Objectifs de la thèse : l’analyse à rupture de structure de type solides et membranes et la modélisation de la rupture quasi-fragile par la méthode des éléments finis à forte discontinuité en cas de solide 2D. Dans ce travail, la méthode de continuation avec une équation de contrainte quadratique est présentée sous sa forme linéarisée. En présence de ruptures locales, la méthode de continuation standard peut échouer. Afin d’améliorer la performance de cette méthode, nous proposons de nouvelles versions plus sophistiquées qui prennent en compte les ruptures locales des matériaux. D’une part, une version est basée sur une équation supplémentaire adaptative imposant une limitation. Cette version est considérée relativement satisfaisante pour les matériaux adoucissants. D’autres versions sont basées sur le contrôle de la dissipation incrémentale pour les matériaux inélastiques. Plusieurs formulations d’éléments finis à forte discontinuité sont présentées en détails pour l’analyse de rupture quasi-fragile. Les approximations discrètes du champ de déplacement sont basées sur des éléments quadrilatéraux isoparamétriques ou des éléments quadrilatéraux enrichis par la méthode des modes incompatibles. Afin de décrire la formation d’une fissure ainsi que son ouverture, la cinématique de l’élément est enrichie par quatre modes de séparation et des paramètres cinématiques. On a également proposé un nouvel algorithme de repérage de fissure pour l’évaluation de la propagation de la fissure à travers le maillage. Plusieurs exemples numériques sont réalisés afin de montrer la performance de l’élément quadrilatéral à forte discontinuité ainsi que l’algorithme de repérage de fissure proposé. / The thesis studies: the methods for failure analysis of solids and structures, and the embedded strong discontinuity finite elements for modelling material failures in quasi brittle 2d solids. As for the failure analysis, the consistently linearized path-following method with quadratic constraint equation is first presented and studied in detail. The derived path-following method can be applied in the nonlinear finite element analysis of solids and structures in order to compute a highly nonlinear solution path. However, when analysing the nonlinear problems with the localized material failures (i.e. materialsoftening), standard path-following methods can fail. For this reason we derived new versions of the pathfollowing method, with other constraint functions, more suited for problems that take into account localized material failures. One version is based on adaptive one-degree-of-freedom constraint equation, which proved to be relatively successful in analysing problems with the material softening that are modelled by the embedded-discontinuity finite elements. The other versions are based on controlling incremental plastic dissipation or plastic work in an inelastic structure. The dissipation due to crack opening and propagation, computed by e.g. embedded discontinuity finite elements, is taken into account. The advantages and disadvantages of the presented path-following methods with different constraint equations are discussed and illustrated on a set of numerical examples. As for the modelling material failures in quasi brittle 2d solids (e.g. concrete), several embedded strong discontinuity finite element formulations are derived and studied. The considered formulations are based either on: (a) classical displacement-based isoparametric quadrilateral finite element or (b) on quadrilateral finite element enhanced with incompatible displacements. In order to describe a crack formation and opening, the element kinematics is enhanced by four basic separation modes and related kinematic parameters. The interpolation functions that describe enhanced kinematics have a jump in displacements along the crack. Two possibilities were studied for deriving the operators in the local equilibrium equations that are responsible for relating the bulk stresses with the tractions in the crack. For the crack embedment, the major-principle-stress criterion was used, which is suitable for the quasi brittle materials. The normal and tangential cohesion tractions in the crack are described by two uncoupled, nonassociative damage-softening constitutive relations. A new crack tracing algorithm is proposed for computation of crack propagation through the mesh. It allows for crack formation in several elements in a single solution increment. Results of a set of numerical examples are provided in order to assess the performance of derived embedded strong discontinuity quadrilateral finite element formulations, the crack tracing algorithm, and the solution methods. / Doktorska disertacija obravnava: (i) metode za porušno analizo trdnih teles in konstrukcij, ter (ii) končne elemente z vgrajeno močno nezveznostjo za modeliranje materialne porušitve v kvazi krhkih 2d trdnih telesih. Za porušno analizo smo najprej preučili konsistentno linearizirano metodo sledenja ravnotežne poti skvadratno vezno enačbo (metoda krožnega loka). Metoda omogoča izračun analize nelinearnih modelov, ki imajo izrazito nelinearno ravnotežno pot. Kljub temu standardne metode sledenja poti lahko odpovedo,kadar analiziramo nelinearne probleme z lokalizirano materialno porušitvijo (mehčanje materiala). Zatosmo izpeljali nove različice metode sledenja poti z drugimi veznimi enačbami, ki so bolj primerne zaprobleme z lokalizirano porušitvijo materiala. Ena različica temelji na adaptivni vezni enačbi, pri katerivodimo izbrano prostostno stopnjo. Izkazalo se je, da je metoda relativno uspešna pri analizi problemov zmaterialnim mehčanjem, ki so modelirani s končnimi elementi z vgrajeno nezveznostjo. Druge različicetemeljijo na kontroli plastične disipacije