an exact finite rotation shell theory its mixed variational formulation and its finite element implementation a non linear shell theory including transverse shear strains with exact description of the kinematical fields is developed the strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor thus in plane rotations also called drilling degrees of freedom are included in a natural way various alternatives of the theory are derived for a special version of the theory with altogether six kinematical fields different mixed variational principles are given a hybrid finite element formulation which does not exhibit locking phenomena is developed numerical examples of shell deformation at finite rotations with excellent element performance are presented comparison with results reported in the literature demonstrates the features of the theory as well as the proposed finite element formulation

numerical implementation of multiplicative elasto plasticity into assumed strain elements with application to shells at large strains alternative formulations of isotropic large strain elasto plasticity are presented which are especially well suited for the implementation into assumed strain elements based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts three distinct eigenvalue problems related to the reference intermediate and current configuration are investigated these eigenvalue problems are connected by similarity transformations which preserve the eigenvalues they play an important role in the subsequent development of alternative constitutive formulations and the corresponding finite element implementation the developed constitutive procedures rely on the right cauchy green tensor or equivalently on the green lagrangian strain tensor rather than the deformation gradient consequently they can be applied directly to assumed strain elements specifically we are concerned with efficient low order shell elements for which the assumed strain method has proven to be extremely powerful to overcome spurious locking effects

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

constrained integration of rigid body dynamics in the present paper rigid body dynamics is formulated as mechanical system with holonomic constraints this approach offers the appealing possibility to deal with finite rotations without employing any specific rotational parameterization the numerical discretization of the underlying system of differential algebraic equations is treated in detail the proposed algorithm obeys major conservation laws of the underlying continuous system such as conservation of energy and angular momentum in addition to that the constraints on the configuration and momentum level are fulfilled exactly two numerical examples are dealt with to assess the performance of the constrained algorithm

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

on the parametrization of finite rotations in computational mechanics a classification of concepts with application to smooth shells this paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure category ii is based on total rotational degrees of freedom leading to an additive update structure whereas category i relies on linearized rotational degrees of freedom leading to a multiplicative update structure based on this classification new formulations are developed among them are the rodrigues formula applied within category ii with additive update structure for the two components of the rotation vector and two successive elementary rotations in the sense of euler angles applied within category i yielding a singularity free formulation the numerical examples confirm that every rotational parametrization when applied within category ii leads to singularities whereas the application within category i enables the calculation of overall rotations unrestricted in size without any singularity

contact between spheres and general surfaces abstract this work presents a mathematical model to establish the weak form and tangent operator contributions due to contact between spherical surfaces and general surfaces normal and friction components are included such as dissipative actions the main concern herein is the proper consideration of the kinematics of the spherical surface and its influence on contact forces including the appropriate description of the spherical motion by gap functions in normal and tangential directions this leads to the possibility of dealing with complex kinematics of rolling or alternating rolling sliding of the spherical surface mapping its motion on another general surface to achieve that the model is based on rotational and translational degrees of freedom used to map the motion of contacting surfaces one may employ such strategy for interactions between spheres and rigid or flexible bodies modeled using finite element meshes as examples parameterizations of rigid and flexible surfaces are proposed and used practical applications are shown with usage of beam and shell finite elements experiencing contact interactions

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

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

geometrically exact sandwich shells the static case the present formulation offers a general method for analyzing the static response of geometrically exact sandwich shells undergoing large deformation the layer directors at a point in the reference surface are connected to each other by universal joints and form a chain of rigid links finite rotations of the directors in every layer are allowed with shear deformation independently accounted for in each layer the thickness and the length of each layer can be arbitrary thus allowing the modeling of an important class of multilayer structures having ply drop offs the present formulation is thus suitable to model shell structures with patches of constrained viscoelastic materials or of piezoelectric materials the nonlinear weak form of the governing equations of equilibrium is constructed here and then the linearization of the weak form and the associated inextensible directors update are derived leading to a symmetric tangent stiffness matrix a galerkin finite element projection of the linearized equilibrium equations results in a system of nonlinear algebraic equations in which the interpolation accounts for the finite rotations of the directors we present extensive numerical examples including sandwich shells with three identical layers and ply drop offs to illustrate the applicability and versatility of the proposed formulation

