non linear stochastic optimal control for coupled structures system of multi degree of freedom coupled structures under random excitation are modelled as a quasi integrable hamiltonian system of multi degree of freedom and the reduced order model in structural mode space is formulated a non linear stochastic optimal control method for the system is presented the non linear optimal control of adjacent tall building structures coupled with supplemental control devices and under random seismic excitation is performed by using the proposed method first applying the stochastic averaging method to the system yields ito stochastic differential equations for modal vibration energy processes so that the system energy control is conducted generally instead of the system state control and the dimension of the control problem is reduced then applying the stochastic dynamical programming principle to the controlled diffusion processes yields a dynamical programming equation taking into account random excitation spectra an explicit polynomial solution to the equation is proposed to determine the non linear optimal control forces furthermore the response statistics of the controlled non linear coupled structures under random seismic excitation are evaluated by using the stochastic averaging method and are compared with those of the uncontrolled structures to determine the control efficacy numerical results illustrate the high control effectiveness and efficiency of the proposed non linear stochastic optimal control method for coupled structures as a quasi integrable hamiltonian system

l p norms and support of eigenfunctions on graphs this article is concerned with properties of delocalization for eigenfunctions of schrodinger operators on large finite graphs more specifically we provide l p norm estimates for such eigenfunctions and show that they have a large support our estimates hold for any fixed possibly irregular graph in prescribed energy regions and also for certain sequences of graphs such as n lifts

numerical solutions of forced vibration and whirling of a non linear string using the theory of a cosserat point abstract o reilly and holmes 1 performed experiments and analysis of non linear motion of a string subjected to vertical oscillation of one end their analysis suggested that differences between theoretical and experimental results were mainly due to uncertainty in the assumed form for the forcing function that was used to satisfy the forced boundary condition here a numerical solution of the non linear string problem is considered which incorporates the forced boundary condition directly and thus avoids the need to assume an arbitrary forcing function specifically the numerical formulation of string problems based on the theory of a cosserat point is used it is shown that the director inertia coefficient in the cosserat theory can be chosen so that the theory predicts the first modal frequency of small deformation free vibration exactly for each level of discretizational also examples of forced in plane motion and free non linear whirling motion are considered to validate the numerical model the resulting simulations indicate that the forcing amplitude for the onset of persistent whirling and aperiodic response is about five times smaller than that observed in the experiments even when the uncertainty in the forcing function is removed from the analysis additional attempts to bridge the gap between theory and experiment suggest that there is still some mechanism in the experiment that is not properly modelled

objective energy momentum conserving integration for the constrained dynamics of geometrically exact beams in this paper the results in s leyendecker p betsch p steinmann energy conserving integration of constrained hamiltonian systems a comparison of approaches comput mech 33 2004 174 185 are extended to geometrically exact beams the finite element formulation for nonlinear beams in terms of directors providing a framework for the objective description of their dynamics is considered geometrically exact beams are analysed as hamiltonian systems subject to holonomic constraints with a hamiltonian being invariant under the action of so 3 the reparametrisation of the hamiltonian in terms of the invariants of so 3 is perfectly suited for a temporal discretisation which leads to energy momentum conserving integration in this connection the influence of alternative procedures the lagrange multiplier method the penalty method and the augmented lagrange method for the treatment of the constraints is investigated for the example of a beam with concentrated masses 2005 elsevier b v all rights reserved

mixed finite element methods for elastic rods of arbitrary geometry the boundary value problem for rods having arbitrary geometry and subjected to arbitrary loading is studied within the context of the small strain theory the basic assumptions underlying the rod kinematics are those corresponding to the timoshenko hypotheses in the plane rectilinear case that is plane sections normal to the line of centroids in the undeformed state remain plane but not necessarily normal the problem is formulated in both the standard and mixed variational forms and after establishing the existence and uniqueness of solutions to these equivalent problems the corresponding discrete problems are studied finite element approximations of the mixed problem are shown to be stable and convergent it is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion though the results of numerical experiments are presented these confirm the convergent behaviour of the mixed problem

stress resultant geometrically nonlinear shell theory with drilling rotations part i a consistent formulation abstract in this work we present a consistent theoretical framework for a novel stress resultant geometrically nonlinear shell theory the main feature of the present shell theory development which stands in contrast with the classical developments in the shell theory is the presence of a rotation component around the shell normal so called drilling rotation in the description of shell finite rotations the relationship of the proposed theory with a finite deformation theory of a three dimensional continuum with independent rotation field is clearly identified the latter is shown to be an important link which facilitates the proper choice of the shell constitutive model and a proper construction of the regularized form of the theory capable of supporting the drilling rotations the corresponding linearized form of the present shell theory is discussed in the closure

on the numerical solution of non linear string problems using the theory of a cosserat point abstract the objective of this paper is to develop an appropriate form of the theory of a cosserat point asme j appl mech 52 368 1985 which can be used to formulate the numerical solution of the three dimensional motion of a non linear elastic string the string is divided into n material parts each of which is modelled as a cosserat point with its own equations of motion and constitutive equations then the motion of each cosserat point is coupled with that of its neighbours and boundary conditions are introduced to obtain a system of ordinary differential equations of time only which describe the motion of the string two examples of a rotating string are considered for each example we show that director inertia rotary inertia is significant and that the cosserat solution converges rapidly to the exact solution developed by rosenau and rubin phys rev a31 3480 1985

