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

finite elements with increased freedom in choosing shape functions starting with a mathematical statement of the convergence requirements for an element stiffness matrix the paper discusses displacement shape functions that may be used in connection with the potential energy principle in short these functions must be force orthogonal and energy orthogonal but they need not be conforming satisfy interelement compatibility it is shown that the requirements to the displacement functions may be greatly relaxed through slight modifications of the coupling stiffness between fundamental and higher order displacement modes several alternative formulations are examined in particular a new free formulation is suggested using this form which is very simple the only requirement to the displacement patterns used is that they should contain the fundamental deformation modes and be linearly independent applications of the theory to triangular and rectangular plate bending elements are shown the simple stiffness matrix for the latter is given explicitly the numerical results compare favourably with other types of finite elements

modelling of superelements in mechanism analysis the paper deals with substructuring for dynamic analysis of flexible multibody systems three different techniques based on component synthesis are discussed corresponding respectively to fully consistent mass discretization lumped mass discretization and corotational approximation of inertia forces to simplify the computer implementation only the lumped mass and corotational approximations have been considered in detail and programmed both approaches are validated on simple examples of rotating beams for which a full elastic model is available using a fully non linear beam element the computational efficiency of the corotational inertia approach is also demonstrated on the deployment of a large flexible satellite antenna

calcul d un invariant de star produit ferme sur une variete symplectique let m be a symplectic manifold over in cfs the authors construct an invariant in the cyclic cohomology of m for any closed star product they compute this invariant in the de rham complex of m when m t v we generalize this result by computing the image of in the de rham complex for any symplectic manifold and any star product and we show how this invariant is related to the general classification of kontsevich the proof uses the riemann roch theorem for periodic cyclic chains of nest tsygan

an o n complexity recursive algorithm for multi flexible body dynamics based on absolute nodal coordinate formulation summary a new algorithm called recursive absolute nodal coordinate formulation algorithm rec ancf is presented for dynamic analysis of multi flexible body system including nonlinear large deformation this method utilizes the absolute nodal coordinate formulation ancf to describe flexible bodies and establishes a kinematic and dynamic recursive relationship for the whole system based on the articulated body algorithm aba in the ordinary differential equations odes obtained by the rec ancf a simple form of the system generalized jacobian matrix and generalized mass matrix is obtained thus a recursive forward dynamic solution is proposed to solve the odes one element by one element through an appropriate matrix manipulation utilizing the parent array to describe the topological structure the rec ancf is suitable for generalized tree multibody systems besides the cutting joint method is used in simple closed loop systems to make sure the o n algorithm complexity of the rec ancf compared with common ancf algorithms the rec ancf has several advantages the optimal algorithm complexity o n under limited processors simple derivational process no location or speed constraint violation problem higher algorithm accuracy the validity and efficiency of this method are verified by several numerical tests this article is protected by copyright all rights reserved

multiple bounded solutions of differential inclusions the nielsen theory approach abstract the generalized nielsen number is defined for self maps which are composed by operators with r values on compact connected anrs then it is applied to caratheodory differential inclusions with constraints for obtaining the multiplicity criteria more precisely such problems are transformed to those for the lower estimate of fixed points of the related operators with the given properties on bounded compact connected neighbourhood retracts of frechet spaces in this way multiple solutions can be proved e g for the multivalued initial value problem on the half line or in the single valued case for boundary value problems to ordinary differential equations

periodic solutions in a given set of differential systems the existence of a periodic solution in a given set of caratheodory differential systems is studied the proof of the main statement is based on the wazewski type approach jointly with the degree arguments an illustrating example is supplied and a comparison with the analogous results of other authors is indicated

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

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

on continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals soit x t f t x t t j x s ou jcr est un intervalle donne f j r n r n est une fonction continue et s un sous ensemble de c j r n l espace des fonctions continues de j dans r n on considere l equation operateur associee x t x ou t est un operateur non lineaire discontinu dans son domaine

a unified formulation of small strain corotational finite elements i theory this paper presents a unified theoretical framework for the corotational cr formulation of finite elements in geometrically nonlinear structural analysis the key assumptions behind cr are i strains from a corotated configuration are small while ii the magnitude of rotations from a base configuration is not restricted following a historical outline the basic steps of the element independent cr formulation are presented the element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms it is shown that this framework permits the derivation of a set of cr variants through selective simplifications this set includes some previously used by other investigators the different variants are compared with respect to a set of desirable qualities including self equilibrium in the deformed configuration tangent stiffness consistency invariance symmetrizability and element independence we discuss the main benefits of the cr formulation as well as its modeling limitations

