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

The paper describes the extension of the critical displacement method (CDM), presented by Onate and Matias in 1996, to the instability analysis of structures with non‐linear material behaviour using a simple damage model. The extended CDM is useful to detect instability points using a prediction of the critical displacement field and a secant load‐displacement relationship accounting for material non‐linearities. Examples of application of CDM to the instability analysis of structures using bar and solid finite elements are presented.

Combination of the critical displacement method with a damage model for structural instability analysis

Institute of Mechanics and Computational Mechanics, University of HannoverAppelstr. 9A, 30167 Hannover, GermanyDiscreteElementsareusedforthesimulationofgranularmaterials(sand, ballast)aswellasformolecularassemblies. Circles(2D) and spheres (3D) are often used in literature on the Discrete Element Method (DEM) however they represent a strongidealisation of the real geometry. Superellipsoids provide the opportunity to generate a wide variety of three-dimensionalgeometrical shapes (e.g. sphere, cube, cylinder).The motion of each particle is described by means of rigid body dynamics. Suitable numerical integration methods arenecessary which are able to conserve the essential physical quantities like momentum energy etc.. Possible choices are e.g.the explicit Verlet-Leapfrog method for the translation and the explicit fourth order Runge-Kutta method for the rotation.The implemented contact formulation takes damping as well as friction into account. Efﬁcient implementation of thecontact search is the main aim of this part of the work. It is subdivided into the neighbourhood search and the local search. Abisection algorithm is used to calculate the gap between two superellipsoids within the search. For the neighbourhood searchtwo binning algorithms were implemented and compared for several packages of particles.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.200610031

Three-dimensional modelling of discrete particles by superellipsoids

A computational approach is presented in this contribution that allows a direct numerical simulation of 3D particulate movements. The given approach is based on the Discrete Element Method (DEM) The particle properties are constitutively described by specific models that act at contact points. The equations of motion will be solved by appropriate time marching algorithms. Additionally coupling schemes with the Finite Element Method (FEM) are discussed for the numerical treatment of particle-solid and particle-fluid interaction. The presented approach will be verified by computational results and compared with those of the literature. Finally, the method is applied for the simulation of different engineering applications using computers with parallel architecture.

Discrete Element Methods: Basics and Applications in Engineering

Meshless methods provide a highly continuous approximation field, convenient for thin structures like shells. Nevertheless, the lack of Kronecker Delta property makes the formulation of essential boundary conditions not straightforward, as the trial and test fields cannot be tailored to boundary values. Similar problem arise when different approximation regions must be joined, in a multi-region problem, such as kinks, folds or joints. This work presents three approaches to impose both kinematic conditions: the well-known Lagrange multiplier method, used since the beginning of the element free Galerkin method; a pure penalty approach; and the recently rediscovered alternative of Nitsche's method. We use the discretization technique for thick Reissner---Mindlin shells and adapt the weak form as to separate displacement and rotational degrees of freedom and obtain suitable and separate stabilization parameters. This approach enables the modeling of discontinuous shells and local refinement on multi-region problems.

Meshless analysis of shear deformable shells: boundary and interface constraints

During sheet bulk metal forming processes both, flat geometries and three-dimensional structures change their shape significantly while undergoing large plastic deformations. As for forming processes, FE-simulations are often done before in situ experiments, a very accurate material model is required, performing well for a huge variety of different geometrical characteristics. Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically observable plastic deformation is traced back to dislocations within considered slip systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level. A finite crystal plasticity model is used to model the behaviour of polycrystalline materials in representative volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the elastic contribution. In order to circumvent singularities stemming from the linear dependency of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the plastic slips and slip resistances. The model is validated with experimental microstructural data under deformation. Through homogenisation and optimisation techniques, effective stress-strain curves are determined and can be compared to results from real forming processes leading to a suitable effective material model.

Peter Wriggers

Papers

Combination of the critical displacement method with a damage model for structural instability analysis

Three-dimensional modelling of discrete particles by superellipsoids

Discrete Element Methods: Basics and Applications in Engineering

Meshless analysis of shear deformable shells: boundary and interface constraints

Computational Homogenisation of Polycrystalline Elastoplastic Microstructures at Finite Deformation