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

intrinsic equations for the nonlinear dynamics of space beams equations are developed for the fully intrinsic nonlinear dynamics of a spatially curved pretwisted beam direct deformation measures are used as variables the equations are not referred to any global frame of references fixed moving or floating large deformations can be accommodated the bernoulli euler assumption is introduced for the gross dynamics but not for the constitutive relations the latter may be also inelastic and time dependent initial and edge conditions force and kinematic types are presented specific equations are also given for the plane case and for the linear case some of the merits and drawbacks of this approach are discussed the extension to a continuum is also briefly discussed

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

dynamic lateral torsional post buckling of a beam mass system theory abstract in this paper the theory of a cosserat point is used as a numerical model of a nonlinear elastic beam mass system and simulations of dynamic lateral torsional buckling are compared with results of experiments the cosserat equations are solved using the newmark time integration scheme and an analytical expression for the tangent stiffness of the resulting newton raphson iteration procedure is developed also the effects of material damping aerodynamic drag and gravity have been included the simulations reproduce the experimental result that two different nonlinear modes of vibration occur at the same excitation amplitude and frequency one mode torsion i is a post buckling mode associated with out of plane motion of the beam mass system which is dominated by oscillating torsion of the beam the second mode bending ii is dominated by a nonlinear second bending mode in the weak bending plane the simulations of torsion i and bending ii and another mode torsion ii are in reasonably good quantitative agreement with the experimental results however the model is not able to accurately simulate the response of torsion iii which was similar to torsion i and ii but with a larger amplitude of vibration comparison is also made with the commercial code ansys

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

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

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

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

three dimensional free vibrations of a circular arch using the theory of a cosserat point small deformation three dimensional free vibrations of a circular arch with uniform rectangular cross section have been investigated by using different theoretical approaches and by experimental verification special emphasis has been focused on a numerical formulation which models each element of the arch using the theory of a cosserat point comparison has been made with accurate three dimensional numerical modeling of the arch using the computer program ansys also three dimensional beam elements in the computer programs ansys and i deas have been considered along with an exact solution of approximate model equations of the arch the results indicate that appropriate modeling of rotary inertia is necessary to predict accurate frequencies associated with torsional out of plane modes moreover in its general form the theory of a cosserat point is a fully nonlinear theory that has already been tested for buckling of beams and arches therefore the success of the cosserat theory for the dynamic problem considered in this paper suggests that the theory of a cosserat point can be used for more complicated nonlinear dynamic problems of thin rod like structures

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

expansion of random graphs new proofs new results we present a new approach to showing that random graphs are nearly optimal expanders this approach is based on recent deep results in combinatorial group theory it applies to both regular and irregular random graphs let gamma be a random d regular graph on n vertices and let lambda be the largest absolute value of a non trivial eigenvalue of its adjacency matrix it was conjectured by alon combinatorica 6 2 83 96 1986 that a random d regular graph is almost ramanujan in the following sense for every varepsilon 0 lambda 0 a a s lambda rho varepsilon where rho is the spectral radius of the universal cover of omega when omega is regular we get a bound of rho 0 84 and for an arbitrary omega we prove a nearly optimal upper bound of sqrt 3 rho this is a substantial improvement upon all known results by friedman linial puder lubetzky sudakov vu and addario berry griffiths

a new 3 d finite element for nonlinear elasticity using the theory of a cosserat point abstract the theory of a cosserat point has been used to formulate a new 3 d finite element for the numerical analysis of dynamic problems in nonlinear elasticity the kinematics of this element are consistent with the standard tri linear approximation in an eight node brick element specifically the cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations for hyperelastic response the constitutive equations for the director couples are determined by derivatives of a strain energy function restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test it is shown that the cosserat balance laws are in one to one correspondence with those obtained using a bubnov galerkin formulation nevertheless there is an essential difference between the two approaches in the procedure for obtaining the strain energy function specifically the cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data in contrast the bubnov galerkin approach determines these constitutive coefficients by integrating the 3 d strain energy function using the kinematic approximation it is shown that the resulting cosserat equations eliminate unphysical locking and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions

on coupled longitudinal and lateral vibrations of elastic rods abstract in this paper the coupled longitudinal and lateral vibrations of a rectangular parallelepiped composed of a uniform isotropic elastic body are examined the one dimensional model for the body was developed by green naghdi and several of their co workers it is based on a cosserat or directed rod theory the frequency spectrum is found to divide naturally into three regions which are separated by degenerate cases when the cross section of the body is square a natural splitting of the modes occurs one half of this split involves motions coupling the axial extension and symmetric lateral deformation of the body the other half involves asymmetric lateral deformations the cases of a free free rod and a cantilevered rod are discussed and the results compared to existing works in the literature which use the three dimensional theory of elasticity to model the vibrations

a reduced order model for optimal design of 2 mdof adjacent structures connected by hysteretic dampers abstract coupling adjacent structures with supplemental control devices appears to be a useful method to mitigate structural response in this work a reduced order model for optimal design of two multi degree of freedom 2 mdof structures connected by hysteretic dampers is studied the seismic input is modeled as a gaussian white noise stationary stochastic process since the passive connection is modeled as a nonlinear hysteretic element represented by the differential bouc wen law a stochastic linearization technique is applied in order to simplify the problem the design procedure is based on replacing the 2 mdof system with a generalized two single degree of freedom 2 sdof system by using the principle of virtual displacements here each structure is represented by an elementary oscillator interconnected by a hysteretic device once the equivalent structural parameters of the generalized 2 sdof system is known it is possible to carry out the optimal design of the connection by using simple spectra obtained by the authors in a previous work where the optimal design of a horizontal hysteretic link connecting a 2 sdof system has been studied and solved illustrative examples confirm the entire methodology and also verify the effectiveness of hysteretic connection on seismic response

on the significance of normal cross sectional extension in beam theory with application to contact problems abstract the main features of contact problems of elastic beams are explored by considering a specific equilibrium problem of a beam in contact with a smooth rigid flat surface solutions of four separate linear theories namely a general theory g and three others which are constructed as constrained theories are considered these constrained theories differ from the general theory only by the degree of exclusion of one or both types of deformation usually referred to as a transverse normal strain and b transverse shear deformation thus with both a and b absent the constrained theory corresponds to the bernoulli euler beam theory be with only b absent the constrained theory n accounts for normal extensional deformation and with only a absent the constrained theory corresponds to the timoshenko beam theory t comparison of the predictions of the solutions of the three constrained theories with that of theory g shows that theory n contains the main physics of the contact problem and correctly predicts the conditions under which the beam loses contact also the contact force is continuous at the end points of the contact region in contrast neither of the other two constrained theories be or t correctly predicts these features

the finite deformation theory for beam plate and shell part i the two dimensional beam theory a finite deformation theory is proposed that can describe precisely the nonlinear geometric behavior of a two dimensional beam structure unlike the existing nonlinear theories this theory does away with simplifications such as the assumptions of small displacement small normal or shearing strain small rotation as well as small loading step the finite element method and its formulation of this theory are deduced numerical examples performed by the theory proved it valid and computed results obtained by the theory are independent of loading steps

eigenvalues of graphs and a simple proof of a theorem of greenberg abstract in his ph d thesis greenberg proved that if x is the spectral radius of the universal cover x of a finite graph x then for each 0 a positive proportion depending only on x and of the eigenvalues of x have absolute value at least x in this paper we show that the same result holds true if we remove absolute from the previous result we also prove an analogue result for the smallest eigenvalues of x

ramanujan coverings of graphs abstract let g be a finite connected graph and let be the spectral radius of its universal cover for example if g is k regular then 2 k 1 we show that for every r there is an r covering a k a an r lift of g where all the new eigenvalues are bounded from above by it follows that a bipartite ramanujan graph has a ramanujan r covering for every r this generalizes the r 2 case due to marcus spielman and srivastava 26 every r covering of g corresponds to a labeling of the edges of g by elements of the symmetric group s r we generalize this notion to labeling the edges by elements of various groups and present a broader scenario where ramanujan coverings are guaranteed to exist in particular this shows the existence of richer families of bipartite ramanujan graphs than was known before inspired by 26 a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups the core argument of this component is taken from 27 another important ingredient of our proof is a new generalization of the matching polynomial of a graph we define the r th matching polynomial of g to be the average matching polynomial of all r coverings of g we show this polynomial shares many properties with the original matching polynomial for example it is real rooted with all its roots inside

invariant manifolds in mechanics ii zero dimensional elastic bodies with directors application de la methode des varietes invariantes a un cas concret etablissement a partir de l elasticite tridimensionnelle d une theorie des corps de dimension 0 avec trois directeurs

