rational points on k3 surfaces a new canonical height a fundamental tenet of diophantine geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties this is certainly true for curves where the sign of the euler characteristic of c determines whether the set of rational points on c is finite x c 0 for higher dimensional varieties there are some precise conjectures due to bombieri lang and vojta 15 which predict when the rational points on a variety should be finite or degenerate i e not zariski dense and some conjectures of manin et al 2 5 on the distribution of rational points in those cases when they are zariski dense but except for abelian varieties their subvarieties and some fano varieties varieties for which the anticanonical bundle is ample there are very few general theorems in this paper we will study the rational points on a certain class of k 3 surfaces defined over a number field k the moduli space of marked algebraic k3 surfaces is a countable union of 19 dimensional quasi projective varieties we are going to look at the 18 dimensional family studied by wehler 1 17 wehler s family consists of k3 surfaces s whose automorphism group aut s contains a subgroup d isomorphic to the free product 7 2 2 of two cyclic groups of order 2 we will use the geometric information provided by this infinite automorphism group to study the k rat ional points on s for any point pes we can look at the orbit of p under the action of d

conservation of angular momentum and energy in the integration of non linear dynamic equations the analysis and implementation of a step by step algorithm conceived to preserve scalar invariants of motion i e angular momentum and energy of a rigid body in the integration of non linear dynamic equations is presented here the considered rigid body is subjected to generic translational and rotational motions with large displacements and finite rotations the algorithm is implemented in a new efficient c f e m code which allows an easy performance of several numerical applications

global format for energy momentum based time integration in nonlinear dynamics summary a global format is developed for momentum and energy consistent time integration of second order dynamic systems with general nonlinear stiffness the algorithm is formulated by integrating the state space equations of motion over the time increment the internal force is first represented in fourth order form consisting of the end point mean value plus a term containing the stiffness matrix increment this form gives energy conservation for systems with internal energy as a quartic function of the displacement components this representation is then extended to general energy conservation via a discrete gradient representation the present procedure works directly with the internal force and the stiffness matrix at the time integration interval end points and in contrast to previous energy conserving algorithms it does not require any special form of the energy function nor use of mean value products at the element level or explicit use of a geometric stiffness matrix an optional monotonic algorithmic damping increasing with response frequency is developed in terms of a single damping parameter in the solution procedure the velocity is eliminated and the nonlinear iterations are based on the displacement components alone the procedure represents an energy consistent alternative to available collocation methods with an equally simple implementation copyright 2014 john wiley sons ltd

exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

an energy decaying scheme for nonlinear dynamics of shells a novel integration scheme for nonlinear dynamics of geometrically exact shells is developed based on the inextensible director assumption the new algorithm is designed so as to imply the strict decay of the system total mechanical energy at each time step and consequently unconditional stability is achieved in the nonlinear regime furthermore the scheme features tunable high frequency numerical damping and it is therefore stiffly accurate the method is tested for a finite element spatial formulation of shells based on mixed interpolations of strain tensorial components and on a two parameter representation of director rotations the robustness of the scheme is illustrated with the help of numerical examples

energy consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints the present work deals with the energy consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints the underlying differential algebraic equations daes with index three are directly discretized this approach makes possible the development of a new energy consistent time stepping scheme for general nonholonomic systems in particular both nonholonomic problems from rigid body dynamics as well as flexible multibody dynamics can be treated in a unified manner in this connection specific constrained formulations of rigid bodies and geometrically exact beams are presented moreover the newly developed discrete null space method is applied to achieve a size reduction and an improved conditioning of the discrete system the numerical examples deal with a nonholonomic rigid body system and a flexible multibody system

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

on non linear dynamics of shells implementation of energy momentum conserving algorithm for a finite rotation shell model continuum and numerical formulations for non linear dynamics of thin shells are presented in this work an elastodynamic shell model is developed from the three dimensional continuum by employing standard assumptions of the first order shear deformation theories motion of the shell directior is described by a singularity free formulation based on the rotation vector temporal discretization is performed by an implicit one step second order accurate time integration scheme in this work an energy and momentum conserving algorithm which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability is used it may be regarded as a modification of the standard mid point rule spatial discretization is based on the four noded isoparametric element particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space in order to achieve a quadratic rate of asymptotic convergence typical for the newton raphson based solution procedures an unconditionally stable time finite element formulation suitable for the long term dynamic computations of flexible shell like structures which may be undergoing large displacements large rotations and large motions is therefore obtained a set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four noded shell finite element in conjunction with the implicit energy and momentum conserving time integration algorithm

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

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions

variational time integrators for finite dimensional thermo elasto dynamics without heat conduction this paper focuses on the formulation of variational integrators vis for finite dimensional thermo elastic systems without heat conduction the dynamics of these systems happen to have a hamiltonian structure after thermal displacements are introduced it is then possible to formulate integrators by taking advantage of standard methods in vi the class of integrators we construct have some remarkable features a they are symplectic b they exactly conserve the entropy of the system or in other words they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes c they nearly exactly conserve the value of the energy for very long times and d they exactly conserve linear and angular momentum we first describe how to adapt any vi for the classical mechanical systems to integrate adiabatic thermo elastic ones and then formulate three new types of integrators the first class based on a generalized trapezoidal rule gives rise to two first order explicit integrators and a second order implicit one by composing then the two first order integrators we construct a second order explicit one finally we formulate a fourth order implicit integrator which is a symplectic partitioned runge kutta method the performance of these new algorithms is showcased through numerical examples copyright 2011 john wiley sons ltd

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

the role of geometric stiffness in momentum and energy conserving time integration a momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic green strain momentum conserving algorithms are formulated from an integral of the equations of motion and energy conservation has traditionally been obtained by evaluating the contribution from the internal forces by use of a combined mean value of stresses and virtual strains on the element level it is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the geometric stiffness matrix over the current time step usually directly available in global form in existing finite element programmes the theory is derived by the use of a state space formulation where this extra term is located in the same position as the viscous damping matrix indicating that the effect of the extra incremental geometric stiffness term in the non linear algorithm is equivalent to a variable damping term depending on the change of the state of stress over a time increment in the actual numerical algorithm the new value of the velocity vector is eliminated leaving a non linear equation for the displacement increment alone followed by an explicit vector update of the velocity increment copyright 2006 john wiley sons ltd

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

explicit momentum conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm we reformulate the midpoint lie algorithm which is implicit in the torque calculation to achieve explicitness in the torque evaluation this is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step thus we obtain two related variants both of which are explicit in the torque calculation but only first order in the time step both variants are momentum conserving and both are symplectic consequently drawing on the properties of the composition of maps we introduce another algorithm that combines the two variants in a single time step the resulting algorithm is explicit momentum conserving symplectic and second order its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators

limiting subdifferentials of indefinite integrals abstract we compute the limiting subdifferential f x of the indefinite integral of the form f x a x f t d t where f is an essentially bounded measurable function or a function continuous on an interval containing x r except for possibly x or a step function which has a countable number of steps around x the related problem of computing the aumann integral of the limiting subdifferential mapping f where f is a lipschitz real function defined on an open set u r n is also investigated

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

conservative fourth order time integration of non linear dynamic systems abstract an energy conserving time integration algorithm with fourth order accuracy is developed for dynamic systems with nonlinear stiffness the discrete formulation is derived by integrating the differential state space equations of motion over the integration time increment and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts this process introduces the time derivatives of the state space variables and these are then substituted from the original state space differential equations the resulting discrete form of the state space equations is a direct fourth order accurate representation of the original differential equations this fourth order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components energy conservation for a force potential of general form is obtained by addition of a higher order secant type correction term the formulation leads to a consistent representation of the motion within a time increment corresponding to cubic hermite interpolation in time this in turn leads to excellent phase representation with only a small fourth order error permitting integration of oscillatory systems with only a few integration points per period three numerical examples demonstrate the high accuracy of the algorithm

