Jan A. Snyman

An Optimization Approach to the Determination of the Boundaries of Manipulator Workspaces

An optimization approach to computing the boundaries of the workspaces of planar manipulators is presented. This numerical method consists of finding a suitable radiating point in the output coordinate space and then determining the points of intersection of a representative pencil of rays, emanating from the radiating point, with the boundary of the accessible set. This is done by application of a novel constrained optimization approach that has the considerable advantage that it may easily be automated. The method is illustrated by its application to two planar mechanisms, namely a planar Stewart platform and a planar redundantly controlled serial manipulator. In addition to the exterior boundaries of the workspace, interior curves that represent configurations at which controllability and mobility may be limited, are also mapped. The optimization methodology, implemented here for the planar case, may readily be extended to spatial Stewart platforms.

Minimizing the effect of automotive pollution in urban geometry using mathematical optimization

Abstract One of the factors that needs to be considered during the layout of new urban geometry (e.g. street direction, spacing and width, building height restrictions) is the effect of the air pollution associated with the automotive transport that would use routes in this urban area. Although the pollution is generated at street level, its effect can be widespread due to interaction of the pollutant dispersion and diffusion with the wind speed and direction. In order to study the effect of a new urban geometry on the pollutant levels and dispersion, a very time-consuming experimental or parametric numerical study would have to be performed. This paper proposes an alternative approach, that of combining mathematical optimization with the techniques of computational fluid dynamics (CFD). In essence, the meteorological information as represented by a wind rose (wind speed and direction), is used to calculate pollutant levels as a function of urban geometry variables: street canyon depth and street canyon width. The pollutant source specified in conjunction with a traffic scenario with CO is used as pollutant. The main aim of the study is to be able to suggest the most beneficial configuration of an idealized urban geometry that minimizes the peak pollutant levels due to assumed traffic distributions. This study uses two mathematical optimization methods. The first method is implemented through a successive maximization–minimization approach, while the second method determines the location of saddle points of the pollutant level, considered as a function of urban geometry and wind rose. Locally, a saddle point gives the best urban geometry for the worst meteorological scenario. The commercial CFD code, STAR-CD, is coupled with a version of the DYNAMIC-Q optimization algorithm of Snyman, first to successively locate maxima and minima in a min–max approach; and then to locate saddle points. It is shown that the saddle-point method is more cost-effective. The methodology presented in this paper can readily be extended to optimize traffic patterns for existing geometry or in the development of geometry modification for pollution control or toxic releases.

A model for the emplacement of the eastern compartment of the bushveld complex

The stratigraphy and geological position of the eastern compartment of the Bushveld Complex are described. A mechanical model for the initiation and growth of the eastern compartment of the Bushveld intrusion has been developed using thin elastic plate theory, assuming linked conical magma chambers. It is shown that the contribution to the pressure at the base of a cell by the restitutional force exerted by the roof of Rooiberg felsites is 104 times as great as that of the layers of host in the cone. Both are minimal compared to the lithostatic pressure exerted by the magma pile. Roof deformation is therefore seen to be a more important process than sagging of the floor during intrusion—a feature which probably occurred during cooling, solidification and isostatic readjustment of the area. A stratigraphie model is proposed in which the intrusion of basic rocks into the Transvaal sequence is discussed in the light of continuous basin subsidence. Early submarine sedimentation in an irregularly-floored basin some 620 km in diameter situated on the Archaean craton gave rise to a 7.7 km thick sedimentary pile, to which was added some 7 km of subaerial basalts and felsites. Depression of the floor of the basin into the regime of maximum horizontal compression induced favourable conditions for the intrusion of a total of 2.5 km of diabase sills which further assisted the subsidence. The 9 km thick Bushveld Complex was intruded into the basal sections at points along a 010° trend in a regime characterised by shear failure. Early magma influxes gave rise to a laminated marginal zone forming a shallow cone, with associated sill activity, whilst continued later influxes filled the conical cell, transgressed the floor and uparched the roof. Partial melting in the regions beneath the Complex, exacerbated by continued crustal depression, gave rise to the late Bushveld granites.

Optimization of Gas Turbine Combustor Mixing for Improved Exit Temperature Profile

In this article, a design optimization technique for mixing in a gas turbine combustor is presented. The technique entails the use of computational fluid dynamics and mathematical optimization to optimize the combustor exit temperature profile. Combustor geometric parameters were used as optimization design variables. This work does not intend to suggest that combustor exit temperature profile is the only performance parameter important for the design of gas turbine combustors. However, it is a key parameter of an optimized combustor that is related to the power output and durability of the turbine. The combustor in this study is an experimental liquid-fuelled atmospheric combustor with a turbulent diffusion flame. The computational fluid dynamics simulations use a standard k-ϵ model. The optimization is carried out with the Dynamic-Q algorithm, which is specifically designed to handle constrained problems where the objective and constraint functions are expensive to evaluate. The optimization leads to a more uniform combustor exit temperature profile than with the original one.

Jets in Crossflow Mixing Analysis Using Computational Fluid Dynamics and Mathematical Optimization

Computational fluid dynamics and mathematical optimization were used to investigate the mixing effectiveness of jets in crossflow. A numerical model was developed, validated, and calibrated against experimental measurements of a temperature distribution at different cross-sectional planes downstream of an orifice injection plane. Good agreement was obtained when the ratio between momentum and species diffusivities was varied according to the jet-to-mainstream momentum flux ratio. Numerical optimization of various double-sided jet configurations followed, using a parametric approach. The results obtained showed that changes in orifice size and spacing at a constant orifice-to-mainstream area ratio and momentum flux ratio have a significant influence on mixing effectiveness. The optimum configuration compared favorably with an empirically defined relationship between orifice spacing and momentum flux ratio. Mathematical optimization was then combined with numerical methods to predict the optimum orifice configuration. The results showed the feasibility of using a gradient-based approximation method to allow, for a given set of parameters, the systematic adjustment of design variables to achieve improvement in performance.

Optimisation of structured grid spacing parameters for separated flow simulation using mathematical optimisation

Abstract This paper describes the use of computational fluid dynamics (CFD) and mathematical optimisation techniques to minimise the error in predicting the recirculation zone for a separated flow topology. Grid spacing parameters are varied in the optimisation process. The accuracy of separated flow solutions is known to be dependent on the grid resolution and clustering. Although general guidelines have been developed for grid generation of separated flow topologies, the flow solutions using the resulting grids often under-predict features like recirculation zones. This study addresses this aspect by providing an automatic tool for optimising the grid for solution accuracy. This approach has until recently been too expensive, but is becoming more viable with ever-increasing computer power. A two-dimensional sinusoidal hill is used as an example of a separated flow topology. The CFD simulation employs the commercial CFD solver STAR-CD to solve the Reynolds-Averaged Navier–Stokes equations with the RNG k – e turbulence model. CFD solution time is drastically reduced by making use of initial field restarts. The optimisation is carried out by means of Snymans DYNAMIC-Q method, which is specifically designed to handle constrained problems where the objective or constraint functions are expensive to evaluate. Six design variables (grid spacing parameters) are considered in this study. The results indicate that the re-attachment point of the recirculation zone is predicted to within 1% of the specified experimental value in four optimisation iterations and therefore represents a cost-effective way to determine grids based on solution accuracy.

A strongly interacting dynamic particle swarm optimizational method

A new direct search method using only function values is proposed for finding a local minimizer of a real valued function f(x), x = [x1, x2, ..., xn]T ? Rn. This method is a synthesis of two unconventional trajectory methods for unconstrained minimization. The first is the dynamic method of Snyman [2], and the second method is the particle swarm optimization (PSO) method of Eberhardt and Kennedy [1]. In the new (DYN-PSO) method, the minimization of a function is achieved through the dynamicmotion of a strongly interacting particle swarm, where each particle in the swarm is simultaneously attracted by all other particles located at positions of lower function value.

A reassessment of the Snyman---Fatti dynamic search trajectory method for unconstrained global optimization

The aim of this paper is to present a thorough reassessment of the Snyman–Fatti (SF) Multi-start Global Minimization Algorithm with Dynamic Search Trajectories, first published twenty years ago. The reassessment is done with reference to a slightly modified version of the original method, the essentials of which are summarized here. Results of the performance of the current code on an extensive set of standard test problems commonly in use today, are presented. This allows for a fair assessment to be made of the performance of the SF algorithm relative to that of the popular Differential Evolution (DE) method, for which test results on the same standard set of test problems used here for the SF algorithm, are also given. The tests show that the SF algorithm, that requires relatively few parameter settings, is a reliably robust and competitive method compared to the DE method. The results also indicate that the SF trajectory algorithm is particularly promising to solve minimum potential energy problems to determine the structure of atomic and molecular clusters.

Application of novel constrained optimization algorithms to the minimum volume design of planar CHS trusses with parallel chords

The present study has two aims: firstly, to determine the nature of the dependence of the optimum truss height and optimum cross-sectional dimensions that give minimum volume, on several truss parameters such as loads, number of fields, etc. Using the optimized height, significant mass and cost savings can be achieved in the design stage. This is in contrast to the current routine design practice, in which the truss height is taken proportional to the span length, and which does not give an optimal solution with minimum mass. The secondary aim of the article is to present a brief overview of the two novel and inter-related constrained gradient-based optimization techniques, namely the leap-frog method and the dynamic-Q method, used in this study. The reliability and relative effectiveness of the two methods are discussed. Finally, the discretization of the computed continuous optima is addressed.

SOME BASIC OPTIMIZATION THEOREMS

This chapter supplies the proofs for a number of theorems on which Chapters 1 to 4 have extensively relied on. In total nineteen proofs are presented of theorems that cover the characterization of unconstrained and constrained minima, saddle point conditions, conjugate gradient and Quasi-Newton methods.