advantages of the mixed format in geometrically nonlinear analysis of beams and shells using solid finite elements summary the paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed stress and displacement format with respect to the more commonly used displacement one i the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and ii the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used the smaller extrapolation error produces a lower number of iterations and larger step length in path following analysis and a greater accuracy in koiter asymptotic method to focus on the origin of the phenomenon the two formats are derived for the same finite element interpolation the reduced polynomial dependence improves the koiter asymptotic strategy in terms of both computational efficiency accuracy and simplicity copyright 2016 john wiley sons ltd

structural instability analyses based on generalised path following this paper describes how quasi static conservative instability problems can be completely described using generalised path following procedures for augmented equilibrium formulations in particul

a class of solutions of the time dependent reaction diffusion equation for the processes a a to a and a a to oe consideration is given to the treatment of the nonlinear partial differential equation describing the diffusion of an assembly of particles which simultaneously coagulate or annihilate it is shown how a class of solutions may be obtained in terms of the solutions of an ordinary differential equation and detailed application is made to the case of an initially localized particle distribution

on improved predictions for structural equilibrium path evaluations the paper describes how several procedures for higher order predictions have been introduced in order to improve the convergence speed in a general finite element program for non linear structural

an efficient isogeometric solid shell formulation for geometrically nonlinear analysis of elastic shells abstract in this work an isogeometric solid shell model for geometrically nonlinear analyses is proposed it is based on a linear interpolation through the thickness and a nurbs interpolation on the middle surface of the shell for both the geometry and the displacement field the green lagrange strains are linearized along the thickness direction and a modified generalized constitutive matrix is adopted to easily eliminate thickness locking without introducing any additional unknowns and to model multi layered composite shells reduced integration schemes which take into account the high continuity of the shape functions are investigated to avoid interpolation locking and to increase the computational efficiency the relaxation of the constitutive equations at each integration point is adopted in the iterative scheme in order to reconstruct the equilibrium path using large steps and a low number of iterations even for very slender structures this strategy makes it possible to minimize the number of stiffness matrix evaluations and decompositions and it turns out to be particularly convenient in isogeometric analyses

element formulation and numerical techniques for stability problems in shells in the context of instability problems in shells or shell like structures the objective of the present paper is twofold primarily the paper describes how quasi static conservative instability problems can be considered in a multi parametric context where generalized path following procedures for augmented equilibrium problems are used as computational tools these allow systematic treatment of the higher dimensional solution sets generated under the variations of certain parameters deemed relevant for the given problem the efficient implementation of the above mentioned procedures requires however as an essential ingredient a non linear finite element which is not only accurate but also inexpensive to this end a systematic view on a corotational total lagrangian formulation is described the tric element of argyris and coworkers is slightly modified and introduced as core element formulation special emphasis is given to the alternative methods for treatment of finite three dimensional rotations with reference to both the element definition and solution algorithms numerical examples verify the element capabilities and the possibility to completely describe instability phenomena of large discretized models

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

tracing the equilibrium path beyond compound critical points a unified presentation of some popular continuation procedures used in the non linear finite element analysis of structural mechanics is introduced an extension of the elliptical constraint equation proposed by crisfield is given it is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur for bifurcation points a modification of rheinboldt s branching procedure is presented post critical response after limit and bifurcation points is determined in some numerical examples

co rotational beam elements with warping effects in instability problems the present paper investigates the formulation of 3d co rotational beam elements for the buckling and post buckling analysis of frame structures following pacoste and eriksson comput methods app

path following analysis of thin walled structures and comparison with asymptotic post critical solutions a path following non linear elastic analysis for structures composed of assemblages of flat slender elastic panels is presented the proposed path following method employs fem technology and a kinematical model to analyse these structures using a koiter asymptotic approach as a result it is possible to verify the accuracy achieved by the asymptotic method the proposed mixed path following formulation is both efficient and robust with regards to the locking extrapolation phenomenon that strongly affects compatible formulations the use of an hc finite element makes it possible to avoid the problem of the finite rotations in the space maintaining a high degree of continuity and making the numeric formulation simple and efficient copyright 2002 john wiley sons ltd

numerical analysis of complex instability behaviour using incremental iterative strategies the paper describes how quasi static conservative instability problems can be analysed in a multi parametric space using generalised path following procedures for augmented equilibrium problems

symplectic structure of the moduli space of flat connection on a riemann surface we consider the canonical symplectic structure on the moduli space of flat g connections on a riemann surface of genus g with a marked points for a being a semisimple lie algebra we obtain an explicit efficient formula for this symplectic form and prove

how to get around a simple quadratic fold we analyse from a theoretical point of view a computational technique previously introduced 2 5 for tracing branches of solutions of nonlinear equations near simple quadratic folds

