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

deterministic extractors for bit fixing sources and exposure resilient cryptography we give an efficient deterministic algorithm that extracts omega n 2 gamma almost random bits from sources where n frac 1 2 gamma of the n bits are uniformly random and the rest are fixed in advance this improves upon previous constructions which required that at least n 2 of the bits be random in order to extract many bits our construction also has applications in exposure resilient cryptography giving explicit adaptive exposure resilient functions and in turn adaptive all or nothing transforms for sources where instead of bits the values are chosen from d for d 2 we give an algorithm that extracts a constant fraction of the randomness we also give bounds on extracting randomness for sources where the fixed bits can depend on the random bits

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

gate elimination circuit size lower bounds and sat upper bounds abstract most of the known lower bounds for binary boolean circuits with unrestricted depth are proved by the gate elimination method the most efficient known algorithms for the sat problem on binary boolean circuits use similar case analyses to the ones in gate elimination chen and kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds that is lower bounds on the size of approximations of an explicit function in this paper we provide a general framework for proving worst average case lower bounds for circuits and upper bounds for sat that is built on ideas of chen and kabanets a proof in such a framework goes as follows one starts by fixing three parameters a class of circuits a circuit complexity measure and a set of allowed substitutions the main ingredient of a proof goes as follows by going through a number of cases one shows that for any circuit from the given class one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount this case analysis immediately implies an upper bound for sat to obtain worst average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser extractor for the class of sources defined by the set of substitutions under consideration we show that many known proofs of circuit size lower bounds and upper bounds for sat fall into this framework using this framework we prove the following new bounds average case lower bounds of 3 24 n and 2 59 n for circuits over u 2 and b 2 respectively though the lower bound for the basis b 2 is given for a quadratic disperser whose explicit construction is not currently known and faster than 2 n sat algorithms for circuits over u 2 and b 2 of size at most 3 24 n and 2 99 n respectively here by b 2 we mean the set of all bivariate boolean functions and by u 2 the set of all bivariate boolean functions except for parity and its complement

deterministic extractors for bit fixing sources by obtaining an independent seed an n k bit fixing source is a distribution x over 0 1 n such that there is a subset of k variables in x 1 ldots x n which are uniformly distributed and independent of each other and the remaining n k variables are fixed a deterministic bit fixing source extractor is a function e 0 1 n rightarrow 0 1 m which on an arbitrary n k bit fixing source outputs m bits that are statistically close to uniform recently kamp and zuckerman proceedings of the 44th annual ieee symposium on foundations of computer science 2003 pp 92 101 gave a construction of a deterministic bit fixing source extractor that extracts omega k 2 n bits and requires k sqrt n in this paper we give constructions of deterministic bit fixing source extractors that extract 1 o 1 k bits whenever k log n c for some universal constant c 0 thus our constructions extract almost all the randomness from bit fixing sources and work even when k is small for k gg sqrt n the extracted bits have statistical distance 2 n omega 1 from uniform and for k le sqrt n the extracted bits have statistical distance k omega 1 from uniform our technique gives a general method to transform deterministic bit fixing source extractors that extract few bits into extractors which extract almost all the bits

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

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

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

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

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

energy consistent algorithms for dynamic finite deformation plasticity abstract this paper addresses the theoretical development and numerical implementation of energy consistent algorithms for dynamic elastoplasticity emphasizing finite strain constitutive formulations so that unconditional stability of the algorithms is assured even in the fully nonlinear regime the key concept behind energy consistency is the requirement that the discretized system obey an a priori stability estimate which may be derived in general using the first and second laws of thermodynamics this approach to computational dynamic plasticity differs from typical application of traditional algorithms such as newmark or hilber hughes taylor methods where local time integration schemes for plasticity laws are developed somewhat independently from the global time integration scheme for the equations of motion without explicit consideration of thermodynamical restrictions two algorithms based on both additive and multiplicative finite deformation plasticity model are formulated within the energy consistent framework both algorithms possess the desirable feature of nonlinear stability of previous energy momentum algorithms for elastodynamics

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

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

universal analytical solution of the steady state response of an infinite beam on a pasternak elastic foundation under moving load abstract in this paper the steady state response of a uniform infinite euler bernoulli elastic beam resting on a pasternak elastic foundation and subjected to a concentrated load moving at a constant velocity along the beam is analytically investigated a universal closed form analytical solution is derived through fourier transform apt to represent the response for all possible beam foundation parameters a rigorous mathematical procedure is formulated for classifying the parametric behavior of the solution including for viscous damping depending on such a classification different types of bending wave shapes are shown to propagate within the beam ahead and behind the moving load position and crucial physical characteristics such as critical velocity and critical damping are reinterpreted into a wholly exact and complete mathematical framework mechanical features of the solution are revealed for the steady state response in terms of normalized deflection cross section rotation bending moment and shear force

