geometrically non linear 3d finite element analysis of micropolar continuum abstract in this work a 3d geometrically non linear micropolar finite element formulation based on biot like tensor representation of stresses couple stresses strains and curvature is presented the discrete approximation is based on hexahedral finite elements with lagrange interpolation for displacements and rotational spins the complete residual derivation linearisation and update are presented in detail the elements are tested against a non linear generalisation of the linear micropolar pure bending problem derived in this work and it is shown that the elements converge to the derived solution the elements are additionally tested on a combined bending and torsion problem and a well known 45 bend as a genuine problem involving large 3d rotations both also analysed in the framework of micropolar elasticity it is observed that the derived finite elements are reliable and robust in modelling finite deformation problems in micropolar elasticity exhibiting large displacements and large rotations and converge to reference geometrically non linear classical elasticity solutions for small micropolar effects with increasing micropolar effects the response becomes stiffer and the presented numerical examples may serve as benchmark problems to test geometrically non linear micropolar finite elements including those involving large 3d rotations

an objective 3d large deformation finite element formulation for geometrically exact curved kirchhoff rods the objective of this work is the development of a new finite element formulation for beams according to the kirchhoff theory of thin rods which includes the deformation states of axial tension torsion and bending the proposed formulation accounts for large deformations in three dimensional problem settings consisting of slender prismatic beams with arbitrarily curved initial geometries and arbitrary cross section shapes like in geometrically exact reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations rotations and strains a novel orthogonal interpolation strategy is applied to the triad field representing the cross section orientation in order to fulfill the kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem the continuity requirements resulting from the corresponding weak form are fulfilled by a cubic hermite interpolation thus leading to a c1 continuous representation of the beam centerline fundamental properties such as objectivity path independence consistency and accuracy of the developed beam element are verified by means of suitable numerical examples

kernels of toeplitz operators via bourgain s factorization theorem using bourgain s factorization theorem we characterize the subspaces of hp 1 p that coincide with the kernels of toeplitz operators this is related to but entirely independent of earlier work of e hayashi one consequence of our characterization is that for some inner functions the class of uniqueness sets for kp depends on p here kp defhp zhp is the star invariant subspace in hp generated by moreover a construction is provided of blaschke products b and b for which the dimension function p dim kpb bhp has prescribed jumps at prescribed points

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

explicit free floating beam element summary a two node free floating beam element capable of undergoing arbitrary large displacements and finite rotations is presented in explicit form the configuration of the beam in three dimensional space is represented by the global components of the position of the beam nodes and an associated set of convected base vectors directors the local constitutive stiffness is derived from the complementary energy of a set of six independent deformation modes each corresponding to an equilibrium state of constant internal force or moment the deformation modes are characterized by generalized strains formed via scalar products of the element related vectors this leads to a homogeneous quadratic strain definition in terms of the generalized displacements whereby the elastic energy becomes at most bi quadratic additionally the use of independent equilibrium modes to set up the element stiffness avoids interpolation of kinematic variables resulting in a locking free formulation in terms of three explicit matrices a set of classic benchmark examples illustrates excellent performance of the explicit beam element copyright 2014 john wiley sons ltd

on a non linear formulation for curved timoshenko beam elements considering large displacement rotation increments summary an incremental total lagrangian formulation for curved beam elements that includes the effect of large rotation increments is developed a complete and symmetric tangent stiffness matrix is obtained and the numerical results show in general an improvement over the standard formulation where the assumption of infinitesimal rotation increments is made in the derivation of the tangent stiffness matrix

bifurcations and internal resonances in space curved rods abstract the paper discusses a geometrically non linear model of space curved beams the timoshenko type model is used which includes the shear effects and rotatory inertia the motion of the system is described by a non linear matrix equation which accounts for non linearities up to the second order to solve the non linear problem discretization methods based on the galerkin method are used and the resulting non linear eigenvalue problem is then solved by the continuation methods the analysis of the discretized problem allows the study of the bifurcations of solutions the post bifurcation analysis is used to explain the phenomenon of internal resonance which is of much importance in the prediction of the dynamic response of the analyzed systems the approach is illustrated by numerical examples which consider the free vibration of beams and circular arches with different boundary conditions

