Mihai Anitescu

Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems

A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates.

A Computational Framework for Uncertainty Quantification and Stochastic Optimization in Unit Commitment With Wind Power Generation

We present a computational framework for integrating a state-of-the-art numerical weather prediction (NWP) model in stochastic unit commitment/economic dispatch formulations that account for wind power uncertainty. We first enhance the NWP model with an ensemble-based uncertainty quantification strategy implemented in a distributed-memory parallel computing architecture. We discuss computational issues arising in the implementation of the framework and validate the model using real wind-speed data obtained from a set of meteorological stations. We build a simulated power system to demonstrate the developments.

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 iterative approach for cone complementarity problems for nonsmooth dynamics

Aiming at a fast and robust simulation of large multibody systems with contacts and friction, this work presents a novel method for solving large cone complementarity problems by means of a fixed-point iteration. The method is an extension of the Gauss-Seidel and Gauss-Jacobi method with overrelaxation for symmetric convex linear complementarity problems. The method is proved to be convergent under fairly standard assumptions and is shown by our tests to scale well up to 500,000 contact points and more than two millions of unknowns.

On Using the Elastic Mode in Nonlinear Programming Approaches to Mathematical Programs with Complementarity Constraints

We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using algorithms and procedures of smooth nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier set to be nonempty. MPCCs that have nonempty Lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter. In this context, the elastic mode transforms MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem, which in turn makes it approachable by sequential quadratic programming (SQP) algorithms. One such algorithm is shown to achieve local linear convergence once the problem is relaxed. Under stronger conditions, we also prove superlinear convergence to the solution of an MPCC using an adaptive elastic mode approach for an SQP algorithm recently analyzed in an MPCC context in [R. Fletcher, S. Leyffer, S. Sholtes, and D. Ralph, Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints, Tech. report NA 210, University of Dundee, Dundee, UK, 2002]. Our assumptions are more general since we do not use a critical assumption from that reference. In addition, we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen.

Optimization-based simulation of nonsmooth rigid multibody dynamics

We present a time-stepping method to simulate rigid multibody dynamics with inelastic collision, contact, and friction. The method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program. We prove that a solution sequence of the method converges to the solution of a measure differential inclusion. We present numerical results for a few examples, and we illustrate the difference between the results from our scheme and previous, linear-complementarity-based time-stepping schemes.

Polynomial regression approaches using derivative information for uncertainty quantification

Abstract In this work we describe a polynomial regression approach that uses derivative information for analyzing the performance of a complex system that is described by a mathematical model depending on several stochastic parameters. We construct a surrogate model as a goal-oriented projection onto an incomplete space of polynomials; find coordinates of the projection by regression; and use derivative information to significantly reduce the number of the sample points required to obtain a good model. The simplified model can be used as a control variate to significantly reduce the sample variance of the estimate of the goal. For our test model, we take a steady-state description of heat distribution in the core of the nuclear reactor core, and as our goal we take the maximum centerline temperature in a fuel pin. For this case, the resulting surrogate model is substantially more computationally efficient than random sampling or approaches that do not use derivative information, and it has greater precision than linear models.

Global Convergence of an Elastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints

We prove that any accumulation point of an elastic mode approach, that approximately solves the relaxed subproblems, is a C-stationary point of the problem of optimizing a parametric mixed P variational inequality. If, in addition, the accumulation point satisfies the MPCC-LICQ constraint qualification, and if the solutions of the subproblem satisfy approximate second-order sufficient conditions, then the limiting point is an M-stationary point. Moreover, if the accumulation point satisfies the upper-level strict complementarity condition, the accumulation point will be a strongly stationary point. If we assume that the penalty function associated with the feasible set of the mathematical program with complementarity constraints has bounded level sets, and if the objective function is bounded below, we show that the algorithm will produce bounded iterates and will therefore have at least one accumulation point. We prove that the obstacle problem satisfies our assumptions for both a rigid and a deformable obstacle. The theoretical conclusions are validated by several numerical examples.

Degenerate Nonlinear Programming with a Quadratic Growth Condition

We show that the quadratic growth condition and the Mangasarian--Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an

Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties

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