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

post critical analysis of structures with a nonlinear pre buckling state in the presence of imperfections in the context of the geometrically nonlinear analysis of structures by continuation methods a technique for the determination of the imperfection sensitivity is carried out the technique is based on galerkin s reduction of equilibrium equations in the subspace of extrapolation branches emanating from the bifurcation points the pre and post critical behaviour in the neighbourhood of bifurcation points however is well approximated provided that extrapolations have a large convergence domain to this end an efficient asymptotic representation of the primary and secondary equilibrium paths and procedures that overcome the ill conditioning of the systems defined in the critical points is used here

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

extrapolation locking and its sanitization in koiter s asymptotic analysis abstract this paper shows that the fem implementation of koiter s asymptotic method w t koiter on the stability of elastic equilibrium 1970 ph d thesis delft 1945 english transl nasa tt f10 883 1967 affdl tr70 25 outlined by casciaro et al finite element asymptotic analysis of slender elastic structures a simple approach int j num meth eng 35 1992 1397 1426 provides accurate and reliable results in the critical and post critical analysis of non linear elastic structures care however does have to be taken in implementing apparently minor details to avoid locking effects which adversely affect accuracy and which can destroy the method reliability as the effects related to the finite element interpolation have been discussed before this paper focuses on the non linear locking due to the use implicit in the method of finite distance extrapolations within this scope it is shown that perturbation algorithms based on compatible formulations can imply a strong critical and post critical locking when analysing structures characterized by high stiffness ratios in the presence of moderate pre critical rotations on the contrary perturbation algorithms based on independent extrapolations of displacements and stresses furnish reliable results in excellent agreement with those provided by step by step analysis at a small fraction of its computational cost

mixed formulation and locking in path following nonlinear analysis abstract the arc length riks strategy has rapidly become a standard tool for path following analysis of nonlinear structures due to its theoretical ability to surpass limit points the aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle locking effect intrinsic to the nonlinear nature of the problem as a consequence its sanitization has to be pursued within a reformulation of the structural model the use of a mixed stress displacement variant of the algorithm in particular appears very promising in this respect the topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in 39 it is shown that with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm

a consistent co rotational finite element formulation for geometrically nonlinear dynamic analysis of 3 d beams abstract a co rotational total lagrangian finite element formulation for the geometrically nonlinear dynamic analysis of spatial euler beam with large rotations but small strain is presented the nodal coordinates displacements rotations velocities accelerations and the equations of motion of the structure are defined in a fixed global set of coordinates the beam element has two nodes with six degrees of freedom per node the element nodal forces are conventional forces and moments the kinematics of beam element are defined in terms of element coordinates which are constructed at the current configuration of the beam element both the element deformation nodal forces and inertia nodal forces are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory using the d alembert principle and the virtual work principle in the current element coordinates an incremental iterative method based on the newmark direct integration method and the newton raphson method is employed here for the solution of the nonlinear equations of motion numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method

on a path following method for non linear solid mechanics with applications to structural and cardiac mechanics subject to arbitrary loading scenarios abstract in computational solid mechanics path following methods have been proven useful when dealing with non monotonously evolving loading magnitudes as encountered e g in instability or softening behaviour the present work derives from a displacement based method a cavity volume based path following method and considers its application specifically to cardiac mechanics problems both methods are able to account for an arbitrary number of simultaneously acting loading conditions when applied to the newton raphson method the corresponding loading increments are computed by means of volume increments or point wise prescribed displacement increments respectively no additional variables are required and the physics of the problem at hands is not altered both methods are implemented in an in house meshfree modelling software and successfully applied to non linear elastic and inelastic problems in structural mechanics the cavity volume control method in particular is demonstrated to accurately predict the highly non linear elastic and anisotropic material behaviour encountered when modelling the heart albeit the proposed methods can be equally used e g in finite element methods they are very well suited for meshfree methods where the kronecker delta property does not apply

performance of axially compressed stiffened panels abstract the paper reviews the theoretical foundations of a new analytical model developed for the study of nonlinear interaction of local and overall instabilities of axially compressed stiffened plates it is shown that the mixed second order field contains within itself all the essential secondary local modes liable to be triggered in the interaction any such mode having an eigenvalue of the same order of magnitude as the primary local critical stress must be treated as a fundamental mode fully participating in the interaction in order that the analytical procedure is not riddled by singularities in the evaluation of the mixed second order field and the technique of amplitude modulation is an effective substitute for accounting for a set of local modes of the same transverse description but of slightly differing wave lengths the computational aspects are briefly touched upon and the form of an explicit potential energy function derived is shown the results obtained using the present model are then compared with the theoretical results obtained by koiter and pignataro and tvergaard for several cases the question of optimality is examined the results confirm the earlier finding of tvergaard that the optimally designed panels would have the local critical stresses higher than the overall buckling ones a comparison of the present theory against the experimental results obtained by thompson et al for panels with stocky and thin stiffeners is then presented in all cases there is found to be very good agreement between the experimental and theoretical results

