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

the discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of poisson equation by using the direct method of lines the numerical simulation for the exterior problem of poisson equation is considered we introduced a polygonal artificial boundary ce and designed a discrete artificial boundary condition on it by using the direct method of lines then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary the finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed furthermore numerical examples show that the discrete artificial boundary condition is very eaective and more accurate than the neumann boundary condition which is often used in engineering literatures 1999 elsevier science s a all rights reserved

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

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

third order complex time step methods for transient analysis abstract using complex time steps and the method for the first order differential equations third order accurate a stable step by step time integration algorithms with controllable numerical dissipation are constructed for transient analysis the results at two sub step locations are evaluated and combined linearly with the corresponding weighting factors to give higher order accurate results at the end of a time step the sub step locations and the weighting factors are complex numbers the amplification factors at infinity are controlled by another algorithmic parameter a fourth order accurate algorithm is obtained when 1 the algorithmic characteristics and the computational procedure for the present third order algorithms are discussed equivalent algorithms and comparisons with other algorithms are given the present algorithms are found to be competitive with the other third and fourth order accurate algorithms

precise integration methods based on lagrange piecewise interpolation polynomials this paper introduces two new types of precise integration methods for dynamic response analysis of structures namely the integral formula method and the homogenized initial system method the applied loading vectors in the two algorithms are simulated by the lagrange piecewise interpolation polynomials based on the zeros of the first chebyshev polynomial developed on the basis of the integral formula and the lagrange piecewise interpolation polynomial and combined with the recurrence relationship of some key parameters in the integral computation suggested in this paper with the solving process of linear algebraic equations the integral formula method has been set up on the basis of the lagrange piecewise interpolation polynomial and transforming the non homogenous initial system into the homogeneous dynamic system the homogenized initial system method without dimensional expanding is presented this homogenized initial system method avoids the matrix inversion operation and is a general homogenized high precision direct integration scheme the accuracy of the presented time integration schemes is studied and is compared with those of other commonly used schemes the presented time integration schemes have arbitrary order of accuracy wider application and are less time consuming two numerical examples are also presented to demonstrate the applicability of these new methods copyright 2008 john wiley sons ltd

mixed finite element formulation for the solution of parabolic problems a mixed formulation of the finite element method is used to establish a higher order incremental method for the solution of parabolic equations the displacement and velocity fields in the terminology of solid mechanics are approximated independently in time using a hierarchical adaptive basis three time bases are tested namely polynomial radial and wavelet bases the time approximation criterion ensures stability and preserves parabolicity the resulting discretization in the time domain leads to an uncoupled governing system in the space dimension this system can be subsequently solved using the alternative formulations for the finite element and boundary element methods the application of the conventional conform displacement formulation of the finite element method is illustrated particular attention is given to the extension of pian and trefftz type hybrid stress finite elements for transient analysis in the time domain the performance of the coupled time space integration method is assessed and the results obtained are compared with the results reported in the literature for alternative time integration procedures

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

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

design spaces measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design illustration to structural dynamics for the first time for time discretized operators we describe and articulate the importance and notion of design spaces and algorithmic measures that not only can provide new avenues for improved algorithms by design but also can distinguish in general the quality of computational algorithms for time dependent problems the particular emphasis is on structural dynamics applications for the purpose of illustration and demonstration of the basic concepts the underlying concepts can be extended to other disciplines as well for further developments in time discretized operators and or for evaluating existing methods from the established measures for computational algorithms the conclusion that the most effective in the sense of convergence namely the stability and accuracy and complexity namely the algorithmic formulation and algorithmic structure computational algorithm should appear in a certain algorithmic structure of the design space amongst comparable algorithms is drawn with this conclusion and also with the notion of providing new avenues leading to improved algorithms by design as an illustration a novel computational algorithm which departs from the traditional paradigm in the sense of lms methods with which we are mostly familiar with and widely used in commercial software is particularly designed into the perspective design space representation of comparable algorithms and is termed here as the forward displacement non linearly explicit l stable fdel algorithm which is unconditionally consistent and does not require non linear iterations within each time step from the established measures for comparable algorithms simply for illustration purposes the resulting design of the fdel formulation is then compared with the commonly advocated explicit central difference method and the implicit newmark average acceleration method alternately the same conclusion holds true against controllable numerically dissipative algorithms which pertain to the class of linear multi step lms methods for assessing both linear and non linear dynamic cases the conclusions that the proposed new design of the fdel algorithm which is a direct consequence of the present notion of design spaces and measures is the most effective algorithm to date to our knowledge in comparison to the class of second order accurate algorithms pertaining to lms methods for routine and general non linear dynamic situations is finally drawn through rigorous numerical experiments

