Aproximace, numerická realizace a kvalitativní analýza kontaktních úloh se třením.

Title: Approximation, numerical realization and qualitative analysis of contact problems with friction Author: Tomáš Ligurský Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Jaroslav Haslinger, DrSc., Department of Numerical Mathe- matics Abstract: This thesis deals with theoretical analysis and numerical realization of dis- cretized contact problems with Coulomb friction. First, discretized 3D static contact prob- lems with isotropic and orthotropic Coulomb friction and solution-dependent coefficients of friction are analyzed by means of the fixed-point approach. Existence of at least one solution is established for coefficients of friction represented by positive, bounded and con- tinuous functions. If these functions are in addition Lipschitz continuous and upper bounds of their values together with their Lipschitz moduli are sufficiently small, uniqueness of the solution is guaranteed. Second, properties of solutions parametrized by the coefficient of friction or the load vector are studied in the case of discrete 2D static contact problems with isotropic Coulomb friction and coefficient independent of the solution. Conditions under which there exists a local Lipschitz continuous branch of solutions around a given reference point are established due to two variants of the...Department of Numerical MathematicsKatedra numerické matematikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Approximation, réalisation numérique et l'analyse qualitative des problèmes de contact avec frottement

Tato práce se zabývá teoretickou analýzou a numerickou realizací diskretizovaných kontaktních úloh s Coulombovým třením. Nejprve je pomocí pevněbodového přístupu provedena analýza diskretizovaných 3D statických kontaktních úloh s izotropním a ortotropním Coulombovým třením a koeficienty tření závislými na řešení. Existence alespoň jednoho řešení je dokázána pro koeficienty tření reprezentované omezenými kladnými spojitými funkcemi. Pokud jsou tyto funkce navíc lipschitzovsky spojité a horní meze jejich hodnot spolu s jejich moduly lipschitzovskosti jsou dostatečně malé, je zaručena jednoznačnost tohoto řešení. Dále jsou v případě 2D statických kontaktních úloh s izotropním Coulombovým třením a koeficientem nezávislým na řešení studovány vlastnosti řešení parametrizovaných koeficientem tření nebo vektorem zatížení. S pomocí dvou variant věty o implicitních funkcích jsou ustaveny podmímky, za nichž existuje lokální lipschitzovská větev řešení na okolí daného referenčního bodu. Následně je navržen algoritmus po částech hladké kontinuace, který nám umožňuje sledovat takové větve řešení numericky. Na závěr je uvedena dobře formulovaná prostorová semidiskretizace dynamických kontaktních úloh s izotropním Coulombovým třením, kde koeficient nezávisí na řešení.This thesis deals with theoretical analysis and numerical realization of discretized contact problems with Coulomb friction. First, discretized 3D static contact problems with isotropic and orthotropic Coulomb friction and solution-dependent coefficients of friction are analyzed by means of the fixed-point approach. Existence of at least one solution is established for coefficients of friction represented by positive, bounded and continuous functions. If these functions are in addition Lipschitz continuous and upper bounds of their values together with their Lipschitz moduli are sufficiently small, uniqueness of the solution is guaranteed. Second, properties of solutions parametrized by the coefficient of friction or the load vector are studied in the case of discrete 2D static contact problems with isotropic Coulomb friction and coefficient independent of the solution. Conditions under which there exists a local Lipschitz continuous branch of solutions around a given reference point are established due to two variants of the implicit-function theorem. Consequently, a piecewise smooth continuation algorithm, which enables us to follow such branches of solutions numerically, is proposed. In the end, a well-posed spatial semi-discretization of dynamic contact problems with isotropic Coulomb friction where the coefficient does not depend on the solution is introduced.Cette thèse est consacrée à l'analyse théorique et la réalisation numérique des problèmes de contact discrétisés avec frottement de Coulomb. Tout d'abord, des problèmes de contact statiques discrétisés tridimensionnels avec frottement de Coulomb isotrope et orthotrope et coefficients de frottement dépendants de la solution sont analysés au moyen de l'approche de point fixe. L'existence d'au moins une solution est établie pour les coefficients de frottement représentés par des fonctions positives, bornées et continues. Si ces fonctions sont en plus Lipschitz continues et si les limites supérieures de leurs valeurs ainsi que leurs modules de Lipschitz sont suffisamment petites, l'unicité de la solution est garantie. Deuxièmement, les propriétés des solutions paramétrées par le coefficient de frottement ou le vecteur de chargement sont étudiées dans le cas des problèmes de contact statique discrets bidimensionnels avec frottement de Coulomb isotrope et le coefficient indépendant de la solution. Les conditions dans lesquelles il existe une branche Lipschitzienne locale des solutions autour d'un point de référence donné sont établies grâce à deux variantes du théorème des fonctions implicites. Après, un algorithme de continuation lisse par morceaux qui nous permet de suivre numériquement telles branches de solutions est proposé. Finalement, une semi-discrétisation spatiale bien posée des problèmes de contact dynamique avec frottement de Coulomb isotrope où le coefficient ne dépend pas de la solution est introduite

Bifurcations in contact problems with Coulomb friction

To explore the bifurcation in this contact problem, we have taken uniform meshes with 4096, 16384, 65536 and 262144 triangles. We shall show that the bifurcation behaviour is more complex here. Branches 1 and 4 approach one another for finer meshes, and they disappear both for the finest mesh. Nevertheless, regarding the branching of the corresponding contact problem with forces h = (h1,h2) over the plane h1-h2, one can find it stable and convergent, again. \

Approximation, numerical realization and qualitative analysis of contact problems with friction