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

improved minimal augmentation procedure for the direct computation of critical points this paper presents a new numerical procedure for the direct computation of critical points for elastic beam structures undergoing large displacements and rotations compared to the approach described by wriggers et al comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155 two main modifications are introduced first following eriksson comput methods appl mech engrg 114 1994 77 comput methods appl mech engrg 156 1998 45 comput methods appl mech engrg 179 1999 265 int j struct stability dynam 1 1 2001 the condition of criticality is expressed by a scalar equation instead of a vectorial one next the present procedure does not use exclusively the extended system obtained from the equilibrium equations and the criticality condition but also introduces intermediate iterations based purely on equilibrium equations under load or displacement control eight numerical examples presenting bifurcation and limit points are used in order to compare the performances of this new method and the one presented in comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155

asymptotic study of the elastic postbuckling behavior of structures by the finite element method abstract within the framework of finite element analysis an asymptotic method is presented for the study of geometrically nonlinear static behavior of thin structures under one parameter conservative loading the method can be applied when the prebuckling behavior is moderately non linear so that the bifurcation analysis is no longer accurate enough an iterative process makes it possible to find suitable deformation modes which enable a good approximation of the structural behavior around the buckling point as in the rayleigh ritz approach a reduced energy can be formulated with only a small number of generalized degrees of freedom the influence of small initial imperfections on the buckling load can easily be analyzed as in koiter s asymptotic method a comprehensive treatment and several improvements of this asymptotic iterative method are given and selected examples illustrate the basic features of the method

rank revealing qr factorizations and the singular value decomposition t chan has noted that even when the singular value decomposition of a matrix a is known it is still not obvious how to find a rank revealing qr factorization rrqr of a if a has numerical rank deficiency this paper offers a constructive proof of the existence of the rrqr factorization of any matrix a of size m x n with numerical rank r the bounds derived in this paper that guarantee the existence of rrqr are all of order f i in comparison with chan s 0 2 it has been known for some time that if a is only numerically rank one deficient then the column permutation l7 of a that guarantees a small rnn in the qr factorization of a n can be obtained by inspecting the size of the elements of the right singular vector of a corresponding to the smallest singular value of a to some extent our paper generalizes this well known result we consider the interplay between two important matrix decompositions the singular value decomposition and the qr factorization of a matrix a in particular we are interested in the case when a is singular or nearly singular it is well known that for any a e r m x n a real matrix with rn rows and n columns where without loss of generality we assume rn n there are orthogonal matrices u and v such that where c is a diagonal matrix with nonnegative diagonal elements we assume that a 2 a2 2 2 on 2 0 the decomposition 0 1 is the singular value decomposition svd of a and the ai are the singular values of a the columns of v are the right singular vectors of a and the columns of u are the left singular vectors of a mathematically in terms of the singular values received december 1 1990 revised february 8 199 1 199 1 mathematics subject classification primary 65f30 15a23 15a42 15a15

beam elements in instability problems this paper deals with the formulation of beam elements for the numerical analysis of instability phenomena in frame type structures total versus co rotational approaches are discussed comparatively for both two dimensional and three dimensional problems and the similarities between the two types are outlined in the context of 3d beam elements special attention is given to the parameterisation of the orthogonal transformation used to define the rotational field of the beam the technique advocated in the paper is based on the so called rotational vector this leads to symmetric stiffness matrices and avoids the need for special updating procedures for the rotational variables a set of test problems for which the critical behaviour is governed by fold cusp and butterfly catastrophes is used to assess the performances of the considered element types it is shown that analytically verified identities in element formulation also hold in numerical application the examples also show how complex instability behaviour can be reproduced by all elements where sufficient accuracy is introduced into the kinematic expressions the analytical derivation of element expressions with symbolic manipulations from stated basic assumptions is consistently used in the paper

generalized deflated block elimination a stable algorithm is presented to solve a nonsingular bordered system of the form left begin array 20 c a b c t d end array right left begin array 20 c x y end array right left begin array 20 c f g end array right where b and c are n by m matrices and the n by n matrix a could be nearly singular with at most mu small singular values the algorithm needs only a solver for a and the solution to an m mu by m mu dense linear system it is thus well suited for problems for which a has easily exploitable structures and m mu ll n such as in continuation methods bifurcation problems and constrained optimization

