Isogeometrical approach for cable type structures allowing large sliding contact

A special combination of two finite elements approaches is considered in order to derive a formulation suitable for cable type structures. First, a finite beam element allowing arbitrarily large rotations together with a high order approximation technique including a C¹-continuous approximation is used to model cable behavior. Afterwards, a covariant approach for a curve-to-curve contact is exploited to incorporate contact between a pair of cable elements. This leads to a cable type element with contact. In numerical examples both “soft” and “hard” cables are represented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An atomic interaction-based continuum model for computational multiscale contact mechanics

A computational multiscale contact mechanics model is presented which describes the interaction between deformable solids based on the interaction of individual atoms or molecules. The contact model is formulated in the framework of large deformation continuum mechanics and combines the approaches of molecular modelling [1] and continuum contact mechanics [2]. In the following a brief overview of the contact model is given. Further details can be found in [3], [4] and [5]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive level set method for shock-driven fluid-structure interaction

The fluid-structure interaction simulation of shock- and detonation-loaded structures requires numerical methods that can cope with large deformations as well as local topology changes. A robust, level-set-based shock-capturing fluid solver is described that allows coupling to any solid mechanics solver. As computational example, the elastic response of a thin steel panel, modeled with both shell and beam theory, to a shock wave in air is considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The contact problem for a geometrically non-linear Timoshenko-type shell

An algorithm is developed for the numerical solution of the contact problem of an elastic Timoshenko-type shell subjected to arbitrarily large displacements and rotations, using mixed finite-element approximations. It is essential that six displacements of the faces of the shell are chosen as the required functions. This enables one, first, to simplify the formulation of contact problems in the mechanics of thin-walled structures, since functions by means of which the conditions for the non-penetration of the bodies are formulated are chosen as the required functions and, second, to obtain relations for the components of the Green-Lagrange strain tensor in curvilinear, orthogonal coordinates which accurately represent arbitrarily large displacements of a shell as a rigid body.     

Isomorphism of the Groups of Vassiliev Invariants of Legendrian and Pseudo-Legendrian Knots

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R³ are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

A consistent model for adhesion: The case of the rod

A one-dimensional curvilinear model of adhesion is established in the framework of continuum mechanics and thermodynamics. In addition to the usual kinematic variables, the contact zone is characterized by an internal variable b\beta, representing the intensity of adhesion. The model is constructed by using a first gradient theory associated with the principle of virtual powers. An application is given showing the qualitative behavior of the debonding process.     

Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm

In this paper, we consider the knot placement problem in B-spline curve approximation. A novel two-stage framework is proposed for addressing this problem. In the first step, the $l_{\infty, 1}$-norm model is introduced for the sparse selection of candidate knots from an initial knot vector. By this step, the knot number is determined. In the second step, knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm — the differential evolution algorithm (DE). The candidate knots selected in the first step are served for initial values of the DE algorithm. Since the candidate knots provide a good guess of knot positions, the DE algorithm can quickly converge. One advantage of the proposed algorithm is that the knot number and knot positions are determined automatically. Compared with the current existing algorithms, the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance. Furthermore, the proposed algorithm is robust to noisy data and can handle with few data points. We illustrate with some examples and applications.     

Then-equivalence of knots and invariants of finite degree

A series of new equivalence relations for classical knots is introduced. Factorization of the semigroup of knots converts it into a finitely generated Abelian group. The n-equivalence class of a knot is its universal invariant of degree n. The techniques described can be successfully applied in a considerably more general situation. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 153–173. Translated by O. A. Ivanov.     

求解摩擦接触问题的一个非内点光滑化算法

给出了一个求解三维弹性有摩擦接触问题的新算法,即基于NCP函数的非内点光滑化算法．首先通过参变量变分原理和参数二次规划法,将三维弹性有摩擦接触问题的分析归结为线性互补问题的求解;然后利用NCP函数,将互补问题的求解转换为非光滑方程组的求解;再用凝聚函数对其进行光滑化,最后用NEWTON法解所得到的光滑非线性方程组．方法具有易于理解及实现方便等特点．通过线性互补问题的数值算例及接触问题实例证实了该算法的可靠性与有效性．     

Block mesh refinement for incompressible flows in curvilinear domains

A method for the solution of the Navier–Stokes equation for the prediction of flows inside domains of arbitrary shaped bounds by the use of Cartesian grids with block-refinement in space is presented. In order to avoid the complexity of the body fitted numerical grid generation procedure, we use a saw tooth method for the curvilinear geometry approximation. By using block-nested refinement, we achieved the desired geometry Cartesian approximation in order to find an accurate solution of the N–S equations. The method is applied to incompressible laminar flows and is based on a cell-centred approximation. We present the numerical simulation of the flow field for two geometries, driven cavity and stenosed tubes. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single grid algorithm. The Cartesian block refinement algorithm can be used in any complex curvilinear geometry simulation, to accomplish a reduction in memory requirements and the computational time effort.     

Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics

Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included. Received February 21, 1995 / Revised version received December 21, 1995     

A fast convergent iterative solver for approximate inverse of matrices

In this paper, a rapid iterative algorithm is proposed to find robust approximations for the inverse of nonsingular matrices. The analysis of convergence reveals that this high‐order method possesses eighth‐order convergence. The interesting point is that, this rate is attained using less number of matrix‐by‐matrix multiplications in contrast to the existing methods of the same type in the literature. The extension of the method for finding Moore–Penrose inverse of singular or rectangular matrices is also presented. Numerical comparisons will be given to show the applicability, stability and consistency of the new scheme by paying special attention on the computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

High Performance Computing Techniques for the FEM Simulation in Structural Mechanics

For the solution of large scale simulations in structural mechanics iterative solving methods are mandatory. The efficiency of such methods can crucially depend on different factors: choice of material parameters, quality of the underlying computational mesh and number of processors in a parallel computing system. We distinguish between three aspects of ‘efficiency’: processor efficiency (degree to which the solving algorithm is able to exploit the processor's computational power), parallel efficiency (ratio between computation and communication times) and numerical efficiency (convergence behaviour). With the new FEM software package Feast we pursue the aim to develop a solver mechanism which at the same time gains high efficiencies in all three aspects, while trying to minimise the mentioned dependencies. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Transverse Knots and Branched Covers

We study contact manifolds that arise as cyclic branched coversof transverse knots in the standard contact 3-sphere. We discussproperties of these contact manifolds and describe them in termsof open books and contact surgeries. In many cases we show thatsuch branched covers are contactomorphic for smoothly isotopictransverse knots with the same self-linking number. These pairsof knots include most of the nontransversely simple knots ofBirman–Menasco and Ng–Ozsváth–Thurston.     

On the Best Least Squares Approximation of Continuous Functions using Linear Splines with Free Knots

Approximations to continuous functions by linear splines cangenerally be greatly improved if the knot points are free variables.In this paper we address the problem of computing a best linearspline L₂-approximant to a given continuous function on a givenclosed real interval with a fixed number of free knots. We describe an algorithm that is currently available and establishthe theoretical basis for two new algorithms that we have developedand tested. We show that one of these new algorithms had goodlocal convergence properties by comparison with the other techniques,though its convergence is quite slow. The second new algorithmis not so robust but is quicker and so is used to aid efficiency.A starting procedure based on a dynamic programming approachis introduced to give more reliable global convergence properties. We thus propose a hybrid algorithm which is both robust andreasonably efficient for this problem.     

On a geometrically exact theory for contact interactions

The focus of the contribution is on the development of the unified geometrical formulation of contact algorithms in a covariant form for various geometrical situations of contacting bodies leading to contact pairs: surface-to-surface, line-to-surface, point-to-surface, line-to-line, point-to-line, point-to-point. The computational contact algorithm will be considered in accordance with the geometry of contact bodies in a covariant form. This combination forms a geometrically exact theory of contact interaction, see [1]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

We show that for a large class of contact three-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J⁺-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle ST∗F of F.On the other hand, we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.     

Quantum algebras,quantum coalgebras,invariants of 1-1 tangles and knots

Quantum coalgebras are defined and studied. A theory of asso­ciated invariants of 1-1 tangles, knots and links is developed. The notion of quantum coalgebra is more general than dual of quantum algebra. Examples of quantum algebras include quasitriangular Hopf algebras and examples of quantum coalgebras include coquasi triangu­lar Hopf algebras.     

An algorithm for automatically selecting a suitable verification method for linear systems

Several methods have been proposed to calculate a rigorous error bound of an approximate solution of a linear system by floating-point arithmetic. These methods are called ‘verification methods’. Applicable range of these methods are different. It depends mainly on the condition number and the dimension of the coefficient matrix whether such methods succeed to work or not. In general, however, the condition number is not known in advance. If the dimension or the condition number is large to some extent, then Oishi–Rump’s method, which is known as the fastest verification method for this purpose, may fail. There are more robust verification methods whose computational cost is larger than the Oishi–Rump’s one. It is not so efficient to apply such robust methods to well-conditioned problems. The aim of this paper is to choose a suitable verification method whose computational cost is minimum to succeed. First in this paper, four fast verification methods for linear systems are briefly reviewed. Next, a compromise method between Oishi–Rump’s and Ogita–Oishi’s one is developed. Then, an algorithm which automatically and efficiently chooses an appropriate verification method from five verification methods is proposed. The proposed algorithm does as much work as necessary to calculate error bounds of approximate solutions of linear systems. Finally, numerical results are presented.