one dimensional random attractor and rotation number of the stochastic damped sine gordon equation abstract this paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine gordon equation with homogeneous neumann boundary condition it is shown that for any positive damping and diffusion coefficients the equation possesses a random attractor and when the damping and diffusion coefficients are sufficiently large the random attractor is a one dimensional random horizontal curve regardless of the strength of noise hence its dynamics is not chaotic it is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful

existence and upper semicontinuity of random attractors for non autonomous stochastic strongly damped sine gordon equation on unbounded domains abstract in this paper we study the asymptotic behavior of solutions of the non autonomous stochastic strongly damped sine gordon equation driven by multiplicative noise defined on an unbounded domain first we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions second we consider the existence and upper semicontinuity of random attractors for the equation

an isogeometric implicit g1 mixed finite element for kirchhoff space rods abstract in the paper we present a multi patch isogeometric mixed element with implicit geometric continuity at the ends for the analysis of non polar kirchhoff space rods rotations are introduced at the ends of the element analogously to the hermitian interpolation performing a change of basis for the configuration space a mixed method in which the internal forces are interpolated by means of an l 2 projection leading to a b formulation is used to cope with the membrane flexural locking that affects the solution the efficiency of the method in avoiding locking and instabilities is investigated the formulation adopted has the advantage with respect to other formulations that a non full global stiffness matrix is obtained this approach appears very appealing for efficient analyses of general assemblies of 3d non polar elements

frame indifferent beam finite elements based upon the geometrically exact beam theory this paper describes a new beam finite element formulation based upon the geometrically exact beam theory in contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame indifference or objectivity of the underlying beam theory the space interpolation of rotational degrees of freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory copyright 2002 john wiley sons ltd

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

nonlinear isogeometric spatial bernoulli beam abstract a new element formulation of a geometrically nonlinear spatial curved beam assuming bernoulli theory including torsion without warping is proposed the element formulation is derived directly from the 3d continuum by means of nonlinear kinematics thus accounting for large displacements the geometric description of the proposed element is adapted from the spatial rod of greco and cuomo 2013 and extended to a nonlinear element formulation the proposed formulation can handle arbitrary orientations of the cross section along the beam in this publication nurbs are used as basis functions for discretization since they can easily provide the required c 1 continuity between elements the presented element formulation has four degrees of freedom three for displacements and one for the rotation around the center line in order to prove the accuracy of the developed spatial bernoulli beam several numerical examples are presented and compared to analytic solutions and other element formulations

velocity based approach in non linear dynamics of three dimensional beams with enforced kinematic compatibility abstract in the paper we present a new finite element formulation for the dynamic analysis of geometrically exact three dimensional beams we limit our studies to implicit time integration schemes and possible approaches for increasing their robustness and numerical stability in contrast to standard displacement rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities to describe the rotational degrees of freedom quaternions are used the time integration scheme and the governing equations of the three dimensional beam are modified accordingly in the numerical implementation the galerkin type discretization is employed to obtain the finite element formulation of the problem the result of our studies is simple but accurate efficient and robust numerical model

a geometrically non linear model of rotating shafts with internal resonance and self excited vibration a geometrically non linear model of the rotating shaft is introduced which includes karman non linearity non linear curvature effects large displacements and rotations as well as gyroscopic effects through applying timoshenko type assumptions the shear effects are also included in the model convenient matrix descriptions are used in order to facilitate the analysis based on galerkin and continuation methods the model is used to analyze the phenomenon of internal resonance the influence of some of the system s parameters on the amplitude and frequency of self excited vibration is investigated

