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2004 | Buch

Mathematical Modelling

Case Studies and Projects

herausgegeben von: Jim Caldwell, Douglas K. S. Ng

Verlag: Springer Netherlands

Buchreihe : Kluwer Texts in the Mathematical Sciences

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Über dieses Buch

Over the past decade there has been an increasing demand for suitable material in the area of mathematical modelling as applied to science, engineering, business and management. Recent developments in computer technology and related software have provided the necessary tools of increasing power and sophistication which have significant implications for the use and role of mathematical modelling in the above disciplines. In the past, traditional methods have relied heavily on expensive experimentation and the building of scaled models, but now a more flexible and cost effective approach is available through greater use of mathematical modelling and computer simulation. In particular, developments in computer algebra, symbolic manipulation packages and user friendly software packages for large scale problems, all have important implications in both the teaching of mathematical modelling and, more importantly, its use in the solution of real world problems. Many textbooks have been published which cover the art and techniques of modelling as well as specific mathematical modelling techniques in specialist areas within science and business. In most of these books the mathematical material tends to be rather tailor made to fit in with a one or two semester course for teaching students at the undergraduate or postgraduate level, usually the former. This textbook is quite different in that it is intended to build on and enhance students’ modelling skills using a combination of case studies and projects.

Inhaltsverzeichnis

Frontmatter

Introduction

Introduction

Ordinary Differential Equations

Frontmatter
Case Study A1. Deterministic Model in Contagious Disease
Summary
This case study extends past work by Caldwell and Ram in the study of a deterministic model in the theory of contagious disease. A more realistic model is considered by introducing a third variable, namely, the number of removals in the population. Numerical results are obtained by using the Runge-Kutta-Fehlberg method including error control. Results and graphs are produced to show the effects of variation of the infection rate and the removal rate on the number of removals from the population over long time periods. These results are validated and are shown to agree well with analytical results.
Case Study A2. Electromagnetic Forces in High Field Magnet Coils
Summary
A limiting design of large high field superconducting magnets is the problem of supporting the electromagnetic forces. It is therefore important to be able to estimate the forces appearing on the windings of magnets. A simple mathematical model is obtained which represents the stress distribution in magnet windings. In deriving the equations a number of simplifying assumptions have been made. This model is validated by checking the accuracy against homogeneous thick cylinder theory which involves the calculation of stresses by solving the Timoshenko stress equations. In this way, values of the circumferential stress have been compared for two separate coil configurations and conclusions are drawn.
Project B1. Mass Balance of a Reactor in Steady State
Summary
The design of a chemical reactor is particularly important in the field of chemical engineering. The design of the system allows the chemical reaction to take place in a safe and efficient way. A mathematical model is built which represents the concentration of the chemical along the reactor in the steady state case. Both numerical and analytical solutions will be presented and compared for accuracy purposes. This kind of study can apply to real life problems, such as waste treatment.
Project B2. The Free and Forced Vibration of an Automobile
Summary
This project considers the free and forced vibrations of an automobile supported by springs and shock absorbers. By using Newton’s second law of motion, a mathematical model is formulated taking into account the damping force and the spring force. The case where the car is subject to periodic force is analyzed in detail. In this way it is possible to examine how the amplitude magnification factor varies with the ratio of the forcing and natural frequencies for a range of damping factors.
The realistic problem of determining the stability of a proposed design which has good comfort on rough roads in then considered. Of course, a number of modelling assumptions, including the use of realistic data, are required to achieve meaningful results. The numerical results are validated using MAPLE and MATLAB.
Project B3. Cantilever Beam Subjected to an End Load
Summary
This project investigates the mathematical modelling of a beam which is either imbedded at both ends or free at one end. The model involves a fourth-order ODE together with boundary conditions which depend on the manner in which the beam is supported. Analytical solutions are obtained for a number of test cases.
The particular case of a steel cantilever beam subjected to an end load is then investigated by using both analytical and numerical techniques. The deflection of the beam obtained by an analytical approach is validated by using finite difference methods and suggestions are given on possible finite element approaches.
ODE Problems

