canonical factorization and diagonalization of baxterized braid matrices explicit constructions and applications braid matrices r corresponding to vector representations are spectrally decomposed obtaining a ratio fi fi for the coefficient of each projector pi appearing in the decomposition this directly yields a factorization f 1f for the braid matrix implying also the relation r r i this is achieved for glq n soq 2n 1 soq 2n spq 2n for all n and also for various other interesting cases including the eight vertex matrix we explain how the limits can be interpreted to provide factorizations of the standard non baxterized braid matrices a systematic approach to diagonalization of projectors and hence of braid matrices is presented with explicit constructions for glq 2 glq 3 soq 3 soq 4 spq 4 and various other cases such as the eight vertex one for a specific nested sequence of projectors diagonalization is obtained for all dimensions the canonical property implemented in the diagonalizers is mutual orthogonality of the rows in each factor f our diagonali

solving the two dimensional constant quantum yang baxter equation a detailed analysis of the constant quantum yang baxter equation rk1k2j1j2 rl1k3k1j3rl2l3k2k3 rk2k3j2j3 rk1l3j1k3rl1l2k1k2 in two dimensions is presented leading to an exhaustive list of its solutions the set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the grobner basis methods each solution was then transformed into a canonical form based on the various trace matrices of r for final elimination of duplicates and subcases if we use homogeneous parametrization the solutions can be combined into 23 distinct cases modulo the well known c p and t reflections and rotations and scalings r q q r q q 1

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

finite deformation frictional mortar contact using a semi smooth newton method with consistent linearization a two dimensional finite deformation frictional contact formulation with coulomb s law is presented the approach considers multibody contact and is based on a mortar formulation the enforcement of contact constraints is realized with dual lagrange multipliers these alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase friction kinematic variables are formulated in an objective way and enter non smooth complementarity functions for expressing the contact constraints an active set strategy is derived by applying a semismooth newton method which treats contact nonlinearities material and geometrical nonlinearities in one single iterative scheme by further carrying out a consistent linearization for both normal and frictional contact forces and constraints a robust and highly efficient algorithm for linear and higher order quadratic interpolation is achieved efficiency of the proposed method and quality of results are demonstrated in serveral exemples

aspects of a new class of braid matrices roots of unity and hyperelliptic q for triangularity l algebra link invariants noncommutative spaces various properties of a class of braid matrices presented before are studied considering n2 n2 n 3 4 vector representations for two subclasses for q 1 the matrices are nontrivial triangularity r2 i corresponds to polynomial equations for q the solutions ranging from roots of unity to hyperelliptic functions the algebras of l operators are studied as a crucial feature one obtains 2n central grouplike homogenous quadratic functions of lij constrained to equality among themselves by the rll equations they are studied in detail for n 3 and are proportional to i for the fundamental 3 3 representation and hence for all iterated coproducts the implications are analyzed through a detailed study of the 9 9 representation for n 3 the turaev construction for link invariants is adapted to our class a skein relation is obtained noncommutative spaces associated to our class of r are constructed the transfer matrix map is implemented with the n 3 case as example for an iterated construction of no

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

duality for exotic bialgebras in the classification of hietarinta three triangular 4 4 r matrices lead via the frt formalism to matrix bialgebras which are not deformations of the trivial one in this paper we find the bialgebras which are in duality with these three exotic matrix bialgebras we note that the l t duality of frt is not sufficient for the construction of the bialgebras in duality we find also the quantum planes corresponding to these bialgebras both by the wess zumino r matrix method and by manin s method