Title: Approximation, numerical realization and qualitative analysis of contact problems with friction Author: Tomáš Ligurský Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Jaroslav Haslinger, DrSc., Department of Numerical Mathe- matics Abstract: This thesis deals with theoretical analysis and numerical realization of dis- cretized contact problems with Coulomb friction. First, discretized 3D static contact prob- lems with isotropic and orthotropic Coulomb friction and solution-dependent coefficients of friction are analyzed by means of the fixed-point approach. Existence of at least one solution is established for coefficients of friction represented by positive, bounded and con- tinuous functions. If these functions are in addition Lipschitz continuous and upper bounds of their values together with their Lipschitz moduli are sufficiently small, uniqueness of the solution is guaranteed. Second, properties of solutions parametrized by the coefficient of friction or the load vector are studied in the case of discrete 2D static contact problems with isotropic Coulomb friction and coefficient independent of the solution. Conditions under which there exists a local Lipschitz continuous branch of solutions around a given reference point are established due to two variants of the..

Approximation and numerical realization of contact problems with given friction and a coefficient of friction depending on the solution in 3D.

Three-dimensional contact problems with given friction and a coeficient of friction depending on the solution are studied. By means of the xed-point approach, the existence of at least one solution is proved provided that the coeficient of friction F is represented by a continuous, positive and bounded function. Under an additional assumption, namely the Lipschitz continuity of F with a suficiently small modulus of the Lipschitz continuity, the uniqueness of the solution is shown. The problem is discretized by the nite element method. The existence and uniqueness of the solution to the discrete problems are investigated in a similar way as it has been done in the continuous setting. Convergence of solutions to the discrete models in an appropriate sense is established. The method of successive approximations is used for nding xed-points. Each iterative step leads to a contact problem with given friction and a coeficient of friction which does not depend on the solution. We introduce a mixed variational formulation of this problem from which the dual formulation used in computations can be derived. Numerical results of model examples are presented

Approximation and numerical realization of contact problems with given friction and a coefficient of friction depending on the solution in 3D.

Three-dimensional contact problems with given friction and a coeficient of friction depending on the solution are studied. By means of the xed-point approach, the existence of at least one solution is proved provided that the coeficient of friction F is represented by a continuous, positive and bounded function. Under an additional assumption, namely the Lipschitz continuity of F with a suficiently small modulus of the Lipschitz continuity, the uniqueness of the solution is shown. The problem is discretized by the nite element method. The existence and uniqueness of the solution to the discrete problems are investigated in a similar way as it has been done in the continuous setting. Convergence of solutions to the discrete models in an appropriate sense is established. The method of successive approximations is used for nding xed-points. Each iterative step leads to a contact problem with given friction and a coeficient of friction which does not depend on the solution. We introduce a mixed variational formulation of this problem from which the dual formulation used in computations can be derived. Numerical results of model examples are presented.Department of Numerical MathematicsKatedra numerické matematikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Approximation and numerical realization of contact problems with given friction and a coefficient of friction depending on the solution in 3D.

Three-dimensional contact problems with given friction and a coefficient of friction depending on the solution are studied. By means of the fixed-point approach, the existence of at least one solution is proved provided that the coefficient of friction F is represented by a continuous, positive and bounded function. Under an additional assumption, namely the Lipschitz continuity of F with a sufficiently small modulus of the Lipschitz continuity, the uniqueness of the solution is shown. The problem is discretized by the finite element method. The existence and uniqueness of the solution to the discrete problems are investigated in a similar way as it has been done in the continuous setting. Convergence of solutions to the discrete models in an appropriate sense is established. The method of successive approximations is used for finding fixed-points. Each iterative step leads to a contact problem with given friction and a coefficient of friction which does not depend on the solution. We introduce a mixed variational formulation of this problem from which the dual formulation used in computations can be derived. Numerical results of model examples are presented

Modelling HM Processes in Porous Media

The report contains an overview of modelling hydro-mechanical (HM) processes in porous media. It starts with an introduction to continuum approach to porous media. Next, a succession of various continuum models is derived, namely, for saturated and unsaturated flow, both with and without coupling to deformation of the porous medium. Each model is developed from balance equations, which are supplied by constitutive relationships

A Method of Piecewise-Smooth Numerical Branching

International audienceA method of numerical branching is proposed for piecewise-smooth steady-state problems when any analytical expressions are not known for the regions of smoothness of the function involved. The performance of the method is shown for model bifurcations in discretised contact problems with Coulomb friction. First, different parameter settings in its input are tested so that the optimum one can be proposed. Then, the method is used to investigate the behaviour of the bifurcations for different meshes. All solution branches seem to be discovered reliably if the parameters are set up properly

Subdifferential-based implicit return-mapping operators in Mohr-Coulomb plasticity

The paper is devoted to constitutive solution, limit load analysis and Newton-like methods in elastoplastic problems containing the Mohr-Coulomb yield criterion. Within the constitutive problem, we introduce a self-contained derivation of the implicit return-mapping solution scheme using a recent subdifferential-based treatment. Unlike conventional techniques based on Koiter's rules, the presented scheme a priori detects a position of the unknown stress tensor on the yield surface even if the constitutive solution cannot be found in a closed form. This eliminates blind guesswork from the scheme and enables to analyze properties of the constitutive operator. It also simplifies the construction of the consistent tangent operator, which is important for the semismooth Newton method when applied to the incremental boundary-value elastoplastic problem. The incremental problem in Mohr-Coulomb plasticity is combined with limit load analysis. Beside a conventional direct method of incremental limit analysis, a recent indirect one is introduced and its advantages are described. The paper contains 2D and 3D numerical experiments on slope stability with publicly available Matlab implementations.Web of Science97121523150