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

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

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

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

variational based energy momentum schemes of higher order for elastic fiber reinforced continua abstract this paper contributes to the improvement of recently published higher order accurate energy momentum schemes for an anisotropic hyperelastic material formulation based on the concept of structural tensors these galerkin based energy momentum schemes are developed by means of time finite elements with the focus on improving numerical stability and robustness in the presence of stiffness and large rotations they show advantages over conventional time stepping schemes in combination with anisotropic materials formulated with polyconvex strain energy density functions in this paper higher order energy momentum schemes are developed by using a differential variational principle in order to remedy the drawbacks of these galerkin based schemes namely a variationally inconsistent time approximation of the strains in the higher order case no coordinate free superimposed stress tensor and an unseparated time approximation of the fibers and the matrix material these new structure preserving time integrators preserve all conservation laws of a free flying hyperelastic anisotropic continuum namely the total linear and the total angular momentum conservation as well as the total energy conservation numerical examples demonstrate dynamic behavior of anisotropic continua discrete conservation properties of flying continua and the effect of higher order accuracy subject to transient loads

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

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

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

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

solution of 3d laplace and helmholtz equations in exterior domains using hp infinite elements abstract this work is devoted to a convergence study for infinite element discretizations for laplace and helmholtz equations in exterior domains the proposed approximation applies to separable geometries only combining an hp fe discretization on the boundary of the domain with a spectral like representation resulting from the separation of variables in the radial direction the presentation includes a convergence proof for the laplace equation and a stability analysis for the variational formulation of the helmholtz equation in weighted sobolev spaces the theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

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

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

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

error estimates for the finite element approximation of linear elastic equations in an unbounded domain in this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain the finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one in fact there are a family of nonlocal approximate boundary conditions with increasing accuracy and computational cost and a family of local ones for a given artificial boundary our error estimates show how the errors of the finite element approximations depend on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary a numerical example for navier equations outside a circle in the plane is presented numerical results demonstrate the performance of our error estimates

an easily implementable fourth order method for the time integration of wave problems abstract we are concerned with the time integration of systems of ordinary differential equations arising from the space discretization of partial differential wave equations with smooth solutions a method is suggested that while being as easily implementable as the standard implicit mid point rule is fourth order accurate the new method is symplectic so that it is very well suited for long time integrations of problems with a hamiltonian structure numerical experiments are reported that refer to a fourth order galerkin space discretization of the korteweg de vries equation and to a pseudospectral space discretization of the same equation

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

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

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

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

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

solving two dimensional second order elliptic equations in exterior domains using the inverted finite elements method in this paper inverted finite element method is used for solving two dimensional second order elliptic equations with a dirichlet boundary condition in an exterior domain after laying down the method and after giving an estimate of the error we detail how its implementation can be accomplished numerical results show the high efficiency and the accuracy of the method especially for equations with infinitely varying coefficients

weighted lp theory of the stokes and the bilaplacian operators in the half space abstract we consider the stokes and the bilaplacian equations in the half space of r n n 2 existence uniqueness and smoothness of weak weighted l p solutions are treated we prove that these problems can be broken down into three or four poisson like problems the importance of this decomposition is displayed through a short discussion on the numerical approximation of solutions by means of inverted finite elements method

