analytical solution of spatial elastica and its application to kinking problem abstract an analytical solution is presented for the spatially large deformation of a thin elastic rod spatial elastica which is naturally straight and uniform with equal principal stiffnesses and is subjected to terminal loads the elastica can suffer not only flexure and torsion as in the classical kirchhoff theory but also extension and shear the present solution is expressed in integral form and described in terms of only four parameters this solution clears the difficulty with the polar singularity in the use of euler angles hence the numerical analysis is possible for various boundary value problems with no limitation in this paper we study the post buckling behavior of an elastica under the terminal twist and uniaxial end shortening and give a theoretical explanation to commonly observed phenomena such as secondary bifurcation formation of a kink snap through behavior the contact problem is analyzed in the case where the elastica contacts with itself and forms a kink these results are available for other analysis e g based on finite element approximations

elliptic integral solutions of plane elastica with axial and shear deformations abstract closed form solutions are derived for elastica with axial and shear deformations using elliptic integrals the theories used here are the timoshenko beam theory of finite displacements with finite strains and that with small strains elliptic integral solutions are further transformed to normal forms to obtain accurate solutions as a result both of the closed form solutions are expressed in normal forms from the first to the third kind of elliptic integrals with these solutions two kinds of structures are analyzed to examine the effect of shear deformations along with the accuracy of the approximate theory of finite displacements with small strains

corotational nonlinear dynamic analysis of thin shell structures with finite rotations a new formulation for corotational nonlinear dynamic analysis of thin shell structures involving large displacements and finite rotations has been presented in this paper by introducing the kinematic description of the geometrically exact quasi static shell model to describe the motion of an arbitrary material point in a triangular shell element the new expressions for the inertial force vector and tangent inertia matrix were derived systematically the elastic deformation was handled by using the consistent symmetrizable equilibrated corotational formulation which is independent with respect to the local finite element modeling an energy momentum conserving algorithm and an energy decaying momentum conserving algorithm were developed for nonlinear dynamic analysis by applying the generalized energy momentum method in the corotational formulation three classic numerical examples including the dynamic response free motion and dynamic buckling of thin shell structures were computed to verify the ac

locally linearized runge kutta method of dormand and prince in this paper the effect that produces the local linearization of the embedded runge kutta formulas of dormand and prince for initial value problems is studied for this embedded locally linearized runge kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations for a variety of well known test equations with different dynamics the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones which implies a substantial reduction of the number of time steps and consequently a sensitive reduction of the overall computational cost of their adaptive implementation

high order w methods implicit methods are the natural choice for solving stiff systems of odes rosenbrock methods are a class of linear implicit methods for solving such stiff systems of odes in the rosenbrock methods the exact jacobian must be evaluated at every step these evaluations can make the computations costly by contrast w methods use occasional calculations of the jacobian matrix this makes the w methods popular among the class of linear implicit methods for numerical solution of stiff odes however the design of high order w methods is not easy because as the order of the w methods increases the number of order conditions of the w methods increases very fast in this paper we describe an approach to constructing high order w methods

co rotational finite element formulation for thin walled beams with generic open section a consistent co rotational total lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin walled beams with generic open section is presented the element developed here has two nodes with seven degrees of freedom per node the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis the deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element the element nodal forces are derived using the virtual work principle the virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces a procedure is proposed to determine the virtual rigid body motion the way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent in element nodal forces all coupling among bending twisting and stretching deformations of the beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory in the derivation of the element tangent stiffness matrix the change of element nodal forces induced by the element rigid body rotations should be considered for the present method thus a stability matrix is included in the element tangent stiffness matrix an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state numerical examples are presented to investigate the accuracy and efficiency of the proposed method the effect of the terms in the element nodal force and tangent stiffness matrix which will converge to zero with the decrease of element size on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples 2005 elsevier b v all rights reserved