modal interaction in composite cylinders under hydrostatic pressure abstract a novel and computationally effective procedure for the study of the overall and local instabilities in composite ring stiffened shells subjected to hydrostatic pressure is presented the key feature of the formulation employed is a judicious combination of koiter s amplitude modulation technique and the byskov hutchinson asymptotic procedure for the interactive buckling analysis a potential energy function is formulated in terms of a comparatively small number of degrees of freedom viz the degrees of freedom depicting the amplitude modulating function and a scaling factor of the overall buckling mode as a result of the amplitude modulation there arise in the potential energy function non vanishing cubic interactive terms which control the process of mode interaction imperfection sensitivity under coincident buckling is examined it appears that a 50 reduction of the buckling capacity must probably be allowed for as a result of modal interaction under near coincident buckling the optimality under interactive buckling is also examined with the total volume of the material kept constant and the critical stress ratio varied

a numerical analysis of infinitesimal mechanisms the paper presents a numerical algorithm based on koiter s theory of the elastic stability for detecting the order of infinitesimal mechanisms i e kinematically indeterminate systems of pin jointed bars in cases of one degree of indeterminacy the algorithm improves in terms of computational simplicity and efficiency an analogous algorithm proposed by salerno in 1992 this is shown to be due to the vanishing of the terms higher than the third order of the asymptotic expressions of the energy owing to the use of the green strain measure and a mixed displacement and stress formulation of the problem moreover the proposed algorithm is able to provide a correct definition of mechanism in cases of several degrees of indeterminacy mainly for structures like those first studied by connelly and servatius in 1994 which the paper will treat in depth

asymptotic post buckling fem analysis using corotational formulation finite element asymptotic post buckling analysis being based on fourth order expansions of the strain energy requires that nonlinear structural modeling be accurate to same order at least with respect to the rigid motions of the elements a corotational description is proposed here as a general tool to satisfy this requirement of objectivity by referring each element to a local frame which moves rotates with the element so filtering its rigid motion in this description nonlinearity of the problem derives essentially from the change of reference from the global fixed frame to the local one the strain energy being governed by their relative rotations in finite kinematics this noticeably complicates the algebra for obtaining exact expressions of its variations quite simple basic expressions for the first four corotational derivatives of the strain energy are provided allowing the set up of a fourth order accurate asymptotic analysis starting from standard finite elements based on linear or simplified nonlinear local modelings the formulation is implemented for the analysis of 3d beam assemblages and several numerical results are presented and discussed showing the effectiveness and robustness of the proposed approach in reproducing the nonlinear equilibrium path in both cases of monomodal and coupled multimodal buckling

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

interactive buckling analysis with finite strips on presente un procede semi analytique pour l etude de phenomenes de flambage interactif dans des structures de plaques prismatiques la methode combine les concepts de bandes finies et la theorie des interactions de modes

finite element asymptotic analysis of slender elastic structures a simple approach an asymptotic method directly derived from koiter s theory and suitable for the solution of elastic buckling problems and its natural adaptation to a numerical solution by means of a finite element technique are presented here the order of the extrapolation of the equilibrium equations has been intentionally kept very low because attention has been entirely devoted to all those features theoretical definitions eigenproblem numerical techniques suitable fem implementation which make such an approach competitive with respect to the classic step by step methods for plane frames and 3d pin jointed trusses the performances of the algorithm numerical accuracy and computational cost are compared with those of riks are length method

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

koiter s analysis of thin walled structures by a finite element approach summary this paper refers to the analysis of the postbuckling behaviour of thin walled structures by means of an asymptotic approach based on a finite element implementation of koiter s non linear theory of instability the analysis has been accomplished by using the following assumptions i the structure is described as an assemblage of flat slender rectangular panels ii a non linear kirchhoff type plate theory is used to model each panel iii hc finite elements discretization is used iv linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths respectively v multimodal buckling is considered and vi imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest descent path criterion several numerical results are presented and discussed comparisons with numerical solution obtained by standard incremental codes are given which show the accuracy and reliability of the proposed approach

an object oriented implementation of structural path following abstract the paper describes an object oriented implementation in c of structural path following the necessary parts leading to the full implementation are shown starting from a matrix class library used as a foundation for all matrix computations the paper continues by describing a module containing common finite element algorithms needed for the modeling of general structures with arbitrary constraints the path following algorithms are contained in a separate module the default algorithms are briefly explained and it is shown how these can be redefined through inheritance by algorithms better suited for a specific problem it is shown how the two modules are integrated into a general path following implementation finally some numerical examples showing the efficiency of the implementation and simplicity of using it are presented

