on the physical significance of higher order kinematic and static variables in a three dimensional shell formulation abstract in recent years considerable attention has been given to the development of higher order plate and shell models these models are able to approximately represent three dimensional effects while pertaining the efficiency of a two dimensional formulation due to pre integration of the structural stiffness matrix across the thickness especially the possibility to use unmodified complete three dimensional material laws within shell analysis has been a major motivation for the development of such models while the theoretical and numerical formulation of so called 7 parameter shell models including a thickness stretch of the shell has been discussed in numerous papers no thorough investigation of the physical significance of the additional kinematic and static variables coming along with the extension into three dimensions is known to the authors however realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures e g concerning loading and kinematic boundary conditions and a correct interpretation of the results in the present paper the significance of kinematic and static variables appearing in a 7 parameter model proposed by buchter and ramm 1992a are discussed it is shown how these quantities refine the model behavior and how they can be related to the classical variables such as curvatures and stress resultants furthermore the special role of the material law within such a formulation is addressed it is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness to ensure correct results in this context it is found that the considered 7 parameter model can be regarded as optimal with respect to the number of degrees of freedom involved

parallel multilevel solution of nonlinear shell structures the analysis of large scale nonlinear shell problems asks for parallel simulation approaches one crucial part of effi cient and well scalable parallel fe simulations is the solver for the system of equations due to the inherent suitability for parallelization one is verymuch directed towards preconditioned iterative solvers however thin walled structures discretized byfinite elements lead to ill conditioned system matrices and therefore performance of iterative solvers is generallypoor this situation further deteriorates when the thickness change of the shell is taken into account a preconditioner for this challenging class of problems is presented combining two approaches in a parallel framework the first approach is a mechanicallymotivated improvement called scaled director conditioning sdc and is able to remove the extra ill conditioning that appears with three dimensional shell formulations as compared to formulations that neglect thickness change of the shell it is introduced at the element level and harmonizes well with the second approach utilizing a multilevel algorithm here a hierarchyof coarse grids is generated in a semi algebraic sense using an aggregation concept therebythe complicated and expensive explicit generation of course triangulations can be avoided the formulation of this combined preconditioning approach is given and the effects on the performance of iterative solvers is demonstrated via numerical examples

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

a mortar segment to segment contact method for large deformation solid mechanics abstract contact modeling is still one of the most difficult aspects of non linear implicit structural analysis most 3d contact algorithms employed today use node on segment approaches for contacting dissimilar meshes two pass node on segment contact approaches have the well known deficiency of locking due to over constraint in this work we develop and demonstrate a segment to segment contact approach based on the mortar method by eliminating the over constraint this method appears to be much more robust than the node on segment approach furthermore node on segment approaches suffer when individual nodes slide out of contact at contact surface boundaries or when contacting nodes slide from facet to facet this causes jumps in the contact forces due to the discrete nature of the constraint enforcement and difficulties in convergence for implicit solution techniques these jumps in forces are avoided in the segment to segment approach since penetration is measured by the occluded weighted volume which varies smoothly with perturbations until now all mortar schemes have been developed for small deformation sliding and have mainly been applied on flat surfaces here we will present the integration scheme and the linearization approach used for handling the contact surface integrals in the event of large sliding the schemes are designed to conserve linear momentum and can be applied to arbitrarily curved 3d surfaces although expensive our numerical examples demonstrate that the approach is far more robust than even smoothed node on segment contact techniques that is problems previously not solvable with node on segment can now be solved with our segment to segment contact

two dimensional mortar contact methods for large deformation frictional sliding this paper presents a mortar based formulation for the solution of two dimensional frictional contact problems involving finite deformation and large sliding as is widely recognized traditional node to surface contact formulations have several drawbacks in solution of deformable to deformable contact problems including lack of general patch test passage degradation of spatial convergence rates and robustness issues associated with the faceted representation of contacting surfaces the mortar finite element method initially proposed as a technique to join dissimilarly meshed domains has been shown to preserve optimal convergence rates in tied contact problems see discretization methods and iterative solvers based on domain decomposition springer verlag heidelberg 2001 for a recent review and is examined here as an alternative spatial discretization method for large sliding contact in particular a novel description for frictional sliding conditions in large deformation mortar formulations is proposed in this work in recent years the mortar element method has already been successfully implemented to solve frictional contact problems with linearized kinematics see int j numer meth engng 1993 36 3451 however in the presence of large deformations and finite sliding one must face difficulties associated with the definition and linearization of contact virtual work in the case where the mortar projection has a direct dependence on the tangential relative motion along the interface in this paper such a formulation is presented with particular emphasis on key aspects of the linearization procedure and on the robust description of the friction kinematics some novel techniques are proposed to treat the non smoothness in the contact geometry and the searching required to define mortar segments a number of numerical examples illustrate the performance and accuracy of the proposed formulation copyright 2004 john wiley sons ltd

a non diagonalizable pure state we construct a pure state on the c algebra formula see text of all bounded linear operators on formula see text which is not diagonalizable i e it is not of the form formula see text for any orthonormal basis formula see text of formula see text and an ultrafilter u on n this constitutes a counterexample to anderson s conjecture without additional hypothesis and improves results of c akemann n weaver i farah and i smythe who constructed such states making additional set theoretic assumptions it follows from results of j anderson and the positive solution to the kadison singer problem due to a marcus d spielman and n srivastava that the restriction of our pure state to any atomic masa formula see text of diagonal operators with respect to an orthonormal basis formula see text is not multiplicative on formula see text

formulation and analysis of conserving algorithms for frictionless dynamic contact impact problems this paper presents the formulation of conserving time stepping algorithms for frictionless dynamic contact of solids a new class of finite element methods is proposed for the solution of these problems that exhibit the same conservation laws as the underlying continuum dynamical system the proposed methods are based on a penalty regularization of the constrained contact problem and lead to full conservation of the total energy of the system including the regularization penalty potential during persistent contact and restoration of the original energy upon release both linear and angular momenta are conserved by the scheme furthermore the newly developed methods have the ability to enforce the associated constraints in the velocity besides the impenetrability constraint in the displacements while preserving the conservation restoration properties of the final numerical scheme a modification of these schemes is described that assures positive energy dissipation if desired even in the highly nonlinear setting of contact impact problems leading to contact schemes with high frequency energy dissipation representative numerical simulations are presented illustrating the performance of the proposed numerical schemes

subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3d this paper presents a strategy to parameterize contact surfaces of arbitrary mesh topology in 3d with at least c1 continuity for both quadrilateral and triangular meshes in the regular mesh domain four quadrilaterals or six triangles meet in one node even c2 continuity is attained therefore we use subdivision surfaces for which non physical pressure jumps are avoided for contact interactions they are usually present when the contact kinematics is based on facet elements discretizing the interacting bodies the properties of subdivision surfaces give rise to basically four different implementation strategies each strategy has specific features and requires more or less efforts for an implementation in a finite element program one strategy is superior with respect to the others in the sense that it does not use nodal degrees of freedom of the finite element mesh at the contact surface instead it directly uses the degrees of freedom of the smooth surface thereby remarkably it does not require an interpolation we show how the proposed method can be used to parameterize adaptively refined meshes with hanging nodes this is essential when dealing with finite element models whose geometry is generated by means of subdivision techniques three numerical 3d problems demonstrate the improved accuracy robustness and performance of the proposed method over facet based contact surfaces in particular the third problem adopted from biomechanics shows the advantages when designing complex contact surfaces by means of subdivision techniques copyright 2004 john wiley sons ltd

design of energy conserving algorithms for frictionless dynamic contact problems this paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum angular momentum and energy in finite element simulations it is seen that a lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties a related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized this penalty method sacrifices the energy conservation property but is dissipative under all conditions of changing contact so that the global algorithm remains stable notably it is also shown that augmented lagrangian iteration utilizing this penalty kernel reproduces the energy conserving i e lagrange multiplier solution to any desired degree of accuracy the result is a robust stable method even in the context of large deformations as is shown by some representative numerical examples in particular the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations 1997 by john wiley sons ltd

a 3d contact smoothing method using gregory patches in this work a method is developed for smoothing three dimensional contact surfaces the method can be applied to both regular and irregular meshes the algorithm employs gregory patches to interpolate finite element nodes and provide tangent plane continuity between adjacent patches the resulting surface interpolation is used to calculate gaps and contact forces in a variationally consistent way such that contact forces due to normal and frictional contact vary smoothly as slave nodes transition from one patch to the next this eliminates the chatter which typically occurs in a standard contact algorithm when a slave node is situated near a master facet edge the elimination of this chatter provides a significant improvement in convergence behaviour which is illustrated by a number of numerical examples furthermore smoothed surfaces also provide a more accurate representation of the actual surface such that resulting stresses and forces can be more accurately computed with coarse meshes in many problems this fact is also demonstrated by the numerical examples

geotensity combining motion and lighting for 3d surface reconstruction this paper is about automatically reconstructing the full 3d surface of an object observed in motion by a single static camera based on the two paradigms structure from motion and linear intensity subspaces we introduce the geotensity constraint that governs the relationship between four or more images of a moving object we show that it is possible in theory to solve for 3d lambertian surface structure for the case of a single point light source and propose that a solution exists for an arbitrary number point light sources the surface may or may not be textured we then give an example of automatic surface reconstruction of a face under a point light source using arbitrary unknown object motion and a single fixed camera