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

an implicit multi patch b spline interpolation for kirchhoff love space rod abstract the paper presents a novel formulation for the isogeometric analysis of assemblies of kirchhoff love space rod elements introducing a multi patch implicit g 1 formulation so that an automatic non singular stiffness operator is obtained without the need of adding continuity conditions the goal is achieved using a polar decomposition of the deformation of the first and last segments of the control polygon that allows to introduce directly the end rotations as degrees of freedom both parametric and geometric continuity can be obtained in this way we use bezier and b spline interpolations and we show that they are able to attain very good accuracy for developing a 3d exact curve element with geometric torsion pre twisted rod in the paper the performance of the multi patch elements is examined comparing the rates of convergence of the l 2 error norm for the multi patch and single patch formulations it is shown that the rate of convergence remains the same although in certain cases the accuracy is lower for the multi patch solutions

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

skew ornstein uhlenbeck processes and their financial applications in this paper we investigate a special class of skew diffusions skew ornstein uhlenbeck abbr ou processes whose scale and speed densities are both piecewise functions the existence and uniqueness of solutions regarding the related stochastic differential equations abbr sdes with local time are established as well as the construction through time changes afterwards we concentrate on three computing issues including the explicit expressions of transition densities the cumulative distributions and laplace transforms of the first hitting times for skew ou processes with the hypothesis on asset dynamics two financial instances in the field of credit risk are illustrated at the end of this paper

b spline interpolation of kirchhoff love space rods the paper deals with the isogeometric analysis via b splines of space rods under kirchhoff love hypotheses the approach was used by gontier and vollmer 12 for developing a plane curve element within the framework of the timoshenko rod model but they adopted only one patch to represent entirely the geometry of the rod furthermore the authors developed their theory only for plane elements in this work we develop an isogeometric approach for the numerical analysis of the 3d kirchhoff love rod theory we use b splines and bezier interpolations and we show that they are able to attain very good accuracy for rod structures particularly for developing a 3d exact curve element with geometric torsion the paper presents an original parametrization of the geometric torsion that proves to be very effective the use of b splines allows to avoid discontinuities on the geometrical quantities and particularly on the normal fields so that even relatively low order interpolation functions are able to yield accurate results

finite element of slender beams in finite transformations a geometrically exact approach this article is devoted to the modelling of thin beams undergoing finite deformations essentially due to bending and torsion and to their numerical resolution by the finite element method the solution proposed here differs from the approaches usually implemented to treat thin beams as it can be qualified as geometrically exact two numerical models are proposed the first one is a non linear euler bernoulli model while the second one is a non linear rayleigh model the finite element method is tested on several numerical examples in statics and dynamics and validated through comparison with analytical solutions experimental observations and the geometrically exact approach of the reissner beam theory initiated by simo the numerical result shows that this approach is a good alternative to the modelling of non linear beams especially in statics copyright 2003 john wiley sons ltd

discrete variational lie group formulation of geometrically exact beam dynamics the goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics by using a lie group variational integrator both spatial and temporal discretization are implemented in a geometry preserving manner the resulting scheme preserves both the discrete momentum maps and symplectic structures and exhibits almost perfect energy conservation comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically

computational aspects of vector like parametrization of three dimensional finite rotations theoretical and computational aspects of vector like parametrization of three dimensional finite rotations which uses only three rotation parameters are examined in detail in this work the relationship of the proposed parametrization with the intrinsic representation of finite rotations via an orthogonal matrix is clearly identified careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of reissner s non linear beam theory pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2 node three dimensional beam element a number of numerical simulations in three dimensional finite rotation analysis are presented in order to illustrate the proposed approach

limiting dynamics for stochastic reaction diffusion equations on the sobolev space with thin domains abstract this paper is devoted to bi spatial random attractors of the stochastic reaction diffusion equation when the terminate space is the sobolev space on a thin domain where the nonlinearity can be decomposed into two functions with p q growth exponents by means of a computation method of induction it is shown that the difference of solutions near the initial time is integrable of k p 2 k 2 order this higher order integrability shows continuity of the solution operator from the square lebesgue space to k p 2 k 2 order lebesgue spaces in particular the 2 p 2 order integrability shows continuity of the solution operator from the square lebesgue space to the sobolev space which further shows existence of a random attractor in the sobolev space when the initial space is the square lebesgue space moreover the higher order integrability can be uniform with respect to all thin domains which provides uniformly asymptotic compactness of the random dynamical systems as a conclusion the upper semi continuity of attractors under the sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

