J.A. Snyman

A new and dynamic method for unconstrained minimization

A new gradient method for unconstrained minimization is proposed. The method differs conceptually from other gradient methods. Here the minimization problem is solved via consideration of the analogous physical problem of the motion of a particle in a conservative force field, where the potential energy of the particle is represented by the function to be minimized. The method simulates the motion of the particle and by monitoring its kinetic energy an interfering strategy is adopted which ensures that the potential energy is systematically reduced. An algorithm, representing this approach, has been extensively tested using seven different kinds of test function. The results are compared with the performance of other quasi-Newton methods. Overall the results are encouraging showing the method to be competitive and particularly robust and reliable. The evidence that the computational time required by the new method for convergence increases roughly linearly with the dimension of the problem is of considerable significance.

An improved version of the original leap-frog dynamic method for unconstrained minimization: LFOP1(b)

Abstract A modified version of the authors original dynamic algorithm for unconstrained minimization is proposed. It employs time step selection procedure which results in a more efficient utilization of the original dynamic algorithm. The performance of the new algorithm is compared with that of a well established conjugate gradient algorithm when applied to three different extended test functions. Based on a comparison of the respective CPU times required for convergence, the new algorithm appears to be competitive.

A dynamic penalty function method for the solution of structural optimization problems

Abstract This paper presents an adaptation of an existing dynamic trajectory method for unconstrained minimization to handle constrained optimization problems. This is done by the application of a dynamic penalty parameter procedure to allow for the constraints. The method is applied to structural optimization problems that involve the determination of minimum weight structures of trusses and frames, subject to stress, displacement, and frequency constraints, under various prescribed load conditions. Because structural problems, in general, require detailed finite-element analyses to evaluate the constraint functions, the direct application of the trajectory method, requiring updated information at each step along the path, would be expensive. This problem is overcome by the successive application of the trajectory method to approximate quadratic subproblems that can be solved economically. The comprehensive new approach is called the DYNAMIC-Q method. The method is successfully applied to a number of truss and frame problems and is found to be both reliable and easy to use.

New Successive Approximation Method for Optimum Structural Design

A new procedure, using only first-order explicit information and consisting of the successive solution of approximate subproblems, is proposed. The method differs from established sequential quadratic programming methods by not utilizing the Lagrangian form and not attempting to store full and explicit Hessian information. Instead each constraint is individually approximated by a quadratic function involving only one coefficient in the quadratic term. This coefficient is determined from a two-point collocation. A novel feature of the method is that each quadratic subproblem is solved by a recently proposed interior feasible direction method. An auxiliary problem is formulated to generate a feasible starting point for each subproblem

Optimisation of road vehicle passive suspension systems. Part 1. Optimisation algorithm and vehicle model

Abstract Numerous problems have in the past been experienced during the development of military vehicle suspension systems. In order to solve some of these problems a two-dimensional multi-body vehicle dynamics simulation model has been developed for computer implementation. This model is linked to a mathematical optimisation algorithm in order to enable the optimisation of vehicle design parameters through the minimisation of a well defined objective function. In part 1 of this paper the concept of multi-disciplinary design optimisation is discussed. This is followed by the presentation of the up to six degrees of freedom vehicle model developed for this study, and a discussion of the specific gradient-based optimisation algorithm selected for the optimisation. In particular the derivation of the set of second-order differential equations, describing the acceleration of the different solid bodies of the vehicle model, is presented. In order to perform the optimisation of the non-linear suspension component characteristics, a six piece-wise continuous and linear approximation is used which is also described in this paper. Part 2 of this study will outline the simulation programme and the qualification of the programme. It will also present a typical case study where the proposed optimisation methodology is applied to improve the damper characteristics of a specific vehicle.

Optimisation of road vehicle passive suspension systems. Part 2. Qualification and case study

Abstract As an aid during the concept design phase, the two-dimensional vehicle simulation programme Vehsim2d has been developed (see part 1 of this paper for the vehicle model). The leap-frog optimisation algorithm for constrained problems (LFOPC) has been linked to the multi-body dynamics simulation code (Vehsim2d) to enable the computationally economic optimisation of certain vehicle and suspension design variables. This paper describes the simulation programme, the qualification of the programme, and gives an example of the application of the Vehsim2d/LFOPC system. In particular it is used to optimise the damper characteristics of an existing 22 ton three axle vehicle, over a typical terrain and at a representative speed. By using this system the optimised damper characteristics with respect to ride comfort for the vehicle are computed. The optimum damper characteristics give a 28.5% improvement in the ride comfort of the vehicle over the specified terrain and prescribed speed. Further optimisation runs were performed considering other terrain and different speed values. From these results final damper characteristics for the vehicle are proposed. Using the proposed characteristics, simulations were performed with the more advanced and proven DADS programme. The results show that the damper suggested by the optimisation study is indeed likely to improve the suspension of the vehicle. This study proves that the Vehsim2d/LFOPC vehicle modelling and optimisation system is indeed a valuable tool for a vehicle design team.

Optimal structural design of a planar parallel platform for machining

Abstract Parallel manipulators have many advantages over traditional serial manipulators. These advantages include high accuracy, high stiffness and high load-to-weight ratio, which make parallel manipulators ideal for machining operations where high accuracy is required to meet the requirements that modern standards demand. Recently, the finite element method has been used by some workers to determine the stiffness of spatial manipulators. These models are mainly used to verify stiffness predicted using kinematic equations, and are restricted to relatively simple truss-like models. In this study, state-of-the-art finite elements are used to determine the out of plane stiffness for parallel manipulators. Euler–Bernoulli beam elements and flat shell elements with drilling degrees of freedom are used to model the platform assembly. The main objective of this study is to quantify the stiffness, particularly the out of plane stiffness, of a planar parallel platform to be used for machining operations. The aim is to obtain a design that is able to carry out machining operations to an accuracy of 10 μm for a given tool force. Reducing the weight of a parallel manipulator used in machining applications has many advantages, e.g. increased maneuverability, resulting in faster material removal rates. Therefore the resulting proposed design is optimized with respect to weight, subject to displacement and stress constraints to ensure feasible stiffness and structural integrity. The optimization is carried out by means of two gradient-based methods, namely LFOPC and Dynamic-Q.

Modified Trajectory Method for Practical Global Optimization Problems

A modification of tbe Snyman-Fatti (SF) stocbastic multistart trajectory metbod for global optimization is developed. In the modified SF algorithm, distinction is made hetween a global and a local phase in the application of the original minimization procedure for a particular starting point. The glohal phase ensures convergence to the neighborhood of a relative low local minimum, whereas in the local phase further accuracy is pursued. Different choices of parameter values are proposed for the individual phases, and a procedure is proposed to deal with simple hound violations. The modifications lead to substantial improvement in the efficiency of the original trajectory method. The performance of the modified algorithm is assessed by its application to a selection of test problems and the results are compared with those of some other methods. The method is also successfully applied to laminate structural prohlems in which optimal sequences of the ply orientations of the layers are determined.

Numerical determination of the configurations of heavy rotating chains

The problem to be considered here is the determination of the possible configurations of a perfectly flexible and inextensible heavy chain rotating uniformly about an axis in the direction of the gravitational acceleration and with both ends fixed, a given distance apart, to the axis of rotation. This problem has recently become of interest in connection with the design of vertical axis wind turbines’ and is also of importance in the study of rotating threads in the spinning industry.2 In spite of comprehensive analyses and sophisticated treatments of this problem in three modern papers2-4 no one has yet succeeded in proposing a method, analytical or numerical, which gives a quantitative description of the configurations of heavy rotating chains. Since the chain may assume nodes on the vertical axis between the two fixed ends more than one possible nodal configuration must be allowed for, given a particular angular velocity of rotation. In this paper we present a simple numerical method by which, through physical discretization of the chain, the different nodal configurations may easily be calculated for any given angular velocity. In addition, in agreement with the work of Kolodner,3 the existence of a critical angular velocity for each nodal configuration, below which it cannot exist, is also demonstrated. To illustrate the basic difference between the conventional approach to the problem and ours we consider first, very briefly, the essentials of the continuous formulation as set out in the definitive article of Kolodner3 and as used by subsequent authors.2,4

Numerical study of linear and nonlinear string vibrations by means of physical discretization

Abstract To solve partial differential equations numerically, discretization of the continuous model is required and may be achieved either mathematically or physically. This paper illustrates how physical discretization of a continuous string may be accomplished by employing discrete model theory which has as its essential substance Newtonian mechanics. Typical examples of wave motion in discretized ‘linear’ and ‘non-linear’ strings are discussed. They include the transverse vibrations of a string after having been subjected to a given initial displacement, reflection and superposition of wave pulses in the string, and resonance of the string when coupled to a harmonic vibrator. The equations that arise after application of discrete model theory to these problems, describe the subsequent motion of the string, and are solved numerically by computer. In all cases the results obtained for the discrete linear string agree remarkably well with those for the corresponding continuous physical string. The stability of the solutions obtained by discretization are also investigated.