variational integrators for the dynamics of thermo elastic solids with finite speed thermal waves this paper formulates variational integrators for finite element discretizations of deformable bodies with heat conduction in the form of finite speed thermal waves the cornerstone of the construction consists in taking advantage of the fact that the green naghdi theory of type ii for thermo elastic solids has a hamiltonian structure thus standard techniques to construct variational integrators can be applied to finite element discretizations of the problem the resulting discrete in time trajectories are then consistent with the laws of thermodynamics for these systems for an isolated system they exactly conserve the total entropy and nearly exactly conserve the total energy over exponentially long periods of time moreover linear and angular momenta are also exactly conserved whenever the exact system does for definiteness we construct an explicit second order accurate algorithm for affine tetrahedral elements in two and three dimensions and demonstrate its performance with numerical examples

a self stabilized algorithm for enforcing constraints in multibody systems a new algorithm is developed for the enforcement of constraints within the framework of nonlinear flexible multibody system modeled with the finite element approach the proposed algorithm exactly satisfies the constraints at the displacement and velocity levels and furthermore it achieves nonlinear unconditional stability by imposing the vanishing of the work done by the constraint forces when combined with specific discretizations of the inertial and elastic forces identical convergence rates are observed for the displacements velocities and lagrange multipliers the proposed algorithm is closely related to the stabilized index 2 or ggl method although no additional multipliers are introduced in the present approach these desirable characteristics are obtained without resorting to numerically dissipative algorithms if high frequencies are present in the system i e the system is physically stiff dissipative schemes become necessary the proposed algorithm is extended to deal with this situation

a new energy and momentum conserving algorithm for the non linear dynamics of shells a numerical time integration scheme for the dynamics of non linear elastic shells is presented that simultaneously and independent of the time step size inherits exactly the conservation laws of total linear total angular momentum as well as total energy the proposed technique generalizes to non linear shells recent work of the authors on non linear elastodynamics and is ideally suited for long term large scale simulations the algorithm is second order accurate and can be immediately extended with no modification to a fourth order accurate scheme the property of exact energy conservation induces a strong notion of non linear numerical stability which manifests itself in actual simulations

energy conservation in newmark based time integration algorithms abstract energy balance equations are established for the newmark time integration algorithm and for the derived algorithms with algorithmic damping introduced via averaging the so called methods the energy balance equations form a sequence applicable to newmark integration of the undamped equations of motion an extended form including structural damping and finally the generalized form including structural as well as algorithmic damping in all three cases the expression for energy appearing in the balance equation is the mechanical energy plus some additional terms generated by the discretization by the algorithm the magnitude and character of these terms as well as the associated damping terms are discussed in relation to energy conservation and stability of the algorithms it is demonstrated that the additional terms in the energy lead to periodic fluctuations of the mechanical energy and are the cause of the phenomenon of response overshoot previously observed empirically in the application of newmark based algorithms to high frequency components it is also demonstrated that the stability limit of the explicit newmark algorithm is reached when the stiffness term in the algorithmic energy vanishes and that energy fluctuations take place for integration intervals close to the stability limit

stabilization and attitude of a triaxial rigid body by lie group methods this paper deals with the construction of algorithms which preserve the first integrals of a class of forced rigid motions which retains a hamiltonian structure using lie group methods such as those due to lewis and simo and munthe kaas for these mechanical systems we also study the reconstruction of dynamical processes to describe the evolution of the orientation of the forced rigid body in space for this we consider different algorithms to solve the kinematic equations a comparison between these algorithms is made finally we illustrate the numerical methods discussed here studying the stabilization of the relative equilibrium corresponding to the stationary rotation about the intermediate axis in the presence of external torques about the minor axis

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

properties of time integration with first order filter damping introduction of algorithmic damping by increasing the parameter values in the newmark algorithm leads to undesirable low frequency damping and reduced order of accuracy it is demonstrated how these effects can be removed by introducing an extra damping term in the form of a first order linear filter when the linear filter is discretized in time the state variable associated with the filter can be eliminated leading to a weighted average of the equations of motion at two consecutive times the filter procedure contains the known versions of alpha weighted newmark methods as special cases but gives a different and improved weighting of the excitation terms a complete analysis of the properties of the algorithm when used on linear systems is given including the frequency response properties it is demonstrated that the effect of overshoot is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system and analytic results are presented for the magnitude of the effect copyright 2005 john wiley sons ltd