sensitivity analysis of flexible multibody systems using composite materials components in this paper a general formulation for the computation of the first order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented the direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables which are the ply orientations of the laminated in order to appraise the benefits of the approach suggested and to highlight the risks of the procedure the analytical sensitivities are compared with the numerical results obtained by using the finite difference method for the beam composite material elements the section properties and their sensitivities are found using an asymptotic procedure that involves a two dimensional 2 d finite element analysis of their cross section the equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems the equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations velocities and the sensitivities of accelerations and velocities are integrated in time using a multi step multi order integration algorithm through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure with respect to finite difference approaches or to the direct implementation of the analytic sensitivities are discussed copyright 2008 john wiley sons ltd

a procedure for the optimal synthesis of planar mechanisms based on non linear position problems the present paper describes a procedure for the optimal dimensional synthesis of mechanisms which is based on the solution of non linear position problems planar mechanisms are modelled very easily by means of finite elements of bi hinged rod type the judicious choice of constraint conditions makes possible the solution of classical position problems initial finite displacement deformed and static equilibrium problems as well as the main types of dimensional synthesis path generation functions rigid body guidance mixed and multiple a single synthesis error function applies to all these types and moreover is general since it is valid for mechanisms of any configuration this function is based on the idea that the best mechanism for the given synthesis conditions is the one that least needs to be deformed when it is obliged to satisfy them exactly to which end the elastic deformation of the elements is permitted 1997 by john wiley sons ltd

bounded solutions in a given set of differential systems the criteria for an entirely bounded solution in a given set of a differential system with a quasi linear part are developed sequentially via an asymptotic boundary value problem this enables us to prove several bounded solutions separated by given functions so the multiplicity results can be also obtained in this way for possible applications in epidemics and population dynamics see e g the references in gaines and santanilla rocky mountain j math 12 1982 669 678

model order reduction for dynamic simulation of a flexible multibody system via absolute nodal coordinate formulation abstract the absolute nodal coordinate formulation ancf can be used to model a flexible multibody system subject to both overall motions and large deformations with an increasing number of finite elements of ancf to mesh flexible bodies however the computation cost will become extremely high to improve the computational efficiency of ancf for a large scaled flexible multibody system a systematic method is proposed in this study for the model order reduction based on the proper orthogonal decomposition and the galerkin projection at first an approach for the selection of reduced constraint equations is proposed to deal with the singularity of the coefficient matrix of the reduced order model rom then the computation of the reduced stiffness matrix and generalized force vector of the rom are parallelized via the openmp directives afterwards two parametric approaches are presented to make the rom be adaptive to the change of system parameters one is the interpolation approach of reduced order basis vectors on a manifold to obtain the parametric rom and the other is the fast to slow rom approach via the reduced order basis vectors extracted from the fast dynamic responses to simulate slow dynamic responses finally four numerical examples are given to validate the efficacy of the proposed method for the dynamic simulations of both rigid and flexible multibody systems

the global modal parameterization for non linear model order reduction in flexible multibody dynamics in flexible multibody dynamics advanced modelling methods lead to high order non linear differential algebraic equations daes the development of model reduction techniques is motivated by control design problems for which compact ordinary differential equations odes in closed form are desirable in a linear framework reduction techniques classically rely on a projection of the dynamics onto a linear subspace in flexible multibody dynamics we propose to project the dynamics onto a submanifold of the configuration space which allows to eliminate the non linear holonomic constraints and to preserve the lagrangian structure the construction of this submanifold follows from the definition of a global modal parameterization gmp the motion of the assembled mechanism is described in terms of rigid and flexible modes which are configuration dependent the numerical reduction procedure is presented and an approximation strategy is also implemented in order to build a closed form expression of the reduced model in the configuration space numerical and experimental results illustrate the relevance of this approach

a unified co rotational framework for solids shells and beams abstract the paper describes a unified framework for applying the co rotational method to the analysis of solids shells and beams the general method stems from an unusual application of the technique which involves solid elements the proposed framework allows a formulation that is simpler than many of the earlier procedures and in addition gives a direct indication of the terms in the tangent stiffness which may be ignored

geometric non linear substructuring for dynamics of flexible mechanical systems a substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non linear effects large relative displacement is permitted between points within bodies that undergo small strain elastic deformation components are divided into substructures on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure compatibility conditions are derived and imposed as constraint equations at boundary points between substructures system equations of motion that include geometric non linear effects of large rotation in terms of generalized co ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure are assembled the method is implemented in an industry standard flexible multibody dynamics code with minimal modification use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

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

flexible translational joint analysis by meshless method element free galerkin method efgm is used to analyze a flexible translational joint a moving constraint condition is a major concern for the translational joint analysis it is shown that the efgm is well suited for an analysis of flexible translational joint original shape function in the efgm is modified so that essential boundary conditions are imposed by the same way as that of finite element method fem the modified shape function possesses physical values at both ends of the element the completeness of the modified shape function is discussed employing the present modified shape function makes easy to implement the moving constraint condition numerical examples for a static cantilever beam are presented for a comparison between the penalty method the lagrange multiplier method and the present one as second example a large deformation problem is solved for a comparison between the fem and the efgm finally a simulation of flexible translational joint is shown

the construction of free free flexibility matrices for multilevel structural analysis abstract this article considers a generalization of the classical structural flexibility matrix it expands on previous papers by taking a deeper look at computational considerations at the substructure level direct or indirect computation of flexibilities as influence coefficients has traditionally required pre removal of rigid body modes by imposing appropriate support conditions mimicking experimental arrangements with the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free free stiffness matrix this generalized inverse preserves the stiffness spectrum the definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid body mode basis the free free flexibility has proven useful in special application areas of finite element structural analysis notably massively parallel processing model reduction and damage localization it can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition if substructures contain thousands of d o f exploitation of the stiffness sparseness is important for that case this paper presents a computation procedure based on an exact penalty method and a projected rank regularized inverse stiffness with diagonal entries inserted by the sparse factorization process these entries can be physically interpreted as penalty springs this procedure takes advantage of the stiffness sparseness while forming the full free free flexibility or a boundary subset and is backed by an in depth null space analysis for robustness

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

a study of optimal membrane triangles with drilling freedoms abstract this article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms the term optimal is used in the sense of exact inplane pure bending response of rectangular mesh units of arbitrary aspect ratio following a comparative summary of element formulation approaches the construction of an optimal three node triangle using the andes formulation is presented the construction is based upon techniques developed by 1991 in student term projects but taking advantage of the more general framework of templates developed since the optimal element that fits the andes template is shown to be unique if energy orthogonality constraints are enforced two other formulations are examined and compared with the optimal model retrofitting the conventional linear strain triangle element by midpoint migrating and congruential transformations is shown unable to produce an optimal element while rank deficiency is inevitable use of the quadratic strain field of the 1988 allman triangle or linear filtered versions thereof is also unable to reproduce the optimal element moreover these elements exhibit serious aspect ratio lock these predictions are verified on benchmark examples

partitioned analysis of flexible multibody systems using filtered linear finite element deformational modes summary this work presents a partitioned finite element formulation for flexible multibody systems based on the floating frame ff approach under the assumption of small deformations but arbitrarily large rotations of the bodies in classical ff of reference methods deformational modes are normally computed by modal analysis in this approach free floating modes are eliminated from the linear model using projection techniques and substituted by a complete set of non linear finite rotations in this way all deformational modes are retained and no modal selection is needed the main difference between this work and a classical ff of reference formulation is an algebraic separation of pure deformational modes from rigid body motions the proposed methodology presents the following advantages first the position and orientation of the ff has no restriction and can be freely located in the body with identical results second the formulation uses only the linear finite element matrices of a classical vibration problem hence they can be easily obtained from linear fem packages third no selection of modes is needed all deformational modes are retained through the filtering process and finally thanks to the use of localized lagrangian multipliers llm a partitioned system is obtained that can be solved iteratively and in a distributed manner by available scalable solvers copyright 2014 john wiley sons ltd

plastic instability of beam structures using co rotational elements in a previous paper comput methods appl mech engrg 191 2002 1755 the authors have presented a 3d co rotational elastic beam element including warping effects this formulation is now further developed in order to incorporate elasto plastic deformations the element possesses seven degrees of freedom at each node and can be used to model beams with arbitrary cross sections thus within the present approach the centroid and shear center of the cross section are not necessarily coincident the main purpose of this element is to model elasto plastic instability problems in this context two methods of branch switching are tested and discussed in the first one the bifurcation point is isolated by successive bisections and the branch switching is operated by using the eigenvector associated to the negative eigenvalue in the second one introduced by petryk an energy approach is used to select automatically the stable post bifurcation path six examples including large displacement and stability problems are used in order to assess the performances of the element

component mode synthesis with constant mass and stiffness matrices applied to flexible multibody systems in flexible multibody systems component mode synthesis is used to reduce the size of system matrices of the single bodies from millions to a few hundred while maintaining the full coupling between body deformation and overall rigid body motion the combination of clamped eigenmodes and static deformation modes results in a dense and non diagonal structure of the system matrices which is computationally costly in the present paper the conventional concept of the floating frame of reference is transformed to a description based on absolute coordinates and a corotational linearization of the deformation the constant mass matrix and the corotated stiffness matrix need to be factorized only once for the whole simulation in a planar example component mode synthesis is applied to this absolute coordinate formulation by using twice as many deformation modes as usual while the solution of the overall system becomes less expensive due to efficient factorization the size of the system matrices needs to be increased considerably for each axis of rotation however the computational complexity for the factorization reduces from cubic to quadratic numerical examples are presented in order to demonstrate the difference between standard procedures and the methodology now proposed copyright 2007 john wiley sons ltd

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

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