a variationally consistent fractional time step integration method for incompressible and nearly incompressible lagrangian dynamics in this paper we present a fractional time step method for lagrangian formulations of solid dynamics problems the method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly energy fluctuations are found to be minimal and stay bounded for long durations in order to handle incompressibility a mixed formulation in which the pressure appears explicitly is adopted the velocity update over a time step is split into deviatoric and volumetric components the deviatoric component is advanced using explicit time marching whereas the pressure correction for each time step is computed implicitly by solving a poisson like equation once the pressure is known the volumetric component of the velocity update is calculated in contrast with standard explicit schemes where the time step size is determined by the speed of the pressure waves the allowable time step for the proposed scheme is found to depend only on the shear wave speed this leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems copyright 2005 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

qualitative and numerical analysis of quasi static problems in elastoplasticity the quasi static problem of elastoplasticity with combined kinematic isotropic hardening is formulated as a time dependent variational inequality vi of the mixed kind that is it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space this vi differs from the standard parabolic vi in that time derivatives of the unknown variable occur in all of its terms the problem is shown to possess a unique solution we consider two types of approximations to the vi corresponding to the quasi static problem of elastoplasticity semidiscrete approximations in which only the spatial domain is discretized by finite elements and fully discrete approximations in which the spatial domain is again discretized by finite elements and the temporal domain is discretized and the time derivative appearing in the vi is approximated in an appropriate way estimates of the errors inherent in the above two types of approximations in suitable sobolev norms are obtained for the quasi static problem of elastoplasticity in particular these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and where appropriate of the time step size k a major difficulty in solving the problems is caused by the presence of the nondifferentiable terms we consider some regularization techniques for overcoming the difficulty besides the usual convergence estimates we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

the midpoint scheme and variants for hamiltonian systems advantages and pitfalls the implicit midpoint scheme like higher order gauss collocation schemes is algebraically stable and symplectic and it preserves quadratic integral invariants it may appear particularly suitable for the numerical solution of highly oscillatory hamiltonian systems such as those arising in molecular dynamics or structural mechanics because there is no stability restriction when it is applied to a simple harmonic oscillator although it is well known that the midpoint scheme may also exhibit instabilities in various stiff situations one might still hope for good results when resonance type instabilities are avoided in this paper we investigate the suitability of the midpoint scheme for highly oscillatory frictionless mechanical systems where the step size k is much larger than the system s small parameter varepsilon in the case that the solution remains bounded as varepsilon rightarrow 0 we show that in general one must require that k 2 varepsilon be small enough or else even the errors in slowly varying quantities like the energy may grow undesirably especially when fast and slow modes are tightly coupled or worse the computation may yield misleading information in some cases this may already happen when k o varepsilon the same holds for higher order collocation at gaussian points the encountered restrictions on k are typically still better than the corresponding ones for explicit schemes

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

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

a time reversible variable stepsize integrator for constrained dynamics this article considers the design and implementation of variable timestep methods for simulating holonomically constrained mechanical systems symplectic variable stepsizes are briefly discussed and we consider time reparameterization techniques employing a time reversible symmetric integration method to solve the equations of motion we give several numerical examples including a simulation of an elastic inextensible unshearable rod undergoing large deformations and collisions with the sides of a bounding box numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation

glauber dynamics of the sk model theory and simulations in the high temperature phase the low temperature glauber dynamics of the sherrington kirkpatrick spin glass model is studied using a combination of theory and computer simulations the theoretical approach follows the spirit of mori s continuous fraction expansion the predicted amplitude of the long time decay of the time dependent equilibrium spin spin correlation function agrees well with the glauber dynamics simulation results the theory predicts a universal dynamic exponent of in the low temperature phase the simulations cannot distinguish between a universal exponent and a temperature dependent one

energy preserving schemes for nonlinear hamiltonian systems of wave equations application to the vibrating piano string the problem of the vibration of a string is well known in its linear form describing the transversal motion of a string nevertheless this description does not explain all the observations well enough nonlinear coupling between longitudinal and transversal modes seams to better model the piano string as does for instance the geometrically exact model gem this report introduces a general class of nonlinear systems nonlinear hamiltonian systems of wave equations in which fits the gem mathematical study of these systems is lead in a first part showing central properties energy preservation existence and unicity of a global smooth solution finite propagation velocity ldots space discretization is made in a classical way variational formulation and time discretization aims at numerical stability using an energy technique a definition of preserving schemes is introduced and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is linear a general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear hamiltonian systems of wave equations class

