a quasi monte carlo method for the boltzmann equation a new quasi monte carlo method for solving the boltzmann equation in a simplified case is described the analysis is restricted to a spatially homogeneous and isotropic gas in addition the molecular model only involves isotropic scattering the scheme makes use of particles and combines an euler scheme with numerical integrations the sequence which is used for the quadratures must possess some symmetry properties which prescribe energy conservation for colliding particles the error of the method is estimated by means of the discrepancy of the sequence which performs the quadratures an algorithm for generating convenient sequences is proposed in an example where an exact solution is known the computation of effective errors is included introduction rarefied gas flows are usually simulated by monte carlo techniques besides the successful direct simulation monte carlo dsmc method of bird 2 another scheme was derived by nanbu 14 from the boltzmann equation itself a drawback of both schemes are numerical fluctuations caused by the use of pseudorandom numbers an improved monte carlo scheme which reduces fluctuations has recently been developed at the university of kaiserslautern 1 it will be referred to as the kmc scheme a fully deterministic method for solving the boltzmann equation is proposed here the constraints on the analysis are the following we consider an infinite spatially homogeneous and isotropic gas the velocity distribution is radially symmetric the molecular model is characterized by isotropic scattering the differential cross section a depends only on the relative speed g and not on the deflection angle in addition we assume that ga g is some nonnegative nondecreasing and bounded function the hypothesis on the cross section allows physically relevant models such as the vhs model of bird 3 when a cutoff is used in earlier communications io i i the simplest choice ga g equals a constant was considered a deterministic version of the scheme of nanbu that we called the low discrepancy ld method was described an error analysis was proposed and it was shown that the ld method outperforms the original scheme of nanbu in a test case where an exact solution is known received march 1 1990 1980 mathematics subject classification 1985 revision primary 65m15 secondary 65c05 llk38 1991 american mathematical society 0025 5718 91 l 00 25 per page

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

a quasi monte carlo approach to particle simulation of the heat equation the convergence of the monte carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi random in this paper the convergence of monte carlo particle simulation is studied when these quasi random sequences are used for the one dimensional heat equation discretized in both space and time convergence is proved for a quasi random simulation using reordering of the particles according to their position experimental results are presented for the spatially continuous heat equation in one and two dimensions the results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard monte carlo simulations for certain low dimensional problems

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part ii second order methods we present in this paper the formulation of a new high frequency dissipative time stepping algorithm for nonlinear elastodynamics that is second order accurate in time the new scheme exhibits unconditional energy dissipation and momentum conservation and thus the given name of energy dissipative momentum conserving second order scheme edmc 2 leading also to the conservation of the relative equilibria of the underlying physical system the unconditional character of these properties applies not only with respect to the time step size but equally important with respect to the considered elastic potential moreover the dissipation properties are fully controlled through an algorithmic parameter reducing to existing fully conserving schemes if desired the design of the new algorithm is described in detail including a complete analysis of its dissipation conservation properties in the fully nonlinear range of finite elasticity to motivate the different constructions that lead to the dissipative properties of the final scheme the same arguments are used first in the construction of new linear time stepping algorithms for the system of linear elastodynamics including first and second order schemes the new schemes exhibit a rigorous decay of the physical energy with the second order schemes formulated in a general two stage formula accommodating the aforementioned control parameter in the dissipation of the energy a complete spectral analysis of the new schemes is presented in this linear range to evaluate their different numerical properties in particular the dissipative character of the proposed schemes in the high frequency range is fully demonstrated in fact it is shown that the new second order scheme is l stable most remarkably the extension of these ideas to the nonlinear range is accomplished avoiding the use of multi stage formulae given the freedom gained in using general nonlinear relations while preserving the conservation laws of the momenta and the corresponding relative equilibria several representative numerical simulations are presented in the context of nonlinear elastodynamics to evaluate the performance of the newly proposed schemes

formulation and analysis of two energy consistent methods for nonlinear elastodynamic frictional contact problems energy conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability furthermore some physical phenomena such as friction can generate dissipation then in this work we present and analyse two energy consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon the first approach permits one to enforce respectively the kuhn tucker and persistency conditions during each time step by combining an adapted continuation of the newton method and a lagrangean formulation in addition the second method which is based on the work in p hauret p le tallec energy controlling time integration methods for nonlinear elastodynamics and low velocity impact comput methods appl mech eng 195 2006 4890 4916 represents a specific penalisation of the unilateral contact conditions some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods

energy dissipative momentum conserving time stepping algorithms for finite strain multiplicative plasticity abstract this paper presents a new energy dissipative momentum conserving algorithm for multiplicative models of finite strain plasticity the proposed scheme preserves exactly the conservation laws of linear and angular momenta and leads to the exact energy dissipation between the two computed solutions in a given time step in particular the energy is exactly conserved if the step is elastic thus recovering existing energy momentum conserving schemes in the purely elastic range extensions accommodating a strictly non negative numerical energy dissipation in the high frequency to handle the numerical stiffness of the problem are also discussed briefly these conservation dissipation properties are proven rigorously in the very nonlinear setting of multiplicative finite strain plasticity in fact the analysis drives the design of the new return mapping algorithm proposed to integrate the plastic evolution equations so the above dissipation conservation properties hold these analyses account for the spatial discretization of the problem in the context of the finite element method several representative numerical simulations are presented to illustrate the performance of the new algorithm

conservation of generalized momentum maps in mechanical optimal control problems with symmetry summary in this paper a structure preserving direct method for the optimal control of mechanical systems is developed the new method accommodates a large class of one step integrators for the underlying state equations the state equations under consideration govern the motion of affine hamiltonian control systems if the optimal control problem has symmetry associated generalized momentum maps are conserved along an optimal path this is in accordance with an extension of noether s theorem to the realm of optimal control problems in the present work we focus on optimal control problems with rotational symmetries the newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem this is true for a variety of one step integrators used for the discretization of the state equations examples are the one step theta method a partitioned variant of the theta method and energy momentum em consistent integrators numerical investigations confirm the theoretical findings copyright 2016 john wiley sons ltd

extended state space time integration with high frequency energy dissipation the introduction of algorithmic dissipation into conservative time integration algorithms is addressed in two steps the problem is formulated in state space and it is demonstrated that the use of two balanced dissipation terms one in the equation of motion and the other in the relation between the state space variables can produce dissipation that increases monotonically with the modal frequency this formulation is only partially satisfactory because it leads to dissipation also in the low frequency range a substantially improved form of algorithmic dissipation is then developed by representing the two balanced dissipation terms via auxiliary state space variables related to the displacement and velocity increments by a first order filter the filter technique retains the high frequency behavior of the original balanced algorithm but reduces the low frequency dissipation from first order in the frequency to third order energy balance equations are derived and the behavior of the two algorithms is described by detailed spectral analysis with explicit asymptotic results directly in terms of the damping parameter of the algorithms both algorithms satisfy full symmetry of displacement and normalized velocity variables for the free algorithmically damped vibration of any single mode and therefore exhibit monotonic transients without overshoot effects it is also demonstrated how the algorithm can be arranged in single solve format where only the displacement update requires a matrix equation solution while the velocity and the auxiliary state space variables are updated subsequently in each time step by simple vector relations with scalar coefficients copyright 2007 john wiley sons ltd

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

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part i low order methods for two model problems and nonlinear elastodynamics abstract we present in this paper the development of a class of time stepping algorithms for nonlinear elastodynamics that exhibits the controllable numerical dissipation in the high frequency range required for the robust solution of the resulting numerically stiff systems to motivate and illustrate better the developments in this general case we present first the formulation and analysis of these methods for two simple model problems namely we consider a nonlinear elastic spring mass system and a simplified model of thin elastic beams as it is discussed in detail in this paper the conservation by the numerical algorithm of the momenta and corresponding relative equilibria of these characteristic hamiltonian systems with symmetry is of the main importance these conservation properties lead for a fixed and finite time step to a correct qualitative picture of the phase space where the discrete dynamics takes place even in the presence of the desired and controlled numerical dissipation of the energy this situation is contrasted with traditional dissipative numerical schemes which are shown through rigorous analyses to not only lose their dissipative character in the general nonlinear range but also the aforementioned conservation properties thus leading to a qualitatively distorted approximation of the phase dynamics the key for a successful algorithm in this context is the incorporation of the numerical dissipation in the internal modes of the motion while not affecting the group motions of the systems the algorithms presented in this work accomplish these goals the focus in this first part is given to first order methods representative numerical simulations ranging from applications in nonlinear structural dynamics to nonlinear continuum three dimensional elastodynamics are presented in the context of the finite element method to illustrate these ideas and results

formulation and analysis of conserving algorithms for frictionless dynamic contact impact problems this paper presents the formulation of conserving time stepping algorithms for frictionless dynamic contact of solids a new class of finite element methods is proposed for the solution of these problems that exhibit the same conservation laws as the underlying continuum dynamical system the proposed methods are based on a penalty regularization of the constrained contact problem and lead to full conservation of the total energy of the system including the regularization penalty potential during persistent contact and restoration of the original energy upon release both linear and angular momenta are conserved by the scheme furthermore the newly developed methods have the ability to enforce the associated constraints in the velocity besides the impenetrability constraint in the displacements while preserving the conservation restoration properties of the final numerical scheme a modification of these schemes is described that assures positive energy dissipation if desired even in the highly nonlinear setting of contact impact problems leading to contact schemes with high frequency energy dissipation representative numerical simulations are presented illustrating the performance of the proposed numerical schemes

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

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

accuracy of one step integration schemes for damped forced linear structural dynamics summary this paper proposes an energy based measure for the evaluation of the local truncation error of two level one step integration schemes the measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses and it segregates the error associated with the free and forced response components of the problem to illustrate the approach two examples associated with the application of the trapezoidal scheme and of a high order scheme proposed in the literature are analyzed the latter reveals the shortcomings of the standard approach that is based on the undamped unforced linear oscillator and therefore highlights the need for the proposed framework indeed the scheme order of accuracy is below expectation when structural damping or external forcing is considered in the numerically dissipative setting developments on the basis of the time discontinuous galerkin tdg method are then proposed to recover the scheme high order accuracy additionally they show the similarity that exists between schemes related to the tdg method and the ones obtained by integration by parts of the equation of motion copyright 2014 john wiley sons ltd

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

transient three dimensional domain decomposition problems frame indifferent mortar constraints and conserving integration the present work deals with transient large deformation domain decomposition problems the tying of dissimilar meshed grids is performed by applying the mortar method in this connection a reformulation of the original linear mortar constraints is proposed which retains frame indifference for arbitrary discretizations of the interface furthermore a specific coordinate augmentation technique is proposed to make possible the design of an energy momentum scheme numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy momentum scheme for three dimensional problems copyright 2009 john wiley sons ltd

a quasi monte carlo scheme using nets for a linear boltzmann equation a quasi monte carlo particle simulation for solving a linear boltzmann equation is constructed and a convergence proof is given the analysis is restricted to the three dimensional equation in the space homogeneous case and the velocity domain is normalized to be i 3 0 1 3 a particle simulation is described it combines a euler scheme in time with quasi monte carlo integration in velocity space the quadratures use 0 m s nets which are sets with a very regular distribution behavior the particles are reordered according to the components of their velocity at each time step a deterministic error bound is obtained finally numerical experiments are presented for some test problems where an exact solution is known the accuracy of the quasi monte carlo scheme is compared with a standard monte carlo algorithm the results show an improvement in both magnitude of error and convergence rate of quasi monte carlo over monte carlo approach

solving tree problems on a mesh connected processor array in this paper we present techniques that result in o n time algorithms for computing many properties and functions of an n node forest stored in an n n mesh of processors our algorithms include computing simple properties like the depth the height the number of descendents the preorder resp postorder inorder number of every node and a solution to the more complex problem of computing the minimax value of a game tree our algorithms are asymptotically optimal since any nontrivial computation will require n time on the mesh all of our algorithms generalize to higher dimensional meshes

low discrepancy sequences for solving the boltzmann equation abstract some new quasi monte carlo methods for the solution of the boltzmann equation in a simplified case are described the errors of the methods are estimated by means of the discrepancies of the sequences used for performing the monte carlo quadratures their accuracy is assessed through computation of effective errors in an example where an exact solution is known

a particle method for some parabolic equations abstract we present a quasi monte carlo particle simulation of some multidimensional linear parabolic equations with constant coefficients we approximate the elliptic operator in space by a finite difference operator we discretize time into intervals of length t the discrete representation of the solution at time t n n t is a sum of dirac delta measures using the explicit euler scheme the resulting approximation at time t n 1 is recovered by a quasi monte carlo integration we make use of a technique involving renumbering the simulated particles in every time step we state and prove a convergence theorem for the method experimental results are presented for some model problems the results suggest that the quasi monte carlo simulation tends to give more accurate solutions than a monte carlo simulation when the correct renumbering technique is used other choices can result in significant loss of efficiency

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

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

list ranking on interconnection networks the list ranking problem is considered for parallel computers which communicate through an interconnection network each pu holds k nodes of a set of linked lists a novel randomized algorithm gives a considerable improvement over earlier ones for a large class of networks and sufficiently large k it takes only twice the number of steps required by a k k routing for hypercubes the condition is k log2 n even better results are achieved for d dimensional meshes we show that the ranking time exceeds the routing time only by lower order terms for all k d2 we also show that list ranking requires at least the time required for k k routing thus the results are within a factor two from optimal those for meshes even match the lower bound up to lower order terms

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

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

multi conformal symplectic pdes and discretizations past work on integration methods that preserve a conformal symplectic structure focuses on hamiltonian systems with weak linear damping in this work systems of pdes that have conformal symplectic structure in time and space are considered meaning conformal symplecticity is fully generalized for pdes using multiple examples it is shown that pdes with this particular structure have interesting applications what it means to preserve a multi conformal symplectic conservation law numerically is explained along with presentation of two numerical methods that preserve such properties then the advantages of the methods are briefly explored through applications to linear equations consideration of momentum and energy dissipation and backward error analysis numerical simulations for two pdes illustrate the properties of the methods as well as the advantages over other standard methods

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

event driven integration of linear structural dynamics models under unilateral elastic constraints abstract this paper proposes an algorithm for the numerical simulation of linear structural dynamics problems under unilateral elastic constraints i e constraints with a linear force displacement characteristic whenever active the presented procedure relies on an event driven strategy for the handling of the contact constraints in combination with one step schemes dedicated to the time integration of the second order equations of motion efficiency of the procedure follows from the use of cubic hermite interpolation to continuously extend the normal gap functions that reflect the openings of the contact interfaces robustness follows from the proper handling of complex numerical situations e g numerical grazing or discontinuity sticking through appropriate algorithm structure and numerical implementation and integration stability is guaranteed by the very nature of the algorithm and that of the one step integration scheme following a detailed coverage of the integration procedure and the countermeasures to the expected numerical difficulties three application examples are treated for illustration purposes a matlab implementation of the procedure is provided online download and usage information are given in appendix a

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

modified newmark family for non linear dynamic analysis a simple modification of the newmark scheme is suggested which keeps the total energy of mechanical systems constant and thereby enhances the unconditional stability in non linear dynamic analysis numerical damping which is formulated on an energy basis is also introduced and the dissipative character of the algorithm is guaranteed in the non linear regime representative examples in support of such a modified newmark family are presented copyright 2004 john wiley sons ltd

computational schemes for non linear elasto dynamics this paper deals with the development of computational schemes for the dynamic analysis of non linear elastic systems the focus of the investigation is on the derivation of unconditionally stable time integration schemes presenting high frequency numerical dissipation for these types of problem at first schemes based on galerkin and time discontinuous galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed though useful these schemes require casting the equations of motion in the symmetric hyperbolic form which is not always possible furthermore this approaches to unacceptably high computational costs next unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form both energy preserving and energy decaying schemes are derived numerical examples are presented to demonstrate the accuracy and efficiency of the proposed schemes copyright 1999 john wiley sons ltd

volume preserving energy momentum schemes for isochoric multiplicative plasticity abstract this paper presents a new energy dissipative momentum conserving algorithm for multiplicative finite strain plasticity f f e f p that also preserves exactly the plastic volume for isochoric plastic models the new algorithm exhibits exactly the conservation laws of linear and angular momentum of the underlying physical problem as well as its energy evolution a strictly positive energy dissipation in fact the exact energy dissipation is obtained by design during plastic steps while enforcing the plastic consistency i e the yield condition on the final stress appearing in the equations of motion exact energy conservation is attained in particular during elastic steps the aforementioned preservation of the plastic volume is obtained by a new treatment of the geometric structure behind the considered multiplicative models of finite strain plasticity namely we present a new approximation of the reference and elastic metrics whose contractions with the incremental total and elastic strains lead exactly to the increment of the total and elastic natural volumetric strains respectively the elastic metric is defined in the intermediate configuration defined itself by the proper discrete approximation in time of the plastic deformation gradient f p the new algorithm extends to the plastic range existing energy momentum conserving schemes for nonlinear elastic problems but incorporating a new modified elastic stress formula consistent with this new geometric setting the inherited conservation laws of momenta and especially the non negative character of the energy dissipation leads to an improved performance over existing more classical schemes showing numerical instabilities in the considered highly nonlinear geometric setting of large deformations and strains several numerical simulations are presented illustrating these properties

energy conserving time integration scheme for geometrically exact beam an energy conserving finite element formulation for the dynamic analysis of geometrically non linear beam like structures undergoing large overall motions has been developed the formulation uses classical displacement based planar beam finite elements described in an inertial frame it takes into account finite axial bending and shear strains a theoretically consistent approach is used to derive a novel and simple energy conserving scheme using the unconventional incremental strain update rather than the standard strong form numerical examples demonstrate perfect energy and momenta conservation stability and robustness of the scheme and good convergence properties in terms of both the newton raphson method and time step size c 2006 elsevier b v all rights reserved

energy conserving and decaying algorithms in non linear structural dynamics a generalized formulation of the energy momentum methodwill be developed within the framework of the generalizedmethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response furthermore the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the constraint energy momentum algorithm the goal of this general approach of implicit energyconserving and decaying time integration schemes is to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non linear structural dynamics among the methods considered will be the newmark method the classical methods the energy momentum methodwith and without numerical dissipation the constraint energymomentum algorithm and the constraint energy method copyright 1999 john wiley sons ltd

numerical integration of the stiff dynamics of geometrically exact shells an energy dissipative momentum conserving scheme this paper presents a new family of time stepping algorithms for the integration of the dynamics of non linear shells we consider the geometrically exact shell theory involving an inextensible director field the so called five parameter shell model the main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid in this context we develop time stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system and allow the introduction of a controllable non negative energy dissipation to handle the high numerical stiffness characteristic of these problems a series of algorithmic parameters for the different components of the deformation of the shell i e membrane bending and transverse shear fully control this numerical dissipation recovering existing energy momentum schemes as a particular choice of these algorithmic parameters we present rigorous proofs of the numerical properties of the resulting algorithms in the full non linear range furthermore it is argued that the numerical dissipation is introduced in the high frequency range by considering the proposed algorithm in the context of a linear problem the finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation dissipation properties we present several representative numerical simulations illustrating the performance of the newly proposed methods the robustness gained over existing methods in these stiff problems is confirmed in particular copyright 2002 john wiley sons ltd

a first order energy dissipative momentum conserving scheme for elasto plasticity using the variational updates formulation abstract in a previous paper l noels l stainier j p ponthot an energy momentum conserving algorithm using the variational formulation of visco plastic updates int j numer methods engrg 65 2006 904 942 the authors demonstrated the efficiency of the variational formulation of elasto plastic updates to develop energy momentum conserving time integration algorithms indeed within such a framework the stress tensor always derives from an incremental potential even when plastic behavior is considered therefore the verification of the conservation of energy in the non linear range can easily be demonstrated the sum of the reversible stored energy and irreversible dissipated energy exactly corresponds to the work of the external forces applied to the structure although this formulation was shown to be accurate and robust the introduction of numerical dissipation for high frequency numerical modes can be necessary to simulate complex phenomena in this work we propose a modification of the variational updates framework to introduce this numerical property leading to a new energy dissipative momentum conserving time integration algorithm for elasto plasticity

asynchronous multi domain variational integrators for nonlinear hyperelastic solids we present the asynchronous multi domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary value problem in hyperelasticity variational time integration schemes based on the principle of minimal action within the lagrangian framework are constructed for the equation of motion and implemented into a variational finite element framework which is systematically derived from the three field de veubeke hu washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly incompressible materials for efficient parallel computing we use the dual domain decomposition method with local lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains the method for time discretization and the multi domain spatial decomposition enable us to use different types of integrators explicit vs implicit and different time steps on different parts of a computational domain and thus efficiently capture the underlying physics with less computational effort the energy conservation of our nonlinear midpoint asynchronous integration scheme is investigated using the energy method and both local and the global energy error estimates are derived we illustrate the performance of proposed variational multi domain time integrators by means of three examples first the method of manufactured solutions is used to examine the consistency of the formulation in the second example we investigate energy conservation and stability finally we apply the method to the motion of a heterogeneous plane domain where different integrators and time discretization steps are used accordingly with disparate material data of individual parts

constraint energy momentum algorithm and its application to non linear dynamics of shells abstract a modified time integration method is developed for non linear structural dynamics combining controllable algorithmic dissipation of higher modes and at the same time conservation of energy as well as linear and angular momentum although these features seem to be actually inconsistent numerical stability with large time steps for dynamic buckling and snap through problems as well as for dynamical systems with smooth solutions is guaranteed arbitrary implicit one step time integration schemes with newmark approximations are suitable for use as basic algorithms for the proposed method the desired conservation attributes are obtained if the basic algorithm is augmented by energy and momentum constraints the potential of the suggested algorithm is demonstrated by selected applications to non linear dynamics of shell structures

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

energy momentum consistent finite element discretization of dynamic finite viscoelasticity this paper is concerned with energy momentum consistent time discretizations of dynamic finite viscoelasticity energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics the discretization is energy momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations the performed approximation is based on a three field formulation in which the deformation field the velocity field and a strain like viscous internal variable field are treated as independent quantities the new non linear viscous evolution equation satisfies a non negative viscous dissipation not only in the continuous case but also in the fully discretized system the initial boundary value problem is discretized by using finite elements in space and time thereby the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments incremental objectivity although the present approach makes possible to design schemes of arbitrary order the focus is on finite elements relying on linear lagrange polynomials for the sake of clearness the discrete energy momentum consistency is based on the collocation property and an enhanced second piola kirchhoff stress tensor the obtained coupled non linear algebraic equations are consistently linearized the corresponding iterative solution procedure is associated with newly proposed convergence criteria which take the discrete energy consistency into account the iterative solution procedure is therefore not complicated by different scalings in the independent variables since the motion of the element is taken into account for solving the viscous evolution equation representative numerical simulations with various boundary conditions show the superior stability of the new time integration algorithm in comparison with the ordinary midpoint rule both the quasi rigid deformations during a free flight and large deformations arising in a dynamic tensile test are considered copyright 2009 john wiley sons ltd

adapting parallel algorithms to the w stream model with applications to graph problems in this paper we show how parallel algorithms can be turned into efficient streaming algorithms for several classical combinatorial problems in the w stream model in this model at each pass one input stream is read one output stream is written and data items have to be processed using limited space streams are pipelined in such a way that the output stream produced at pass i is given as input stream at pass i 1 we first introduce a simulation technique that allows turning efficient pram algorithms into optimal w stream ones for many classical combinatorial problems including list ranking and euler tour of a tree for other problems most notably graph problems however this technique leads to suboptimal algorithms to overcome this difficulty we introduce the relaxedpram rpram computational model as an intermediate model between pram and w stream rpram allows every processor to access a non constant number of memory cells per parallel round albeit with some restrictions the rpram model while being more powerful than the pram model can be simulated in w stream within the same asymptotic bounds the extra power provided by rpram allows us in many cases to substantially reduce the number of processors while maintaining the same number of parallel rounds leading to more efficient w stream simulations of parallel algorithms our rpram technique gives new insights on developing streaming algorithms and yields efficient algorithms for several classical problems in this model including sorting connectivity minimum spanning tree biconnected components and maximal independent set in addition to allowing smooth space passes tradeoffs our algorithms are also shown by proving almost tight communication complexity based lower bounds in w stream to be optimal up to polylog factors

a comparative study of pseudo and quasi random sequences for the solution intergral equations a numerical study for the solution of integral equations using random numbers from congruential generators and the halton and the faure sequences is presented the results indicate that for high precision monte carlo quasi random sequences yield better results in general the relative performance of the quasi random sequences is better with monte carlo schemes which reduce the dimensionality of the problem

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

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

an objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics we present in this paper a new finite element formulation of geometrically exact rod models in the three dimensional dynamic elastic range the proposed formulation leads to an objective or frame indifferent under superposed rigid body motions approximation of the strain measures of the rod involving finite rotations of the director frame in contrast with some existing formulations this goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod s cross section furthermore the proposed framework allows the development of time stepping algorithms that preserve the conservation laws of the underlying continuum hamiltonian system the conservation laws of linear and angular momenta are inherited by construction leading to an improved approximation of the rod s dynamics several numerical simulations are presented illustrating these properties copyright 2002 john wiley sons ltd

an energy consistent material point method for dynamic finite deformation plasticity energy consistency for the material point method mpm is examined for thermodynamically consistent hyperelastic plastic materials it is shown that mpm can be formulated with implicit three field variational finite element algorithms which dissipate energy and conserve momentum for that class of material models with a consistent mass matrix the resulting overall numerical method inherits the energy dissipative and momentum conserving properties of the mesh solution thus the proposed mpm algorithm satisfies by construction a time discrete form of the second law of thermodynamics properties of the method are illustrated in numerical examples

thermodynamically consistent time stepping algorithms for non linear thermomechanical systems we present the basic theory for developing novel monolithic and staggered time stepping algorithms for general non linear coupled thermomechanical problems the proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics for isolated systems they preserve the total energy and the entropy never decreases furthermore if the governing equations of the problem have symmetries the proposed integrators preserve them too the formulation of such methods is based on two ideas expressing the evolution equation in the so called general equations for non equilibrium reversible irreversible coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields the new methods can be considered as an extension of the energy momentum integration algorithms to coupled thermomechanical problems to which they reduce in the purely hamiltonian case in the article the new ideas are applied to a simple coupled problem a double thermoelastic pendulum with symmetry numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability which stems precisely from their ability to preserve the structure of the evolution equations they discretize copyright 2009 john wiley sons ltd

energy controlling time integration methods for nonlinear elastodynamics and low velocity impact it is now well established that discrete energy conservation dissipation plays a key role for the unconditional stability of time integration schemes in nonlinear elastodynamics in this paper from a rigorous conservation analysis of the hilber hughes taylor time integration scheme h hilber t hughes r taylor improved numerical dissipation for time integration algorithms in structural dynamics earthquake engrg struct dynam 5 1977 283 292 we propose an original way of introducing a controllable energy dissipation while conserving momenta in conservative strategies like j simo n tarnow the discrete energy momentum method conserving algorithms for nonlinear elastodynamics z angew math phys 43 1992 757 792 moreover we extend the technique proposed in o gonzalez exact energy and momentum conserving algorithms for general models in nonlinear elasticity comput methods appl mech engrg 190 13 14 2000 1763 1783 to provide energy controlling time integration schemes for frictionless contact problems enforcing the standard kuhn tucker conditions at time discretization points we also extend this technique to viscoelastic models numerical tests involving the impact of incompressible elastic or viscoelastic bodies in large deformation are proposed to confirm the theoretical analysis

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