Meshless methods provide a highly continuous approximation field, convenient for thin structures like shells. Nevertheless, the lack of Kronecker Delta property makes the formulation of essential boundary conditions not straightforward, as the trial and test fields cannot be tailored to boundary values. Similar problem arise when different approximation regions must be joined, in a multi-region problem, such as kinks, folds or joints. This work presents three approaches to impose both kinematic conditions: the well known Lagrange Multiplier method, used since the beginning of the Element Free Galerkin method; a pure penalty approach; and the recently rediscovered alternative of Nitsche\'s Method. We use the EFG discretization technique for thick Reissner-Mindlin shells and adapt the weak form as to separate displacement and rotational degrees of freedom and obtain suitable and separate stabilization parameters. This approach enables the modeling of discontinuous shells and local refinement on multi-region problems. / Métodos sem malha geram campos de aproximação com alta continuidade, convenientes para estruturas finas como cascas. No entanto, a ausência da propriedade de Delta de Kronecker dificulta a formulação de condições essenciais de contorno, já que os campos de aproximação e teste não podem ser moldados aos valores de contorno. Um problema similar aparece quando diferentes regiões de aproximação precisam ser juntadas em um problema multi-regiões como dobras, vincos ou junções. Este trabalho apresenta três métodos de imposição ambas condições cinemáticas: o já conhecido método dos multiplicadores de Lagrange, usado desde o começo do método de Galekin sem elementos (EFG); uma abordagem de penalidade pura; e o recentemente redescoberto método de Nitsche. Nós usamos a técnica de discretização com EFG para cascas espessas de Reissner-Mindlin e adaptamos a forma fraca de forma a separar graus de liberdade de deslocamento e rotação e obter coeficientes de estabilização diferentes e apropriados. Essa abordagem permite a modelagem de cascas discontínuas e o refinamento local em problemas multi-regiões.

Avec l'avancé des méthodes numériques, de nouvelles méthodes dites « sans maillage » sont apparues pour remédier à certaines limitations de la méthode des éléments finis. Ces méthodes ont la particularité de n’employer aucun maillage prédéfini : elles utilisent un ensemble de nœuds dispersés dans le domaine considéré et sur ses frontières. L’objectif de cette étude est de montrer l’intérêt de l’application des méthodes sans maillage basées sur les fonctions de base radiale pour la simulation des procédés de mise en forme en général et de poinçonnage rapide en particulier. Une attention particulière sera portée sur la méthode RPIM qui offre l’avantage de proposer une interpolation nodale. La démarche proposée dans ce document consiste: à rappeler succinctement les principes essentiels des méthodes sans maillage en précisant leurs avantages par rapport aux méthodes classiques, à présenter et à mettre en œuvre la technique numérique de la méthode sans maillage RPIM avec un calibrage des paramètres caractéristiques, et enfin, à traiter des exemples numériques de procédés de mise en forme avec amorçage et propagation de fissure qui confirmeront ces avantages. / In recent years, new methods named Meshfree methods have been developed to surmount limitations of the finite element method. The main characteristic of these methods is to not employ any pre-defined mesh: they use a set of nodes scattered within the problem domain as well as sets of nodes scattered on the boundaries of the domain. A particular attention is paid to the RPIM method, which proposes a nodal interpolation. The followed steps are: a presentation of « Meshfree methods » and their advantages compared to the traditional methods, an introduction to the meshfree RPIM method with a calibration of its associated parameters, and finally, a discussion on results obtained with the RPIM in forming processes exhibiting initiation and propagation of a crack showing the interest of the approach.

Le procédé de soudage par friction et malaxage est un procédé récent qui a été développé au sein de l'institut de soudure britannique "The Welding Institute" au début des années 90. Ce procédé, utilisé généralement en aéronautique, est sans apport de matière et permet de souder principalement des alliages d'aluminium difficilement soudables par les procédés classiques de soudage. Il consiste à malaxer le matériau de base à l'aide d'un outil constitué d'un pion et d'un épaulement frottant sur les faces supérieures des tôles à souder. La modélisation de ce procédé est très complexe puisque ce dernier implique des couplages entre des phénomènes mécaniques, thermiques et métallurgiques. Le malaxage dans le procédé de soudage FSW est difficile à simuler à l'aide de la méthode des éléments finis (en lagrangien) puisque la zone proche de l'outil de soudage est le siège de grandes déformations. Donc le remaillage est nécessaire. Cependant, le remaillage est cher et très difficile pour les problèmes tridimensionnels. Par ailleurs, après un remaillage, il est nécessaire d'interpoler les champs (vitesses, contraintes,...) correspondant à la solution courante, ce qui peut introduire des erreurs supplémentaires dans le calcul (on parle de diffusion numérique). Nous proposons dans ce travail des modèles basés sur la méthode sans maillage dite "Smoothed Particle Hydrodynamics SPH" et la méthode des moindres carrés mobiles (Moving Least Square MLS) pour la simulation de ce procédé. Ces modèles sont formulés dans le cadre lagrangien et utilisent la forme forte des équations aux dérivées partielles. Le premier modèle basé sur SPH considère la zone de soudure comme un fluide non newtonien faiblement compressible et dont la viscosité dépend de la température. Ce modèle est proposé pour la simulation numérique du comportement thermomécanique d'un matériau soudé par le procédé FSW. Dans le deuxième modèle, un algorithme itératif implicite de premier ordre a été proposé, pour simuler le malaxage de la matière dans le cas d'un matériau viscoplastique, en utilisant la méthode MLS et la technique de collocation. Le troisième modèle est un algorithme implicite d'ordre élevé basée sur le couplage de la méthode MLS et la Méthode Asymptotique Numérique MAN. Cet algorithme permet de réduire le temps de calcul par rapport à l'algorithme itératif implicite de premier ordre. La validation de ces trois modèles proposés a été faite par le code industriel Fluent / Friction stir welding is a recent process that has been developed by the British Welding Institute TWI "The Welding Institute" since 1990s. This process, generally used in aerospace, does not need additional material and allows mainly joining plates of aluminum alloys which are difficult to weld by the classical welding processes. It consists in mixing the base material using a tool comprising a pin and a shoulder which heats the plates to be welded by friction. The modeling of this process is very complex since it involves the coupling between mechanical, thermal and metallurgical phenomena. The mixing in welding process FSW is difficult to simulate using finite element method in lagrangian framework since the area near the welding tool is submitted to large deformations. So remeshing procedure is often required. However, remeshing can be very expensive and difficult to perform for three-dimensional problems. Moreover, after remeshing step, it is necessary to interpolate the fields (velocities, constraints ...) corresponding to the current solution, which may lead to additional errors in the calculation (called numerical diffusion). We propose in this work models based on meshless methods called "Smoothed Particle Hydrodynamics SPH" and Moving Least Square method for the simulation of this welding process. These models are formulated in lagrangian framework and use the strong form of partial differential equations. The first model based on SPH considers the welding zone as a weakly compressible non-newtonian fluid and whose viscosity depends on the temperature. This model is proposed for the numerical simulation of thermo-mechanical behavior of a welded material by FSW process. The second model is a first order implicit iterative algorithm proposed to simulate material mixing in the case a visco-plastic behavior using the MLS method and the collocation technique. The third model is a high order implicit algorithm based on the coupling of MLS method and Asymptotic Numerical Method (ANM). This algorithm allows reducing the computation time by comparison with the first order implicit iterative algorithm. The validation of these three proposed models was done by the industrial code Fluent

Numerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however, has begun to outstrip efforts to reduce analysis overhead. As such, most commercially available software packages are now limited by the human effort required to prepare, develop, and initialize the necessary computational models. Primarily due to the mesh-based analysis methods utilized in these software packages, the dependence on model preparation greatly limits the accessibility of these analysis tools. In response, the so-called meshless or mesh-free methods have seen considerable interest as they promise to greatly reduce the necessary human interaction during model setup. However, despite the success of these methods in areas demanding high degrees of model adaptability (such as crack growth, multi-phase flow, and solid friction), meshless methods have yet to gain notoriety as a viable alternative to more traditional solution approaches in general solution domains. Although this may be due (at least in part) to the relative youth of the techniques, another potential cause is the lack of focus on developing robust methodologies. The failure to approach development from a practical perspective has prevented researchers from obtaining commercially relevant meshless methodologies which reach the full potential of the approach. The primary goal of this research is to present a novel meshless approach called MIMS (Model Integrated Meshless Solver) which establishes the method as a generalized solution technique capable of competing with more traditional PDE methodologies (such as the finite element and finite volume methods). This was accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input (small features, poorly formed faces, etc.) to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.

Composite materials and hybrid materials such as fiber-metal laminates, and functionally graded materials are increasingly common in application in aerospace structures. However, adhesive bonding of dissimilar materials makes these materials susceptible to delamination. The use of integrated Sinc methods for predicting interlaminar failure in laminated composites and hybrid material systems was examined. Because the Sinc methods first approximate the highest-order derivative in the governing equation, the in-plane derivatives of in-plane strain needed to obtain interlaminar stresses by integration of the equilibrium equations of 3D elasticity are known without post-processing. Interlaminar stresses obtained with the Sinc method based on Interpolation of Highest derivative were compared for the first-order and third-order shear deformable theories, the refined zigzag beam theory and the higher-order shear and normal deformable beam theory. The results indicate that the interlaminar stresses by the zigzag theory compare well with those obtained by a 3D finite element analysis, while the traditional equivalent single layer theories perform well for some laminates. The philosophy of the Sinc method based on Interpolation of Highest Derivative was extended to create a novel weak form based approach called the Integrated Local Petrov-Galerkin Sinc Method. The Integrated Local Petrov-Galerkin Sinc Method is easily utilized for boundary-value problem on non-rectangular domains as demonstrated for analysis of elastic and elastic-plastic plane-stress panels with elliptical notches. The numerical results showed excellent accuracy compared to similar results obtained with the finite element method. The Integrated Local Petrov-Galerkin Sinc Method was used to analyze interlaminar debonding of composite and fiber-metal laminated beams. A double-cantilever beam and a fixed-ratio mixed mode beam were analyzed using the Integrated Local Petrov-Galerkin Sinc Method and the results were shown to correlate well with those by the finite element method. An adaptive Sinc point distribution technique was implemented for the delamination analysis which significantly improved the methods accuracy for the present problem. Delamination of a GLARE, plane-strain specimen was also analyzed using the Integrated Local Petrov-Galerkin Sinc Method. The results correlate well with 2D, plane-strain analysis by the finite element method, including interlaminar stresses obtained by through-the-thickness integration of the equilibrium equations of 3D elasticity. / Ph. D.

A partial differential equation in space and time represents the physics of rotating beams. Mostly, the numerical solution of such an equation is an available option as analytical solutions are not feasible even for a uniform rotating beam. Although the numerical solutions can be obtained with a number of combinations (in space and time), one tries to seek for a better alternative. In this work, various numerical techniques are applied to the rotating beam problems: finite element method, meshless methods, and B-spline finite element methods. These methods are applied to the governing differential equations of a rotating Euler-Bernoulli beam, rotating Timoshenko beam, rotating Rayleigh beam, and cracked Euler-Bernoulli beam. This work provides some elegant alternatives to the solutions available in the literature, which are more efficient than the existing methods: the p-version of finite element in time for obtaining the time response of periodic ordinary differential equations governing helicopter rotor blade dynamics, the symmetric matrix formulation for a rotating Euler-Bernoulli beam free vibration problem using the Galerkin method, and solution for the Timoshenko beam governing differential equation for free vibration using the meshless methods. Also, the cracked Euler-Bernoulli beam free vibration problem is solved where the importance of higher order polynomial approximation is shown. Finally, the overall response of rotating blades subjected to aerodynamic forcing is obtained in uncoupled trim where the response is independent of the overall helicopter configuration. Stability analysis for the rotor blade in hover and forward flight is also performed using Floquet theory for periodic differential equations.

Avec l'augmentation de l'utilisation des mousses polyuréthanes dans l'industrie automobile et la complexité des formes des pièces, plusieurs défauts de production qui influent sur la qualité des produits finaux peuvent apparaître. L'utilisation des outils numériques pour la simulation du procédé de moussage du polyuréthane est une solution pour prédire le comportement de la mousse pendant le remplissage du moule et détecter les défauts à l'avance. Dans ce contexte, les travaux de cette thèse avaient pour objectif de modéliser et de simuler le procédé de moussage du polyuréthane. Des modèles prenant en compte les deux principales réactions chimiques de la formation du polyuréthane, l'effet exothermique de ces réactions ainsi que le couplage thermo-rhéo-cinétique caractérisant ce procédé ont été proposés. Ces modèles ont été implémentés dans le logiciel NOGRID-points basé sur une méthode sans maillage (FPM) qui donne plus de flexibilité en termes de simulation des écoulements à surface libre. Une technique d'identification inverse qui permet de minimiser l'écart entre les résultats numériques et les résultats expérimentaux obtenus suite à un travail expérimental de caractérisation a permis de déterminer les paramètres des modèles utilisés. Les résultats numériques ont été validés en réalisant des incomplets avec un sous-système de forme simple et un moule d'une pièce industrielle de forme complexe. Cette validation a consisté à comparer les positions des fronts de la mousse obtenues expérimentalement avec celles obtenues numériquement. Nous avons également proposé une prédiction de certaines caractéristiques acoustiques de la mousse en se basant sur les résultats de la simulation numérique du moussage et un modèle semi-phénoménologique. / With the increasing use of polyurethane foam in the automotive industry and the complexity of the shapes of the parts, several production defects that affect the quality of the final products may occur. The use of numerical simulation tools is a valuable method to control the mold filling during the foaming process and to detect defects at an early stage. In this context, this work aimed to model and to simulate the polyurethane foaming process. Models taking into account the two main chemical reactions of the formation of polyurethane, the exothermic effect of these reactions as well as the thermo-rheo-kinetic coupling characterizing this process have been proposed. These models have beenimplemented in the software NOGRID-points based on a meshless method (FPM) which gives more flexibility in terms of simulation of free surface flows. The parameters of the used models were identified by an inverse analysis method which minimizes the difference between the numerical and the experimental results obtained by an experimental characterization work. The numerical resultswere validated by carrying out a set of short shot foams using a panel mold cavity and an automotive underlay carpet cavity. The validation was to compare the flow front positions obtained experimentally with the numerical ones. We have also proposed a prediction of some acoustic foam characteristics based on the results of the numerical simulation of the foaming process and a semiphenomenological model.