geometrically exact 3d beam theory implementation of a strain invariant finite element for statics and dynamics geometrically exact 3d beam theory has been used as a basis for development of a variety of finite element formulations it has recently become apparent that the important requirement of objectivity of adopted strain measures although provided by the theory itself does not automatically extend to a finite element formulation in this paper we present a new finite element formulation of the geometrically exact 3d beam theory specifically designed to preserve the objectivity of the adopted strain measures in order to do so the current local rotations are interpolated in a manner similar to that adopted in co rotational approaches however no approximations typical for co rotational approaches are introduced into the procedure so in contrast to co rotational formulations the present formulation fully preserves the geometric exactness of the theory a range of numerical examples serves to illustrate the problem and to assess the formulation

a kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle the present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by simo and vu quoc a geometrically exact rod model incorporating shear and torsion warping deformation int j solids structures 27 3 1991 371 3931 along the lines of the work on the planar beam theory presented by saje a variational principle for finite planar deformation of straight slender elastic beams internat j solids and structures 26 1990 887 9001 we apply exact non linear kinematic relationships of the space finite strain beam theory assuming the bernoulli hypothesis and neglecting the warping deformations of the cross section finite displacements and rotations as well as finite extensional shear torsional and bending strains are accounted for in the formulation a deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame rigidly attached to the cross section of the beam the position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step also the incremental rotational vector is introduced which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points because in 3 d space both the axial and the follower moments are non conservative not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure similar to that of lagrangian multipliers this makes possible the elimination of the displacement vector field from the principle so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle other researchers on the other hand employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field as a result more accurate and efficient family of beam finite elements for the non linear analysis of space frames has been obtained a one field formulation results in the fact that in the present finite elements the locking never occurs any combination of deformation states is described equally precisely this is in contrast with the elements developed in literature where in order to avoid the locking a reduced numerical integration has to be applied which unfortunately diminishes the accuracy of the solution polynomials have been chosen for the approximation of the components of the rotational vector in this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value the newton method is used for the iterative solution of the non linear equilibrium equations in an non equilibrium configuration the tangent stiffness matrix obtained by the linearization of governing equations using the directional derivative is non symmetric even for conservative loadings only upon achieving an equilibrium state the tangent stiffness matrix becomes symmetric thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the newton method for non conservative loadings however the tangent stiffness matrix is always non symmetric the numerical examples demonstrate capability of the present formulation to determine accurately the non linear behaviour of space frames in numerical examples the out of plane buckling loads are determined and the whole pre and post critical load displacement paths of a cantilever and a right angle frame are traced these in the analysis of space beams standard verification example problems show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself the rotations of 2n and extensional strains much beyond the realistic values of linear elastic material

direct computation of critical equilibrium states for spatial beams and frames summary in this paper explicit formulas for second order derivatives of the residual vector with respect to the state variables for a geometrically exact 3d beam element based on the reissner s model are presented these derivatives are required when a direct non linear stability eigenvalue problem is solved by the newton s method if the external load is parametrized by a single parameter such an eigenvalue problem consists of solving the critical state variables the eigenmode and the critical load parameter from the equation system consisting of the equilibrium equations the criticality condition and some auxiliary conditions depending on the type of a critical point copyright 2011 john wiley sons ltd

on the geometrically exact beam model a consistent effective and simple derivation from three dimensional finite elasticity the goal of the present work is to emphasize some intrinsic features of the three dimensional beam model originally proposed by simo simo j c 1985 a finite strain beam formulation the three dimensional dynamic problem part i comput methods appl mech eng 49 55 70 and to derive the model equations for the general finite deformation case as well as for the finite deformation small strain case in a consistent but simpler way with respect to what it is generally done in the literature

