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

2The present work is primarily concerned with the characterization of the material response of the heterogeneous microstructure of hierarchical composite materials. Many natural and man-made materials exhibit a structured composition over more than one length scale. Examples of such materials with structural hierarchy include laminated composites [1], particulate composites [2], foams [3], cellular materials [4], as well as biological materials such as the human bones [5], tendon [6], wood [7], marine gastropods [8] and the Gecko’s adhesive tissues [9]. With respect to standard composites, where only two length scales are present, viz. the macro- and the meso-scale, hierarchical materials show at least three structural levels (see the sketch in Fig. 1). The macro-level is characterized by the structural size-scale, which corresponds to the size of the component. Then, a meso-level composed of hard meso-granules embedded into a tougher matrix is observed, as happens in classical fibre-reinforced composites. The novelty here is given by the fact that the meso-granules are no longer homogeneous, but show themselves a heterogeneous internal microstructure. This corresponds to the micro-level, where polycrystalline micro-grains are embedded into a tough binder. Experimental [4] and theoretical insights [10] have recently shown that structural hierarchy plays a major role in determining the global material properties and permits to achieve impressive mechanical strength and toughness, in spite of the low fracture toughness of their individual constituents. This is the result of the capability of the material to withstand severe damage phenomena occurring at the micro-level, thanks to the important role of the interfaces between the meso-granules and the matrix, which act as crack arresters and confine the damage to the micro-level. From the computational point of view, despite advances in computational techniques and computing power, direct simulation of such heterogeneous materials is still not a viable option. The characteristic size of the microstructure is in fact at least one order of magnitude less than the diameter of the meso-granules, which is in turn at least one order of magnitude less than the structural sizescale. Hence, multi-scale techniques have to be pursued. The target is to describe the material at the macro-level as homogeneous with effective properties based on the information obtained from numerical analyses at smaller length scales. Hierarchical modelling [11], domain decomposition methods [12] or concurrent multi-level methods [13] can be used for the macro-meso scale connection, as for standard two-level composite materials. The connection between the meso- and the micro-structure, which is the main focus of this study, is herein investigated following a computational homogenization approach. As in previous studies [14-16], the idea is to apply a bottom-up approach and to pass the information from the lower to the higher scale in the form of material properties. The homogenized response of the meso-level is determined from microscopic analyses of a representative volume element (RVE) of the material microstructure. The average elasticity tensor relating the average stress and strain of the RVE should take into account the effect of microscale property degradation due to subcontinuum scale microcracks, interface debonding, crack closure effects and irregular rough crack paths due to branching phenomena. This scenario is much more complex than the simpler interface debonding failure in fibre-reinforced composites. To simplify the matter, the binder between the polycrystalline grains will be replaced by a zero-thickness interface whose response is governed by a thickness-dependent cohesive zone model. Materials science studies have shown that the thickness of the binder and the degree of contiguity of the grains is in fact a fundamental property [17]. This new cohesive zone formulation will be able to modulate the phenomenon of strain localization (dissipation over a surface instead over a volume), as a function of the real interface thickness. With this model in hand, incrementally loaded tests conducted on the RVE

A computational homogenization approach for a three-level hierarchical composite material

This paper is a further step towards the full analysis of the complex behaviour of contacting beams in the coupled thermo-electro-mechanical field. Now we add to our previous work the coupling with the thermal field. The coupling between electric, mechanical and thermal fields is manifested in: dependence of material parameters on the temperature, frictional heating, heat generation by the electric current flow, dependence of the contact area on the temperature, dependence of thermal and electric fields on the change of the contact area and the contact point position. In the present preliminary analysis we take into account only the last aspect. We also consider the indirect influence of thermal contact on the mechanical field using the thermo-mechanical beam element. The contact constraints are enforced with the penalty method within the finite element technique. It is assumed that the heat flow in the contact area is unlimited which leads to the equalling of temperatures between two contacting bodies. This constraint is also introduced by the penalty method. The set of governing equations including the coupling is solved by the monolithic scheme. The problem is nonlinear, hence the linearized version of equations is derived. The consistent linearization leading to the consistent tangent stiffness matrix and the corresponding residual vector is performed to apply efficiently the Newton-Raphson method and to ensure the quadratic convergence. Some numerical examples are presented to show the efficiency of the suggested approach.

Peter Wriggers

Papers

A computational homogenization approach for a three-level hierarchical composite material

Thermo-electro-mechanical coupling in beam-to-beam contact

Material modeling of ferritic steel on microscopic length scale under cyclic loading

Spline-based smooth beam-to-beam contact model

Examining errors and correction techniques for SPH