In this thesis the variation of a functional defined on a variable domain has been studied and applied to the problem of finding the optimum shape of the domain in which some performance criterion has an extreme. The method most frequently used is one due to Gelf and Fomin. It is applied to problems governed by first and second order partial differential equations, unsteady one dimensional gas movements and the problem of minimum drag on a body with axial symmetry in Stokes flow.

The problem of optimizing a nonlinear function of one or more variables in the sense of locating the values of the variables which give the greatest or least value of the function, is considered from two points of view. First, the development of two new and improved techniques for optimization is described. Second, the ways in which the available techniques can be applied are discussed with reference to case studies of practical significance. The two new techniques are for unconstrained optimization problems of a type which frequently occur in curve-fitting and modelling applications and also in the solution of sets of nonlinear equations. The first of these is a new two-part algorithm for minimizing a sum of squares objective function; it uses a new descent method in combination with a modified Gauss-Newton search to give an algorithm which has proved extremely reliable even when applied to difficult problems. The second technique is a hybrid algorithm for minimizing a sum of moduli objective function; it makes novel use of the methods of parametric linear programming. Ten case studies of the application of optimization techniques are described, ranging from problems involving a single variable up to a problem with several hundred variables. The areas from which the applications are drawn include biochemistry, engineering, statistics and theoretical physics; the problems themselves are mainly concerned with curve-fitting or the solution of nonlinear equations.

Lofncal unconstrained numerical optimisation techniques are applied to a vast number of problems from various branches of engineering. The available methods may be divided into two classes: those that assume no special form for the objective function and those that require an objective function in the form of a sum of squares. While there exist a number of methods of the first class that use second derivatives, until now there has been a lack of second derivative methods of the second class. Although methods of the first class can be applied to an objective function in the form of a sum of squares, it is generally recognized that if the sum is zero at the solution methods of the second class exhibit better terminal convergence. This is demonstrated here using several examples, including a transistor model problem where the objective function is defined to be the sum of the squares of the residuals of a set of highly non-linear simultaneous equations. Problems of this type are prevalent in methods for the design of electrical circuits. The main objective of this research was to determine whether the use of second derivatives could be of benefit in the solution of these problems. A number of second derivative sum of squares optimisation algorithms were devised, investigated and assessed using the transistor model problem as a standard test case. The most successful methods were then incorporated into a program for the synthesis of three-terminal lumped linear networks comprising resistors and capacitors. The development of the algorithms and their performance on these and various other trials is described; based on the results obtained some conclusions are drawn regarding the areas where the new algorithms are likely to be of benefit.

The thesis in concerned with two problems from the prediction theory of continuous parameter stationary stochastic processes, and the related questions concerning the measurent on the real line which is associated with the process via Bochner's theorem. In Section 1 of Chapter 1, we describe briefly the background required from the theory of Hardy spaces in the upper half-plane, and some facts about entire functions of exponential type are given. Then, in Section 2, we discuss stationary processes and describe the main problems, motivating their study by a brief description of the classical prediction problems of Wiener and Kolmogorov and the work of Nelson, Sarason and Szegb. Chapter 2 is devoted to the proof of two representation theorems for weight functions satisfying the strong mixing condition pl -- 0 and the positive angle criterion pl < 1. The proof uses a result on analytic continuation and a characterisation of the algebra H4 + BUC. These results generalise the known results for discrete parameter processes. Chapter 3 consists of a discussion of the spaces BMO and VMO and their relationship to the strong mixing condition; and the Heleon-Szegb condition of Chapter 2. We prove a result characterising those positive functions f on R for which log f E.VMO, and derive a connection between BMO, the condition pl t 1,and the boundedness of the con3ugat+enoperator on a subset of L2(p), depending on A. This generalises the discrete, version which is due to Nelson and Szegb. In Chapter 4, we consider the mixing conditions for a multivariate stationary process. The main result is an example of the mitian 2 x 2 matrix G, all of whose entries are real VMD functions, which is such that exp G does not satisfy the strong mixing condition pl '110. The proof depends on the construction of a VMO function which goes off to infinity at the origin, and the fact that no VMO function can have a jump discontinuity.

Algorithms have been developed which perform least squares adaptive filtering with great computational efficiency. Unfortunately, the fast recursive least squares (RLS) algorithms all exhibit numerical instability due to finite precision computational errors, resulting in their failure to produce a useful solution after a short number of iterations. In this thesis, a new solution to this instability problem is considered, making use of interval arithmetic. By modifying the algorithm so that upper and lower bounds are placed on all quantities calculated, it is possible to obtain a measure of confidence in the solution calculated by a fast RLS algorithm and if it is subject to a high degree of inaccuracy due to finite precision computational errors, then the algorithm may be rescued, using a reinitialisation procedure. Simulation results show that the stabilised algorithms offer an accuracy of solution comparable with the standard recursive least squares algorithm. Both floating and fixed point implementations of the interval arithmetic method are simulated and long-term stability is demonstrated in both cases. A hardware verification of the simulation results is also performed, using a digital signal processor(DSP). The results from this indicate that the stabilised fast RLS algorithms are suitable for a number of applications requiring high speed, real time adaptive filtering. A design study for a very large scale integration (VLSI) technology coprocessor, which provides hardware support for interval multiplication, is also considered. This device would enable the hardware realisation of a fast RLS algorithm to operate at far greater speed than that obtained by performing interval multiplication using a DSP. Finally, the results presented in this thesis are summarised and the achievements and limitations of the work are identified. Areas for further research are suggested.

Within the Finite-Element Plasticity Research Group at the University of Birmingham the interest in finite elements is principally confined to their use in the modelling of elastic-plastic problems (specifically metal-forming processes). The methods employed allow a process to be studied as it develops through time. A number of problems in the modelling of such processes give rise to errors, most notably due to singularities and other complex geometrical anomalies which are the result of complicated die surfaces and other boundaries. These boundary conditions can give rise to severe mesh distortions. The results of this thesis will show that improvements may be made to the modelling of such processes, by using a computer tool to allow remeshing to be carried out where such inaccuracies occur. A number of investigations have been made into such tools in other areas of study and it is hoped, eventually, that those techniques, together with ones developed here may be applied to metal forming in 3-dimensions. The aims of this thesis are to formulate the initial steps towards such an analysis. The work demonstrates how the basics of such techniques might be implemented, concentrating specifically on the transfer of data from an old mesh to a new mesh and simple error measures. Once this data transfer has been implemented it is hoped that future work will produce a fully self-adaptive process, ensuring that remeshing occurs based on algorithms continuously monitoring the potential for errors throughout the process The initial work in this thesis was confined to plane-strain applications. Its methods and implementation are discussed and some results shown. This gave some understanding of the difficulties which would be involved when transferring the techniques developed to a fully 3-dimensional solution. In the 3-dimensional application area are more quantitative work was also undertaken to assess the errors arising from element degeneracy and the ability of the mesh to model steep strain gradients. It should be noted by that full 3-dimensional remeshing, when applied to general metal forming problems, will involve far more sophisticated mesh generation software than is currently, or in the short term future, available.

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This thesis realises the need for describing computer curves and surfaces in terms of intrinsic quantities and certain properties relative to the Euclidean space in which they are embedded. Chapter 1 introduces some of the ideas and problems involved in what can be termed computational differential geometry. Chapter 2 presents some analysis of the major types of computer curves in terms of a number of shape control parameters. Chapter 3 gives a similar analysis of computer surfaces. Chapter 4 considers the' calculus of variations in connection with the minimal immersion and a particular invariantly defined functional analogous to energy. Chapter 5 applies the energy functional to a class of computer curves. Chapter 6 looks at a number of surfaces in relation to surface mappings and distortion. Some mappings are also derived. This generally involves the solution of non linear differential equations the linearisation of which will almost certainly remove the salient features of the theory. A bibliography and a number of figures are provided following chapter 6.

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