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

the total domination and total bondage numbers of extended de bruijn and kautz digraphs in this paper we consider the total domination number and the total bondage number for digraphs the total bondage number defined as the minimum number of edges whose removal enlarges the total domination number measures to some extent the robustness of a network where a minimum total dominating set is required we determine the total domination number and total bondage number of the extended de burijn digraph and the extended kautz digraph proposed by shibata and gonda in 1995 which generalize the classical de bruijn digraph and the kautz digraph

stress resultant geometrically exact form of classical shell model and vector like parameterization of constrained finite rotations summary in this work we consider the geometrically exact shell model subjected to 7nite rotations making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells a modi7cation of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed copyright 2001 john wiley sons ltd

treatment of reissner mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework this work presents a framework for the computation of complex geometries containing intersections of multiple patches with reissner mindlin shell elements the main objective is to provide an isogeometric finite element implementation which neither requires drilling rotation stabilization nor user interaction to quantify the number of rotational degrees of freedom for every node for this purpose the following set of methods is presented control points with corresponding physical location are assigned to one common node for the finite element solution a nodal basis system in every control point is defined which ensures an exact interpolation of the director vector throughout the whole domain a distinction criterion for the automatic quantification of rotational degrees of freedom for every node is presented an isogeometric reissner mindlin shell formulation is enhanced to handle geometries with kinks and allowing for arbitrary intersections of patches the parametrization of adjacent patches along the interface has to be conforming the shell formulation is derived from the continuum theory and uses a rotational update scheme for the current director vector the nonlinear kinematic allows the computation of large deformations and large rotations two concepts for the description of rotations are presented the first one uses an interpolation which is commonly used in standard lagrange based shell element formulations the second scheme uses a more elaborate concept proposed by the authors in prior work which increases the accuracy for arbitrary curved geometries numerical examples show the high accuracy and robustness of both concepts the applicability of the proposed framework is demonstrated

four node semi eas element in six field nonlinear theory of shells we propose a new four node c finite element for shell structures undergoing unlimited translations and rotations the considerations concern the general six field theory of shells with asymmetric strain measures in geometrically nonlinear static problems the shell kinematics is of the two dimensional cosserat continuum type and is described by two independent fields the vector field for translations and the proper orthogonal tensor field for rotations all three rotational parameters are treated here as independent hence as a consequence of the shell theory the proposed element has naturally six engineering degrees of freedom at each node with the so called drilling rotation this property makes the element suitable for analysis of shell structures containing folds branches or intersections to avoid locking phenomena we use the enhanced assumed strain eas concept we derive and linearize the modified hu washizu principle for six field theory of shells what makes the present approach original is the combination of eas method with asymmetric membrane strain measures based on literature we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry to gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches

a four node corotational quadrilateral elastoplastic shell element using vectorial rotational variables four node corotational quadrilateral elastoplastic shell element summary a four node corotational quadrilateral elastoplastic shell element is presented the local coordinate system of the element is defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product this local coordinate system rotates rigidly with the element but does not deform with the element as a result the element rigid body rotations are excluded in calculating the local nodal variables from the global nodal variables the two smallest components of each nodal orientation vector are defined as rotational variables leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure different from other existing corotational finite element formulations the resulting element tangent stiffness matrix is symmetric owing to the commutativity of the local nodal variables in calculating the second derivative of strains with respect to these variables for elastoplastic analyses the maxwell huber hencky von mises yield criterion is employed together with the backward euler return mapping method for the evaluation of the elastoplastic stress state the consistent tangent modulus matrix is derived to eliminate locking problems we use the assumed strain method several elastic patch tests and elastoplastic plate shell problems undergoing large deformation are solved to demonstrate the computational efficiency and accuracy of the proposed formulation copyright 2013 john wiley sons ltd