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

conservation properties of a time fe method part iii mechanical systems with holonomic constraints a galerkin based discretization method for index 3 differential algebraic equations pertaining to finite dimensional mechanical systems with holonomic constraints is proposed in particular the mixed galerkin mg methodis introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained hamiltonian system namely total energy and angular momentum in addition to that the constraints on the configuration level and on the velocity momentum level are fulfilled exactly the application of the mg method to specific mechanical systems such as the pendulum rigid body dynamics and the coupled motion of rigid and flexible bodies is presented related numerical examples are investigated to evaluate the numerical performance of the mg 1 and mg 2 method

quaternion based rigid body rotation integration algorithms for use in particle methods the resolution of translational motion in discrete element method and molecular dynamics applications is a straightforward task however resolving rotational motion is less obvious many applications update rotation using an explicit integration involving products of matrices which has well known drawbacks although rigid body rotation has received attention in large angle rotation applications relatively little attention has been dedicated to the unique requirements of particle methods using explicit time stepping algorithms this paper reviews existing explicit algorithms and shows the benefits of using a quaternion based re parameterization of both the central difference algorithm and the approach of munjiza et al int j numer meth engng 2003 56 36 55 the improvement not only provides guaranteed orthonormality of the resulting rotation but also allows the assumption of small angle rotation to be relaxed and the use of a more accurate taylor expansion instead the current and quaternion based algorithms are compared for accuracy and computational efficiency copyright 2007 john wiley sons ltd

algorithms for coupled problems that preserve symmetries and the laws of thermodynamics part i monolithic integrators and their application to finite strain thermoelasticity abstract we propose discretizations for a general class of nonlinear coupled thermomechanical problems of evolution the most salient feature of the new methods is that they rigorously preserve the two laws of thermodynamics as well as the symmetries of the systems they model to formulate such methods we exploit the geometric structure afforded by the generic formalism of non equilibrium thermodynamics and we follow a systematic methodology that results in discrete evolution equations which mimic the generic structure as an illustration a complete discretization of finite strain thermoelasticity is presented using finite elements in space and a monolithic integrator in time simulations are provided which demonstrate the conservation features of the algorithm as well as its remarkable robustness

geometrically exact intrinsic theory for dynamics of curved and twisted anisotropic beams a formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams when the applied loads at the ends of and distributed along the beam are independent of the deformation neither displacement nor rotation variables appear an intrinsic formulation like well known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two advantages of such a formulation are demonstrated with a simple example when the initial curvature and twist are constant along the beam two space time conservation laws are shown to exist one being a work energy relation and the other a generalized impulse momentum relation these laws can be used for example as benchmarks to check the accuracy of any proposed solution including time marching and finite element schemes the structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization a particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end loaded beams that by virtue of the kirchhoff analogy leads to a time marching scheme for the dynamics of a pivoted rigid body in a gravity field this time marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy whereas the analogous spatial discretization scheme for the nonlinear static behavior of end loaded beams satisfies analogous integrals of deformation along the beam span remarkably a straightforward generalization of these discretization schemes is shown to satisfy both space time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space time element

a formulation of conserving impact system based on localized lagrange multipliers this paper proposes a new finite element method of frictional impact of elastic bodies the formulation introduces a contact frame that is placed in between contacting bodies and represents the contact surface the nonpenetration condition and the slip stick condition are defined between the contacting body and the contact frame with the aid of the independent localized lagrange multipliers representing the contact forces the position of the contact frame and the local coordinate of the contacting node along the contact frame are also treated as the independent variable which enables the exact satisfaction of the constraint conditions without the deficiency or redundant constraint the energy and momentum conservation algorithm is applied to the proposed impact system for the case of frictional impact the linear momentum is exactly conserved and the angular momentum is approximately conserved with negligible error copyright 2006 john wiley sons ltd