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

a well posed problem for the exterior stokes equations in two and three dimensions this paper treats the stokes problem in exterior lipschitz continuous domains of 2 and 3 using the weighted sobolev spaces of hanouzet in 3 and giroire in 2 we establish the inf sup condition between the velocity and pressure spaces this fundamental result shows that the variational stokes problem is well posed in those spaces in the last paragraph we obtain additional regularity of the solution when the data are smoother

a new family of exponential infinite elements for the analysis of lossless electromagnetic waveguides in this paper the finite element method is employed for the analysis of electromagnetic lossless waveguides the curl curl equation deduced from maxwell s equations is solved for the electric field we have studied the appearance of parasitic solutions and formulated a sufficient condition for their elimination which puts a contraint on the finite element spaces for the approximation of the field components a new family of discontinuous exponential infinite elements is proposed for the analysis of open waveguides they are employed together with edge elements and a non linear eigenvalue problem is deduced we have presented numerical results in order to prove the validity of such infinite elements versus the virtual boundary technique

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

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

high order local artificial boundary conditions for problems in unbounded domains in this paper we present error estimates for the finite element approximation of poisson and modified helmholtz equations outside an obstacle or in a semi infinite strip in the plane the finite element approximation is formulated on a bounded domain using a local approximate artificial boundary condition in fact there is a sequence of local approximate boundary conditions for a given artificial boundary our error estimates are based on the mesh size and the location of the artificial boundary the numerical stability and robustness of the method are discussed numerical experiments are presented to demonstrate the performance of the method and our error estimates

weighted hilbert spaces for the stationary exterior stokes problem with navier slip boundary conditions abstract in this paper we analyze the three dimensional exterior stokes problem with the navier slip boundary conditions describing the flow of a viscous and incompressible fluid past an obstacle where it is assumed that the fluid may slip at the boundary because the flow domain is unbounded we set the problem in weighted spaces in order to control the behavior at infinity of the solutions this functional framework also allows to prescribe various behaviors at infinity of the solutions growth or decay existence and uniqueness of solutions are shown in a hilbert setting which gives the tools for a possible numerical analysis of the problem weighted korn s inequalities are the key point in order to study the variational problem

fictitious domains and level sets for moving boundary problems applications to the numerical simulation of tumor growth in this work we present a new strategy for solving numerically a relatively simple model of tumor growth in principle this is devoted to describe avascular growth although by choosing the parameters appropriately it also permits to give an idea of the behavior after vascularization the numerical methods rely on fictitious domain and level set techniques with a combination of quadratic finite elements and finite differences approximations we present a collection of numerical results that essentially coincide with others previously obtained with other techniques

well posedness and stability of a multi dimensional tumor growth model we study a moving boundary problem modeling the growth of in vitro tumors this problem consists of two elliptic equations describing the distribution of the nutrient and the internal pressure respectively and a first order partial differential equation describing the evolution of the moving boundary an important feature is that the effect of surface tension on the moving boundary is taken into account we show that this problem is locally well posed for a large class of initial data by using analytic semi group theory we also prove that if the surface tension coefficient is larger than a threshold value then the unique flat equilibrium is asymptotically stable whereas in the case this flat equilibrium is unstable

error estimates for the finite element approximation of problems in unbounded domains in this paper we present error estimates for the finite element approximation of linear elliptic problems in unbounded domains that are outside an obstacle and a semi infinite strip in the plane the finite element approximation is formulated on a bounded domain using a nonlocal approximate artificial boundary condition in fact there is a family of approximate boundary conditions with increasing accuracy and computational cost for a given artificial boundary our error estimates are based on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary numerical examples for poisson s problem outside a circle and in a semi infinite strip are presented numerical results demonstrate the performance of our error estimates