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

Preface. Introduction. Introduction to Contact Mechanics. Continuum Solid Mechanics and Weak Forms. Contact Kinematics. Constitutive Equations for Contact Interfaces. Contact Boundary Value Problem and Weak Form. Discretization of the Continuum. Discretization, Small Deformation Contact. Discretization, Large Deformation Contact. Solution Algorithms. Thermo--mechanical Contact. Beam Contact. Adaptive Finite Element Methods for Contact Problems. Computation of Critical Points with Contact Constraints. Appendix A: Gauss Integration Rules. Appendix B: Convective Coordinates. Appendix C: Parameter Identification for Friction Materials. References. Index.

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Nonlinear Phenomena.- Basic Equations of Continuum Mechanics.- Spatial Discretization Techniques.- Solution Methods for Time Independent Problems.- Solution Methods for Time Dependent Problems.- Stability Problems.- Adaptive Methods.- Special Structural Elements.- Special Finite Elements for Continua.- Contact Problems.- Automation of the Finite Element Method by J. Korelc.

Nonlinear Finite Element Methods

Mesoscale models for concrete: homogenisation and damage behaviour

Some Basics of the Mechanics of Solid Continua.- Fundamental Weak Formulations.- Fundamental Micro-Macro Concepts.- A Basic Finite Element Implementation.- Computational/Statistical Testing Methods.- Various Extensions and Further Interpretations of Partitioning.- Domain Decomposition Analogies and Extensions.- Nonconvex-Nonderivative Genetic Material Design.- Modeling Coupled Multifield Processes.- Closing Comments.

Peter Wriggers

Papers

Computational Contact Mechanics

Computational Contact Mechanics

Nonlinear Finite Element Methods

Mesoscale models for concrete: homogenisation and damage behaviour

An Introduction to Computational Micromechanics