stochastic flow map of fourth-order (<i>M=4</i>), the maximum number of random basis vectors to consider within the FSC scheme is just 7, independent whether the dimensionality of the probability space is as low as 1 or as high as 10,000.</div><div><br></div><div>With the aim of reducing the complexity of the presentation, this dissertation includes three levels of abstraction for the FSC method, namely: a <i>specialized version</i> of the FSC method for dealing with structural dynamical systems subjected to uncertainties (Chapter 2), a <i>generalized version</i> of the FSC method for dealing with dynamical systems governed by (nonlinear) stochastic ODEs of arbitrary order (Chapter 3), and a <i>multi-element version</i> of the FSC method for dealing with dynamical systems that exhibit discontinuities over the probability space (Chapter 4). This dissertation also includes an implementation of the FSC method to address the dynamics of large-scale stochastic structural systems more effectively (Chapter 5). The implementation is done via a modal decomposition of the spatial function space as a means to reduce the number of degrees of freedom in the system substantially, and thus, save computational runtime.</div>