Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

Distribution of melamine in polyester–melamine surface coatings cured under nonisothermal conditions

The influence of experimental cure parameters on the diffusion of reactive species in polyester–melamine thermoset coatings during curing has been investigated with X‐ray photoelectron spectroscopy and attenuated total reflectance Fourier transform infrared. The diffusion of melamine plays a vital role in the curing process and, therefore, in the ultimate properties of coatings. At a low (<20%) hexamethoxymethylmelamine (HMMM) crosslinker concentration, the matrix composition is uniform, but at high HMMM concentrations, excess HMMM rapidly segregates to the air–coating interface. The rate of migration is governed by the difference in the surface free energies of polyester and HMMM and the concentration gradient of HMMM between the bulk and the surface. An increased rate of energy absorption also increases the rate of migration of HMMM to the surface. A physical model has been proposed to explain this surface segregation phenomenon in terms of cocondensation and self‐condensation reactions. It suggests that an appropriate amount of melamine can be segregated on the surface and allowed to self‐condense to form a desired thickness of a melamine topcoat through the control of the binder composition and cure conditions. This technique can be implemented to apply a melamine topcoat during cure. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 83–91, 2004     

Mineral and chemical compositions and structure-sorption properties of bottom sediments as the basic components of enterosorbents and application materials

The mineral and chemical compositions, as well as some structure-sorption characteristics were determined for clay minerals selected for examination. The integrated coefficients of contamination of bottom sediments with each of the toxicant metals (Fe, Ni, Co, Cu, Zn, Pb, Mn) were calculated. The minerals examined presumably can be used in medicine as efficient application materials, cosmetic agents, and sorbents (enterosorbents).     

Transformations of cobalt trimethylacetate complexes with pyrazole upon oxidation by air in THF

Synthetic routes to oxidation of cobalt atoms in cobalt trimethylacetate complexes containing coordinated pyrazole molecules have been considered.     

Two-dimensional on-line particle imaging velocimetry

A high-performance algorithm is described for the on-line computation of two-dimensional velocity maps using particle displacement imaging. It relies on the computation of direct correlations in a multiple-scale spatial hierarchy. Layered rejection criteria based on correlation quality and conditional averaging techniques are employed to achieve the robustness required for continuous, unsupervised operation. Implementations are described using a UNIX-based workstation and a DOS-based personal computer with a real-time image processing/correlator subsystem. The cooperation of the algorithm with other real-time visualization techniques is demonstrated using the example of colour-coded streak integration.The authors wish to thank Dr. Philippe Georis of the Université Libre de Bruxelles (ULB) for making available to us video sequences of his Marangoni convection experiment.     

Decay of confined,two-dimensional,spatially periodic arrays of vortices: A numerical investigation

The disarrangement of a perturbed lattice of vortices was studied numerically. The basic state is an exponentially decaying, exact solution of the Navier-Stokes equations. Square arrays of vortices with even numbers of vortex cells along each side were perturbed and their evolution was investigated. Whether the energy in the perturbation grows somewhat before it decays or decays monotonically depends on the initial strength of the vortices of the basic state, the extent of lateral confinement and the structure of the perturbation. The critical condition for temporally local instability, i.e. the critical amplitude of the basic state that must be exceeded to allow energy transfer from the basic state to the perturbation, is discussed. In the strongly confined case of a square lattice of four vortices the appearance of enchancement of global rotation is the result of energy transfer from the basic state to a temporally local unstable mode. Energy is transferred from the basic state to larger-scaled structures (inverse cascade) only if the scales of the larger structures are inherently contained in the initial structure of the perturbation. The initial structure of the double array of vortices is not maintained except for a very special form of perturbation. The facts that large scales decay more slowly than small scales and that, when non-linearities are sufficiently strong, energy is transferred from one scale to another explain the differences in the disarrangement process for different initial strengths of the vortices of the basic state. The stronger vortices, i.e. the vortices perturbed in a manner that increases their strength, tend to dominate the weaker vortices. The pairing and subsequent merging (or capture) of vortices of like sense into larger-scale vortices are described in terms of peaks in the evolution of the square root of the palinstrophy divided by the enstrophy.     

Application of the variation principle for the problem of buckling of a nonlinearly elastic ring nonuniform along its thickness

There has been considerable recent attention given to the stressed and buckled states of items with complicated configuration made of different nonlinearly elastic materials joined by complete adhesion. However, effective analytical solutions for such problems have been hindered by mathematical difficulties. Approximate methods have thus been developed for such problems. A variational combined principle has been formulated in this communication. A nonlinear geometrical approach has been used for formulating a mixed-type functional with physical relationships given by Euler equations, nonlinear equilibrium equations, and nonlinear boundary conditions for a piecewise-nonuniform nonlinearly elastic body composed of finite elements (particles). As an example, buckling along the nonuniform thickness of nonlinearly elastic rings was analyzed hypothetically assuming plane cross-sections. Options for two-, three-, four-, five-, and six-layered rings in a periodical structure have been reviewed. The critical buckling forces for an even number of layers have been found to be equal to each other. The ratios of the critical forces, elasticity moduli, and proportionality levels were determined for all five variants by the Runge-Kutta method.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October, 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 262–268, March–April, 1995.