ali plastičnega dela v neelastičnem trdnem telesu ali konstrukciji.Upoštevana je tudi disipacija zaradi širjenja razpok v elementih z vgrajeno nezveznostjo. Prednosti inslabosti predstavljenih metod sledenja ravnotežnih poti z različnimi veznimi enačbami so predstavljeni naštevilnih numeričnih primerih. Za modeliranje porušitve materiala v kvazi krhkih 2d trdnih telesih (npr. betonskih) smo izpeljali različne formulacije končnih elementov z vgrajeno močno nezveznostjo v pomikih. Obravnavane formulacije temeljijo bodisi (a) na klasičnem izoparametričnem štirikotnem končnem elementu bodisi (b) na štirikotnem končnem elementu, ki je izboljšan z nekompatibilnimi oblikami za pomike. Nastanek in širjenje razpoke opišemo tako, da kinematiko v elementu dopolnimo s štirimi osnovnimi oblikami širjenja razpoke in pripadajočimi kinematičnimi parametri. Interpolacijske funkcije, ki opisujejo izboljšano kinematiko, zajemajo skoke v pomikih vzdolž razpoke. Obravnavali smo dva načina izpeljave operatorjev, ki nastopajo v lokalni ravnotežni enačbi in povezujejo napetosti v končnem elementu z napetostmi na vgrajeni nezveznosti. Kriterij za vstavitev nezveznosti (razpoke) temelji na kriteriju največje glavne napetosti in je primeren za krhke materiale. Normalne in tangentne kohezijske napetosti v razpoki opišemo z dvema nepovezanima, poškodbenima konstitutivnima zakonoma za mehčanje. Predlagamo novi algoritem za sledenje razpoki za izračun širjenja razpoke v mreži končnih elementov. Algoritem omogoča formacijo razpok v več končnih elementih v enem obtežnem koraku. Izračunali smo številne numerične primere, da bi ocenili delovanje izpeljanih formulacij štirikotnih končnih elementov z vgrajeno nezveznostjo in algoritma za sledenje razpoki kot tudi delovanje metod sledenja ravnotežnih poti.

The Support Vector Machine (SVM) is a popular binary classification model due to its superior generalization performance, relative ease-of-use, and applicability of kernel methods. SVM training entails solving an associated quadratic programming (QP) that presents significant challenges in terms of speed and memory constraints for very large datasets; therefore, research on numerical optimization techniques tailored to SVM training is vast. Slow training times are especially of concern when one considers that re-training is often necessary at several values of the models regularization parameter, C, as well as associated kernel parameters. The active set method is suitable for solving SVM problem and is in general ideal when the Hessian is dense and the solution is sparse–the case for the `1-loss SVM formulation. There has recently been renewed interest in the active set method as a technique for exploring the entire SVM regularization path, which has been shown to solve the SVM solution at all points along the regularization path (all values of C) in not much more time than it takes, on average, to perform training at a single value of C with traditional methods. Unfortunately, the majority of active set implementations used for SVM training require positive definite kernels, and those implementations that do allow semi-definite kernels tend to be complex and can exhibit instability and, worse, lack of convergence. This severely limits applicability since it precludes the use of the linear kernel, can be an issue when duplicate data points exist, and doesn’t allow use of low-rank kernel approximations to improve tractability for large datasets. The difficulty, in the case of a semi-definite kernel, arises when a particular active set results in a singular KKT matrix (or the equality-constrained problem formed using the active set is semidefinite). Typically this is handled by explicitly detecting the rank of the KKT matrix. Unfortunately, this adds significant complexity to the implementation; and, if care is not taken, numerical iii instability, or worse, failure to converge can result. This research shows that the singular KKT system can be avoided altogether with simple modifications to the active set method. The result is a practical, easy to implement active set method that does not need to explicitly detect the rank of the KKT matrix nor modify factorization or solution methods based upon the rank. Methods are given for both conventional SVM training as well as for computing the regularization path that are simple and numerically stable. First, an efficient revised simplex method is efficiently implemented for SVM training (SVM-RSQP) with semi-definite kernels and shown to out-perform competing active set implementations for SVM training in terms of training time as well as shown to perform on-par with state-of-the-art SVM training algorithms such as SMO and SVMLight. Next, a new regularization path-following algorithm for semi-definite kernels (Simple SVMPath) is shown to be orders of magnitude faster, more accurate, and significantly less complex than competing methods and does not require the use of external solvers. Theoretical analysis reveals new insights into the nature of the path-following algorithms. Finally, a method is given for computing the approximate regularization path and approximate kernel path using the warm-start capability of the proposed revised simplex method (SVM-RSQP) and shown to provide significant, orders of magnitude, speed-ups relative to the traditional grid search where re-training is performed at each parameter value. Surprisingly, it also shown that even when the solution for the entire path is not desired, computing the approximate path can be seen as a speed-up mechanism for obtaining the solution at a single value. New insights are given concerning the limiting behaviors of the regularization and kernel path as well as the use of low-rank kernel approximations.