theory and approximation of quasistatic frictional contact problems abstract a variational formulation of a class of quasistatic frictional contact problems is derived for which a compliant interface model of the contact surface is assumed proof of existence and uniqueness of solutions is given finite element approximations of a regularized version of the fractional problem are developed and a priori error estimates are derived the theory is implemented and used to solve representative test problems of frictional contact the numerical simulations show that frictional stress distributions can be very sensitive to load histories and surface properties of contact interfaces

regularized friction and continuation comparison with coulomb s law abstract periodic solutions of systems with friction are difficult to investigate because of the non smooth nature of friction laws this paper examines periodic solutions and most notably stick slip on a simple one degree of freedom system mass spring damper and belt with coulomb s friction law and with a regularized friction law i e the friction coefficient becomes a function of relative speed with a stiffness parameter with coulomb s law the stick slip solution is constructed step by step which gives a usable existence condition with the regularized law the asymptotic numerical method and the harmonic balance method provide bifurcation diagrams with respect to the belt speed or normal force and for several values of the regularization parameter formulations from the coulomb case give the means of a comparison between regularized solutions and a standard reference with an appropriate definition regularized stick slip motion exists its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force

inherently energy conserving time finite elements for classical mechanics in this paper we develop a finite element method for the temporal discretization of the equations of motion the continuous galerkin method is based upon a weighted residual statement of hamilton s canonical equations we show that the proposed finite element formulation is energy conserving in a natural sense a family of implicit one step algorithms is generated by specifying the polynomial approximation in conjunction with the quadrature formula used for the evaluation of time integrals the numerical implementation of linear quadratic and cubic time finite elements is treated in detail for the model problem of a circular pendulum in addition to that concerning dynamical systems with several degrees of freedom we address the design of nonstandard quadrature rules which retain the energy conservation property our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time stepping schemes

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

conservative unconditionally stable discretization methods for hamiltonian equations applied to wave motion in lattice equations modeling protein molecules abstract a new approach is described for generating exactly energy momentum conserving time discretizations for a wide class of hamiltonian systems of des with quadratic momenta including mechanical systems with central forces it is well suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the davydov system modeling energetic pulse propagation in protein molecules the method is unconditionally stable making it well suited to equations of broadly discrete nls form including many arising in nonlinear optics key features of the resulting discretizations are exact conservation of both the hamiltonian and quadratic conserved quantities related to continuous linear symmetries preservation of time reversal symmetry unconditional stability and respecting the linearity of certain terms the last feature allows a simple efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability avoiding the need for full newton type solvers one distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations based on a discrete gradient calculus with product rule that mimics the essential properties of partial derivatives this numerical method is then used to study the davydov system revealing that previously conjectured continuum limit approximations by nls do not hold but that sech like pulses related to nls solitons can nevertheless sometimes arise

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

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

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

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

analysis of dynamic nonlinear thermoviscoelastic beam problems abstract in this paper we study a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles thermal effects are also taken into account and the contact is modeled using the classical signorini condition the existence of solutions is proved by considering approximate problems from a penalization procedure proving some a priori estimates and passing to the limit an exponential decay property is also obtained then fully discrete approximations of the approximate problem are provided using the finite element method for the spatial approximation and the implicit euler scheme for the discretization of the time derivatives a stability property is obtained and a priori error estimates are proved from which under some additional regularity conditions the linear convergence of the algorithm is deduced fully discrete approximations of the signorini problem are then introduced proceeding in a similar way for which a stability property is obtained finally some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution

an energy momentum conserving algorithm using the variational formulation of visco plastic updates in this paper we use the variational formulation of elasto plastic updates proposed by ortiz and stainier comput methods appl mech eng 1999 171 419 444 in the context of consistent time integration schemes we show that such a formulation is well suited to obtain a general expression of energy momentum conserving algorithms moreover we present numerical examples that illustrate the efficiency of our developments copyright 2005 john wiley sons ltd

riemann solutions and spacetime discontinuous galerkin method for linear elastodynamic contact abstract we derive riemann solutions for stick slip and separation contact modes in a linear elastic material with an isotropic coulomb friction relation and explore their numerical implementation the riemann solutions preserve the characteristic structure of the underlying elastodynamic system and imply dynamic contact conditions that are distinct from the quasi static conditions used in some numerical models nonphysical discontinuities in the standard coulomb model at stick slip transitions can cause contact mode chatter in numerical simulations we restate the coulomb relation to remove these artificial discontinuities and eliminate the need for algorithmic remedies discontinuous response at abrupt separation to contact transitions is physically reasonable and we propose a regularization scheme to address this case we implement the riemann contact solutions within an adaptive spacetime discontinuous galerkin sdg code and report numerical results that demonstrate the model s efficacy

