Force‐stepping integrators in Lagrangian mechanics

We formulate an integration scheme for Lagrangian mechanics, referred to as the force‐stepping scheme, which is symplectic, energy conserving, time‐reversible, and convergent with automatic selection of the time‐step size. The scheme also conserves approximately all the momentum maps associated with the symmetries of the system. The exact conservation of momentum maps may additionally be achieved by recourse to the Lagrangian reduction. The force‐stepping scheme is obtained by replacing the potential energy by a piecewise affine approximation over a simplicial grid or regular triangulation. By taking triangulations of diminishing size, an approximating sequence of energies is generated. The trajectories of the resulting approximate Lagrangians can be characterized explicitly and consist of piecewise parabolic motion, or free fall. Selected numerical tests demonstrate the excellent long‐term behavior of force‐stepping, its automatic time‐step selection property, and the ease with which it deals with constraints, including contact problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Time integrators based on approximate discontinuous Hamiltonians

We introduce a class of time integration algorithms for finite dimensional mechanical systems whose Hamiltonians are separable. By partitioning the system's configuration space to construct an approximate potential energy, we define an approximate discontinuous Hamiltonian (ADH) whose resulting equations of motion can be solved exactly. The resulting integrators are symplectic and precisely conserve the approximate energy, which by design is always close to the exact one. We then propose two ADH algorithms for finite element discretizations of nonlinear elastic bodies. These result in two classes of explicit asynchronous time integrators that are scalable and, because they conserve the approximate Hamiltonian, could be considered to be unconditionally stable in some circumstances. In addition, these integrators can naturally incorporate frictionless contact conditions. We discuss the momentum conservation properties of the resulting methods and demonstrate their performance with several problems, such as rotating bodies and multiple collisions of bodies with rigid boundaries. Copyright © 2011 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.     

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.     

Conservation properties of a time FE method. Part I: time‐stepping schemes for N‐body problems

In the present paper one‐step implicit integration algorithms for the N‐body problem are developed. The time‐stepping schemes are based on a Petrov–Galerkin finite element method applied to the Hamiltonian formulation of the N‐body problem. The approach furnishes algorithmic energy conservation in a natural way. The proposed time finite element method facilitates a systematic implementation of a family of time‐stepping schemes. A particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals. The influence of various standard as well as non‐standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements. Copyright © 2000 John Wiley & Sons, Ltd.     

A time‐stepping method for stiff multibody dynamics with contact and friction

We define a time‐stepping procedure to integrate the equations of motion of stiff multibody dynamics with contact and friction. The friction and non‐interpenetration constraints are modelled by complementarity equations. Stiffness is accommodated by a technique motivated by a linearly implicit Euler method. We show that the main subproblem, a linear complementarity problem, is consistent for a sufficiently small time step h. In addition, we prove that for the most common type of stiff forces encountered in rigid body dynamics, where a damping or elastic force is applied between two points of the system, the method is well defined for any time step h. We show that the method is stable in the stiff limit, unconditionally with respect to the damping parameters, near the equilibrium points of the springs. The integration step approaches, in the stiff limit, the integration step for a system where the stiff forces have been replaced by corresponding joint constraints. Simulations for one‐ and two‐dimensional examples demonstrate the stable behaviour of the method. Published in 2002 by John Wiley & Sons, Ltd.     

Chaotic Scattering and Escape Times of Marginally Trapped Ultracold Neutrons

We compute classical trajectories of Ultracold neutrons (UCNs) in a superconducting Ioffe-type magnetic trap using a symplectic integration method. We find that the computed escape time for a particular set of initial conditions (momentum and position) does not generally stabilize as the time step parameter is reduced unless the escape time is short (less than approximately 10 s). For energy intervals where more than half of the escape times computed for UCN realizations are numerically well determined, we predict the median escape time as a function of the midpoint of the interval.     

An explicit asynchronous contact algorithm for elastic body–rigid wall interaction

The use of multiple‐time‐step integrators can provide substantial computational savings over traditional one‐time‐step methods for the simulation of solid dynamics, while maintaining desirable properties, such as energy conservation. Contact phenomena generally require either the adoption of an implicit algorithm or the use of unacceptably small time steps to prevent large amount of numerical dissipation from being introduced. This paper introduces a new explicit dynamic contact algorithm that, by taking advantage of asynchronous time stepping, delivers an outstanding energy performance at a much more acceptable computational cost. We demonstrated the performance of the numerical method with several three‐dimensional examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A variationally consistent fractional time‐step integration method for incompressible and nearly incompressible Lagrangian dynamics

In this paper we present a fractional time‐step method for Lagrangian formulations of solid dynamics problems. The method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly. Energy fluctuations are found to be minimal and stay bounded for long durations. In order to handle incompressibility, a mixed formulation in which the pressure appears explicitly is adopted. The velocity update over a time step is split into deviatoric and volumetric components. The deviatoric component is advanced using explicit time marching, whereas the pressure correction for each time step is computed implicitly by solving a Poisson‐like equation. Once the pressure is known, the volumetric component of the velocity update is calculated. In contrast with standard explicit schemes, where the time‐step size is determined by the speed of the pressure waves, the allowable time step for the proposed scheme is found to depend only on the shear wave speed. This leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical integration of the stiff dynamics of geometrically exact shells: an energy‐dissipative momentum‐conserving scheme

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

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.     

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.     

Time finite element based Moreau‐type integrators

With the postulation of the principle of virtual action, we propose, in this paper, a variational framework for describing the dynamics of finite dimensional mechanical systems, which contain frictional contact interactions. Together with the contact and impact laws formulated as normal cone inclusions, the principle of virtual action directly leads to the measure differential inclusions commonly used in the dynamics of nonsmooth mechanical systems. The discretization of the principle of virtual action in its strong and weak variational form by local finite elements in time provides a structured way to derive various time‐stepping schemes. The constitutive laws for the impulsive and nonimpulsive contact forces, ie, the contact and impact laws, are treated on velocity‐level by using a discrete contact law for the percussion increments in the sense of Moreau. Using linear shape functions and different quadrature rules, we obtain three different stepping schemes. Besides the well‐established Moreau time‐stepping scheme, we can present two alternative integrators referred to as symmetric and variational Moreau‐type stepping schemes. A suitable benchmark example shows the superiority of the newly proposed integrators in terms of energy conservation properties, accuracy, and convergence.     

Step size adjustment and extrapolation for time‐stepping schemes in non‐smooth dynamics

In this paper we use step size adjustment and extrapolation methods to improve Moreau's time‐stepping scheme for the numerical integration of non‐smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non‐smooth systems in the smooth parts, we show how the time‐stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time‐stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability and accuracy analysis of a discrete model reference adaptive controller without and with time delay

Adaptive control techniques can be applied to dynamical systems whose parameters are unknown. We propose a technique based on control and numerical analysis approaches to the study of the stability and accuracy of adaptive control algorithms affected by time delay. In particular, we consider the adaptive minimal control synthesis (MCS) algorithm applied to linear time‐invariant plants, due to which, the whole controlled system generated from state and control equations discretized by the zero‐order‐hold (ZOH) sampling is nonlinear. Hence, we propose two linearization procedures for it: the first is via what we term as physical insight and the second is via Taylor series expansion. The physical insight scheme results in useful methods for a priori selection of the controller parameters and of the discrete‐time step. As there is an inherent sampling delay in the process, a fixed one‐step delay in the discrete‐time MCS controller is introduced. This results in a reduction of both the absolute stability regions and the controller performance. Owing to the shortcomings of ZOH sampling in coping with high‐frequency disturbances, a linearly implicit L‐stable integrator is also used within a two degree‐of‐freedom controlled system. The effectiveness of the methodology is confirmed both by simulations and by experimental tests. Copyright © 2009 John Wiley & Sons, Ltd.     

Variational integrators – A continuous time approach

This article presents a family of variational integrators from a continuous time point of view. A general procedure for deriving symplectic integration schemes preserving an energy‐like quantity is shown, which is based on the principle of virtual work. The framework is extended to incorporate holonomic constraints without using additional regularization. In addition, it is related to well‐known partitioned Runge–Kutta methods and to other variational integration schemes. As an example, a concrete integration scheme is derived for the planar pendulum using both polar and Cartesian coordinates. Copyright © 2016 John Wiley & Sons, Ltd.     

A priori error estimator of the generalized‐α method for structural dynamics

An a priori error estimator for the generalized‐α time‐integration method is developed to solve structural dynamic problems efficiently. Since the proposed error estimator is computed with only information in the previous and current time‐steps, the time‐step size can be adaptively selected without a feedback process, which is required in most conventional a posteriori error estimators. This paper shows that the automatic time‐stepping algorithm using the a priori estimator performs more efficient time integration, when compared to algorithms using an a posteriori estimator. In particular, the proposed error estimator can be usefully applied to large‐scale structural dynamic problems, because it is helpful to save computation time. To verify efficiency of the algorithm, several examples are numerically investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

Acceleration of material‐dominated calculations via phase‐space simplicial subdivision and interpolation

We develop an acceleration method for material‐dominated calculations based on phase‐space simplicial interpolation of the relevant material‐response functions. This process of interpolation constitutes an approximation scheme by which an exact material‐response function is replaced by a sequence of approximating response functions. The terms in the sequence are increasingly accurate, thus ensuring the convergence of the overall solution. The acceleration ratio depends on the dimensionality, the complexity of the deformation, the time‐step size, and the fineness of the phase‐space interpolation. We ascertain these trade‐offs analytically and by recourse to selected numerical tests. The numerical examples with piecewise‐quadratic interpolation in phase space confirm the analytical estimates. Copyright © 2015 John Wiley & Sons, Ltd.