2d contact smooth formulation based on the mortar method abstract in this paper we apply a smoothing technique to solve contact problems using a segment to segment formulation based on the mortar method in contact smoothing the c 0 finite element description of the geometry is replaced by an improved smoothed one usually with c 1 continuity so far these techniques have been applied to solve contact problems based on node to segment interaction some numerical examples are solved to compare the performance of the smoothing and non smoothing formulations the examples show that the non smoothed mortar formulation is improved mainly for problems with matching curved surfaces discretized with non conforming meshes

a contact smoothing method for arbitrary surface meshes using nagata patches abstract this paper presents a contact surface smoothing method combined with the node to segment discretization technique to solve large deformation frictional contact problems between deformable bodies the nagata patch interpolation is used to smooth the surface mesh providing a master surface with quasi g 1 continuity between patches moreover the local support of the interpolation method allows to deal with surface meshes of arbitrary topology regular and irregular finite element discretizations as well as hybrid meshes the non physical oscillations in the contact force evolution induced by the faceted contact surface representation are reduced using the proposed smoothing method furthermore the smooth representation of the master surface allows a more accurately evaluation of the resulting stresses and forces while providing an important improvement in convergence behaviour four representative numerical examples are used to demonstrate the advantages of the proposed contact smoothing method the results show a significant improvement in the accuracy robustness and performance of the numerical simulations using the smoothing approach when compared with the piecewise faceted contact surface description

transient three dimensional domain decomposition problems frame indifferent mortar constraints and conserving integration the present work deals with transient large deformation domain decomposition problems the tying of dissimilar meshed grids is performed by applying the mortar method in this connection a reformulation of the original linear mortar constraints is proposed which retains frame indifference for arbitrary discretizations of the interface furthermore a specific coordinate augmentation technique is proposed to make possible the design of an energy momentum scheme numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy momentum scheme for three dimensional problems copyright 2009 john wiley sons ltd

modified braid equations for soq 3 and noncommutative spaces general solutions of the rtt equation with a maximal number of free parameters in the specrtal decomposition of vector soq 3 r matrices are implemented to construct modified braid equations mbe these matrices conserve the given standard group relations of the nine elements of t but are not constrained to satisfy the standard braid equation be apart from q and a normalization factor our r contains two free parameters instead of only one such parameter for deformed unitary algebras studied in a previous paper math qa 0009178 where the nonzero right hand side of the mbe had a linear term proportional to r 12 r 23 in the present case the rhs is in general nonlinear several particular solutions are given sec ii and the general structure is analyzed appendix a our formulation of the problem in terms of projectors yields also two new solutions of standard nonmodified braid equation sec ii which are further discussed appendix b the noncommutative three spaces obtained by i

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

modified braid equations baxterizations and noncommutative spaces for the quantum groups glq n soq n and spq n modified braid equations satisfied by generalized r matrices for a given set of group relations obeyed by the elements of t matrices are constructed for q deformed quantum groups glq n soq n and spq n with arbitrary values of n the baxterization of r matrices treated as an aspect complementary to the modification of the braid equation is obtained for all these cases in particularly elegant forms a new class of braid matrices is discovered for the quantum groups soq n and spq n the r matrices of this class while being distinct from the restrictions of the universal r matrix to the corresponding vector representations satisfy the standard braid equation the modified braid equation and the baxterization are obtained for this new class of r matrices diagonalization of the generalized r matrices is studied the diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way giving directly the eigenvalues of the corresponding r matrices applications of such

rtt relations a modified braid equation and noncommutative planes with the known group relations for the elements a b c d of a quantum matrix t as input a general solution of the rtt relations is sought without imposing the yang baxter yb constraint for r or the braid equation for r pr for three biparametric deformatios gl p q 2 gl g h 2 and gl q h 1 1 the standard the nonstandard and the hybrid one respectively r or r is found to depend apart from the two parameters defining the deformation in question on an extra free parameter k such that r 12 r 23 r 12 r 23 r 12 r 23 k k1 1 k k2 1 r 23 r 12 with k1 k2 1 p q 1 1 and 1 1 q respectively only for k k1 or k k2 one has the braid equation arbitray k corresponds to a class conserving the group relations independent of k of the mqybe or modified quantum yb equations studied by gerstenhaber giaquinto and schack various properties of the triparametric r k p q r k g h and r k q h are studied in the larger space of the modified braid equation mbe even r k p