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

energy conserving schemes for the simulation of musical instrument contact dynamics abstract collisions are an innate part of the function of many musical instruments due to the nonlinear nature of contact forces special care has to be taken in the construction of numerical schemes for simulation and sound synthesis finite difference schemes and other time stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a newtonian description of the system however because impact forces are non analytic functions of the phase space variables algorithm stability can rarely be established this way this paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling the proposed numerical formulations follow from discretising hamilton s equations of motion generally leading to an implicit system of nonlinear equations that can be solved with newton s method the approach is first outlined for point mass collisions and then extended to distributed settings such as vibrating strings and beams colliding with rigid obstacles stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples the methodology is exemplified through a case study on tanpura string vibration with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument

a variational principle for dynamic contact with large deformation abstract dynamic contact between a deformable body with either large or small deformation and a rigid obstacle can be modeled with signorini contact condition stated in terms of the normal component of the displacement as long as the relative positions are clearly defined the geometrical relation however is unpredictable when large motion or large deformation and complex surface geometry are involved the use of normal velocity is more suitable a variational inequality is proposed for the dynamic contact problem the penalty method is introduced and implemented in explicit finite element software numerical examples are presented to demonstrate the robustness of the algorithm

assumed strain finite element methods for conserving temporal integrations in non linear solid dynamics this paper presents a new class of assumed strain finite elements to use in combination with general energy momentum conserving time stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time the case of interest corresponds to nearly incompressible material responses in the fully non linear finite strain elastic and elastoplastic ranges the new elements consider the classical scaling of the deformation gradient with an assumed jacobian its determinant defined locally through a weighted averaging procedure at the element level the key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or b bar operator the developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta and the conservation dissipation law of energy for elastic and inelastic problems respectively care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration with these developments at hand a new general b bar operator is introduced that satisfies these conditions the new operator not only accounts for the spatial interpolations e g bilinear displacements with piece wise constant volume but also depends on the discrete structure of the equations in time the aforementioned conservation dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes unconditionally of the time step size and the material model elastic or elastoplastic different finite elements are considered in this framework including quadrilateral and triangular elements for plane problems and brick elements for three dimensional problems several representative numerical simulations are presented involving in particular the use of energy dissipating momentum conserving time stepping schemes recently developed by the author and co workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements including these conservation dissipation properties in time and their locking free response in the quasi incompressible case copyright 2007 john wiley sons ltd

a numerical analysis of a class of problems in elastodynamics with friction on utilise les methodes d elements finis pour analyser le comportement dynamique de corps lineairement elastiques ayant une partie de leurs frontieres soumise au frottement et a des tractions normales prescrites on etablit des estimations d erreur on presente un algorithme de resolution des problemes a deux dimensions et on decrit plusieurs essais numeriques

kinetic theory approach to the sk spin glass model with glauber dynamics a new method to analyse glauber dynamics of the sherrington kirkpatrick sk spin glass model is presented the method is based on ideas used in the classical kinetic theory of fluids it is applied to study spin correlations in the high temperature phase of the sk model at zero external field the zeroth order theory is equivalent to a disorder dependent local equilibrium approximation its predictions agree well with computer simulation results the first order theory involves coupled evolution equations for the spin correlations and the dynamic excess parts of the local field distributions it qualitatively accounts for the error made in the zeroth approximation

energetic pulses in exciton phonon molecular chains and conservative numerical methods for quasilinear hamiltonian systems the phenomenon of coherent energetic pulse propagation in exciton phonon molecular chains such as helix protein is studied using an ode system model of davydov scott type both with numerical studies using a new unconditionally stable fourth order accurate energy momentum conserving time discretization and with analytical explanation of the main numerical observations physically natural impulsive initial data associated with the energy released by atp hydrolysis are used and the best current estimates of physical parameter values in contrast to previous studies

a linear finite element approach to the solution of the variational inequalities arising in contact problems of structural dynamics the present paper deals with the theoretical and numerical treatment of dynamic unilateral problems the governing equations are formulated as an equivalent variational inequality expressing d alembert s principle in its inequality form the discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

coupling of finite elements and boundary elements methods for study of the frictional contact problem this paper presents the numerical study of the elastic contact problem with coulomb friction between both deformable sphere and support using the finite element method we solve this problem governed by a variational inequality with a successive approximations method each iteration of which being constituted by the solution of an intermediate problem with normal pressure fixed to compute stresses in contact we couple the finite elements with the boundary elements our results are conform to the hertz theory and the analytical solution of spence moreover our study shows the influence of friction on parameters such as the contact radius and the normal pressure