behaviour of the extensible elastica solution the general form of the virtual work expression for the large strain euler bernoulli beam theory is derived using the nominal strain biot s tensor from the equilibrium equations derived from the virtual work expression it turns out that a linear relation between biot s stress tensor and the biot nominal strain tensor forms the differential equation used to derive the elastica solution moreover in the differential equation one additional term enters which is related to the extensibility of the beam axis as a special application the well known problem of an axially loaded beam is analysed due to the extensibility of the beam axis it is shown that the buckling load of the extensible elastica solution depends on the slenderness and it is of interest that for small slenderness the bifurcation point becomes unstable this means the bifurcation point changes from being supercritical which always hold for the inextensible case i e the classical elastica solution to being a subcritical point in addition higher order singularities are found as well as nonbifurcating isolated branches

finite rotations in dynamics of beams and implicit time stepping schemes we examine theoretical and computational aspects of three dimensional finite rotations pertinent to the dynamics of beams the model problem chosen for consideration is the reissner beam theory capable of modelling finite strains and finite rotations in geometrically exact manner special emphasis is placed on clarifying the geometry aspects finite rotation updates and the associated linearization procedure pertaining to different choices of rotation parameters the latter is shown to play an important role in constructing the optimal implementation of a time stepping scheme 1998 john wiley sons ltd

three dimensional nonlinear dynamics of slender structures cosserat rod element approach in this paper the modelling strategy of a cosserat rod element cre is addressed systematically for three dimensional dynamical analysis of slender structures we employ the nonlinear kinematic relationships in the sense of cosserat theory and adopt the bernoulli hypothesis the kirchoff constitutive relations are adopted to provide an adequate description of elastic properties in terms of a few elastic moduli a deformed configuration of the rod is described by the displacement vector of the deformed centroid curves and an orthonormal moving frame rigidly attached to the cross section of the rod the position of the moving frame relative to the inertial frame is specified by the rotation matrix parametrized by a rotational vector the approximate solutions of the nonlinear partial differential equations of motion in quasi static sense are chosen as the shape functions with up to third order nonlinear terms of generic nodal displacements based on the lagrangian constructed by the cosserat kinetic energy and strain energy expressions the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements a simple example is presented to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure the corresponding nonlinear dynamical responses of the structures have been presented through numerical simulations by matlab software

the non backtracking spectrum of the universal cover of a graph a non backtracking walk on a graph h is a directed path of directed edges of h such that no edge is the inverse of its preceding edge non backtracking walks of a given length can be counted using the non backtracking adjacency matrix b indexed by h s directed edges and related to ihara s zeta function we show how to determine b s spectrum in the case where h is a tree covering a finite graph we show that when h is not regular this spectrum can have positive measure in the complex plane unlike the regular case we show that outside of b s spectrum the corresponding green function has periodic decay ratios the existence of such a ratio system can be effectively checked and is equivalent to being outside the spectrum we also prove that the spectral radius of the non backtracking walk operator on the tree covering a finite graph is exactly gr where gr is the growth rate of the tree this further motivates the definition of the graph theoretical riemann hypothesis proposed by stark and terras st finally we give experimental evidence that for a fixed finite graph h a ran dom lift of large degree has non backtracking new spectrum near that of h s universal cover this suggests a new generalization of alon s second eigenvalue conjecture

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

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

a simplified implicit newmark integration scheme for finite rotations implicit newmark schemes for integration of finite rotations in structural and continuum mechanics typically are more complicated than those used for translational motion using euler s representation of the rotation tensor l in terms of a rotation vector f numerical integration connecting the values f n l n and f n 1 l n 1 at the beginning and end of the time step is expressed in terms of an incremental rotation vector q and the associated incremental rotation tensor l q here it is shown using backward differentiation that neglecting third order terms in q the approximation for the angular velocity w in terms of q and its time derivative has the same form as that between velocity and displacement consequently the simplified newmark scheme in terms of q w email protected has the same form as that for updating translations details of the newmark scheme and an analytical expression for the tangent stiffness tensor for the associated newton raphson iteration procedure have been presented for rigid body dynamics the resulting integration scheme has been tested on a nontrivial problem of three dimensional motion of a rigid body using a constant time step the results justify the use of the simplified newmark scheme for finite rotations

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

cylindrical element isogeometric model of continuum rod a geometric mapping between a canonical cylindrical domain and a physical rod like body is established and is utilized to analyze rod like structures undergoing large deformations a rod is described as a full three dimensional continuum yet parameterized only by surface control points this description preserves the geometric smoothness and allows for the consideration of lateral deformation in analysis numerical examples are presented to demonstrate the method