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

accurate and efficient simulation of rigid body rotations abstract this paper introduces efficient and accurate algorithms for simulating the rotation of a three dimensional rigid object and compares them to several prior methods first we introduce a second order accurate method that incorporates a third order correction then we introduce a third order accurate method and finally we give a fourth order accurate method these methods are single step and the update operation is only a single rotation the algorithms are derived in a general lie group setting second we introduce a near optimal energy correction method which allows exact conservation of energy this algorithm is faster and easier to implement than implicit methods for exact energy conservation our third order method with energy conservation is experimentally seen to act better than a fourth order accurate method these new methods are superior to naive runge kutta or predictor corrector methods which are only second order accurate for sphere valued functions the second order symplectic mclachlan reich methods are observed to be excellent at approximate energy conservation but are not as good at long term accuracy as our best methods finally we present comparisons with fourth order accurate symplectic methods which have good accuracy but higher computational cost

lie poisson integration for rigid body dynamics abstract in this paper the splitting midpoint rule is presented and proved to be the lie poisson integrators to the rigid body systems further discussions are also given numerical experiments show that this method has well properties comparing with the runge kutta method and ordinary midpoint rule

state space time integration with energy control and fourth order accuracy for linear dynamic systems a fourth order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented it is derived by integrating the phase space representation and evaluating the resulting displacement and velocity integrals via integration by parts substituting the time derivatives from the original differential equations the resulting algorithm has an exact energy equation in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix this implies unconditional stability of the algorithm and the relative phase error is of fourth order an optional high frequency algorithmic damping is constructed by optimal combination of three different damping matrices each proportional to either the mass or the stiffness matrix this leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth order in the frequency thus the low frequency response is virtually undamped and the algorithm remains third order accurate even when algorithmic damping is included the accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium where the algorithmic damping is used to reduce dispersion due to the spatial discretization leading to a smooth solution with a clearly defined wave front copyright 2005 john wiley sons ltd

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

dynamic analysis of rigid and deformable multibody systems with penalty methods and energy momentum schemes abstract a multibody formulation for the nonlinear dynamics of mechanical systems composed of both rigid and deformable bodies is proposed in this work focusing on its conservation properties for basic magnitudes such as total energy and momentum the approach is based on the use of dependent variables cartesian coordinates of selected points and the enforcement of the constraints through the penalty method this choice has the advantage of providing a simple overall structure that allows the inclusion of both rigid bodies discrete model and elastic bodies continuum model discretised with the finite element method under the same framework in order to build a single set of ordinary differential equations the elastic bodies are represented by general hyperelastic models and may undergo large displacements rotations and strains an energy momentum time integration method has been employed achieving remarkable stability and robustness with exact conservation of total energy this approach effectively overcomes drawbacks associated with penalty formulations in other time integration algorithms this important result in fact proves to be the main conclusion of this work some representative numerical simulations are presented for mechanical systems comprised of rigid and deformable bodies

on characterization of convexity for regularly locally lipschitz functions we use a second order upper dini directional derivative in order to characterize convexity for regularly locally lipschitz functions this improves in a certain sense the previously obtained results of cominetti correa and yang

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

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

canonical vector heights on k3 surfaces with picard number three an argument for nonexistence in this paper we investigate a k3 surface with picard number three and present evidence that strongly suggests a canonical vector height cannot exist on this surface

a 3 d discrete element method for dry granular flows of ellipsoidal particles abstract we use 3 d discrete element method dem to simulate dry granular flows of non spherical particles in our simulation code non spherical particles are represented by clusters of spheres for contact detection and for contact force calculation for particles in contact the normal force displacement nfd model is simulated by a partially latching spring system whereas the tangential force displacement tfd model is simulated by an incremental relation that corresponds to a simplification of the mindlin and deresiewicz r d mindlin h deresiewicz asme j appl mech 20 1953 327 344 contact mechanics theory to establish the validity of the simulation procedure benchmark tests such as superball and superbean simulations were carried out we present various results obtained from the simulation of ellipsoidal particles flowing down a chute having a bumpy bottom these ellipsoidal particles had their physical properties obtained from experiments