invariant hermitian finite elements for thin kirchhoff rods ii the linear three dimensional case abstract this paper considers the formulation of finite elements for thin kirchhoff rods in the three dimensional case extending our work on plane rods presented in 5 the new issue to be addressed besides the basic consideration of the three dimensional geometry is the approximation of the rod s torsion still in the geometrically linear setting of infinitesimal strains in particular it is the goal of this paper to identify approximations that preserve the basic invariance properties of the underlying structural theory in the sense that the fundamental rigid body modes of translations and infinitesimal rotations are represented exactly without straining accounting fully for the rod s curvature and associated kirchhoff s kinematics in contrast to common straight framework element approximations based on the simplified euler bernoulli beam kinematics to this purpose different basic and new interpolations of the angle of twist rotation as well as interpolations of directors increments in the setting provided by a cosserat treatment of the rod s kinematics are explored and studied in detail including rigorous analytical characterizations of the aforementioned rigid body modes these different interpolations are combined with the new c 1 continuous hermitian interpolation developed in that reference for the axis displacements extended to general three dimensionally curved rods as well as with an assumed strain mixed treatment to improve the overall performance of the finite elements several representative numerical simulations are presented to illustrate the properties and the numerical performance of the different new finite elements

on the differentiation of the rodrigues formula and its significance for the vector like parameterization of reissner simo beam theory in this paper we present a systematic way of differentiating up to the second directional derivative i the rodrigues formula and ii the spin rotation vector variation relationship to achieve this goal several trigonometric functions are grouped into a family of scalar quantities which can be expressed in terms of a single power series these results are then applied to the vector like parameterization of reissner simo beam theory enabling a straightforward derivation and leading to a clearer formulation in particular and in contrast with previous formulations a relatively compact and obviously symmetric form of the tangent operator is obtained the paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples copyright 2002 john wiley sons ltd

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

interpolation of rotational variables in nonlinear dynamics of 3d beams the formulation of dynamic procedures for three dimensional 3 d beams requires extensive use of the algebra pertaining to the non linear character of the rotation group in space the corresponding extraction procedure to obtain the rotations that span a time step has certain limitations which can have a detrimental eect on the overall stability of a time integration scheme the paper describes two algorithms for the dynamics of 3 d beams which dier in their manifestation of the above limitation the rst has already been described in the literature and involves the interpolation of iterative rotations while an alternative formulation which eliminates the above eect by design requires interpolation of incremental rotations theoretical arguments are backed by numerical results similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one 1998 john wiley sons ltd

a locking free finite element formulation and reduced models for geometrically exact kirchhoff rods in this work we suggest a locking free geometrically exact finite element formulation incorporating the modes of axial tension torsion and bending of thin kirchhoff beams with arbitrary initial curvatures and cross section shapes the proposed formulation has been designed in order to represent general load cases and three dimensional problem settings in the geometrically nonlinear regime of large deformations from this comprehensive theory we not only derive a general beam model but also several reduced formulations which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads the advantages of these reduced models arise for example in terms of simplified finite element formulations less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models a second core topic of this publication is the treatment of membrane locking which is a locking phenomenon predominantly occurring in highly slender curved structures thus exactly in the prime field of application for kirchhoff theories in order to address the membrane locking effect we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as assumed natural strains ans or reduced integration the effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples

objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite element implementation the paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional 3d beams which has been at the heart of most recent

kernels and spectrum of toeplitz operators on the dirichlet space abstract in this paper we study the spectrum of toeplitz operators on the dirichlet space d with bounded conjugate analytic symbols and give a characterization of the kernels of toeplitz operators with harmonic symbols in p m d using this characterization we show that the spectrum of a toeplitz operator with symbol the sum of an analytic polynomial with degree at most two and the conjugate of linear polynomial is connected but there are toeplitz operators with general harmonic polynomial symbols having disconnected spectrum