Partial Differential Equations

Frontmatter
Case Study A3. Cylindrical and Spherical Solidification in Heat Transfer
Summary
This case study uses mathematical modelling to describe, develop and compare several effective methods for the numerical solution of one-dimensional Stefan problems. It is not intended to be an exhaustive treatment but is restricted to a range of problems and geometries including melting in the half-plane, outward cylindrical solidification and outward spherical solidification. The methods used include the enthalpy method, boundary immobilization method, perturbation method, nodal integral method and heat balance integral method. From the comparison of numerical results obtained the models can be validated and some helpful comments can be made which may prove valuable in the future use of these methods for problems of this type.
Case Study A4. Elastic Analysis of a Square Plate with Circular Holes
Summary
The problem under consideration involves the elastic analysis of a square plate subjected to a uniform pressure. In the first instance a square section with a central hole is considered using generalized plane strain element. The pressure is applied to the surface of a circular hole located at the centre of the section. This problem is then generalized to that of a square section with nine holes subjected to internal pressure. Results involving stresses and displacements are obtained for both cases using the BEASY Boundary Element software package from Computational Mechanics Ltd. A check on the accuracy is obtained by using Lamé thick cylinder theory at selected points. The good agreement obtained gives confidence in the use of the boundary element method for problems of this type.
Project B4. Motion of Fluid Layers
Summary
This project investigates the flow of two parallel layers of oil and water which are located between two plates. Starting from the rest position the top plate is moved at a constant velocity which immediately affects the motion of the neighbouring oil layer. Eventually this will affect the motion of the lower water layer. A mathematical model is formulated which involves parabolic PDEs. Of course, the equations cannot be solved separately in the two layers because of the boundary conditions at the oil-water interface. A numerical solution is obtained using an implicit finite divided difference scheme in terms of time and space. In this way the velocity of the two fluid layers is obtained at various distances from the plates at different times. Clearly the effects of motion will be more noticeable as time proceeds. The results are validated against the steady state solution for large time.
Project B5. Mass Balance of a Reactor with Time Dependency
Summary
Project B1 considered the one-dimensional mass balance of a cylindrical chemical reactor in steady state. This meant that the governing parabolic PDE reduced to an ODE. This project now extends the work in Project B1 to include time dependency which involves the full PDE. An analytical solution is no longer possible but a numerical solution is obtained using finite difference methods. As a result computer graphs are plotted which show the variation of concentration of the chemical with distance along the longitudinal axis of the reactor at different times. For model validation purposes, these results for large time are compared with the exact results obtained in Project B1 for the steady state case.
Project B6. Flow Through Porous Media
Summary
A rectangular plate with fixed boundary conditions is an ideal context for demonstrating how elliptic PDEs can be solved numerically. However, more realistic problems involve geometries with irregular shape. This project considers the flow of liquid through porous media. The geometry to be considered involves a rectangular region with an irregular edge. Values of the head and its partial derivatives are specified on the boundary of the region. A mathematical model is formulated after making some simplifying assumptions. Numerical methods are then used to solve the governing Laplace’s equation and including irregular boundary conditions. Consequently numerical results are obtained for the distribution of the head by using MATLAB.
PDE Problems

Optimization

Frontmatter
Case Study A5. Linear Programming Problem Involving Wine Production
Summary
This case study involves the formulation of a wine production problem as a linear programming problem. A vintner producing two types of wine (M and D) to sell to the local shop knows the profit figures ($/gal) for each type. The requirements of each type of wine in terms of the ingredients, namely, grapes, sugar and extract are also known. As the vintner has some constraints on these ingredients, he wishes to know how best to proceed. A mathematical solution is obtained using the simplex method and sensitivity analysis is used to study the effects of changes in the key parameters on the optimal solution. In this way the vintner obtains important information on how to use his resources to maximize profit. The solution is validated by using the linear programming computer package LINDO and the mathematical software package MAPLE.
Case Study A6. Transportation Problem Involving Breweries and Hotels
Summary
In the business world, a Manager must recognize the typical task of allocating units from sources of supply to destinations of demand to minimize cost and that various transportation methods can be applied to effect this allocation of units. How to distribute products in such a manner as to minimize the total cost of their distribution constitutes a good example of an everyday problem that transportation methods can be used to solve.
Project B7. Profit from an Engineering Plant
Summary
This project considers the manufacture of three major products (A, B and C) by an engineering plant. Figures are available on the resources required for the manufacture of each product together with the total resource availability. The problem is formulated as a linear programming (L.P.) problem which requires the maximization of the objective function (which is profit in this case) subject to three linear constraints which involve raw materials, production time and warehouse space. The L.P. problem is solved by hand using the simplex method and the results are validated using the L.P. software package LINDO. An extension is included to show the flexibility in the use of such software packages.
Project B8. Optimization of Manufacture of Personal Computers
Summary
In this project a manufacturer of personal computers is planning the introduction of two new products, a basic model and an enhanced model. Using unconstrained optimization which involves standard solution methods of multivariable calculus, an analytical model is built to determine the production levels of the two types of computer. Constraints are then introduced based on the available production capacity and a model is developed using Lagrange multiplier methods. The problem is then reformulated as a linear programming problem by introducing some simplifying assumptions and solved to obtain the optimum production levels. Sensitivity analysis is also introduced involving both the profit and optimal production levels.
Project B9. Air Freight Transportation Problem
Summary
This project involves the shipping of cargo by air. Apart from the weight constraints, the company has limited volume of aircraft storage compartments. Full details are available on an average daily basis of three types of cargo in terms of weight (tons) and volume (ft3 / ton ). The amount of each type of cargo which should be shipped by air each day is found in order to maximize revenue ($). The problem is modelled using constrained optimization techniques and solved using Lagrange multipliers. The shadow prices are calculated for each constraint and the results are interpreted. An extension is considered which involves the company in reconfiguring some of its older planes to help increase the size of the cargo areas. In this way decisions can be made on whether or not to include alterations and to what extent. The numerical results are validated by solving as a linear programming problem using the computer software package LINDO and the mathematical software package MAPLE.
Optimization Problems
Backmatter
Metadaten
Titel
Mathematical Modelling
herausgegeben von
Jim Caldwell
Douglas K. S. Ng
Copyright-Jahr
2004
Verlag
Springer Netherlands
Electronic ISBN
978-1-4020-1993-7
Print ISBN
978-1-4020-1991-3
DOI
https://doi.org/10.1007/1-4020-1993-9