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

a spatially curved beam element with warping and wagner effects this paper presents a new spatially curved beam element with warping and wagner effects that can be used for the non linear large displacement analysis of members that are curved in space the non linear behaviour of members curved in space shows that the wagner effects are substantial in the large twist rotation analysis most existing finite beam element models such as abaqus and ansys cannot predict the non linear large displacement response of members curved in space correctly because the wagner effects viz the wagner moment and the corresponding finite strain terms have not been considered in these finite beam elements as a consequence these finite beam elements do not provide correct predictions for the out of plane buckling and postbuckling behaviour of arches as well in this paper the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements the warping and wagner effects both the wagner moment and the corresponding finite strain terms and their constitutive relationship are included in the spatially curved beam element two components of the initial curvature the initial twist and their interactions with the displacements are also considered in the spatially curved beam element this ensures that the large twist rotation analysis for the members curved in space is accurate comparisons with existing experimental analytical and numerical results show that the spatially curved beam element is accurate and efficient for the non linear elastic analysis of curved members buckling and postbuckling analysis of arches and in its ability to predict large deflections and twist rotations in more arbitrarily curved members copyright 2005 john wiley sons ltd

an improved class of generalized runge kutta methods for stiff problems part ii the separated system case in order to study a generalization of the explicit p stage runge kutta methods we introduce a new family of p stage formulas for the numerical integration of some special systems of odes that provides better order and stability results with the same number of stages in our recent paper entitled an improved class of generalized runge kutta methods for stiff problems part i the scalar case we studied new schemes for the numerical integration of scalar autonomous odes in this second part we will show that it is possible to generalize our grk methods so that they can be applied to some non autonomous scalar odes and systems obtaining linearly implicit a stable and l stable methods these methods do not require jacobian evaluations in their implementation some numerical examples are discussed in order to show the good performance of the new schemes

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

optimal contact process on complex networks contact processes on complex networks are a recent subject of study in nonequilibrium statistical physics and they are also important to applied fields such as epidemiology and computer and communication networks a basic issue concerns finding an optimal strategy for spreading we provide a universal strategy that when a basic quantity in the contact process dynamics the contact probability determined by a generic function of its degree w k is chosen to be inversely proportional to the node degree i e w k ensuremath sim k ensuremath 1 spreading can be maximized computation results on both model and real world networks verify our theoretical prediction our result suggests the determining role played by small degree nodes in optimizing spreading in contrast to the intuition that hub nodes are important for spreading dynamics on complex networks

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

a study of rosenbrock type methods of high order this paper deals with the solution of nonlinear stiff ordinary differential equations the methods derived here are of rosenbrock type this has the advantage that they area stable or stiffly stable and nevertheless do not require the solution of nonlinear systems of equations we derive methods of orders 5 and 6 which require one evaluation of the jacobian and onelu decomposition per step we have written programs for these methods which use richardson extrapolation for the step size control and give numerical results

topology optimization of nonlinear heat conduction problems involving large temperature gradient abstract in this work we develop a topology optimization method for steady state nonlinear heat conduction problems involving large temperature gradient to minimize the maximum structural temperature temperature dependent material properties tdmps are taken into account to break the widely used assumption of constant material properties cmps in conventional topology optimization the kreisselmeier steinhauser function is adopted as an aggregated measure of the maximum temperature over a specific region and the adjoint method is used to derive the sensitivity expressions to effectively solve the well known unsymmetric adjoint problem caused by material nonlinearity an engineering oriented stationary iterative method esim is constructed to transform the unsymmetric adjoint problem into a symmetric one with a series of right hand sides that can be efficiently solved with mature linear system solvers both 2d and 3d numerical examples are provided to illustrate the validity and utility of the proposed method including the accuracy and convergence of the esim the results show that the assumption of cmps can result in significant inaccuracy in the analysis and design of heat conduction systems working under large temperature gradient on the contrary by considering tdmps and employing the proposed method reasonable designs are achieved and thermal performance is greatly improved