invariant hermitian finite elements for thin kirchhoff rods i the linear plane case abstract the objective of this work is to develop finite elements for the simulation of thin kirchhoff rods that is rods where the influence of transverse shear strain can be neglected leading to governing equations that require c 1 continuous finite element interpolations we start in this paper with the formulation of new finite elements for the infinitesimal plane deformations of otherwise generally curved rods addressing in particular the lack of invariance observed in related traditional treatments of plane arches by non invariance we mean that the classical rigid body modes of translations and infinitesimal rotations of the underlying structural theory lead to non zero strains in the element an effect sometimes referred to as self straining this phenomenon occurs in curved configurations of the elements and in the infinitesimal setting of interest in this paper it results in the elements stiffness matrix not to exhibit the correct zero energy modes associated with these physical rigid body motions as a consequence the basic equilibrium relations of balance of forces and moments are not satisfied for the associated nodal forces and moments in non invariant finite elements in this context we propose a new framework for the formulation of c 1 continuous finite elements based on hermitian shape functions that do lead to the satisfaction of this fundamental invariance property accounting fully for the rod s curvature and associated kirchhoff s kinematics in contrast to common straight framework element approximations based on the simplified euler bernoulli beam kinematics the formulation relies crucially on a consistent treatment of the kinematics and the assumed interpolation of the rods geometry an additional assumed strain treatment is considered to avoid limitations of membrane locking complete rigorous analyses and several numerical simulations are presented to prove and illustrate the properties and performance of the newly proposed finite element methods

a consistent 3d corotational beam element for nonlinear dynamic analysis of flexible structures abstract the purpose of the paper is to present a corotational beam element for the nonlinear dynamic analysis of 3d flexible frames the novelty of the formulation lies in the use of the corotational framework i e the decomposition into rigid body motion and pure deformation to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix as a consequence cubic interpolations are adopted to formulate both inertia and internal local terms in the derivation of the dynamic terms an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained to enhance the efficiency of the iterative procedure an approximate expression of the tangent dynamic matrix is adopted four numerical examples are considered to assess the performance of the new formulation against the one suggested by simo and vu quoc 1988 48 it was observed that the proposed formulation proves to combine accuracy with efficiency in particular the present approach achieves the same level of accuracy as the formulation of simo and vu quoc but with a significantly smaller number of elements

modeling and numerical treatment of elastic rods with frictionless self contact in this paper we present a hyperelastic rod model that takes into account self contact forces the model is based on cosserat rod theory that incorporates shear elongation flexure and twist deformations the problem of avoiding self penetration of parts of the rod is handled by the introduction of a contact distance function and the incorporation of associated contact forces we present a penalty method for the treatment of the multi valued and non differentiable contact law we also describe an augmented lagrangian formulation of this problem we then give the details of the finite element discretization of the elastic rod self contact problem as well as some numerical examples

the quaternion based three dimensional beam theory this paper presents the equations for the implementation of rotational quaternions in the geometrically exact three dimensional beam theory a new finite element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam the formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form a strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented several numerical examples demonstrate the validity performance and accuracy of the proposed approach c 2009 elsevier b v all rights reserved

toeplitz kernels and the backward shift abstract in this paper we study the kernels of toeplitz operators on both the scalar and the vector valued hardy space for 1 p we show existence of a minimal kernel for any element of the vector valued hardy space and we determine a symbol for the corresponding toeplitz operator in the scalar case we give an explicit description of a maximal function for a given toeplitz kernel which has been decomposed in to a certain form in the vectorial case we show not all toeplitz kernels have a maximal function and in the case of p 2 we find the exact conditions for when a toeplitz kernel has a maximal function for both the scalar and vector valued hardy space we study the minimal toeplitz kernel containing multiple elements of the hardy space which in turn allows us to deduce an equivalent condition for a function in the smirnov class to be cyclic for the backward shift

