https://hal.archives-ouvertes.fr/hal-01513346/document

Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

Introduction to linear and nonlinear programming

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed.


Keywords:

contact mechanics;
friction;
penalty method;
Lagrange multiplier method;
contact algorithms;
finite element method;
finite deformations;
discretization methods

Computational Contact Mechanics

Abstract We present a general framework for the analysis and modelling of frictional contact involving composite materials. The study has focused on composite materials formed by a matrix of rubber and synthetic or metallic fibres, which is the case of standard tires. We detail the numerical treatment of incompressibility at large deformations that rubber can experience, as well as the stiffening effect that properly oriented fibres will induce within the rubber. To solve the frictional contact between solids, a Dual Augmented Lagrangian Multiplier Method is used together with the Mortar method. This ensures a variationally consistent estimation of the contact forces. A modified Serial-Parallel Rule of Mixtures is employed to model the behaviour of composite materials. This is a simple and novel methodology that allows the blending of constitutive behaviours as diverse as rubber (very low stiffness and incompressible behaviour) and steel (high stiffness and compressible behaviour) taking into account the orientation of the fibres within the material. The locking due to the incompressibility constraint in the rubber material has been overcome by using Total Lagrangian mixed displacement-pressure elements. A collection of numerical examples is provided to show the accuracy and consistency of the methodology presented when solving frictional contact, incompressibility and composite materials under finite strains. 

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A numerical framework for modelling tire mechanics accounting for composite materials, large strains and frictional contact

Finite element methods (FEMs) are flexible tools which can be applied to solve contact problems with arbitrary geometries. As in any numerical method, the solutions obtained with FEM are only approximate. Errors occur e.g. due to the choice of the ansatz space and the approximation of the geometry. In general it is necessary to automatically control the error inherited in the method to obtain reliable solutions. This is especially true in case of non-linear contact problems since geometry and contact surface can change substantially during the deformation process. Thus the refinement process needed for an accurate analysis cannot be controlled by the user beforehand. Here an adaptive FEM is developed for large strain problems of two or more deformable bodies being in contact. The main focus is the comparison of different error indicators and error estimators related to the contact problem. In detail residual based, error estimators, error indicators relying on superconvergence properties and error estimat...

Comparison of Different Error Indicators for Contact Problems Involving Large Elastic Strains

Die mathematische Formulierung mechanischer Probleme fuhrt auf Gleichungen, die fur konkrete Aufgabenstellungen gelost werden mussen. Diese Gleichungen konnen je nach Fragestellung von ganz unterschiedlichem Typ sein. Sie schliesen algebraische Beziehungen, Differentialgleichungen oder Variationsgleichungen ein. Beispiele dafur finden sich in den ersten drei Banden der Lehrbuchreihe und in den vorangegangenen Kapiteln dieses Buches. In der Elastostatik konnen wir die Gleichgewichtsbedingungen (algebraische Gleichungen) oder die Gleichung der Biegelinie eines Balkens (Differentialgleichung) nennen. In der Kinetik wird die Bewegung des Massenpunktes durch gewohnliche Differentialgleichungen beschrieben. Die Gleichungen fur die Scheibe in Kapitel 2 oder fur die Membran in Kapitel 3 stellen partielle Differentialgleichungen dar. Variationsgleichungen fur den Stab und den Balken sind in Abschnitt 2.7.3 angegeben.

https://www.springer.com/cda/content/document/cda_downloaddocument/9783642243981-c1.pdf?SGWID=0-0-45-1294637-p174310062

Peter Wriggers

Papers

A numerical framework for modelling tire mechanics accounting for composite materials, large strains and frictional contact

Comparison of Different Error Indicators for Contact Problems Involving Large Elastic Strains

Numerische Methoden in der Mechanik

Mixed virtual element formulations for incompressible and inextensible problems

Failure of high-speed bearing at cyclic impact-sliding contacts: Numerical and experimental analysis