genuinely resultant shell finite elements accounting for geometric and material non linearity the underlying theory is statically and geometrically exact and it naturally includes small strain and finite strain problems of thin as well as thick shells this paper examines the development of computational procedures employing the newton kantorovich linearization process and the galerkin type discretization method the treatment of finite rotations through an arbitrary parametrization of the rotation group the interpolation procedure of so 3 valued functions underlying the construction of finite element basis

a discontinuous galerkin formulation of non linear kirchhoff love shells discontinuous galerkin dg methods provide a means of weakly enforcing the continuity of the unknown field derivatives and have particular appeal in problems involving high order derivatives this feature has previously been successfully exploited comput methods appl mech eng 2008 197 2901 2929 to develop a formulation of linear kirchhoff love shells considering only the membrane and bending responses in this proposed one field method the displacements are the only unknowns while the displacement field is continuous the continuity in the displacement derivative between two elements is weakly enforced by recourse to a dg formulation it is the purpose of the present paper to extend this formulation to finite deformations and non linear elastic behaviors while the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple stress from the displacement field at the mid surface of the shell the non linear formulation considered implies the evaluation of the general stress tensor across the shell thickness leading to a reformulation of the internal forces of the shell nevertheless since the interface terms resulting from the discontinuous galerkin method involve only the resultant couple stress at the edges of the shells the extension to non linear deformations is straightforward copyright 2008 john wiley sons ltd

a finite strain quadrilateral shell element based on discrete kirchhoff love constraints this paper improves the 16 degrees of freedom quadrilateral shell element based on pointwise kirchhoff love constraints and introduces a consistent large strain formulation for this element the model is based on classical shell kinematics combined with continuum constitutive laws the resulting element is valid for large rotations and displacements the degrees of freedom are the displacements at the corner nodes and one rotation at each mid side node the formulation is free of enhancements it is almost fully integrated and is found to be immune to locking or unstable modes the patch test is satisfied in addition the formulation is simple and amenable to efficient incorporation in large scale codes as no internal degrees of freedom are employed and the overall calculations are very efficient results are presented for linear and non linear problems copyright 2005 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

large rotations revisited application of lie algebra abstract the paper presents an elementary introduction and application of lie algebra to compound groups of large rotations following an elementary overview of the foundations of lie algebra the authors discuss the rotational groups lo 3 and the exponential form of the rotation matrix these yield a practical link between the elegance of lie group theory and engineering kinematics some illustrative examples as the standard stanford manipulator indicate interesting possibilities on enlarged applications concerning robotic mechanisms vehicles and biosystems

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

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

tensor based finite element formulation for geometrically nonlinear analysis of shell structures in the present study a finite element computational model for the nonlinear analysis of shell structures is presented a tensor based finite element formulation is presented to describe the mathematical model of a shell in a natural and simple way by using curvilinear coordinates in addition a family of high order elements with lagrangian interpolations is used to avoid membrane and shear locking and no mixed interpolations are employed a first order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the lagrangian description this approach takes into account thickness stretching and therefore 3d constitutive equations are utilized numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites as well as functionally graded shells are found to be excellent and show the validity of the developed finite element model moreover the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation of shells

thin shells with finite rotations formulated in biot stresses theory and finite element formulation a bending theory for thin shells undergoing finite rotations is presented and its associated finite element model is described the kinematic assumption is based on a shear elastic reissner mindlin theory the starting point for the derivation of the strain measures are the resultant equilibrium equations and the associated principle of virtual work within this formulation the polar decomposition of the shell material deformation gradient leads to symmetric strain measures the associated work conjugate stress resultants and stress couples are integrals of the biot stress tensor this tensor is invariant with respect to rigid body motions and therefore appropriate for the formulation of constitutive equations finite rotations are introduced via eulerian angles the finite element discretization of arbitrary shells is based on the isoparametric concept formulated with respect to a plane reference configuration the numerical model is applied to different non linear plate and shell problems and compared with existing formulations due to a consistent linearization the step size of a load increment is only limited by the local convergence behaviour of newton s method