Ο στόχος της παρούσας διδακτορικής διατριβής είναι διττός: η ανάπτυξη και η εφαρμογή αριθμητικών τεχνικών για την επίλυση προβλημάτων που εμπίπτουν στην περιοχή του Μη-Καταστροφικού Ελέγχου. Συγκεκριμένα αναπτύχθηκαν η Μέθοδος των Συνοριακών Στοιχείων (ΜΣΣ) και η Μέθοδος των Τοπικών Ολοκληρωτικών Εξισώσεων χωρίς Διακριτοποίηση για την αριθμητική ανάλυση στατικών και μεταβατικών προβλημάτων στο πεδίο της ελαστικότητας και της αλληλεπίδρασης ελαστικού με ακουστικό μέσο στις δύο διαστάσεις. Σημαντικό μέρος της διδακτορικής διατριβής αποτέλεσε η ανάπτυξη προγράμματος ηλεκτρονικού υπολογιστή, το οποίο επιλύει τα προβλήματα στα οποία πραγματεύεται το παρόν σύγγραμμα. Η διδακτορική διατριβή αποτελείται από τρεις ενότητες. Στην πρώτη ενότητα γίνεται πλήρης περιγραφή της απαραίτητης θεωρίας για την κάλυψη και κατανόηση των αριθμητικών ΜΣΣ αλλά και των Τοπικών Μεθόδων χωρίς Διακριτοποίηση (ΤΜχΔ). Στη δεύτερη ενότητα εφαρμόζονται οι προαναφερθείσες αριθμητικές μέθοδοι για την επίλυση στατικών και δυναμικών (στο πεδίο συχνοτήτων) διδιάστατων προβλημάτων, ώστε να πιστοποιηθεί η ακρίβεια και η αξιοπιστία των εν λόγω μεθοδολογιών. Τέλος, στην τρίτη ενότητα οι αριθμητικές ΜΣΣ και ΤΜχΔ εφαρμόζονται για την επίλυση προβλημάτων κυματικής διάδοσης που εμπίπτουν στο πεδίο του Μη-Καταστροφικού Ελέγχου. Πιο συγκεκριμένα μελετήθηκε η κυματική διάδοση σε ελεύθερες επίπεδες πλάκες και σε κυλινδρικές δεξαμενές αποθήκευσης υγρών καυσίμων. / The aim of this doctoral thesis is twofold: the development and implementation of numerical techniques for solving wave propagation problems in Non-Destructive Testing applications. Particularly, the Boundary Element Method (BEM) and the Local Boyndary Integral Equation Method are developed, so as to numerically solve static and transient problems on the field of elasticity and fluid-structure interaction in two dimensions. A major part of the present research is the construction of a computer program for solving such kind of problems. This textbook consists of three sections. In the first section, a thorough description on the theory of the BEM and the Local Meshless Methods (LMM) is done. The second section is dedicated for the numerical implementation of the BEM and LMM for solving steady state and time-harmonic two dimensional elastic and acoustic problems, in order to verify the accuracy and the ability of the proposed methodologies to solve the above-mentioned problems. Finally in the third section, the wave propagation problems of traction-free plates and cylindrical fuel storage tanks is studied, from the perspective of Non-Destructive Testing. The numerical methods of BEM and LMM are implemented, as well as spectral methods are utilized, for drawing useful conclusions on the wave propagation phenomena.