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

a dynamic thermoviscoelastic contact problem with the second sound effect abstract this paper deals with a contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the cattaneo law relating the heat flux to the gradient of the temperature the beam is rigidly clamped at its left end whereas the right end of the beam moves vertically between reactive stops like a nonlinear spring existence and uniqueness of the solution is proved as well as the exponential decay of the related energy then fully discrete approximations are introduced by using the classical finite element method and the implicit euler scheme to approximate the spatial variable and to discretize the time derivatives respectively an a priori error estimates result is proved from which the linear convergence of the algorithm is deduced the case where the two stops are rigid is also studied from the point of view of the existence and longtime behavior of the solutions finally some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution

dynamic linear response of the sk spin glass coupled microscopically to a bath the dynamic linear response theory of a general ising model weakly coupled to a heat bath is derived by employing the quantum statistical theory of mori treating the hamiltonian of the spin bath coupling as a perturbation and applying the markovian approximation both the dynamic susceptibility and the relaxation function are expressed in terms of the static susceptibility and the static internal field distribution function for the special case of the sk spin glass this internal field distribution can be related to the solutions of the tap equations in the entire temperature region application of this new relation and the use of numerical solutions of the modified tap equations lead for finite but large systems to explicit results for the distribution function and for dynamic linear response functions a detailed discussion is presented which includes finite size effects due to the derived temperature dependence of the onsager casimir coefficients a frequency dependent shift of the cusp temperature of the real part of the dynamic susceptibility is found

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

symplectic algorithms based on the principle of least action and generating functions summary in this paper the principle of least action and generating functions are used to construct symplectic numerical algorithms for finite dimensional autonomous hamiltonian systems the approximate action is obtained by approximating the generalized coordinates and momentums by lagrange polynomials and performing gaussian quadrature based on the principle of least action and the requirements of a canonical transformation different types of symplectic algorithms have been constructed by choosing different types of independent variables at two ends of the time step the symmetric property of the four types of symplectic algorithms proposed in this paper is discussed and the exact linear stability domain for small m n and g is discussed the linear stability and precision of different types of symplectic algorithms are tested using numerical examples copyright 2011 john wiley sons ltd

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

a family of noniterative integration methods with desired numerical dissipation a new family of unconditionally stable integration methods for structural dynamics has been developed which possesses the favorable numerical dissipation properties that can be continuously controlled in particular it can have zero damping this numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes whereas the low frequency modes are almost unaffected the most important improvement of this family method is that it involves no nonlinear iterations for each time step and thus it is very computationally efficient when compared with a general second order accurate integration method such as the constant average acceleration method copyright 2014 john wiley sons ltd

nonlinear stability analysis of area preserving integrators linear stability analysis is inadequate for integrators designed for nondissipative systems such as hamiltonian systems in which nonlinear effects are often decisive mathematical theory exists kam theory for rigorous analysis of small perturbations from equilibria but it needs to be expressed in a form that is more easily applicable to the study of area preserving maps we have pursued this obtaining a completely rigorous nonlinear stability analysis for elliptic equilibria based on the moser twist theorem and a result of cabral and meyer nonlinearity 12 1999 pp 1351 1362 together with the theory of normal forms for hamiltonian systems the result is a determination of necessary and sufficient conditions for stability these conditions are sharpened for the case of reversible maps and applied to the symplectic members of the newmark family of integrators which includes the leapfrog the implicit midpoint and the stormer cowell methods nonlinear stability limits are more severe than those of linear theory as an example the leapfrog scheme actually has a step size limitation of 71 of that predicted by linear analysis

irreducible memory function for dissipative stochastic systems with detailed balance abstract we show quite generally that the memory function for purely dissipative stochastic systems with detailed balance can be reduced further and can be expressed in terms of the irreducible memory function this generalizes the similar finding for the smoluchowski equation the irreducible memory function is intimately connected with the lifetime renormalization and is shown to provide a useful starting point for introducing approximations such as the mode coupling approximation

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