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

the shrinkage exponent of de morgan formulas is 2 we prove that if we hit a de morgan formula of size l with a random restriction from rp then the expected remaining size is at most o p 2 log frac 1 p 3 2 l p sqrt l as a corollary we obtain an omega n 3 o 1 formula size lower bound for an explicit function in p this is the strongest known lower bound for any explicit function in np

improved average case lower bounds for de morgan formula size matching worst case lower bound we give an explicit function h 0 1 n to 0 1 such that every de morgan formula of size n 3 o 1 r 2 agrees with h on at most a fraction of frac 1 2 2 omega r of the inputs our technical contributions include a theorem that shows that the expected shrinkage result of h aastad siam j comput 27 1998 pp 48 64 actually holds with very high probability where the restrictions are chosen from a certain distribution that takes into account the structure of the formula using ideas of impagliazzo meka and zuckerman proceedings of focs 2012 pp 111 119

steady state response of an elastically supported infinite beam to a moving load the steady state response of a uniform beam placed on an elastic foundation and subjected to a concentrated load moving with a constant speed has been investigated the foundation is modeled by using one and two parameters the mathematical form of the solution is justified by fourier transform it is observed that the steady state is not attained at supercritical speed of the load in the ideal undamped case numerical results are presented for maximum settlement uplift and bending moment in the beam the effect of difference in the modeling of the foundation is shown to be insignificant

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

a monolithic computational approach to thermo structure interaction in the present work a monolithic solution approach for thermo structure interaction problems motivated by the challenging application of the behaviour of rocket nozzles is proposed structural and thermal fields are independently discretised via finite elements the resulting system of equations is solved via a monolithic thermo structure interaction scheme which is constructed by a block gauss seidel preconditioner in combination with algebraic multigrid methods the proposed method is tested for three numerical examples the second danilovskaya problem a simplified rocket nozzle configuration and an internal loaded hollow sphere good agreement of the numerical results with results from the literature is observed furthermore it is shown that the monolithic solution algorithm can handle the complete range of the parameter spectrum whereas partitioned algorithms are limited to a certain parameter range only moreover the monolithic algorithm exhibits improved efficiency and robustness compared to partitioned algorithms

quantum lower bounds by polynomials we examine the number of queries to input variables that a quantum algorithm requires to compute boolean functions on 0 1 n in the black box model we show that the exponential quantum speed up obtained for partial functions i e problems involving a promise on the input by deutsch and jozsa simon and shor cannot be obtained for any total function if a quantum algorithm computes some total boolean function f with small error probability using t black box queries then there is a classical deterministic algorithm that computes f exactly with o ts6 queries we also give asymptotically tight characterizations of t for all symmetric f in the exact zero error and bounded error settings finally we give new precise bounds for and or and parity our results are a quantum extension of the so called polynomial method which has been successfully applied in classical complexity theory and also a quantum extension of results by nisan about a polynomial relationship between randomized and deterministic decision tree complexity

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

exact analysis of beams on two parameter elastic foundations abstract efficient beams on two parameter elastic foundation finite elements have recently been developed the stiffness matrix and nodal loud vector of these elements have been derived on the basis of the exact displacement function obtained from the solution of the governing differential equation most of the existing elements are however either limited to certain combinations of beam and foundation parameters or provide only the solution of the homogeneous form of the governing equation in this paper a new finite element is derived which eliminates these limitations the stiffness matrix nodal load vector and shape function of the clement are derived using the differential equation of a beam on a two parameter elastic foundation the complete solution of the equation corresponding to the most common types of load is also presented this permits the determination of the deflections and internal forces anywhere along a simple or continuous beam on two parameter foundations

analysis of the critical velocity of a load moving on a beam supported by a finite depth foundation abstract a detailed analysis of the critical velocity of a uniformly moving load is carried out in this paper it is assumed that the load is traversing an infinite beam supported by a finite depth foundation under plane strain condition the problem is solved analytically in the fourier domain respecting two formulations of the interface condition the critical velocity is determined as the velocity under which the undamped beam deflection tends to infinity i e by identifying double poles in the fourier image of the beam deflection situated on the real fourier variable axis results obtained are compared with the previously published results of this author where simplifying assumptions were imposed on the shear contribution of the foundation based on the results an addition to the previous formula for critical velocity estimate is proposed it is confirmed that there are several critical velocities but the lowest one which is the dominating one smoothly changes from the classical value to the lowest wave velocity of propagation in the foundation according to the mass ratio defined as the square root of the fraction of the foundation mass to the beam mass particular situations where the first double pole cannot be formed in this model are discussed in detail in such cases a significant increase of deflection is observed in the vicinity of the expected location defining thus a pseudo critical velocity attention is paid to the cases where there are two or more very close critical velocities which causes an increased risk for practical applications in addition several options of damping are compared

discrete variational hamiltonian mechanics the main contribution of this paper is to present a canonical choice of a hamiltonian theory corresponding to the theory of discrete lagrangian mechanics we make use of lagrange duality and follow a path parallel to that used for construction of the pontryagin principle in optimal control theory we use duality results regarding sensitivity and separability to show the relationship between generating functions and symplectic integrators we also discuss connections to optimal control theory and numerical algorithms

finite element dynamic analysis of finite beams on a bilinear foundation under a moving load abstract the present paper is concerned with the behaviour of finite elastic beams acted by a moving transverse concentrated load interacting with elastic foundations of different stiffnesses in compression and in tension using finite element analyses the displacement amplitudes and the critical velocities of the load on a uic 60 rail are computed and their dependence with respect to the difference between the foundation s moduli in compression and in tension is evaluated the limit case of a tensionless foundation is as well analyzed the numerical algorithm relies on the internal force vectors and tangent stiffness matrices computed exactly with automatic symbolic manipulation

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

entropic thermoelasticity at finite strains aspects of the formulation and numerical implementation abstract this paper presents a formulation of isotropic finite strain thermoelasticity and addresses some aspects of its numerical implementation on the theoretical side a eulerian setting of isotropic thermoelasticity is discussed based on the finger tensor as a strain measure novel aspects are a direct representation of the eulerian thermoelastic moduli in terms of the finger tensor and a rigorous decomposition of the thermoelastic response functions into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts an algorithmic procedure for the computation of the stress and thermoelastic moduli based on a representation of the free energy in terms of eigenvalues of the unimodular part of the finger tensor is developed and applied to model problems of strictly entropic and modified entropic thermoelasticity furthermore two algorithms for the solution of the coupled problem are discussed based on operator splits of the global field equations of thermoelasticity the paper concludes with some representative numerical simulations of thermoelastic processes in rubber like materials

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

stable and time dissipative finite element methods for the incompressible navier stokes equations in advection dominated flows we examine a new galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1 d scalar advection diffusion and that leads to non oscillatory solutions as the streamline upwinding algorithms for 2 d scalar advection diffusion and incompressible navier stokes this method retains the satisfaction of the babuska brezzi condition and thus leads to optimal performance in the incompressible limit this method when combined with the recently proposed linear unconditionally stable algorithms of simo and armero yields a method for solution of the incompressible navier stokes equations ideal for either diffusive or advection dominated flows examples from scalar advection diffusion and the solution of the incompressible navier stokes equations are presented

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

energy and entropy preserving numerical approximations of thermodynamically consistent crystal growth models abstract we present a numerical scheme that preserves the total energy and the entropy production rate termed the energy and entropy production rate preserving scheme for a general class of thermodynamically consistent phase field models for dendritic crystal growth derived from the first and second law of thermodynamics the scheme is second order in time linear and energy and entropy production rate preserving for any time steps the scheme is first discretized in time aided by the energy quadratization eq method and then in space using compact finite difference methods the linear system resulting from the scheme is shown to be uniquely solvable at both the semi discrete and the fully discrete level mesh refinement tests are performed to show the second order time convergence rate in the scheme several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the scheme the effects of various model parameters on growth patterns of the crystal are further investigated in details with the numerical solver the approach to developing the energy and entropy production rate preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

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

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

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

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

a new unconditionally stable fractional step method for non linear coupled thermomechanical problems a new class of time stepping algorithms for strongly coupled thermomechanical problems is presented which retains the computational convenience of traditional staggered algorithms without upsetting the unconditional stability property characteristic of fully implicit monolithic schemes 11 the proposed schemes are fractional step methods associated with a two phase operator split of the full non linear system of thermoelasticity into an adiabatic elastodynamics phase followed by a heat conduction phase at fixed configuration this operator split is shown to inherit the contractivity property of the full problem of evolution thus leading to unconditionally b stable product formula algorithms in sharp contrast with this stability result traditional staggered algorithms based on an isothermal mechanical phase followed by a heat conduction phase with an effective heat source are shown to lead at best to conditionally stable schemes it is further shown that the actual implementation of these two classes of schemes is essentially identical the numerical simulations presented include both dynamic and quasi static problems and are shown to closely replicate the stability estimates and non linear stability results derived for these two classes of staggered schemes

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

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