This is a documentation of the software package COCOS. The purpose of COCOS is the computation of corner singularities of elliptic equations in polyhedral corners and crack tips. COCOS provides a self-contained library for the generation of structured 2D finite element meshes, including various routines for mesh manipulation, as well as several algorithms for the solution of quadratic eigenvalue problems with Hamiltonian structure. These and further features will be described in this documentation.

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

The constant gamma in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for constructing of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.

This is a documentation of the software package COCOS. The purpose of COCOS is the computation of corner singularities of elliptic equations in polyhedral corners and crack tips. COCOS provides a self-contained library for the generation of structured 2D finite element meshes, including various routines for mesh manipulation, as well as several algorithms for the solution of quadratic eigenvalue problems with Hamiltonian structure. These and further features will be described in this documentation.

An elastic FGM layer bonded to a semi-infinite linear elastic, isotropic, homogeneous half plane is considered. The half plane contains periodic cracks perpendicular to the interface. Mechanical loading is applied through crack surface pressure, resulting in a mode I crack problem. The plane elasticity problem described above is formulated by using Fourier transforms and Fourier series. A singular integral equation is obtained for the auxiliary variable, namely derivative of the crack surface displacement. Solution is obtained, and stress intensity factors are calculated for various values of crack period, crack length, crack location, layer thickness and material gradation.

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

The constant gamma in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for constructing of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.

L’enrichissement des éléments finis standard est un outil performant pour améliorer la qualité d’approximation. L’idée principale de cette approche est d’ajouter aux fonctions de base un ensemble de fonctions censées améliorer la qualité des solutions approchées. Le choix de ces dernières est crucial et est en grande partie basé sur la connaissance a priori de quelques informations telles que les caractéristiques de la solution, de la géométrie du problème à résoudre, etc. L’efficacité de cette approche pour résoudre une équation aux dérivées partielles dans un maillage fixe, sans avoir recours au raffinement, a été prouvée dans de nombreuses applications dans la littérature. La clé de son succès repose principalement sur le bon choix des fonctions de base et plus particulièrement celui des fonctions d’enrichissement. Une question importante se pose alors : quelles conditions faut-il imposer sur les fonctions d’enrichissement afin qu’elles génèrent des éléments finis bien définis ?Dans cette thèse sont abordés différents aspects d’une approche générale d’enrichissement d’éléments finis. Notre première contribution porte principalement sur l’enrichissement de l’élément fini du type Q_1. Par contre, notre seconde contribution, certainement la plus importante, met l’accent sur une approche plus générale pour enrichir n’importe quel élément fini qu’il soit P_k, Q_k ou autres, conformes ou non conformes. Cette approche a conduit à l’obtention des versions enrichies de l’élément de Han, l’élément de Rannacher-Turek et l’élément de Wilson, qui font maintenant partie des codes d’éléments finis les plus couramment utilisés en milieu industriel. Pour établir ces extensions, nous avons eu recours à l’élaboration de nouvelles formules de quadrature multidimensionnelles appropriées généralisant les formules classiques bien connues en dimension 1, dites du “point milieu,” des “trapèzes” et de leurs versions perturbées, ainsi que la formule de Simpson. Elles peuvent être vues comme des extensions naturelles de ces formules en dimension supérieure. Ces dernières, en plus de leurs tests numériques implémentés sous MATLAB, version R2016a, ont fait l’objet de notre troisième contribution. Nous mettons particulièrement l’accent sur la détermination explicite des meilleures constantes possibles apparaissant dans les estimations d’erreur pour ces formules d’intégration. Enfin, dans la quatrième contribution nous testons notre approche pour résoudre numériquement le problème d’élasticité linéaire à l’aide d’un maillage rectangulaire. Nous effectuons l’analyse numérique aussi bien l’analyse de l’erreur d’approximation et résultats de convergence que l’analyse de l’erreur de consistance. Nous montrons également comment cette dernière peut être établie à n’importe quel ordre, généralisant ainsi certains travaux menés dans le domaine. Nous réalisons la mise en œuvre de la méthode et donnons quelques résultats numériques établis à l’aide de la bibliothèque libre d’éléments finis GetFEM++, version 5.0. Le but principal de cette partie sert aussi bien à la validation de nos résultats théoriques, qu’à montrer comment notre approche permet d’élargir la gamme de choix des fonctions d’enrichissement. En outre, elle permet de montrer comment cette large gamme de choix peut aider à avoir des solutions optimales et également à améliorer la validité et la qualité de l’espace d’approximation enrichie. / The enrichment of standard finite elements is a powerful tool to improve the quality of approximation. The main idea of this approach is to incorporate some additional functions on the set of basis functions. These latter are requested to improve the accuracy of the approximate solution. Their best choice is crucial and is based on the knowledge of some a priori information, such as the characteristics of the solution, the geometry of the problem to be solved, etc. The efficiency of such an approach for finding numerical solutions of partial differential equations using a fixed mesh, without recourse to refinement, was proved in numerous applications in the literature. However, the key to its success lies mainly on the best choice of the basis functions, and more particularly those of enrichment functions.An important question then arises: How to suitably choose them, in such a way that they generate a well-defined finite element ?In this thesis, we present a general approach that enables an enrichment of the finite element approximation. This was the subject of our first contribution, which was devoted to the enrichment of the classical Q_1 element, as a first step. As a second step, in our second contribution, we have developed a more general framework for enriching any finite element either P_k, Q_k or others, conforming or nonconforming. As an illustration of how to use this framework to build new enriched finite elements, we have introduced the extensions of some well-known nonconforming finite elements, notably, Han element, Rannacher-Turek element and Wilson element, which are now part of the main code of finite element methods. To establish these extensions, we have introduced a new family of multivariate versions of the classical trapezoidal, midpoint and Simpson rules. These latter, in addition to their numerical tests under MATLAB, version R2016a, have been the subject of our third contribution. They may be viewed as an extension of the well-known trapezoidal, midpoint and Simpson’s one-dimensional rules to higher dimensions. We particularly pay attention to the explicit expressions of the best possible constants appearing in the error estimates for these cubatute formulas. Finally, in the fourth contribution we apply our approach to numerically solving the linear elasticity problem based on a rectangular mesh. We carry out the numerical analysis of the approximation error and also for the consistency error, and show how the latter can be established to any order. This constitutes a generalization of some work already done in the field. In addition to our theoretical results, we have also made some numerical tests, which were achieved by using the GetFEM++ library, version 5.0. The aim of this contribution was not only to confirm our theoretical predictions, but also to show how the new developed framework allows us to expand the range of choices of enrichment functions. Furthermore, we have shown how this wide choices range can help us to improve some approximation properties and to get the optimal solutions for the particular problem of elasticity.