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

kitaev lattice models as a hopf algebra gauge theory we prove that kitaev s lattice model for a finite dimensional semisimple hopf algebra h is equivalent to the combinatorial quantisation of chern simons theory for the drinfeld double d h this shows that kitaev models are a special case of the older and more general combinatorial models this equivalence is an analogue of the relation between turaev viro and reshetikhin turaev tqfts and relates them to the quantisation of moduli spaces of flat connections we show that the topological invariants of the two models the algebra of operators acting on the protected space of the kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism are isomorphic this is established in a gauge theoretical picture in which both models appear as hopf algebra valued lattice gauge theories we first prove that the triangle operators of a kitaev model form a module algebra over a hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism both algebras can be viewed as the algebra of functions on gauge fields in a hopf algebra gauge theory the isomorphism between them induces an algebra isomorphism between their subalgebras of invariants which are interpreted as gauge invariant functions or observables it also relates the curvatures in the two models which are given as holonomies around the faces of the lattice this yields an isomorphism between the subalgebras obtained by projecting out curvatures which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models

postbuckling analysis using a general purpose code a new capability for solving postbuckling problems in shell structures is described the matrix theory to adapt newton s method to nonlinear finite element shell analysis is outlined first the matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular the matrix theory suggests a change of variables as part of the usual iterative procedure in newton s method the change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures numerical results from a new option that has been programmed in an existing general purpose code are presented the analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

characterization and computation of singular points with maximum rank deficiency characterizations are given of points at which a mapping has a rank drop possibly greater than one multiple bifurcation points can then be computed without needing a full unfolding of the singularity nondegeneracy conditions are studied in detail

stokes matrices poisson lie groups and frobenius manifolds the purpose of this paper is to point out and then draw some consequences of the fact that the poisson lie group g dual to g gln c may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin g will be fully described in sect 2 the key feature of this point of view is that there is a holomorphic map g g from the dual of the lie algebra to the group g for each choice of diagonal matrix a0 with distinct eigenvalues the irregular type this map is essentially the riemann hilbert map or de rham morphism for such connections we will call it the monodromy map it is generically a local analytic isomorphism the main result is

quantum conjugacy classes of simple matrix groups let g be a simple complex classical group and mathfrak g its lie algebra let mathcal u hbar mathfrak g be the drinfeld jimbo quantization of the universal enveloping algebra mathcal u mathfrak g we construct an explicit mathcal u hbar mathfrak g equivariant quantization of conjugacy classes of g with levi subgroups as the stabilizers

an accurate and efficient implementation of initial geometrical imperfections in the predictor corrector reduced order modeling method abstract a predictor corrector reduced order modeling method based on koiter theory is proposed for nonlinear buckling analysis of thin walled structures with initial geometric imperfections the imperfections are implemented into the von karman kinematics and are considered using an a priori and a posteriori manners respectively in buckling analysis since the nonlinear predictor obtained from the reduced order model with imperfections provides a fairly large step size in path tracing the a priori account of imperfections still shows advantages in computational efficiency the advantages are more significant when considering a posteriori the effects of imperfections in the reduced order model of the structure without imperfections which means building once and for all by simply adding some independent imperfection terms in the reduced system the a posteriori accounts of imperfections with the first second order accuracies are developed respectively to deal with the linear nonlinear prebuckling behaviors and small large imperfection amplitudes the good performance of the proposed method in terms of reliability accuracy and computational effort is demonstrated with several examples using the linear buckling modes or measured geometric shapes as whole field geometric imperfections

spatial fluctuations in reaction diffusion systems a model for exponential growth the spatial fluctuations in an exactly soluble model for the irreversible aggregation of clusters are treated the model is characterized byrate constants k ij i j for the clustering of ani and aj mer anddiffusion constants d j d it is assumed thatd 1 reaction limited aggregation explicit expressions for the correlation functions at equal and at different times are calculated the equal time correlation functions show scaling behavior in the scaling limit the correlation length remains finite ast and the fluctuations becomelarge at large times t t d inany dimension the crossover timet d at which the mean field theory smoluchowski s equation breaks down is given byt d ind these exact results imply that the upper critical dimension of this model isd c and hence that there isno unique upper critical dimension in models for the irreversible aggregation of clusters

poisson lie interpretation of trigonometric ruijsenaars duality a geometric interpretation of the duality between two real forms of the complex trigonometric ruijsenaars schneider system is presented the phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard heisenberg double of u n the collections of commuting hamiltonians of the systems in duality are shown to descend from two families of free hamiltonians on the double which are dual to each other in a poisson lie sense our results give rise to a major simplification of ruijsenaars proof of the crucial symplectomorphism property of the duality map

