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 continuum based framework for the finite element analysis of proximal arteries

In this paper, the finite element formulation of a beam and shell element for numerical modeling of thin-walled aluminum made beam-like structures has been described. Modern thin-walled beam-like members made up of aluminum are used extensively in the construction industry because of their characteristic features like lightness, recyclability, corrosion resistance and, ease of workability. The constitutive behavior of aluminum alloys is different from the structural steel, especially in plastic deformation range. In the case of aluminum alloys, progressive yielding is seen which can not be modeled with normal elasto-plastic material model with isotropic hardening. Having a better constitutive model leads to a more accurate structural behavior in case of extreme loading and failure of the structure and also determination of the load carrying capacity. The Ramberg-Osgood constitutive law is most popularly used to model the elasto-plastic behavior of aluminum material. Here the material model based on Ramberg-Osgood law is implemented in a beam and a shell element. results are compared with the available experimental results illustrating the validity and the application of the formulation in the analysis of aluminum structures.

Beam and shell elements for thin-walled aluminium structures

Special Issue on Fracture and Contact Mechanics for Interface Problems

A small-sliding node-to-node formulation for beam-to-beam frictional contact with large displacements

Contact between bodies is most commonly analyzed using quadrilateral contact elements that are based on 8-node brick (hexahedral) continuum finite elements. As a quadrilateral contact surface, in comparison to a triangular contact surface (tetrahedral continuum elements), is not necessarily flat, or it deforms as deformable body deforms, contact formulation turns to be a complex problem. Recent developments in contact routines based on the Moving Friction Cone (MFC) approach for flat triangular contact elements enabled significant simplifi-cations in the element formulation, what is used herein. The MFC formulation of contact is based on the single gap vector, instead of two vectors (slip and stick one). The curved contact surface is defined in a parametric form, thus enabling fi-nite deformations and a Lagrangian definition of contact.

Peter Wriggers

Papers

A continuum based framework for the finite element analysis of proximal arteries

Beam and shell elements for thin-walled aluminium structures

Special Issue on Fracture and Contact Mechanics for Interface Problems

A small-sliding node-to-node formulation for beam-to-beam frictional contact with large displacements

The quadrilateral parametric contact element based on the moving friction cone formulation