Florian A. Potra

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.

Interior-point methods

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semi-definite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semi-definite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.

The kinetic preprocessor KPP*/a software environment for solving chemical kinetics

Abstract The kinetic preprocessor (KPP) is a software tool that assists the computer simulation of chemical kinetic systems. The concentrations of a chemical system evolve in time according to the differential law of mass action kinetics. A computer simulation requires the implementation of the differential system and its numerical integration in time. KPP translates a specification of the chemical mechanism into fortran or c simulation code that implement the concentration time derivative function and its Jacobian, together with a suitable numerical integration scheme. Sparsity in Jacobian is carefully exploited in order to obtain computational efficiency. KPP incorporates a library with several widely used atmospheric chemistry mechanisms and users can add their own chemical mechanisms to the library. KPP also includes a comprehensive suite of stiff numerical integrators. The KPP development environment is designed in a modular fashion and allows for rapid prototyping of new chemical kinetic schemes as well as new numerical integration methods.

Benchmarking stiff ODE solvers for atmospheric chemistry problems II: Rosenbrock solvers

Abstract In the numerical simulation of atmospheric transport-chemistry processes, a major task is the integration of the stiff systems of ordinary differential equations describing the chemical transformations. It is therefore of interest to systematically search for stiff solvers which can be identified as close to optimal for atmospheric applications. In this paper we continue our investigation from Sandu et al. (1996, CWI Report NM-R9603 and Report in Comput. Math., No. 85) and compare eight solvers on a set of seven box-models used in present day models. The focus is on Rosenbrock solvers. These turn out to be very well suited for our application when they are provided with highly efficient sparse matrix techniques to economize on the linear algebra. Two of the Rosenbrock solvers tested are from the literature, viz. rodas and Ros 4, and two are new and specially developed for air quality applications, viz. rodas 3 and ros 3.

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.

The current state and future direction of Eulerian models in simulating the tropospheric chemistry and transport of trace species: a review

Abstract Limitations on comprehensive tropospheric chemistry/transport models are discussed within the context of a set of issues currently facing the environmental scientific and policy-making communities. A number of central improvements are discussed in a prioritized manner, with consideration of the key progress necessary to include feedback processes between meteorology and chemistry, aerosol formation, in cloud development with subsequent effects on wet removal, dry deposition and surface exchange processes, and impacts of chemical perturbations on radiation, climate, and weather. These improvements would result in a “third-generation model”. The computational framework for this code is outlined, and estimates of required computer resources presented.

Benchmarking stiff ode solvers for atmospheric chemistry problems-I. implicit vs explicit

Abstract In many applications of atmospheric transport-chemistry problems, a major task is the numerical integration of the stiff systems of ordinary differential equations describing the chemical transformations. This paper presents a comprehensive numerical comparison between five dedicated explicit and four implicit solvers for a set of seven benchmark problems from actual applications. The implicit solvers use sparse matrix techniques to economize on the numerical linear algebra overhead. As a result they are often more efficient than the dedicated explicit ones, particularly when approximately two or more figures of accuracy are required. In most test cases, sparse RODAs, a Rosenbrock solver, came out as most competitive in the 1% error region. Of the dedicated explicit solvers, TWOSTEP came out as best. When less than 1% accuracy is aimed at, this solver performs very efficiently for tropospheric gas-phase problems. However, like all other dedicated explicit solvers, it cannot efficiently deal with gas-liquid phase chemistry. The results presented may constitute a guide for atmospheric modelers to select a suitable integrator based on the type and dimension of their chemical mechanism and on the desired level of accuracy. Furthermore, we would like to consider this paper an open invitation for other groups to add new representative test problems to those described here and to benchmark their numerical algorithms in our standard computational environment.

Sharp error bounds for Newton's process

SummaryThe method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].

A Superlinearly Convergent Primal-Dual Infeasible-Interior-Point Algorithm for Semidefinite Programming

A primal-dual infeasible-interior-point path-following algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithm finds an optimal solution in at most

OnQ-order andR-order of convergence

O(\sqrt{n}L)