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

numerical solution procedures for nonlinear elastic rods using the theory of a cosserat point the theory of a cosserat point is developed as a continuum model which is inherently nonlinear and is valid for arbitrary constitutive equations here attention is confined to nonlinear elastic response which is hyperelastic with a strain energy function but large displacements deformations and rotations are allowed it is shown that the theory of a cosserat point can be used to formulate a numerical solution procedure for the dynamic three dimensional motion of nonlinear curved rods by modeling the rod as a set of n connected cosserat points like finite elements specifically the cosserat model allows for axial extension tangential shear deformation normal cross sectional extension normal cross sectional shear deformation and rotary inertia the cosserat approach ensures that the global forms of the balances of linear and angular momentum are satisfied and the hyperelastic nature of the constitutive equations is preserved since the response functions are determined by derivatives of a strain energy function a number of static example problems have been considered which examine the influence of shear deformation by comparing cosserat solutions with nonlinear solutions of an elastica

on finite element implementation of geometrically nonlinear reissner s beam theory three dimensional curved beam elements abstract finite element implementation of the three dimensional finite strain beam theory of reissner is considered in this work in contrast with some earlier works on the subject discussed here are the beam elements whose reference axes are arbitrary space curved lines we have shown that an improved representation of curved reference geometry significantly increases the accuracy of the results however it also makes the choice of non locking finite element interpolations somewhat more delicate a hierarchical displacement interpolation proposed here is proved to be capable of eliminating both shear and membrane locking phenomena a very satisfying non locking performance is demonstrated for a set of problems in nonlinear elasto statics

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

kinematically exact curved and twisted strain based beam the paper presents a formulation of the geometrically exact three dimensional beam theory where the shape functions of three dimensional rotations are obtained from strains by the analytical solution of kinematic equations in general it is very demanding to obtain rotations from known rotational strains in the paper we limit our studies to the constant strain field along the element the relation between the total three dimensional rotations and the rotational strains is complicated even when a constant strain field is assumed the analytical solution for the rotation matrix is for constant rotational strains expressed by the matrix exponential despite the analytical relationship between rotations and rotational strains the governing equations of the beam are in general too demanding to be solved analytically a finite element strain based formulation is presented in which numerical integration in governing equations and their variations is completely omitted and replaced by analytical integrals some interesting connections between quantities and non linear expressions of the beam are revealed these relations can also serve as useful guidelines in the development of new finite elements especially in the choice of suitable shape functions

spectra of lifted ramanujan graphs abstract a random n lift of a base graph g is its cover graph h on the vertices n v g where for each edge uv in g there is an independent uniform bijection and h has all edges of the form i u i v a main motivation for studying lifts is understanding ramanujan graphs and namely whether typical covers of such a graph are also ramanujan let g be a graph with largest eigenvalue 1 and let be the spectral radius of its universal cover friedman 2003 12 proved that every new eigenvalue of a random lift of g is o 1 2 1 1 2 with high probability and conjectured a bound of o 1 which would be tight by results of lubotzky and greenberg 1995 15 linial and puder 2010 17 improved friedman s bound to o 2 3 1 1 3 for d regular graphs where 1 d and 2 d 1 this translates to a bound of o d 2 3 compared to the conjectured 2 d 1 here we analyze the spectrum of a random n lift of a d regular graph whose nontrivial eigenvalues are all at most in absolute value we show that with high probability the absolute value of every nontrivial eigenvalue of the lift is o log this result is tight up to a logarithmic factor and for d 2 3 e it substantially improves the above upper bounds of friedman and of linial and puder in particular it implies that a typical n lift of a ramanujan graph is nearly ramanujan

physically nonlinear and related approximate theories of elasticity and their invariance properties the existing developments of physically nonlinear elasticity have several shortcomings with the aim of remedying these deficiencies a number of approximate theories of elasticity are discussed in the present paper and in particular a theory of physically nonlinear elasticity is systematically developed more specifically stress strain relations for anisotropic physically nonlinear materials are derived the method of casey naghdi 1 is then applied to obtain properly invariant results this method involves the use of auxiliary motions obtained by removing from any given motion the translation and rotation at any one particle called a pivot the auxiliary motions represent the original motions in the approximate theory the connection between the transformation of fields under a change of pivot and invariance requirements associated with the auxiliary motions is investigated

finite element analysis of linear and non linear planar deformations of elastic initially curved beams reference epfl article 102719view record in web of science record created on 2007 04 23 modified on 2016 08 08

isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using jaumann strain formulation abstract using an isogeometric approach a continuum based configuration design sensitivity analysis dsa method is developed for curved kirchhoff beams with multi patch junctions under the total lagrangian formulation large deformations considering the initial curvature of curved beams are described by geometrically exact beam theory gebt and jaumann strain formulation in the isogeometric approach the higher order continuity and the exact description of initial geometry are naturally embedded using nurbs basis functions in multi patch models c 0 continuity of physical displacement or c 1 continuity of displacement component at junction is weakly imposed using the lagrange multiplier method the superior accuracy of isogeometric analysis iga is verified through the comparison with the results of finite element analysis fea using cubic hermite interpolation in the dsa a material derivative is utilized and the kinematical description of gebt is consistently employed to express orientation design variations contrary to the iga based dsa the hermite basis function explicitly depends on design in the fea based dsa due to its element length parameter moreover since the design velocity field is approximated using the nodal velocity imposed at nodal tangential vector the amount of design perturbations should be very small to obtain precise design sensitivity

measure preserving words are primitive a word w 2 fk the free group on k generators is called primitive if it belongs to some basis of fk associated with w and a finite group g is the word map w g g g defined on the direct product of k copies of g we call w measure preserving if given uniform measure on g g the image of this word map induces uniform measure on g for every finite group g it is easy to see that every primitive word is measure preserving and several authors have conjectured that the two properties are in fact equivalent here we prove this conjecture the main ingredients of the proof include random coverings of stallings graphs algebraic extensions of free groups and mobius inversions our methods yield the stronger result that a subgroup of fk is measure preserving iff it is a free factor as an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of fk form a closed set in this topology

determination of hourglass coefficients in the theory of a cosserat point for nonlinear elastic beams abstract the theory of a cosserat point has recently been used int j solids struct 38 2001 4395 to formulate the numerical solution of problems of nonlinear elastic beams in that theory the constitutive equations for inhomogeneous elastic deformations included undetermined constants associated with hourglass modes which can occur due to nonuniform cross sectional extension and nonuniform torsion the objective of this paper is to determine these hourglass coefficients by matching exact solutions of pure bending and pure torsion applied in different directions on each of the surfaces of the element it is shown that the resulting constitutive equations in the cosserat theory do not exhibit unphysical stiffness increases due to thinness of the beam mesh refinement or incompressibility that are present in the associated bubnov galerkin formulation also example problems of a bar hanging under its own weight and a bar attached to a spinning rigid hub are analyzed

a timoshenko beam reduced order model for shape tracking with a slender mechanism abstract we consider a flexible bio inspired slender mechanism modeled as a timoshenko beam it is coupled to the environment by a continuous distribution of compliant elements we derive a reduced order model by projecting the governing partial differential equations along the linear modal basis of the timoshenko beam the coupling with the substrate allows us to formulate the problem in a control framework and eventually to treat the system as a sensor to reconstruct the profile of the substrate through the deformation of the body the coupling is modeled in the framework of two parameters elastic foundations the convergence of the reduced order model with increasing number of basis functions is addressed in a suitable h1 error norm a closed loop force control is simulated for shape morphing when the system is coupled with a smooth substrate

random seismic response analysis of adjacent buildings coupled with non linear hysteretic dampers abstract coupling adjacent buildings with supplemental damping devices is a practical and effective approach to mitigating structural seismic response this paper is intended to develop a method for analyzing the random seismic response of a structural system consisting of two adjacent buildings interconnected by non linear hysteretic damping devices by modelling the buildings as multi degree of freedom elastic structures and representing the hysteretic dampers with the versatile bouc wen bw differential model an augmented state differential equation is formulated to describe the vibration of the coupled structural system under non white random seismic excitation after dealing with the non linear hysteretic dampers using stochastic linearization technique a non linear algebraic lyapunov equation is derived from which the system mean square response is iteratively solved the developed method is applicable to the structural system with an arbitrary number of storeys and with connecting dampers at arbitrary storeys the results of the analysis demonstrate that non linear hysteretic dampers are effective even if they are placed on a few floor levels in particular this type of damping device offers a wideband vibration suppression for earthquake attacks with either low or high dominant exciting frequencies parametric studies also show that optimum damper parameters and numbers exist which minimize the random seismic response

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

tangent operators sensitivity expressions and optimal design of non linear elastica in contact with applications to beams the tangent operator and design sensitivity expressions for non linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method the sensitivity computations are then combined with a numerical programming package to create an optimal design environment to exemplify the optimal design environment a beam s contact surface is contoured to minimize bending stress this work combines the research on the sensitivity analysis of non linear elastic bodies subject to constraints finite strain non linear elastic beam analysis and contact analyses the analysis is valid for any smooth contact surface and specialized for the case in which the surface is represented by a cubic spline the direct differentiation method is utilized to perform the sensitivity analysis in an example problem the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve

numerical formulations of elasto viscoplastic response of beams accounting for the effect of shear the finite deformation elasto viscoplastic response of beams is considered from a numerical point of view in this paper formulations in terms of stress resultants are discussed and extended to include viscoplastic material response by introducing rate equations for the evolution of the finite inelastic deformations special attention is given to the objectivity of the rates and to the form of the plastic flow potential an alternative one dimensional formulation obtained by introducing a kinematic constraint and numerically integrating over the cross section of the beam is also presented both formulations account for the effect of warping which arises as a result of shear deformation a numerical algorithm for the time dependent problem is presented in which the numerically integrated nonlinear rate equations are exactly satisfied at each iteration of the newton raphson scheme the implications of the formulations discussed in this paper are illustrated through a set of numerical examples

invariant manifolds in mechanics i the general construction of coarse theories from fine theories la theorie dite fine fournit potentiellement des informations tres detaillees sur le comportement du materiau et generalement plus d informations que necessaire tandis que la theorie dite grossiere donne des informations brutes plus accessibles qui peuvent etre suffisantes dans des circonstances donnees presentation d une methode permettant de deduire les theories grossieres des theories fines

shape optimal design of elastic space frames with non linear response this paper describes an approach to shape optimal design of elastic planar frames with non linear response the foundation of the proposed approach forms an appropriate strategy of shape representation of the structure based on the design element technique a frame structure is treated as to be assembled from several frame design elements which in turn may consist of several appropriately joined beam finite elements the shape of each frame design element is defined by convenient functions involving bezier blending polynomials the original formulation of the beam finite element proposed by saje is modified in order to fit nicely into the context of the frame design element technique the formulation of the shape optimal design problem in a form of a problem of non linear mathematical programming and its solution by employing gradient based methods of mathematical programming are discussed briefly the theory is illustrated with two numerical examples

nonlinear dynamics of elastic rods using the cosserat theory modelling and simulation the method of cosserat dynamics is employed to explore the nonplanar nonlinear dynamics of elastic rods the rod which is assumed to undergo flexure about two principal axes extension shear and torsion are described by a general geometrically exact theory based on the cosserat theory a set of governing partial differential equations of motion with arbitrary boundary conditions is formulated in terms of the displacements and angular variables thus the dynamical analysis of elastic rods can be carried out rather simply the case of doubly symmetric cross section of the rod is considered and the kirchoff constitutive relations are adopted to provide an adequate description of elastic properties in terms of a few elastic moduli a cantilever is given as a simple example to demonstrate the use of the formulation developed the nonlinear dynamic model with the corresponding boundary and initial conditions are numerically solved using the femlab matlab software packages the corresponding nonlinear dynamical responses of the cantilever under external harmonic excitations are presented through numerical simulations

on steady motions of an elastic rod with application to contact problems based on the directed theory of an elastic rod developed by green naghdi and several of their co workers this paper presents a formulation which is convenient to use when considering steady motions of elastic rods a properly invariant linear theory of this formulation is also developed and used to analyze several systems these systems involve surface constraints and find application in various technologies where elastic rod like bodies are present

boundary control of a timoshenko beam attached to a rigid body planar motion a flexible spacecraft modelled as a rigid body which rotates in an inertial space is considered a light flexible beam is clamped to the rigid body at one end and free at the other end the equations of motion are obtained by using the geometrically exact beam model for the flexible beam and it is then shown that under planar motion assumption linearization of this model yields the timoshenko beam model it is shown that suitable boundary controls applied to the free end of the beam and a control torque applied to the rigid body stabilize the system the proof is obtained by using a lyapunov functional based on the energy of the system

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

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

formulation and solution of inverse spaghetti problem application to beam deployment dynamics a finite element model of the dynamics of axially moving highly flexible beams is presented the model is based on a geometrically nonlinear beam formulation that allows for large overall motions to account for the varying length of a deploying beam a moving finite element reference grid is incorporated within the nonlinear beam formulation such that the number of finite element nodes remains fixed and the finite element length is allowed to vary hamilton s law is used to formulate the equations of motion and a transient integration solution procedure is derived from a space time finite element discretization of the hamiltonian variational principle computational results of the methodology are presented for a planar inverse spaghetti problem he dynamics of axially moving beamlike structures are becoming increasingly important for the analysis of spacecraft and large space structures that deploy flexible appendages such as antenna stabilizing booms solar arrays and long trusslike structures as well as for other applications such as magnetic tape drives printing machines traveling cables and band saws an extensive amount of research has focused on the modeling of axially moving continua to analyze the dynamic behavior of such structures one of the most common models of axially moving continua that has received the most attention is the traveling string problem in what he termed the spaghetti problem carrier1 analyzed the motion of a string being accelerated upward into a fixed orifice in other works since then analytical expressions representing both linear and nonlinear vibrations of axially moving strings subject to various support conditions have been derived 2 9 the dynamics of beamlike structures traveling between or over fixed supports have also been studied to address industrial applications such as high speed tape drives and band saws 10 13 to address spacecraft applications in which antenna or other long flexible structures are deployed from a host satellite the dynamics of a cantilever beam that is ejected from a single fixed guide are modeled using bernoulli euler beam theory in ref 14 in a quite different approach the extrusion of a beam from a rotating base is modeled using a series of elastically connected rigid links in which the number of rigid links are continuously increased decreased to account for deployment retrieval of the beam 15 general formulations of axially moving bernoulli euler beam models cantilevered to a rotating rigid host body have also been developed to study the effect of appendage deployment on the attitude dynamics of orbiting spacecraft 16 20 all of these given analyses are based on an assumption of small elastic deformations with the exception of the discrete spring mass model of ref 15 all of the other models that have been referenced thus far use linear combinations of space depende nt admissible functions of a time varying length weighted by time dependen t generalized coordi

finite element formulation of geometrically exact three dimensional beam theories based on interpolation of strain measures this paper presents a new finite element formulation of the geometrically exact finite strain beam theory the governing equations of the beam element are derived in which the strain vectors are the only unknown functions the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal is enforced to be satisfied at chosen points the solution is found by a collocation algorithm the linearity of the strain space not only simplifies the application of newton s method on the non linear configuration space but also leads to the strain objectivity of the proposed method the accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples c 2003 elsevier b v all rights reserved

a unified procedure for construction of theories of deformable media ii generalized continua in contrast to classical theories discussed in the accompanying paper part i the present paper deals exclusively with generalized continua a term that refers to a body embedded in a euclidean three dimensional space with each of its material points endowed with additional kinematic structure such generalized continua are conveniently identified here by cosserat or directed continua in which the additional structure is represented by independent director fields and the material domain of the body manifold is classified according to the four categories a a three dimensional volume b a two dimensional surface c a one dimensional space curve and d a point in the derivation of a thermomechanical theory for each of these categories we use the same unified procedure as in part i but now the various energies that enter the balance of energy must be modified to include additional kinetic ingredients new theories of this kind have increasingly provided over the last three decades effective means of formulating and studying new behaviour of materials in both fluid and solid mechanics that were not previously possible by the classical theories our attention is first focused in the development of the theory for the first category and this is discussed in two stages first for a cosserat continuum with a single director and then for n directors our derived results include for the first time a derivation from the balance of energy of balances of director momenta and balances of entropies in the presence of more than one independent temperature field the treatments of categories b to d are somewhat shorter since their developments are formally similar to that of category a

an energy preserving description of nonlinear beam vibrations in modal coordinates abstract conserved quantities are identified in the equations describing large amplitude free vibrations of beams projected onto their linear normal modes this is achieved by writing the geometrically exact equations of motion in their intrinsic or hamiltonian form before the modal transformation for nonlinear free vibrations about a zero force equilibrium it is shown that the finite dimensional equations of motion in modal coordinates are energy preserving even though they only approximate the total energy of the infinite dimensional system for beams with constant follower forces energy like conserved quantities are also obtained in the finite dimensional equations of motion via casimir functions the duality between space and time variables in the intrinsic description is finally carried over to the definition of a conserved quantity in space which is identified as the local cross sectional power numerical examples are used to illustrate the main results

total lagrangian reissner s geometrically exact beam element without singularities in this paper we introduce a new reissner s geometrically exact beam element which is based on a total lagrangian updating procedure the element has the rotation vector as the dependent variable and the singularity problems at the rotation angle 2 and its multiples are passed by the change of parametrization on the rotation manifold the beam formulation has several benefits such as all the unknown vectors belong to the same tangential vector space no need for secondary storage variables the path independence in the static case any standard time integration algorithm may be used and the symmetric stiffness copyright 2006 john wiley sons ltd

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

on constitutive relations for elastic rods this paper explores the relationships among several existing procedures for specifying linear and nonlinear constitutive relations for hyperelastic cosserat or directed curves from three dimensional considerations these procedures are designed to ensure that exact solutions of the balance laws of the directed curve are in approximate agreement with the corresponding exact solutions of classical continuum mechanics the particular case of the linearized bernoulli euler beam theory is also examined in addition a new procedure is proposed in this paper this procedure is used to facilitate comparisons among various existing procedures although it is conceptually simple it illustrates the inherent arbitrariness in specifying constitutive relations for elastic rods

circular element isogeometric elements of smooth boundary this article introduces a family of elements of smooth curved geometry unlike traditional approaches based on finite element geometry the proposed method employs rational bezier functions for boundary description and as a result common shapes such as circles and ellipses can be exactly described the interior of the element is parameterized in terms of the boundary control points via basis functions that are smooth extensions of the boundary bezier bases the interior basis functions are constructed over the unit circular domain and applied in an isoparametric setting to other convex domains that are topologically circular this family of elements is particularly suited for analyzing discrete bodies undergoing more or less uniform or regular deformations an example involving the compression of a circular elastic disk in which the disk is modeled as a single 16 point element is presented to demonstrate the application