postbuckling analysis using a general purpose code a new capability for solving postbuckling problems in shell structures is described the matrix theory to adapt newton s method to nonlinear finite element shell analysis is outlined first the matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular the matrix theory suggests a change of variables as part of the usual iterative procedure in newton s method the change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures numerical results from a new option that has been programmed in an existing general purpose code are presented the analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples

deformation and vibration of a spatial elastica with fixed end slopes abstract in this paper we study the static deformations and vibration frequencies of a clamped clamped spatial elastica one clamp is fixed in space and the other is attached to a slider which is allowed to slide without friction on a linear track emphasis is placed on the effect of the slope difference between the two clamps on the response of the elastica when the slider is under the action of an edge thrust the equations of motion of the elastica are formulated within the framework of director theory and the static deformations and vibration frequencies are calculated by shooting method an experimental set up is designed to examine the theoretical predictions when the slope difference between the two clamps is smaller than 68 two separate load deflection curves can be found one for spatial deformation and the other for purely planar deformation in force control the elastica slider assembly may jump between planar and spatial deformations as the edge thrust increases or decreases monotonically in displacement control on the other hand no jump will occur in the case when the slope difference is greater than 68 only planar deformation exists

dynamical robustness analysis of weighted complex networks robustness of weighted complex networks is analyzed from nonlinear dynamical point of view and with focus on different roles of high degree and low degree nodes we find that the phenomenon for the low degree nodes being the key nodes in the heterogeneous networks only appears in weakly weighted networks and for weak coupling for all other parameters the heterogeneous networks are always highly vulnerable to the failure of high degree nodes this point is the same as in the structural robustness analysis we also find that with random inactivation heterogeneous networks are always more robust than the corresponding homogeneous networks with the same average degree except for one special parameter thus our findings give an integrated picture for the dynamical robustness analysis on complex networks c 2013 elsevier b v all rights reserved

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

koiter analysis of folded structures using a corotational approach the paper deals with geometrically nonlinear finite element analysis of folded plate and shell structures a koiter asymptotic approach is proposed based on the reuse of a linear element in the nonlinear context through a corotational formulation the corotational approach represents a simple and effective way to satisfy the basic requirement of koiter analysis i e full objectivity in the finite element modeling in fact starting simply from a suitable linear finite element and implementing the corotational algebra proposed in garcea et al 2009 zagari 2009 lead to objective explicit expressions for the first four variations of the strain energy which are needed by asymptotic analysis the shell element used here is the flat shell quadrangular element with 4 nodes and 6 dofs per node proposed in madeo et al 2012 and called miss 4 a mixed element based on the reissner mindlin plate theory with an allman like quadratic interpolation for displacements and an equilibrated isostatic interpolation for the stress resultants the element is free from locking and spurious zero energy modes so it appears a suitable candidate for nonlinear corotational analysis the results of the numerical validation show the effectiveness and accuracy of the proposed approach and its excellent overall robustness for both mono and multi modal buckling problems also in the presence of strong nonlinear pre critical behavior

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

finite rotation analysis and consistent linearization using projectors abstract a systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis this procedure is a generalization of the work presented in computers and structures volume 30 pp 257 267 the basis of our approach is the element independent co rotational algorithm where the element rigid body motion translations and rotations is separated from the deformational part of its total motion the variation of this co rotational relation results in a projector matrix with the property that a consistent internal force vector is invariant under its action the consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector this results in an unsymmetric tangent matrix whose anti symmetric part depends on the out of balance force vector in this paper we prove that using the symmetric part of the tangent matrix the newton iteration retains its quadratic rate of convergence this approach has been used to solve a number of large rotation test example problems the results demonstrate that it is possible to analyze structures undergoing large rotations within a general co rotational framework using simple and economical finite elements the resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre and post processors to the element routines

on the choice of the linear element for corotational triangular shells the corotational formulation for triangular thin shell elements presented in a eriksson c pacoste element formulation and numerical techniques for stability problems in shells comput methods

an improved class of generalized runge kutta methods for stiff problems part i the scalar case a new family of p stage methods for the numerical integration of some scalar equations and systems of odes is proposed these methods can be seen as a generalization of the explicit p stage runge kutta ones while providing better order and stability results we will show in this first part that at the cost of losing linearity in the formulas it is possible to obtain explicit a stable and l stable methods for the numerical integration of scalar autonomous odes scalar autonomous odes are of very little interest in current applications however be begin studying this kind of problems because most of the work can be easily extended to a more general situation in fact we will show in a second part entitled the separated system case that it is possible to generalize our methods so that they can be applied to some non autonomous scalar odes and systems we will obtain linearly implicit l stable methods which do not require jacobian evaluations in both parts some numerical examples are discussed in order to show the good performance of the new schemes

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 large strain elasto plastic shell formulation using the morley triangle the paper describes a large strain elasto plastic formulation for shells which is based on the faceted morley triangle with rotation variables only being provided at each mid side in the form of a rotation about the side a co rotational formulation is adopted with the local deformation gradient being obtained from a polar decomposition using f ru the material description is then based around logeu with u being decomposed into principal directions plasticity is treated with the aid of the multiplicative f fefp in contrast to many earlier formulations with the morley triangle the current formulation is invariant to the node numbering in order to achieve this invariance it is necessary to re visit the origins of the morley triangle and to re cast the formulation as a special form of discrete kirchhoff approach the associated constraint is now applied in relation to the current configuration copyright 2001 john wiley sons ltd

a c0 triangular plate element with one point quadrature on developpe un element de plaque triangulaire du type de mindlin qui utilise des champs lineaires pour les rotations et les deflexions transversales l element ne necessite qu un seul point de quadrature ce qui est tres interessant pour l analyse non lineaire

a simplified co rotational method for quadrilateral shell elements in geometrically nonlinear analysis summary this paper presents a simplified co rotational formulation for quadrilateral shell elements inheriting the merit of element independence from the traditional co rotational approach in literature with the objective of application to nonlinear analysis of civil engineering structures the authors further simply the formulation of the geometrical stiffness using the small strain assumption which is valid in the co rotational approach with the warping effects considered as eccentricities compared with the traditional element independent co rotational method the projector is neglected both in the tangent stiffness matrix and the internal force vector for simplicity in formulation meanwhile a quadrilateral flat shell element allowing for drilling rotations is adopted and incorporated into this simplified co rotational algorithm for geometrically nonlinear analysis involved with large displacements and large rotations several benchmark problems are presented to confirm the efficiency and accuracy of the proposed method for practical applications

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

avoiding the exactness of the jacobian matrix in rosenbrock formulae abstract a new class of methods for solving stiff systems which avoids the exactness of the jacobian matrix is introduced the order conditions for methods of order p 5 are given the linear stability properties for such methods are analysed numerical testing are also included

weighted k shell decomposition for complex networks based on potential edge weights abstract identifying influential nodes in complex networks has attracted much attention because of its great theoretical significance and wide application existing methods consider the edges equally when designing identifying methods for the unweighted networks in this paper we propose an edge weighting method based on adding the degree of its two end nodes and for the constructed weighted networks we propose a weighted k shell decomposition method w k s further investigations on the epidemic spreading process of the susceptible infected recovered sir model and susceptible infected si model in real complex networks verify that our method is effective for detecting the node influence

variational formulation in the geometrically nonlinear thin elastic shell theory abstract the variational formulation for thin elastic shells undergoing large deflections is discussed no restrictions are imposed on the magnitude of the rotational part of the deformation gradient yet it is assumed that the strains remain small everywhere in the shell ten functional and associated variational principles are derived including the hu washizu the hellinger reissner the generalized complementary energy and the stationary potential energy principles the total lagrangian description tld is exclusively used