unilateral non linear dynamic contact of thin walled structures using a primal dual active set strategy the efficient modelling of three dimensional contact problems is still a challenge in non linear implicit structural analysis we use a primal dual active set strategy siam j optim 2003 13 865 888 based on dual lagrange multipliers siam j numer anal 2000 38 989 1012 to handle the non linearity of the contact conditions this allows us to enforce the contact constraints in a weak integral sense without any additional parameter due to the biorthogonality condition of the basis functions the lagrange multipliers can be locally eliminated we perform a static condensation to achieve a reduced system for the displacements the lagrange multipliers representing the contact pressure can be easily recovered from the displacements in a variationally consistent way for the application to thin walled structures we adapt a three dimensional non linear shell formulation including the thickness stretch of the shell to contact problems a reparametrization of the geometric description of the shell body gives us a surface oriented shell element which allows the application of contact conditions directly to nodes lying on the contact surface shell typical locking phenomena are treated with the enhanced assumed strain method and the assumed natural strain method the discretization in time is done with the implicit generalized method j appl mech 1993 60 371 375 and the generalized energy momentum method comp methods appl mech eng 1999 178 343 366 to compare the development of energies within a frictionless contact description in order to conserve the total energy within the discretized frictionless contact framework we follow an approach from laursen and love int j numer methods eng 2002 53 245 274 who introduced a discrete contact velocity to update the velocity field in a post processing step various examples show the good performance of the primal dual active set strategy applied to the implicit dynamic analysis of thin walled structures

improved implicit integrators for transient impact problems geometric admissibility within the conserving framework the value of energy and momentum conserving algorithms has been well established for the analysis of highly non linear systems including those characterized by the nonsmooth non linearities of an impact event this work proposes an improved integration scheme for frictionless dynamic contact seeking to preserve the stability properties of exact energy and momentum conservation without the heretofore unavoidable compromise of violating geometric admissibility as established by the contact constraints the physically motivated introduction of a discrete contact velocity provides an algorithmic framework that ensures exact conservation locally while remaining independent of the choice of constraint treatment thus making full conservation equally possible in conjunction with a penalty regularization as with an exact lagrange multiplier enforcement the discrete velocity effects are incorporated as a post convergence update to the system velocities and thus have no direct effect on the non linear solution of the displacement equilibrium equation the result is a robust implicit algorithmic treatment of dynamic frictionless impact appropriate for large deformations and fully conservative for a range of geometric constraints copyright 2001 john wiley sons ltd

stable lagrange multipliers for quadrilateral meshes of curved interfaces in 3d for the discretization of coupled problems by means of mortar finite elements the choice of the discrete lagrange multiplier space is of crucial importance for the performance of the resulting numerical scheme dual lagrange multipliers offer several advantages since they can be locally eliminated in this paper the lowest order dual lagrange multipliers are extended to arbitrary quadrilateral or triangular surface grids in a first step two alternatives are provided for the case of planar interfaces for the approximation of vector fields two more modifications are given for use on curvilinear interfaces a priori results are obtained and several numerical examples are shown

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

energy conserving and decaying algorithms in non linear structural dynamics a generalized formulation of the energy momentum methodwill be developed within the framework of the generalizedmethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response furthermore the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the constraint energy momentum algorithm the goal of this general approach of implicit energyconserving and decaying time integration schemes is to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non linear structural dynamics among the methods considered will be the newmark method the classical methods the energy momentum methodwith and without numerical dissipation the constraint energymomentum algorithm and the constraint energy method copyright 1999 john wiley sons ltd

generalized energy momentum method for non linear adaptive shell dynamics in the present study a generalization of the energy momentum method denoted by generalized energy momentum method applied to the non linear dynamics of shells will be developed within the framework of the generalized a method this algorithmic environment contains the unconditionally stable energy momentum method and its numerically damped version as well as the classical newmark and a methods as special cases in order to control the size of the time steps of the integration scheme with respect to accuracy and eaciency an adaptive time stepping procedure based on local a posteriori error estimation will be improved for non linear dynamical systems and applied to the proposed class of algorithms the spatial discretization is realized by an eight noded finite shell element of reissner mindlin type including an extensible shell director field permitting the application of three dimensional material laws the original formulation of this finite element will be developed for non linear dynamic analysis and adapted for the employment within the introduced energy conserving decaying time integration scheme 1999 elsevier science s a all rights reserved

monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems nonconforming domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations we use a generalized mortar method based on dual lagrange multipliers for the discretization of a nonlinear contact problem between linear elastic bodies in the case of unilateral contact problems pointwise constraints occur and monotone multigrid methods yield efficient iterative solvers here we generalize these techniques to nonmatching triangulations where the constraints are realized in terms of weak integral conditions the basic new idea is the construction of a nested sequence of nonconforming constrained spaces we use suitable basis transformations and a multiplicative correction in contrast to other approaches no outer iteration scheme is required the resulting monotone method is of optimal complexity and can be implemented as a multigrid method numerical results illustrate the performance of our approach in two and three dimensions

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

a primal dual active set strategy for non linear multibody contact problems abstract non conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a non linear multibody contact problem we use the mortar approach with a dual lagrange multiplier space to handle the non linearity of the contact conditions we apply a primal dual active set strategy to find the actual contact zone the algorithm can be easily generalized to multibody contact problems a suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case using an inexact primal dual active set strategy in combination with a multigrid method yields an efficient iterative solver different numerical examples for one and multibody contact problems illustrate the performance of the method

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects