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

on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

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

revenue utility tradeoff in assortment optimization under the multinomial logit model with totally unimodular constraints we examine the revenue utility assortment optimization problem with the goal of finding an assortment that maximizes a linear combination of the expected revenue of the firm and the expected utilit

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

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

energy conserving decaying implicit time stepping scheme for nonlinear dynamics of three dimensional beams undergoing finite rotations in this work we present an extension of time integration energy conserving scheme which introduces desirable properties of controllable energy decay as well as numerical dissipation of high frequency contribution to total response finite element implementation details are given for the chosen model problem of geometrically exact beam undergoing finite rotations several numerical simulations illustrate a very satisfying performance of the proposed time stepping scheme

saddle point principles and numerical integration methods for second order hyperbolic equations abstract this work describes a family of functionals whose stationarity often saddle point condition leads to well known so called variational formulations for structural dynamics such as the weak hamilton ritz formulation and the continuous discontinuous galerkin formulation and in turn to methods for the numerical integration of the equations of motion it is shown that all the time integration methods based on variational formulations do descend from such functionals moreover starting from the described family of functionals it is possible to construct new families of time integration methods which might exhibit computational advantages over the corresponding ones derived from variational formulations only

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

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

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

an efficient time discontinuous galerkin procedure for non linear structural dynamics this paper presents an effective time discontinuous galerkin procedure for non linear dynamics the procedure is based on a cheap iterative algorithm which is rather different from previous efforts in fact the algorithm is designed so that the corrected solutions inherit the desired stability and dissipative properties and the iterations serve only to improve accuracy some numerical tests illustrate the good performance of the present procedure which appears to be competitive with the available implementations of time discontinuous galerkin methods

time discontinuous galerkin methods with energy decaying correction for non linear elastodynamics in this paper a new time discontinuous galerkin tdg formulation for non linear elastodynamics is presented the new formulation embeds an energy correction which ensures truly energy decaying thus allowing to achieve unconditional stability that as shown in the paper is not guaranteed by the classical tdg formulation the resulting method is simple and easily implementable into existing finite element codes moreover it inherits the desirable higher order accuracy and high frequency dissipation properties of the classical formulation numerical results illustrate the very good performance of the proposed formulation copyright 2010 john wiley sons ltd

robust integration schemes for flexible multibody systems a framework for the development of robust time integration schemes for nonlinear flexible multibody systems is presented the proposed schemes are designed to meet four specific requirements nonlinear unconditional stability of the scheme a rigorous treatment of both geometric and material nonlinearities exact satisfaction of the constraints and the presence of high frequency numerical dissipation specific algorithms are presented for rigid bodies cables beams shells and elasto dynamics the kinematic nonlinearities are treated in a rigorous manner for all elements and the material nonlinearities can be handled when the constitutive laws stem from the existence of a strain energy density function the treatment of the constraint equations associated with the six lower pair joints are presented as well the efficiency and robustness of the proposed approach is demonstrated with specific numerical examples

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

technical note a conic integer optimization approach to the constrained assortment problem under the mixed multinomial logit model we consider the constrained assortment optimization problem under the mixed multinomial logit model even moderately sized instances of this problem are challenging to solve directly using standard

feti domain decomposition methods for scalar advection diffusion problems abstract in this paper we show that iterative substructuring methods of finite element tearing and interconnecting type can be successfully employed for the solution of linear systems arising from the finite element approximation of scalar advection diffusion problems using similar ideas as those of a recently developed neumann neumann method we propose a one level algorithm and a class of two level algorithms obtained by suitably modifying the local problems on the subdomains we present some numerical results for some significant test cases our methods appear to be optimal for flows without closed streamlines and possibly very small values of the viscosity they also show very good performances for rotating flows and moderate reynolds numbers therefore the algorithms proposed appear to be well suited for many convection dominated problems of practical interest

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

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

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

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

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

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

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

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

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

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

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

a general framework for conservative single step time integration schemes with higher order accuracy for a central force system a general framework for algorithms that conserve angular momentum for single body central force problems is presented it is shown that any family of momentum conserving algorithms can have at most three free parameters one of which may be used to ensure energy conservation and hence will be configuration dependent further restrictions can be made that enable the algorithms to recover the orbits of relative equilibria of the underlying physical problem in addition the algorithms can be made time reversible whilst still leaving two parameters unspecified the order of accuracy of a general momentum conserving family is analysed and it is shown that energy momentum algorithms that preserve the underlying physical relative equilibria can have unlimited accuracy if the two remaining parameters are appropriately chosen functions of the configuration and the time step this does not require any additional degrees of freedom extra stages of calculation or information from past solutions numerical examples are given that show the performance of some representative higher order schemes when applied to stiff and non stiff problems and the issue of newton raphson convergence is discussed

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

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

energy dissipative time finite elements for classical mechanics abstract based upon the temporal discretisation of hamilton s canonical equations by means of the continuous galerkin method a petrov galerkin time finite element formulation is presented to analyse non linear mechanical systems for long time integration with the presence of dissipation the time stepping scheme features a correct estimation of physical dissipative energy for mechanical systems with weak dissipation this enables to introduce a moderate numerical dissipation to damp out undesired spurious high frequency modes in a controllable manner numerical examples are performed to demonstrate that the proposed scheme allows us to obtain a correct amount by which the energy changes over the integration run

a domain decomposition preconditioner for an advection diffusion problem abstract we propose a generalization of the neumann neumann algorithm for advection diffusion problems the neumann conditions are replaced by suitable robin conditions the method is first introduced and analysed in the case where the domain is partitioned into vertical strips then generalized to an arbitrary decomposition a coarse space procedure is proposed finally numerical results for bidimensional and tridimensional tests are given

bddc preconditioning for high order galerkin least squares methods using inexact solvers a high order galerkin least squares gls finite element discretization is combined with a balancing domain decomposition by constraints bddc preconditioner and inexact local solvers to provide an efficient solution technique for large scale convection dominated problems the algorithm is applied to the linear system arising from the discretization of the two dimensional advection diffusion equation and euler equations for compressible inviscid flow a robin robin interface condition is extended to the euler equations using entropy symmetrized variables the bddc method maintains scalability for the high order discretization of the diffusion dominated flows and achieves low iteration count in the advection dominated regime the bddc method based on inexact local solvers with incomplete factorization and p 1 coarse correction maintains the performance of the exact counterpart for the wide range of the peclet numbers considered