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

invariant isogeometric formulations for three dimensional kirchhoff rods abstract in the isogeometric formulation for three dimensional kirchhoff rods the invariance is required such that rigid body modes associated with translations and infinitesimal rotations can be exactly represented by enforcing the invariance in the isogeometric formulation of the rod the equilibrium of forces and moments at fixed ends is preserved and the performance is improved considerably for elements of finite sizes we recognize that not all existing isogeometric formulations are invariant for general geometry of rods though an alternative formulation based on the frenet serret frame and rotation angle representation was shown to be invariant it does not work for rods with vanishing curvature while its accuracy is affected by the low continuity of the angle of twist used motivated by this two new invariant isogeometric formulations are developed through different interpolations for the angle of twist one uses the derivatives of the nurbs functions as the basis for the angle of twist interpolation and the other uses a mixed interpolation incorporating both the lagrange and nurbs functions several representative numerical examples were prepared to verify the accuracy and robustness of the two new formulations particularly for rods of variable curvatures simulated by elements of finite or large sizes the proposed formulations are demonstrated to outperform the two existing ones mentioned in predicting the displacements end reactions and stress resultants

weak compactness in certain star shift invariant subspaces abstract the context of much of the work in this paper is that of a backward shift invariant subspace of the form k b h 2 d bh 2 d where b is some infinite blaschke product we address but do not fully answer the question for which b can one find a convergent sequence f n n 1 in kb such that the sequence of real measures log f n d n 1 converges weak star to some nontrivial singular measure on d we show that in order for this to hold kb must contain functions with nontrivial singular inner factors and in a rather special setting we show that this is also sufficient much of the paper is devoted to finding conditions on b that guarantee that kb has no functions with nontrivial singular inner factors our primary result in this direction is based on the geometry of the zero set of b

geometrically exact thin walled beam including warping formulated on the special euclidean group se 3 abstract based on a formulation on the special euclidean group s e 3 a geometrically exact thin walled beam with an arbitrary open cross section is proposed to deal with the finite deformation and rotation issues the beam strains are based on a kinematic assumption where warping deformation and wagner effects are included such that the nonlinear behavior of a thin walled beam is predicted accurately particular under large torsion to reduce the nonlinearity of rigid motion static and dynamic equations are derived in the s e 3 framework based on the local frame approach as the value of the iteration matrix including the jacobian matrix of inertial and internal forces is invariable under arbitrary rigid motion the number of updates required during the computation process decreases sharply which drastically improves the computational efficiency furthermore the isogeometric analysis iga based on the non uniform rational b splines nurbs basis functions which promotes the integration of computer aided design cad and computer aided engineering cae is adopted to interpolate the displacement rotation and warping fields separately the interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point to obtain the symmetric jacobian matrix of internal forces the linearization operation is conducted based on the previously converged configuration a lie group s e 3 extension of the generalized time integration method is utilized to solve the equations of motion for thin walled beams finally the proposed formulation is successfully tested and validated in several static and dynamic numerical examples

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

backwards compact attractors and periodic attractors for non autonomous damped wave equations on an unbounded domain abstract we study the backwards dynamics for the wave equation defined on the whole 3d euclid space with a positively bounded coefficient of the damping and a time dependent force we introduce a backwards compact attractor which is the minimal one among the backwards compact and pullback attracting sets we prove that a backwards compact attractor is equivalent to a pullback attractor invariant that is backwards compact i e the union of the attractor over the past time is pre compact we also establish a sufficient and necessary criterion of the existence of a backwards compact attractor and show the relationship of a periodic attractor as an application of these abstract results we prove that the non autonomous wave equation has a backwards compact attractor under some backwards assumptions of the non autonomous force moreover we establish the backwards compactness from some periodicity assumptions more precisely if the force is assumed only to be periodic then a backwards compact attractor exists and if the damped coefficient is further assumed to be periodic then the attractor is both periodic and backwards compact

extremal functions as divisors for kernels of toeplitz operators in this paper an extremal function of a banach space of analytic functions in the unit disk not all functions vanishing at 0 is a function solving the extremal problem supref 0 for functions f of norm 1 we study extremal functions of kernels of toeplitz operators on hardy spaces hp 1 2 modulo p dependent multiplicative constants we give examples showing that the extremal function may fail to be a contractive divisor when p 2 and also fail to be an expansive divisor when p 2 we discuss to what extent these results characterize the toeplitz operators via invariant subspaces for the backward shift

a co rotational element time integration strategy for non linear dynamics new procedures are proposed for implicit dynamic analysis using the finite element method the main aim is to give stable solutions with significant rigid body motions in particular rotations in contrast to most conventional approaches the time integration strategy is closely linked to the element technologies with the latter involving a form of co rotational procedure for the undamped situation one of the solution procedures leads to an algorithm that exactly conserves energy when constant external forces are applied i e with gravity loading

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

on the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach abstract in finite element methods that are based on position and slope coordinates a representation of axial and bending deformation by means of an elastic line approach has become popular such beam and plate formulations based on the so called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories examining the existing planar absolute nodal coordinate element which uses a curvature proportional bending strain expression it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and even more serious the normal force is incorrect a correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived the strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the st venant kirchhoff material therefore the strain energy is derived for the latter case and compared to classical nonlinear rod theories the error in the original absolute nodal coordinate formulation is documented by numerical examples the numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications the numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre curved static and dynamic problems including buckling and eigenvalue analysis the resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy momentum conservation

an objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics we present in this paper a new finite element formulation of geometrically exact rod models in the three dimensional dynamic elastic range the proposed formulation leads to an objective or frame indifferent under superposed rigid body motions approximation of the strain measures of the rod involving finite rotations of the director frame in contrast with some existing formulations this goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod s cross section furthermore the proposed framework allows the development of time stepping algorithms that preserve the conservation laws of the underlying continuum hamiltonian system the conservation laws of linear and angular momenta are inherited by construction leading to an improved approximation of the rod s dynamics several numerical simulations are presented illustrating these properties copyright 2002 john wiley sons ltd

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 1 beam concept and geometrically exact nonlinear formulation a geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed it is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three dimensional solid translational and rotational displacements of the cross sections warping of their plane and distortion of their contours the principle of virtual work is applied in a straightforward manner to all non zero six components of the strain and stress tensors expressions are given for tangent matrices of elastic inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non inertial frames finally the numerical properties of the finite element models are demonstrated through examples c 1998 elsevier science s a all rights reserved

a large rotation finite element analysis of 3d beams by incremental rotation vector and exact strain measure with all the desirable features abstract different strategies based on rotation vector and exact strain measure have been proposed over the years for analyzing flexible bodies undergoing arbitrary large rotations to avoid the singularity of the vector like parametrization the interpolation of the incremental rotation vector is the most popular approach in this context even if this leads to path dependence and numerical instability i e error accumulation it is also non objective although both objectivity and path independence are recovered with h and p refinement corotational approaches do not have these drawbacks even though the geometrically exact model is achieved by mesh refinement in this work we develop a novel strategy which uses the incremental nodal rotation vectors to define corotational nodal rotations which are then interpolated for the evaluation of the nonlinear strains this choice makes the approach singularity free allows for additive updates within each increment and preserves all the features of the theoretical problem for any mesh and interpolation rotational variables objectivity exact strain measure path independence and symmetric stiffness matrix for conservative loads this last property is a consequence of the direct differentiation of the relation between local and global rotations whose compact form also makes a simple and general definition of the internal forces and the tangent stiffness for any order of interpolation possible in addition we show how the common approach of interpolating incremental vectors can be made stable by a simple updating procedure based on local rotations carried out at the end of each increment in order to avoid cumulative errors geometrically exact 3d beams are considered as a demonstrative example an iterative strategy based on mixed integration points is used to solve the nonlinear discrete equations efficiently

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells