The graph-theoretic field model— II. application of multi-terminal representations to field problems

This paper is a sequel to a paper entitled “The Graph-Theoretic Field Model—I: Modelling and Formulations” (1). Herein, the Theory of Multi-Terminal Representations is applied to the Graph-Theoretic Field Model to provide mathematical models of finite elements. The element models are obtained solely from the algebraic building blocks of the Graph-Theoretic Field Model, without recourse to any functional mathematics. The theory of Multi-Terminal Representations is developed for both linear and non-linear problems. Examples of the application of the theory to one- and two-dimensional field problems are presented from heat conduction and electrostatics.     

An On-line Search for Optimum Operating Conditions of Noisy Systems

Optimum process operating conditions are often achieved by on-line computer programming. A general method has been developed which is capable of optimizing processes whose responses are corrupted by random error. The method is particularly useful for application to processes whose response functions contain sharp ridges. Application to ridge and non-ridge systems gives the same results in each case showing that the rate of convergence of the search is totally unaffected by the presence of a ridge. An accelarated gradient method is used which obtains its derivatives from a quadratic interpolation. The inability of the gradient method to move along a ridge is overcome by projecting a straight line through points on the ridge to obtain new starting points for a gradient search. Points on the ridge are located by finding the stationary point of the polynomial which is constructed by regression through points found by the gradient search. The method is applied to several test functions known to the literature—a fast convergence rate is obtained for a wide range of noise levels.     

The physics of brass wind instruments

The design of musical instruments owes more to art than to science and assessment of their quality is essentially subjective. Nevertheless, the physical principles of their operation are very interesting and may be of practical importance. This article reviews the acoustics of brass wind instruments, regarding the player and his instrument as an entity, and compares experimental and theoretical results.     

Three-axis platform simulation: Bond graph and Lagrangian approach

A bond graph model is derived for the geometric constraints of a three-axis flight table. Gimbal dynamics are easily added even in asymmetrical and unbalanced cases. A method is introduced to make the local dependent inertias computable. The bond graph compares favourably to the Lagrangian approach as to modelling effort and accessibility of intermediate variables as well as having computational advantages.     

The graph-theoretic field model—I. Modelling and formulations

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure.This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.     

Ladder realizations of multivariable positive real functions

Necessary and sufficient conditions are obtained for the realization of an m-variable positive real function (PRF) as the impedence function of a resistively-terminated ladder network of m lossless two-ports connected in cascade. Each two-port is a single-variable lossless ladder with all of its transmission zeros either at the origin or at finity. Conditions are also obtained when each of the two-ports is a Fujisawa-type lowpass ladder.