quantum dressing orbits on compact groups the quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum iwasawa decompositon in the general case quantum dressing orbits are described explicitly as algebras the dual coalgebras consisting of differential operators are related to the quantum weyl elements besides the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible representations of the algebras of quantum functions representation spaces then consist of analytic functions on classical phase spaces these representations are also interpreted in the framework of quantization in the spirit of berezin applied to symplectic leaves on classical compact groups convenient coherent states are introduced and a correspondence between classical and quantum observables is given

non linear analysis of thin walled structures using plate elements this paper presents a simple and efficient rectangular thin plate element for elastic geometric non linear analysis of thin walled structures based on the updated lagrangian formulation and small strain hypothesis this element can accurately predict the non linear pre and post buckling load deflection paths of a plate structure using symbolic manipulation the linear and geometric stiffness matrices for the rectangular thin plate element are derived explicitly thereby eliminating the need for numerical integration the element contains 30 degrees of freedom d o f 14 d o f for the in plane membrane action and 16 d o f for the out of plane bending action several examples involving the large deflection behaviour of thin walled structures are presented to demonstrate the accuracy efficiency and versatility of the method

mixed formulation in koiter analysis of thin walled beams abstract the paper extends the asymptotic mixed formulation proposed in g garcea g salerno sanitizing of locking in koiter perturbation analysis through mixed formulation fourth world congress on computational mechanics buenos aires argentina 1988 g garcea g salerno r casciaro comput methods appl mech engrg 180 1 2 1999 137 167 to structures composed of an assemblage of flat slender elastic panels the same type of structure has already been investigated in a d lanzo g garcea r casciaro int j numer methods engrg 38 1995 2325 2345 a d lanzo g garcea int j numer methods engrg 39 1996 3007 3031 with a compatible formulation and using the usual hypothesis of negligible precritical rotations in order to avoid the extrapolation locking see also g garcea g a trunfio r casciaro comput methods appl mech engrg 165 1 4 1998 137 167 produced by the high axial to flexural stiffness ratio of the panels the use of a mixed formulation provides a consistent way of eliminating the extrapolation locking without introducing ad hoc approximations it is thus both reliable and accurate when combined with a quadratic strain displacement relationship as presented here it gives zero fourth order energy variations further computational simplifications and a natural linearization of the eigenvalue problem required by the buckling analysis an extensive numerical testing shows the effectiveness and robustness of the proposed approach

coagulation diffusion systems derivation and existence of solutions for the diffuse interface structure equations abstract this paper considers an infinite system of partial differential equations the coagulation diffusion equations which add spatial diffusion to the classical coagulation equations the main emphasis is placed on deriving an infinite system of ordinary differential equations which described the structured interface between reacting coagulation and dilute concentration existence of solutions to interfacial equaitons is proven under spatial boundary conditions

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

asymptotic post buckling analysis of rectangular plates by hc finite elements the present paper extends the finite element perturbation approach already presented for pin jointed and framed structures15 to rectangular thin plates koiter s asymptotic strategy2 is coupled with a high continuity finite element discretization of the plate 22 the consistency of the discrete model is discussed from the kinematical and numerical points of view and several numerical tests are reported it appears that use of the hc elements makes the perturbation algorithm insensitive to the locking phenomenon occurring in the evaluation of the postbuckling behaviour it also allows the use of very fine discretization meshes at low computational cost

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

finite rotation analysis and consistent linearization using projectors abstract a systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis this procedure is a generalization of the work presented in computers and structures volume 30 pp 257 267 the basis of our approach is the element independent co rotational algorithm where the element rigid body motion translations and rotations is separated from the deformational part of its total motion the variation of this co rotational relation results in a projector matrix with the property that a consistent internal force vector is invariant under its action the consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector this results in an unsymmetric tangent matrix whose anti symmetric part depends on the out of balance force vector in this paper we prove that using the symmetric part of the tangent matrix the newton iteration retains its quadratic rate of convergence this approach has been used to solve a number of large rotation test example problems the results demonstrate that it is possible to analyze structures undergoing large rotations within a general co rotational framework using simple and economical finite elements the resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre and post processors to the element routines

quasi hamiltonian geometry of meromorphic connections for each connected complex reductive group g we find a family of new examples of complex quasi hamiltonian g spaces with g valued moment maps these spaces arise naturally as moduli spaces of suitably framed meromorphic connections on principal g bundles over a disk and they generalise the conjugacy class example of alekseev malkin and meinrenken 3 which appears in the simple pole case using the fusion product in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy stokes data of meromorphic connections over arbitrary genus riemann surfaces together with a new proof of the symplectic nature of isomonodromic deformations of such connections