an energy momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells summary in this paper a mixed variational formulation for the development of energy momentum consistent emc time stepping schemes is proposed the approach accommodates mixed finite elements based on a hu washizu type variational formulation in terms of displacements green lagrangian strains and conjugated stresses the proposed discretization in time of the mixed variational formulation under consideration yields an emc scheme in a natural way the newly developed methodology is applied to a high performance mixed shell finite element the previously observed robustness of the mixed finite element formulation in equilibrium iterations extends to the transient regime because of the emc discretization in time copyright 2016 john wiley sons ltd

an investigation of a finite rotation four node assumed strain shell element the paper presents a shell formulation based on the degenerated solid approach the theory employs covariant strains and performs explicit integration through the shell thickness the rigid body motion is exactly represented the consistent tangent stiffness matrix is evaluated for the four node quadrilateral it is shown in the final part that this type of element which distinguishes itself by a very simple and easily understandable theory gives good answers for linear as well as non linear applications

a q1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh abstract in this paper we study a q 1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh it is known that the coercivity is the basement for some other theoretical results stability h 1 and l 2 error estimates etc and the existing results were mainly obtained on h 1 parallelogram meshes with scalar diffusion coefficients for the cases of full diffusion tensors and arbitrary convex quadrilateral meshes we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form based on this result a sufficient condition is suggested to guarantee the coercivity of the scheme more interesting is that this sufficient condition covers the traditional h 1 parallelogram mesh assumption and has an explicit expression by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h 0 moreover an h 1 error estimate is obtained without the h 1 parallelogram assumption and some numerical results are also provided to validate the theoretical results

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

time dependent fibre reorientation of transversely isotropic continua finite element formulation and consistent linearization transverse isotropy is realized by one characteristic direction for instance the fibre direction in fibre reinforced materials commonly the characteristic direction is assumed to be constant but in some cases for instance in the constitutive description of biological tissues liquid crystals grain orientations within polycrystalline materials or piezoelectric materials as well as in optimization processes it proves reasonable to consider reorienting fibre directions various fields can be assumed to be the driving forces for the reorientation process for instance mechanical electric or magnetic fields in this work we restrict ourselves to reorientation processes in hyper elastic materials driven by principal stretches the main contribution of this paper is the algorithmic implementation of the reorientation process into a finite element framework therefore an implicit exponential update of the characteristic direction is applied by using the rodriguez formula to express the exponential term the non linear equations on the local and on the global level are solved by means of the newton raphson scheme accordingly the local update of the characteristic direction and the global update of the deformation field are consistently linearized yielding the corresponding tangent moduli through implementation into a finite element code and some representative numerical simulations the fundamental characteristics of the model are illustrated copyright 2007 john wiley sons ltd

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

theory and numerics of three dimensional beams with elastoplastic material behaviour a theory of space curved beams with arbitrary cross sections and an associated finite element formulation is presented within the present beam theory the reference point the centroid the centre of shear and the loading point are arbitrary points of the cross section the beam strains are based on a kinematic assumption where torsion warping deformation is included each node of the derived finite element possesses seven degrees of freedom the update of the rotational parameters at the finite element nodes is achieved in an additive way applying the isoparametric concept the kinematic quantities are approximated using lagrangian interpolation functions since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load carrying capacity of beam structures copyright 2000 john wiley sons ltd