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

how to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties abstract for a fairly general class of second order accurate algorithms a sub stepping procedure is described which achieves fourth order accuracy while retaining the stability conservation properties and implementation of the underlying second order method two features render the proposed approach especially attractive from a computational perspective over conventional fourth order methods first the improved fourth order accuracy is attained merely by two additional residual evaluations with no changes in the implementation of the underlying second order method in particular no additional storage or extra computations of high order gradients are required second for hamiltonian systems the stability properties and possible conservation properties of the underlying second order method such as energy momentum and symplectic character are preserved the new technique is applied to a number of standard algorithms for elastodynamics structural and rigid body dynamics in particular when applied to the recently proposed second order accurate energy momentum method this approach results in a fourth order accurate energy linear and angular momentum conserving algorithm for general hamiltonian systems

second order variational analysis and characterizations of tilt stable optimal solutions in infinite dimensional spaces abstract the paper is devoted to developing second order tools of variational analysis and their applications to characterizing tilt stable local minimizers of constrained optimization problems infinite dimensional spaces with many results new also in finite dimensional settings the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization based on second order generalized differentiation we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second order growth the results obtained are applied to deriving new necessary and sufficient conditions for tilt stable minimizers in problems of nonlinear programming with twice continuously differentiable data in hilbert spaces

k3 surfaces with picard number three and canonical vector heights in this paper we construct the first known explicit family of k3 surfaces defined over the rationals that are proved to have geometric picard number 3 this family is dense in one of the components of the moduli space of all polarized k3 surfaces with picard number at least 3 we also use an example from this family to fill a gap in an earlier paper by the first author in that paper an argument for the nonexistence of canonical vector heights on k3 surfaces of picard number 3 was given based on an explicit surface that was not proved to have picard number 3 we redo the computations for one of our surfaces and come to the same conclusion

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

generalized midpoint finite element dynamic analysis of elastoplastic systems the application of a four parameter generalized midpoint integration method to small displacement elastoplastic dynamic problems is considered the integration of the governing relations over a time step is formulated as a non linear algebraic problem finite step problem by making use of generalized variables finite element modelling uniqueness and an extremum characterization of the solution of the finite step problem are discussed sufficient conditions for convergence of various iterative algorithms including newton raphson are given the b stability of the proposed procedure is assessed

sensitivity analysis for dynamic mechanical systems with finite rotations this paper presents a sensitivity analysis for dynamic systems with large rotations using a semi analytical direct differentiation technique the choice of a suitable time integration strategy for the rotation group appears to be critical for the development of an efficient sensitivity analysis three versions of the generalized time integration scheme are considered a residual form a linear form and a geometric form their consistency is discussed and we show that the residual form which is the most direct extension of the generalized algorithm defined in structural dynamics should not be used for problems with large rotations the sensitivity analysis is performed and close connections are highlighted between the structure of the sensitivity equations and of the linearized dynamic equations hence algorithms developed for the original problem can be directly reused for the sensitivities finally a numerical example is analysed in detail copyright 2007 john wiley sons ltd

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

high order integration of smooth dynamical systems theory and numerical experiments this paper describes a new class of algorithms for integrating linear second order equations and those containing smooth non linearities the algorithms are based on a combination of ideas from standard newmark integration methods and extrapolation techniques for the algorithm to work the underlying newmark method must be stable second order accurate and produce asymptotic error expansions for response quantities containing only even ordered terms it is proved that setting the newmark parameter equal to 1 2 gives a desirable asymptotic expansion irrespective of the setting for numerical experiments are conducted for one linear and two non linear applications

computing the canonical height on k3 surfaces let s be a surface in p 2 p 2 given by the intersection of a 1 1 form and a 2 2 form then s is a k3 surface with two noncommuting involutions x and y in 1991 the second author constructed two height functions h and h which behave canonically with respect to x and y and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights we discuss how the geometry of the surface s is related to formulas for the local heights and we give practical algorithms for computing the involution x y the local heights and the canonical heights h h