element behavior in post critical plane frame analysis depending on the type of uni modal cuspoid catastrophe which governs the local behavior of a structure at a simple critical point correct prediction of the post critical path requires an accurat

on the existence and computation of lu factorizations with small pivots let a be an n by n matrix which may be singular with a one dimensional null space and consider the lu factorization of a when a is exactly singular we show conditions under which a pivoting strategy will produce a zero n th pivot when a is not singular we show conditions under which a pivoting strategy will produce an nth pivot that is o g or o k a where is the smallest singular value of a and k a is the condition number of a these conditions are expressed in terms of the elements of a in general but reduce to conditions on the elements of the singular vectors corresponding to when a is nearly or exactly singular they can be used to build a 2 pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices as an example we exhibit an lu factorization of the n by n upper triangular matrix

a new model in the calogero ruijsenaars family hamiltonian reduction is used to project a trivially integrable system on the heisenberg double of su n n to obtain a system of ruijsenaars type on a suitable quotient space this system possesses bcn symmetry and is shown to be equivalent to the standard three parameter bcn hyperbolic sutherland model in the cotangent bundle limit

structures in bcn ruijsenaars schneider models we construct the classical r matrix structure for the lax formulation of bcn ruijsenaars schneider systems proposed in commun math phys 115 127 1988 the r matrix structure takes a quadratic form similar to the an ruijsenaars schneider poisson bracket behavior although the dynamical dependence is more complicated commuting hamiltonians stemming from the bcn ruijsenaars schneider lax matrix are shown to be linear combinations of particular koornwinder van diejen external fields ruijsenaars schneider models for specific values of the exponential one body couplings uniqueness of such commuting hamiltonians is established once the first of them and the general analytic structure are given

deformed integrable σ models classical r matrices and classical exchange algebra on drinfel d doubles we describe a unifying framework for the systematic construction of integrable deformations of integrable models within the hamiltonian formalism it applies equally to both the yang baxter type as well as gauged wzw type deformations which were considered recently in the literature as a byproduct these two families of integrable deformations are shown to be poisson lie t dual of one another

interactive buckling in thin walled beams i theory abstract a method is derived for the analysis of mode interaction in thin walled elastic beams a nonlinear plate theory is employed for the plate segments of the beam in which the exact nonlinear expressions are used for the middle surface strain measures but the bending measures are linearized the beam is subjected to a combination of axial compression and a constant bending moment and it is assumed to be simply supported at the ends in the calculation of the total potential energy the influence of the prebuckling deformations is neglected the finite strip method is used with the transverse variation of all three displacement components described by cubic polynomials in the arc length the nonlinear mode interaction is analysed by means of koiter s asymptotic theory of stability some applications of the method to representative problems are presented in a subsequent paper by the authors this shows that significant mode interaction and imperfection sensitivity occur in these structures

mode jumping and attractive paths in multimode elastic buckling this paper summarizes a part of the first author s ph d thesis completely devoted to multimode elastic buckling within an fem strategy the theoretical arguments unfold among critical points on radial paths the unique post critical paths variationally defined algebraic characterizations proposition demonstrations and so on by aiming to prove that the complexity of the phenomenon of multimode buckling secondary bifurcations post critical attractive paths can be theoretically explained 1997 by john wiley sons ltd

on nonlinear beam models from the point of view of computational post buckling analysis abstract the buckling and post buckling analysis of elastic planar frames is considered and the use of geometrically exact beam models is thereby advocated it is shown that usual technical beam models fail to predict correctly the curvature of the post buckling curve at bifurcation even for standard problems of elastic stability theory it is also argued that versatile and efficient computational procedures for bifurcation analysis of general planar frames are to be based on unconstrained beam models some remarks on finite element representation of nonlinear beam models are passed in conclusion

non coboundary poisson lie structures on the book group all possible poisson lie pl structures on the 3d real lie group generated by a dilation and two commuting translations are obtained their classification is fully performed by relating these pl groups to the corresponding lie bialgebra structures on the corresponding book lie algebra by construction all these poisson structures are quadratic poisson hopf algebras for which the group multiplication is a poisson map in contrast to the case of simple lie groups it turns out that most of the pl structures on the book group are non coboundary ones moreover from the viewpoint of poisson dynamics the most interesting pl book structures are just some of these non coboundaries which are explicitly analysed in particular we show that the two different q deformed poisson versions of the sl 2 r algebra appear as two distinguished cases in this classification as well as the quadratic poisson structure that underlies the integrability of a large class of 3d lotka volterra equations finally the quantization problem for these pl groups is sketched

