Energy-dissipative momentum-conserving time-stepping algorithms for the dynamics of nonlinear Cosserat rods

 This paper presents the development of energy-dissipative momentum-conserving algorithms for the numerical integration of the dynamics of nonlinear Cosserat rods. The proposed numerical schemes exhibit a non-negative energy dissipation, controllable through the appropriate algorithmic parameters including an energy-conserving scheme as a particular case. These conservation/dissipation properties are proven rigorously in the general nonlinear setting, accounting specifically for the finite element implementation of the rotational degrees of freedom associated to the motion of the rod's cross-sections. In particular, we consider a direct parameterization of the director fields defining these sections, hence leading to frame-indifferent approximations of the strain measures defining the rod's mechanical response. The robustness added by these considerations when comparing the proposed numerical schemes with existing conserving schemes is illustrated with several representative numerical simulations. RID="†" ID="†" Our motivation behind the developments presented in this paper started from a number of very instructive conversations with Professor M.A. Crisfield. His insight in the numerical treatment of the structural problems considered here was unique. It was for us a great privilege to interact with him and enjoy of his friendship. These interactions were always very rewarding, given especially how contagious his enthusiasm for his work was. We would like to dedicate this modest contribution to his memory. Currently at: E.T.S.I.C.C.P., Universidad Politécnica de Madrid, Spain Dedicated to the memory of Prof. mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years. Financial support for this research was provided by the AFOSR under contract no. F49620-00-1-0360 with UC Berkeley. This support is gratefully acknowledged.     

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.     

A series of energy conserving algorithms for structural dynamics

Abstract

A series of step‐by‐step integration methods has been effectively developed which does not increase the total number of equations of motion and avoids the use of the derivatives of external force. The well‐known Newmark β method [16] with β = 1/4 is the lowest order of accuracy of this series of methods. All the algorithms of this series are unconditionally stable, without overshoot in displacement or in velocity, and they do not possess any numerical dissipations. The rapid changes of dynamic loading can be automatically overcome. It is also verified that the higher the order of the integration method, the more accurate. Consequently, the higher‐order algorithms of this series allow the use of a large time step in step‐by‐step dynamic analysis. Thus, they are competitive in dynamic analysis, especially when the response of a long duration is of interest.     

Interpolation of rotational variables in nonlinear dynamics of 3D beams

The formulation of dynamic procedures for three-dimensional (3-D) beams requires extensive use of the algebra pertaining to the non-linear character of the rotation group in space. The corresponding extraction procedure to obtain the rotations that span a time step has certain limitations, which can have a detrimental effect on the overall stability of a time-integration scheme. The paper describes two algorithms for the dynamics of 3-D beams, which differ in their manifestation of the above limitation. The first has already been described in the literature and involves the interpolation of iterative rotations, while an alternative formulation, which eliminates the above effect by design, requires interpolation of incremental rotations. Theoretical arguments are backed by numerical results. Similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one. © 1998 John Wiley & Sons, Ltd.     

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.     

An energy and momentum conserving method for rigid–flexible body dynamics

In this work, an energy and momentum conserving method is developed for doing coupled flexible and rigid body dynamics. The main focus is on the bilateral connection of flexible finite elements to rigid bodies. The coupling of rigid bodies at joints is also introduced. Existing conserving algorithms for individual (un‐coupled) rigid and flexible bodies are exploited and modified for the coupled system. By using the appropriate rigid body rotational update and generalized force definitions, the resulting rigid–flexible and rigid–rigid systems are unconditionally stable and conserve linear and angular momentum. The conservation and stability properties are demonstrated in numerical simulation. Published in 2001 by John Wiley & Sons, Ltd.     

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 Generalized‐α Methodwhich 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 energy‐conserving 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 α‐methods, the Energy‐Momentum Methodwith and without numerical dissipation, the Constraint Energy‐Momentum Algorithm and the Constraint Energy Method. Copyright © 1999 John Wiley & Sons, Ltd.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

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. The superior performance of the proposed scheme method relative to conventional time-integrators is demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion. These numerical experiments show that symplectic schemes often regarded as unconditionally stable, such as the mid-point rule, can exhibit a dramatic blow-up in finite time while the present method remains perfectly stable.     

An energy–momentum conserving algorithm for non‐linear hypoelastic constitutive models

This paper presents an extension of the energy momentum conserving algorithm, usually developed for hyperelastic constitutive models, to the hypoelastic constitutive models. For such a material no potential can be defined, and thus the conservation of the energy is ensured only if the elastic work of the deformation can be restored by the scheme. We propose a new expression of internal forces at the element level which is shown to verify this property. We also demonstrate that the work of plastic deformation is positive and consistent with the material model. Finally several numerical applications are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Design,analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics

The primary objectives of the present exposition are to: (i) provide a generalized unified mathematical framework and setting leading to the unique design of computational algorithms for structural dynamic problems encompassing the broad scope of linear multi‐step (LMS) methods and within the limitation of the Dahlquist barrier theorem (Reference [3], G. Dahlquist, BIT 1963; 3 : 27), and also leading to new designs of numerically dissipative methods with optimal algorithmic attributes that cannot be obtained employing existing frameworks in the literature, (ii) provide a meaningful characterization of various numerical dissipative/non‐dissipative time integration algorithms both new and existing in the literature based on the overshoot behavior of algorithms leading to the notion of algorithms by design, (iii) provide design guidelines on selection of algorithms for structural dynamic analysis within the scope of LMS methods. For structural dynamics problems, first the so‐called linear multi‐step methods (LMS) are proven to be spectrally identical to a newly developed family of generalized single step single solve (GSSSS) algorithms. The design, synthesis and analysis of the unified framework of computational algorithms based on the overshooting behavior, and additional algorithmic properties such as second‐order accuracy, and unconditional stability with numerical dissipative features yields three sub‐classes of practical computational algorithms: (i) zero‐order displacement and velocity overshoot (U0‐V0) algorithms; (ii) zero‐order displacement and first‐order velocity overshoot (U0‐V1) algorithms; and (iii) first‐order displacement and zero‐order velocity overshoot (U1‐V0) algorithms (the remainder involving high‐orders of overshooting behavior are not considered to be competitive from practical considerations). Within each sub‐class of algorithms, further distinction is made between the design leading to optimal numerical dissipative and dispersive algorithms, the continuous acceleration algorithms and the discontinuous acceleration algorithms that are subsets, and correspond to the designed placement of the spurious root at the low‐frequency limit or the high‐frequency limit, respectively. The conclusion and design guidelines demonstrating that the U0‐V1 algorithms are only suitable for given initial velocity problems, the U1‐V0 algorithms are only suitable for given initial displacement problems, and the U0‐V0 algorithms are ideal for either or both cases of given initial displacement and initial velocity problems are finally drawn. For the first time, the design leading to optimal algorithms in the context of a generalized single step single solve framework and within the limitation of the Dahlquist barrier that maintains second‐order accuracy and unconditional stability with/without numerically dissipative features is described for structural dynamics computations; thereby, providing closure to the class of LMS methods. Copyright © 2003 John Wiley & Sons, Ltd.     

Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.     

Explicit momentum‐conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

We reformulate the midpoint Lie algorithm, which is implicit in the torque calculation, to achieve explicitness in the torque evaluation. This is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Thus, we obtain two related variants, both of which are explicit in the torque calculation, but only first order in the time step. Both variants are momentum conserving and both are symplectic. Consequently, drawing on the properties of the composition of maps, we introduce another algorithm that combines the two variants in a single time step. The resulting algorithm is explicit, momentum conserving, symplectic, and second order. Its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators. Copyright © 2005 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.     

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.     

The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics

In the present work, rigid bodies and multibody systems are regarded as constrained mechanical systems at the outset. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. Concerning external constraints lower kinematic pairs such as revolute and prismatic pairs are treated in detail. Both internal and external constraints are dealt with on an equal footing. The present approach thus circumvents the use of rotational variables throughout the whole time discretization. After the discretization has been completed a size‐reduction of the discrete system is performed by eliminating the constraint forces. In the wake of the size‐reduction potential conditioning problems are eliminated. The newly proposed methodology facilitates the design of energy–momentum methods for multibody dynamics. The numerical examples deal with a gyro top, cylindrical and planar pairs and a six‐body linkage. Copyright © 2006 John Wiley & Sons, Ltd.     

Assumed strain finite element methods for conserving temporal integrations in non‐linear solid dynamics

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

Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element

The paper presents aspects of the finite element formulation of momentum and energy conserving algorithms for the non‐linear dynamic analysis of shell‐like structures. The key contribution is a detailed analysis of the implementation of a Simó–Tarnow‐type conservation scheme in a recently developed new mixed finite shell element. This continuum‐based shell element provides a well‐defined interface to strain‐driven constitutive stress updates algorithms. It is based on the classic brick‐type trilinear displacement element and is equipped with specific gradient‐type enhanced strain modes and shell‐typical assumed strain modifications. The excellent performance of the proposed dynamic shell formulation with respect to conservation properties and numerical stability behaviour is demonstrated by means of three representative numerical examples of elastodynamics which exhibit complex free motions of flexible structures undergoing large strains and large rigid‐body motions. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulation of a confined polymer in solution using the dissipative particle dynamics method

The dynamics of a bead-and-spring polymer chain suspended in a sea of solvent particles are examined by dissipative particle dynamics (DPD) simulations. The solvent is treated as a structured medium, comprised of particles subject to both solvent-solvent and solvent-polymer interactions and to stochastic Brownian forces. Thus hydrodynamic interactions among the beads of the polymer evolve naturally from the dynamics of the solvent particles. DPD simulations are about two orders of magnitude faster than comparable molecular dynamics simulations. Here we report the results of an investigation into the effects of confining the dissolved polymer chain between two closely spaced parallel walls. Confinement changes the polymer configuration statistics and produces markedly different relaxation times for chain motion parallel and perpendicular to the surface. This effect may be partly responsible for the gap width-dependent theological properties observed in nanoscale rheometry.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.     

Analyzing Fe–Zn system using molecular dynamics, evolutionary neural nets and multi-objective genetic algorithms

Failure behavior of Zn coated Fe is simulated through molecular dynamics (MD) and the energy absorbed at the onset of failure along with the corresponding strain of the Zn lattice are computed for different levels of applied shear rate, temperature and thickness. Data-driven models are constructed by feeding the MD results to an evolutionary neural network. The outputs of these neural networks are utilized to carry out a multi-objective optimization through genetic algorithms, where the best possible tradeoffs between two conflicting requirements, minimum deformation and maximum energy absorption at the onset of failure, are determined by constructing a Pareto frontier.