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

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

a discrete mechanics approach to the cosserat rod theory part 1 static equilibria a theory of discrete cosserat rods is formulated in the language of discrete lagrangian mechanics by exploiting kirchhoff s kinetic analogy the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the lagrangian function of a dynamical system the equilibrium equations are derived from a variational principle using a formulation that involves null space matrices in this formulation no lagrange multipliers are necessary to enforce orthonormality of the directors noether s theorem relates first integrals of the equilibrium equations to lie group actions on the configuration bundle so called symmetries the symmetries relevant for rod mechanics are frame indifference isotropy and uniformity we show that a completely analogous and self contained theory of discrete rods can be formulated in which the arc length is a discrete variable ab initio in this formulation the potential energy density is defined directly on pairs of points along the arc length of the rod in analogy to veselov s discrete reformulation of lagrangian mechanics a discrete version of noether s theorem then identifies exact first integrals of the discrete equilibrium equations these exact conservation properties confer the discrete solutions accuracy and robustness as demonstrated by selected examples of application

locking free isogeometric collocation formulation for three dimensional geometrically exact shear deformable beams with arbitrary initial curvature abstract this paper presents a displacement based and a mixed isogeometric collocation formulations for three dimensional geometrically exact shear deformable beams with arbitrarily curved initial geometry strong forms of both formulations are consistently linearized with respect to the underlying configuration manifold high efficiency is achieved by using a geometrically consistent kinematic model in which incremental rotations are parametrized through the spatial rotation vector several numerical tests demonstrate that the proposed mixed formulation is totally locking free for any choice of approximation degree mixed formulation also exhibits higher robustness in the incremental iterative algorithm elimination of shear locking and low computational cost assured by the combination of isogeometric collocation with a consistent kinematic model result in a very efficient accurate and geometrically powerful method able to handle geometrically complex beam problems

a mixed hybrid finite beam element with an interface to arbitrary three dimensional material models abstract a theory for spatial beams and an associated mixed finite element formulation are presented the basic kinematic assumption allows for finite rotations and transverse shear strains the relation of the green lagrangean strains to the beam strains is derived a three field variational formulation with independent displacements rotations stress resultants and beam strains is developed using the kinematic and static field equations and the constitutive equations higher order stress resultants are set to zero within the euler lagrange equations the formulation in this paper is restricted to rectangular cross sections the interpolation of the beam strains consists of two parts the first part is identical to the stress resultant interpolation the second part describes cross section warping and allows for transverse normal strains within this approach an interface to arbitrary three dimensional constitutive laws can easily be realized due to the mixed interpolation technique full gauss integration in length direction of the beam element can be applied to obtain a locking free element formulation the effectiveness of the method is demonstrated by means of several numerical examples we present results for elasticity inelasticity and stability problems and evaluate the load carrying capacities of spatial beam structures the essential feature of the new element is the robustness in nonlinear applications it allows very large load steps in comparison to other element formulations

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 a stress resultant geometrically exact shell model part vi conserving algorithms for non linear dynamics in parts i to v of the present work the formulation and finite element implementation of a non linear stress resultant shell model are considered in detail this paper is concerned with the extension of these results to incorporate completely general non linear dynamic response of special interest here is the dynamics of very flexible shells undergoing large overall motion which conserves the total linear and angular momentum and for the hamiltonian case the total energy a main goal of this paper is the design of non linear time stepping algorithms and the construction of finite element interpolations which preserve exactly these fundamental constants of motion it is shown that only a very special class of algorithms namely a formulation of the mid point rule in conservation form exactly preserves the total linear and angular momentum for the hamiltonian case a somewhat surprising result is proved regardless of the degree of non linearity in the stored energy function a generalized mid point rule algorithm always exists which exactly conserves energy the conservation properties of a time stepping algorithm need not and in general will not be preserved by the spatial discretization precise conditions which ensure preservation of these conservation properties are derived a number of numerical simulations are presented which illustrate the exact conservation properties of the proposed methodology

a general high order finite element formulation for shells at large strains and finite rotations for hyperelastic shells with finite rotations and large strains a p finite element formulation is presented accommodating general kinematic assumptions interpolation polynomials and particularly general three dimensional hyperelastic constitutive laws this goal is achieved by hierarchical high order shell models the tangent stiffness matrices for the hierarchical shell models are derived by computer algebra both non hierarchical nodal as well as hierarchical element shape functions are admissible numerical experiments show the high order formulation to be less prone to locking effects copyright 2003 john wiley sons ltd

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