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

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

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

flexible body dynamics in a local frame with explicitly predicted motion this paper deals with formulation of dynamics of a moving flexible body in a local frame of reference in a conventional approach the local frame is normally fixed to the corresponding body and always represents the positions and angles of the body the positions and angles are represented by cartesian coordinates and euler angles or euler parameters respectively the elastic degrees of freedom are expressed by e g nodal coordinates in a finite element analysis modal coordinates etc however the choice of these variables as the generalized coordinates makes the resulting equations of motion extremely complicated this is because the representation of the rotation of a body is highly non linear and this non linearity makes the coefficient matrices dependent on the coordinates themselves in this paper we propose an alternative way of treating the issue by explicitly predicting the body motions and regularly updating the local frame first the motion of the local frame is assumed to explicitly follow the associated moving body then the equations of motion are derived in a set of generalized coordinates that express both rigid body and elastic degrees of freedom in the local frame these equations are solved by a time integration with a given time interval the motion of the local frame in the interval is estimated from a prediction of the rigid body motions then the gap between the predicted and the actual motions is evaluated finally the predictions are iteratively corrected by the obtained responses in the rigid body motions so that the gap should remain within an imposed tolerance copyright 2009 john wiley sons ltd

generalized differentiation of piecewise linear functions in second order variational analysis abstract the paper is devoted to a comprehensive second order study of a remarkable class of convex extended real valued functions that is highly important in many aspects of nonlinear and variational analysis specifically those related to optimization and stability this class consists of lower semicontinuous functions with possibly infinite values on finite dimensional spaces which are labeled as piecewise linear ones and can be equivalently described via the convexity of their epigraphs in this paper we calculate the second order subdifferentials generalized hessians of arbitrary convex piecewise linear functions together with the corresponding geometric objects entirely in terms of their initial data the obtained formulas allow us in particular to justify a new exact equality type second order sum rule for such functions in the general nonsmooth setting

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

on the design of energy preserving and decaying schemes for flexible nonlinear multi body systems abstract energy preserving schemes achieve unconditional stability for nonlinear systems by establishing discrete energy preservation statements several schemes have been presented by various authors within drastically different frameworks finite difference schemes based on a mid point approximation galerkin and time discontinuous galerkin approximations of the equations of motion written in the symmetric hyperbolic form finite elements in time and 2 stage fsal runge kutta methods furthermore different types of parameterization of finite rotations were used in the various formulations this paper presents a unified finite difference framework which readily allows comparing the various schemes and their respective properties numerical examples are presented and show that the predictions of two of these schemes are in very close agreement with each other

on lower bounds of the second order directional derivatives of ben tal zowe and chaney let f be a regular locally lipschitz real valued function defined on an open convex subset of a normed space we show that at any unit direction u the upper second order derivative d 2f u 0 in the sense of dem yanov and pevnyi dem yanov v f a b pevnyi 1974 expansion with respect to a parameter of the extremal values of game problems ussr computational math and math phys 14 33 45 ben tal and zowe ben tal a j zowe 1982 necessary and sufficient optimality conditions for a class of nonsmooth minimization problems math programming24 70 91 has the same lower bounds as the lower second order derivatives d 2f u 0 consequently one can characterize the convexity of f in terms of these derivatives we also obtain the corresponding results in terms of chaney s second order directional derivatives

a discrete momentum conserving explicit algorithm for rigid body dynamics analysis a discrete momentum conserving explicit time integration is presented the accurate feature and simplicity of the present algorithm are realized by a mid point implicit formula for integrating the euler parameters and a second order discrete momentum conserving form of the central difference algorithm respectively the accuracy and robustness of the algorithm is demonstrated by example problems which exhibit large overall rigid motions under holonomic constraints

lie group methods for rigid body dynamics and time integration on manifolds recently there has been an increasing interest in time integrators for ordinary differential equations which use lie group actions as a primitive in the design of the methods these methods are usually phrased in an abstract sense for arbitrary lie groups and actions we show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general an important part of the lie group methods involves the computation of a coordinate map and its derivative various options are available and they vary in cost accuracy and ability to approximately conserve invariants we discuss how the computation of these maps can be optimized for the rigid body case and we provide numerical experiments which give an idea of the performance of lie group methods compared to other known integration schemes

second order subdifferentials and convexity of real valued functions abstract in this paper we show that the positive semi definiteness psd of the frechet and or mordukhovich second order subdifferentials can recognize the convexity of c 1 functions however the psd is insufficient for ensuring the convexity of a locally lipschitz function in general a complete characterization of strong convexity via the second order subdifferentials is also given