a c0 triangular plate element with one point quadrature on developpe un element de plaque triangulaire du type de mindlin qui utilise des champs lineaires pour les rotations et les deflexions transversales l element ne necessite qu un seul point de quadrature ce qui est tres interessant pour l analyse non lineaire

how to recognize the order of infinitesimal mechanisms a numerical approach structures presenting kinematical inderterminacy are usually called mechanisms this paper is entirely concerned with assemblies which reveal themselves to be mechanisms at a null value of the load among them a first distinction is made between infinitesimal and finite ones the former being characterized by one or several directions of lower but not zero stiffness whereas the latter show at least one finite admissible displacement for which none of the bars undergoes any elongation moreover there exists the possibility to make a further distinction among the infinitesimal mechanisms according to which is the order of the stiffness along the direction considered above the way of evaluating this order is to perform a local analysis of the strain energy of the assembly once the displacement field is parametrized in terms of a suitable parameter by means of a finite element technique this analysis can be easily performed through the numerical approach presented in this report

how to improve efficiency and robustness of the newton method in geometrically non linear structural problem discretized via displacement based finite elements abstract in this paper we show how to significantly improve the robustness and the efficiency of the newton method in geometrically non linear structural problems discretized via displacement based finite elements the strategy is based on the relaxation of the constitutive equations at each integration point this leads to an improved iterative scheme which requires a very low number of iterations to converge and can withstand very large steps in step by step analyses the computational cost of each iteration is the same as the original newton method the impressive performances of the proposal are shown by many numerical tests in geometrically non linear analysis the proposed strategy called mip newton seems worthy to replace the standard newton method in any finite element code based on displacement formulations its implementation in existing codes is very easy

on linear constraints for newton raphson corrections and critical point searches in structural f e problems the paper discusses the introduction of constraining equations in the tangential stiffness relation used to calculate the responses to different load cases in solution algorithms for non linear mechanical finite element f e problems an alternative to the normal two phase solution method is discussed this method is used to represent different iteration constraints and in conjunction with the search for critical solution points numerical tests are presented evaluating the efficiency of different iteration constraints for a model problem practically useful criteria for critical points are discussed the basic methods for search of such points and some numerical aspects are discussed and evaluated for three different problems

co rotational flat facet triangular elements for shell instability analyses abstract this paper investigates the formulation of co rotational flat facet triangular elements for the numerical analysis of instability phenomena in shell structures the elements have three nodes with six degrees of freedom at each node the term co rotational relates here to the provision of a local system that continuously rotates and translates with the element following mainly nour omid and rankin b nour omid and c c rankin finite rotation analysis and consistent linearization using projectors comput methods appl mech engrg 93 1991 353 384 the definition of an element resorts to a change of variables from the local frame to the global one this is done through the use of a projector matrix which relates the variations of the local displacements to the variations of the global ones by extracting the rigid body modes from the latter the main difference from the original formulation lies in the parameterization of 3d finite rotations in contrast to the paper by nour omid and rankin a parameterization based on the rotational vector is here adopted and thus an additional change of variables has to be performed as a result the rotational variables become additive and the necessity of a special updating procedure is avoided the main feature of the adopted formulation is its independence of the local assumptions used to derive the internal forces and tangent stiffness in local coordinates for a certain class of elements i e elements with the same number of nodes and degrees of freedom the main co rotational framework is the same using this property three types of local formulations are considered a set of carefully chosen test problems is used in order to assess the performances of the three element types

numerical non linear analysis of secondary buckling in stability problems abstract in this paper we discuss the numerical calculation of secondary buckling of structures this kind of stability problem contains two parameters one is conservative load parameter the other is auxiliary one and satisfies z 2 symmetry by using the symmetry and introducing a new parameter we construct an augmented system the system avoids the singularities of usual ones adjusting the newly introduced parameter one can calculate a double buckling load and two corresponding buckling modes secondary buckling load and secondary buckling mode directly without tracing primary post buckling path here the use of the augmented system yields a quadratically convergent iteration scheme we also discuss the implementation of newton s method solving the system a partitioning procedure with this algorithm the amount of computation is largely saved

variational formulation in the geometrically nonlinear thin elastic shell theory abstract the variational formulation for thin elastic shells undergoing large deflections is discussed no restrictions are imposed on the magnitude of the rotational part of the deformation gradient yet it is assumed that the strains remain small everywhere in the shell ten functional and associated variational principles are derived including the hu washizu the hellinger reissner the generalized complementary energy and the stationary potential energy principles the total lagrangian description tld is exclusively used