resolution of fourth order problems by the mortar element method we present a resolution of fourth order problems by the mortar element method in the case of spectral discretizations we prove that this approach leads to a well posed discrete problem and the numerical analysis gives an optimal error estimate

geometrical tools for quantum euclidean spaces we apply one of the formalisms of noncommutative geometry to n q the quantum space covariant under the quantum group so q n over n q there are two so q n covariant differential calculi for each we find a frame a metric and two torsion free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature this generalizes results found in a previous article for the case of 3 q as in the case n 3 one has to slightly enlarge the algebra n q for n odd one needs only one new generator whereas for n even one needs two as in the particular case n 3 there is a conformal ambiguity in the natural metrics on the differential calculi over n q while in our previous article the frame was found by hand here we disclose the crucial role of the quantum group covariance and exploit it in the construction as an intermediate step we find a homomorphism from the cross product of n q with u q so n into n q an interesting result in itself

a continuum based finite element formulation for the implicit solution of multibody large deformation frictional contact problems in this paper a formulation is presented for the finite element treatment of multibody large deformation frictional contact problems the term multibody is used to mean that when two bodies mechanically contact both may be deformable a novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first before deriving the corresponding finite element equations this feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general in particular the approach yields a characterization of the frictional constraint assuming a coulomb law suitable for arbitrary discretizations in either two or three dimensions a geometric framework is constructed within which both frictionless and frictional response are naturally described making subsequent finite element discretization a straightforward substitution of finite dimensional solution spaces for their continuum counterparts to our knowledge this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature the development includes exact linearization of the statement of virtual work which enables optimal convergence properties for newton raphson solution strategies and which appears to be highly desirable if not essential for the general robustness of implicit finite element techniques since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non linear behaviour this assertion is substantiated through presentation of a variety of examples in both two and three dimensions

numerical implementation of two nonconforming finite element methods for unilateral contact we consider the finite element approximation of the unilateral contact problem between elastic bodies we are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact it follows that the nodes of both bodies located on the contact surface do not fit together we present two different approaches in order to define unilateral contact on nonmatching meshes the first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way on the contrary the second one expresses local node on segment contact conditions in both cases the theoretical approximation properties are given then we implement and compare the two methods

a spectral element methodology tuned to parallel implementations abstract we present in this paper a modification of the conforming spectral element method for the approximation of the poisson equation this new procedure allows the implementation on a parallel machine in an easier and faster way the resulting method is nonconforming but the numerical analysis shows that the error between the exact solution and the discrete solution is of the same asymptotic order as the error of the original conforming method

connection coefficients for basic harish chandra series basic harish chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems the associated connection coefficients are explicitly computed in terms of jacobi theta functions we interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of cherednik s root system analogs of the quantum knizhnik zamolodchikov equations they thus give rise to explicit elliptic solutions of root system analogs of dynamical yang baxter and reflection equations applications to quantum c functions basic hypergeometric functions reflectionless difference operators and multivariable baker akhiezer functions are discussed

a segment to segment mortar contact method for quadratic elements and large deformations abstract this paper describes a new extension of mortar based segment to segment algorithms for large deformation contact analysis permitting their use when quadratic interpolations are used in the underlying finite element method the work builds upon results previously described by puso and laursen m a puso t a laursen a mortar segment to segment contact method for large deformation solid mechanics comput methods appl mech engrg 193 2004 601 629 m a puso t a laursen a mortar segment to segment frictional contact method for large deformations comput methods appl mech engrg 193 2004 601 629 where methods were developed for quasistatic and implicit dynamic analysis in the context of linear element interpolations here we see that many algorithmic ideas from the linear case can be adopted for use with quadratic elements along with different alternative interpolations of the mortar contact multipliers the result is a class of methods appropriate for use for either mesh tying true large sliding simulation frictionless or frictional and readily admits mixing of element types several numerical examples are given to demonstrate the effectiveness of the approach

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully