Deceleration of a porous rotating disk in a viscous fluid

Numerical linear algebra aspects of globally convergent homotopy methods

Abstract: Probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one These methods are theoretically powerful, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical The concomitant numerical linear algebra problems deal with rectangular matrices, and good algorithms require a delicate balance (not always achieved) of accuracy, robustness, and efficiency in both space and time The author's experience with globally convergent homotopy algorithms is surveyed here, and some of the linear algebra difficulties for dense and sparse problems are discussed

Globally convergent homotopy methods: a tutorial

Abstract: The basic theory for probability one globally convergent homotopy algorithms was developed in 1976, and since then the theory, algorithms, and applications have considerably expanded. These are algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point. Thus they are globally convergent with probability one. They are applicable to Brouwer fixed point problems, certain classes of zero-finding problems, unconstrained optimization, linearly constrained optimization, nonlinear complementarity, and the discrezations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements. A mathematical software package, HOMPACK, exists that implements several different strategies and handles both dense and sparse problems. Homotopy algorithms are closely related to ODE algorithms, and make heavy use of ODE techniques. Homotopy algorithms for some classes of nonlinear systems, such as polynomial systems, exhibit a large amount of coarse grain parallelism. These and other topics are discussed in a tutorial fashion.

Globally convergent homotopy algorithms for nonlinear systems of equations

Abstract: Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, constrained and unconstrained optimization problems, discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper introduces, in a tutorial fashion, the theory of globally convergent homotopy algorithms, deseribes some computer algorithms and mathematical software, and presents several nontrivial engineering applications.

Globally convergent homotopy methods

Abstract: The basic theory for probability one globally convergent homotopy algorithm.q was developed in 1976, and since then the theory, algorithms, and applications have considerably expanded. These are algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for p)most all choices of starting point. Thus they are globally convergent with probability one. They a,-~ applicable to Brouwer fixed point problems, certain classes of zero-finding problems, unconstrained optimization, linearly constrained optimization, nonlinear complementarity, and the dlscretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements. A mathematical software package, HOMPACK, e~sts that implements several different strategies and handles both dense and sparse problems. Homotopy algorith~.q are closely related to ODE algorithms, and make heavy use of ODE techniques. Homotopy algorithrnn for some classes of non|;near systems, such as polynomial systems, exhibit a large amount of coarse grain parallelism. These and other topics are discussed in a tutorial fashion.

Unsteady viscous flow over a rotating stretchable disk with deceleration

Abstract: In this work, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated. The three dimensional Navier–Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method. Mathematically, two solution branches are found for the similarity equations. The lower solution branch may not be physically feasible due to a negative velocity in the circumferential direction. For the physically feasible solution branch, namely the upper solution branch, the fluid behavior is greatly influenced by the disk stretching parameter and the unsteadiness parameter. With disk stretching, the disk can be friction free in both the radial and the circumferential directions, depending on the values of the controlling parameters. The results provide an exact solution to the whole unsteady NS equations with new nonlinear phenomena and multiple solution branches.

Finding zeroes of maps: homotopy methods that are constructive with probability one

Abstract: We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive with probability one" and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.

On the flow due to a rotating disk

Abstract: The von Karman (1921) rotating disk problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution. Exact representations of the non-steady velocity field and pressure are given by suitable power-series expansions in the angle of rotation, Ωt, with coefficients that are functions of a similarity variable. The first four equations for velocity coefficient functions are solved exactly in closed form, and the next six by numerical integration. This gives four terms in the series for the primary flow and three terms in each series for the secondary flow.The results indicate that the asymptotic steady state is approached after about 2 radians of the disk's motion and that it can be approximately obtained from the initial-value, time-dependent analysis. Furthermore, the non-steady flow has three phases, the first two of which are accurately and fully described with the terms computed. During the first-half radian (phase 1), the velocity field is essentially similar in time, with boundary-layer thickening the only significant effect. For 0·5 [lsim ] Ωt [lsim ] 1·5 (phase 2), boundary-layer growth continues at a slower rate, but simultaneously the velocity profiles adjust towards the shape of the ultimate steady-state profiles. At about Ωt = 1·5, some flow quantities overshoot the steady-state values by small amounts. In analogy with the ‘Greenspan-Howard problem’ (1963) it is believed that the third phase (Ωt > 1·5) consists of a small amplitude decaying oscillation about the steady-state solution.

Unsteady flows in a semi-infinite contracting or expanding pipe

Abstract: Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.

A globally convergent algorithm for computing fixed points of C2 maps

Abstract: Chow, Mallet-Paret, and Yorke have recently proposed in abstract terms an algorithm for computing Brouwer fixed points of C^2 maps. A numerical implementation of that algorithm is presented here. Careful attention has been paid to computational efficiency, accuracy, and robustness. Some convergence theorems and results of numerical tests are also included.