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

A model for simulating the deterioration of structural-scale material responses of microheterogeneous solids

Thermomechanical contact—a rigorous but simple numerical approach

The numerical solution of contact problems via the penalty method yields approximate satisfaction of contact constraints. The solution can be improved using augmentation schemes. However their efficiency is strongly dependent on the value of the penalty parameter and usually results in a poor rate of convergence to the exact solution. In this paper we propose a new method to perform the augmentations. It is based on estimated values of the augmented Lagrangians. At each augmentation the converged state is used to extract some data. Such information updates a database used for the Lagrangian estimation. The prediction is primarily based on the evolution of the constraint violation with respect to the evolution of the contact forces. The proposed method is characterised by a noticeable efficiency in detecting nearly exact contact forces, and by superlinear convergence for the subsequent minimisation of the residual of constraints. Remarkably, the method is relatively insensitive to the penalty parameter. This allows a solution which fulfils the constraints very rapidly, even when using penalty values close to zero.

A superlinear convergent augmented Lagrangian procedure for contact problems

Unilateral Contact: Mechanical Modelling (A. Curnier).- Contact, Friction, Discrete Mechanical Structures and Mathematical Programming (A. Klarbring).- Quasistatic Signorini Problem with Coulomb Friction and Coupling to Adhesion (M. Raous).- Finite Elements for Thermomechanical Contact and Adaptive Finite Element Analysis of Contact Problems (P. Wriggers).

New developments in contact problems

In engineering applications often stability or postbuckling behaviour of structural members like thin shells and beams have to be investigated. This can only be done by considering the geometrically nonlinear behaviour. Finite deflections and rotations have to be included for the post-buckling range. In some engineering applications the structures will undergo contact. This adds another type of nonlinearity to the formulation due to the inequality constraints. Here, different contact algorithms will be investigated and compared. Furthermore, the coupling of these types of nonlinearities and the related algorithms are the main aspects of the paper. The contents of the paper may be outlined as follows.

Peter Wriggers

Papers

A model for simulating the deterioration of structural-scale material responses of microheterogeneous solids

Thermomechanical contact—a rigorous but simple numerical approach

A superlinear convergent augmented Lagrangian procedure for contact problems

New developments in contact problems

Algorithms for non-linear contact constraints with application to stability problems of rods and shells