David E. Stewart

AN IMPLICIT TIME-STEPPING SCHEME FOR RIGID BODY DYNAMICS WITH INELASTIC COLLISIONS AND COULOMB FRICTION

In this paper a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks. Unlike other methods which take an instantaneous point of view, this method does not need to identify explicitly impulsive forces. Instead, the treatment is similar to that of J. J. Moreau and Monteiro-Marques, except that the numerical formulation used here ensures that there is no inter-penetration of rigid bodies, unlike their velocity-based formulation. Numerical results are given for the method presented here for a spinning rod impacting a table in two dimensions, and a system of four balls colliding on a table in a fully three-dimensional way. These numerical results also show the practicality of the method, and convergence of the method as the step size becomes small.

Differential variational inequalities

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions.

Time-stepping for three-dimensional rigid body dynamics

Traditional methods for simulating rigid body dynamics involves determining the current contact arrangement (e.g., each contact is either a “rolling” or “sliding” contact). The development of this approach is most clearly seen in the work of Haug et al. [Mech. Machine Theory 21 (1986) 401–425] and Pfeiffer and Glocker [Multibody Dynamics with Unilateral Contacts (Wiley, 1996)]. However, there has been a controversy about the status of rigid body dynamics as a theory, due to simple problems in the area which do not appear to have solutions; the most famous, if not the earliest is due to Paul Painleve [C.R. Acad. Sci. Paris 121 (1895) 112–115]. Recently, a number of time-stepping methods have been developed to overcome these difficulties. These time-stepping methods use integrals of the forces over time-steps, rather than the actual forces. This allows impulsive forces without the need for a separate formulation, or special procedures, to cover this case. The newest of these methods are developed in terms of complementarity problems. The complementarity problems that define the time-stepping procedure are solvable unlike previous methods for simulating rigid body dynamics with friction. Proof of the existence of solutions to the continuous problem can be shown in the sense of measure differential inclusions in terms of these methods. In this paper, a number of these variants will be discussed, and their essential properties proven.

An implicit time-stepping scheme for rigid body dynamics with Coulomb friction

In this paper a new time-stepping method for simulating systems of rigid bodies is given. Unlike methods which take an instantaneous point of view, our method is based on impulse-momentum equations, and so does not need to explicitly resolve impulsive forces. On the other hand, our method is distinct from previous impulsive methods in that it does not require explicit collision checking and it can handle simultaneous impacts. Numerical results are given for one planar and one three dimensional example, which demonstrate the practicality of the method, and its convergence as the step size becomes small.

A high accuracy method for solving ODEs with discontinuous right-hand side

SummaryOrdinary Differential Equations with discontinuities in the state variables require a differential inclusion formulation to guarantee existence [8]. From this formulation a high accuracy method for solving such initial value problems is developed which can give any order of accuracy and “tracks” the discontinuities. The method uses an “active set” approach, and determines appropriate active sets from solutions to Linear Complementarity Problems. Convergence results are established under some non-degeneracy assumptions. The method has been implemented, and results compare favourably with previously published methods [7, 21].

Note on the end game in homotopy zero curve tracking

Homotopy algorithms to solve a nonlinear system of equations <italic>f(x)</italic> = 0 involve tracking the zero curve of a homotopy map <italic>p(a, λ, x)</italic> from λ = 0 until λ = 1. When the algorithm nears or crosses the hyperplane λ = 1, an “end game” phase is begun to compute the solution <italic>x¯</italic> satisfying <italic>p(a, λ, x¯) = f(x¯)</italic> = 0. This note compares several end game strategies, including the one implemented in the normal flow code FIXPNF in the homotopy software package HOMPACK.

Existence of solutions to rigid body dynamics and the Painlevé paradoxes

Abstract The existence of solutions of rigid body dynamics with impact, shock and Coulomb friction in the sense of measure differential inclusions is announced. For the first time, such results include the famous counter-examples of Painleve. The resolution of the paradox involves impulsive forces without collisions, and the use of the post-impact time in the formulation of Coulombs law.

A unified approach to discrete frictional contact problems

We present a unified treatment of discrete, single-step, time-discretized contact problems with Coulomb friction that include quasistatic and dynamic problems involving rigid or elastic bodies undergoing small or large displacements. A general existence theory for these finite-dimensional problems is established under broad assumptions that are easily satisfied by many special models. The proof is based on a homotopy argument. This result extends many existence results known to date for discrete contact problems.

Algebraic multigrid (AMG) for saddle point systems from meshfree discretizations

Meshfree discretizations construct approximate solutions to partial differential equation based on particles, not on meshes, so that it is well suited to solve the problems on irregular domains. Since the nodal basis property is not satisfied in meshfree discretizations, it is difficult to handle essential boundary conditions. In this paper, we employ the Lagrange multiplier approach to solve this problem, but this will result in an indefinite linear system of a saddle point type. We adapt a variation of the smoothed aggregation AMG method of Vaněk et al. to this saddle point system. We give numerical results showing that this method is practical and competitive with other methods with convergence rates that are ∼c/logN. Copyright

Optimal control of systems with discontinuous differential equations

In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right-hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such a right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. We show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize. This means that the strategy of “optimize the discretization” will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that Euler’s method is used where the step size is “sufficiently small” in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing that involves Coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity.