Boundary Element Methods in Heat Transfer

Transcription

1 Boundary Element Methods in Heat Transfer

2 International Series on Computational Engineering Aims: Computational Engineering has grown in power and diversity in recent years, and for the engineering community the advances are matched by their wider accessibility through modern workstations. The aim of this series is to provide a clear account of computational methods in engineering analysis and design, dealing with both established methods as well as those currently in a state of rapid development. The series will cover books on the state-of-the-art development in computational engineering and as such will comprise several volumes every year covering the latest developments in the application of the methods to different engineering topics. Each volume will consist of authored work or edited volumes ofseveral chapters written by the leading researchers in the field. The aim will be to provide the fundamental concepts of advances in computational methods as well as outlining the algorithms required to implement the techniques in practical engineering analysis. The scope of the series covers almo>t the entire spectrum of engineering analysis. As such, it will cover Stress Analysis, Inelastic Problems, Contact Problems, Fracture Mechanics, Optimization and Design Sensitivity Analysis, Plate and Shell Analysis, Composite Materials, Probabilistic Mechanics, Fluid Mechanics, Groundwater Flow, Hydraulics, Heat Transfer, Geomechanics, Soil Mechanics, Wave Propagation, Acoustics, Electromagnetics, Electrical Problems, Bioengineering, Knowledge Based Systems and Environmental Modelling. Series Editor: Dr C.A. Brebbia Wessex Institute of Technology Computational Mechanics Institute Ashurst Lodge Ashurst Southampton S04 2AA Associate Editor: Dr M.H. Aliabadi Wessex Institute of Technology Computational Mechanics Institute Ashurst Lodge Ashurst Southampton S04 2AA Editorial Board: Professor H. Antes Institut fur Angewandte Mechanik Technische Universitii.t Braunschweig Postfach 3329 D-3300 Braunschweig Germany Professor H.D. Bui Laboratoire de Mecanique des Solides Ecole Polytechnique Palaiseau Cedex France Professor A.H-D. Cheng University of Delaware College of Engineering Department of Civil Engineering 137 Dupont Hall Newark, Delaware Professor D. Beskos Civil Engineering Department School of Engineering University of Patras GR-261l0 Patras Greece Professor D. Cartwright Department of Mechanical Engineering Bucknell University Lewisburg University Pensylvania Professor J.J. Connor Department of Civil Engineering Massachusetts Institute of Technology Cambridge MA 02139

3 Professor J. Dominguez Escuela Superior de Ingenieros Industriales Av. Reina Mercedes Sevilla Spain Professor G.S. Gipson School of Civil Engineering Engineering South 207 Oklahoma State University Stillwater, OK Professor S. Grilli The University of Rhode Island Department of Ocean Engineering Kingston, RI Professor D.B. Ingham Department of Applied Mathematical Studies School of Mathematics The University of Leeds Leeds LS2 9JT Professor P. Molinaro Ente Nazionale per l'energia Elettrica Direzione Degli Studi e Ricerche Centro di Ricerca Idraulica e Strutturale Via Ornato 90/ Milano Italy Professor Dr. K. Onishi Department of Mathematics II Science University of Tokyo Wakamiya-cho 26 Shinjuku-ku Tokyo 162 Japan Professor H. Pina Instituto Superior Tecnico Av. Rovisco Pais 1096 Lisboa Codex Portugal Dr. A.P.S. Selvadurai Department of Civil Engineering Room 277, C.J. Mackenzie Building Carleton University Ottawa Canada K1S 5B6 Professor A. Giorgini Purdue University School of Civil Engineering West Lafayette, IN Professor W.G. Gray Department of Civil Engineering and Geological Sciences University of Notre Dame Notre Dame, IN Dr. S. Hernandez Department of Mechanical Engineering University of Zaragoza Maria de Luna Zaragoza Spain Professor G.D. Manolis Aristotle University of Thessaloniki School of Engineering Department of Civil Engineering GR-54006, Thessaloniki Greece Dr. A.J. Nowak Silesian Technical University Institute of Thermal Technology Gliwice Konarskiego 22 Poland Professor P. Parreira Departamento de Engenharia Civil Avenida Rovisco Pais 1096 Lisboa Codex Portugal Professor D.P. Rooke DRA (Aerospace Division) Materials and Structures Department R50 Building RAE Farnborough Hampshire GU14 GTD Professor R.P. Shaw S.U.N.Y. at Buffalo Department of Civil Engineering School of Engineering and Applied Sciences 212 Ketter Hall Buffalo, New York 14260

4 Professor P. Skerget University of Maiibor Faculty of Technical Sciences YU Maribor Smetanova 17 P.O. Box 224 Yugoslavia Professor M.D. Trifunac Department of Civil Engineering, KAP 216D University of Southern California Los Angeles, CA Dr P.P. Strona Centro Ricerche Fiat S.C.p.A. Strada Torino, Orbassano (TO) Italy Professor N.G. Zamani University of Windsor Department of Mathematics and Statistics 401 Sunset Windsor Ontario Canada N9B 3P4 Acknowledgement is made to Professor N. Tosaka for the use of figure 8.18 (isotherms) on page 262, which appears on the front cover of this book.

5 Boundary Element Methods in Heat Transfer Editors: L.C. Wrobel and C.A. Brebbia Computational Mechanics Publications Southampton Boston Co-published with Elsevier Applied Science London New York CMP

6 L.C. Wrobel Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton S04 2AA C.A. Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton S04 2AA Co-published by Computational Mechanics Publications Ashurst Lodge, Ashurst, Southampton, Computational Mechanics Publications Ltd Sole Distributor in the and Canada: Computational Mechanics Inc. 25 Bridge Street, Billerica, MA 01821, and Elsevier Science Publishers Ltd Crown House, Linton Road, Barking, Essex IGll 8JU, Elsevier's Sole Distributor in the and Canada: Elsevier Science Publishing Company Inc. 655 Avenue of the Americas, New York, NY 10010, British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN Elsevier Applied Science, London, New York ISBN Computational Mechanics Publications, Southampton ISBN Computational Mechanics Publications, Boston, Library of Congress Catalog Card Number No responsibility is assumed by the Publishers for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material Mechanics Publications 1992 Printed and bound by Bookcraft Ltd, Bath All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

7 CONTENTS Preface xi Chapter 1 - Solving Heat Transfer Problems by the Dual Reciprocity BEM C.A. Brebbia, A.J. Nowak 1.1 Introduction Steady-State Problems with Heat Sources Transient Heat Conduction Numerical Examples and Conclusions 18 Acknowledgement 25 References 26 Chapter 2 - Transient Problems using Time-Dependent Fundamental Solutions R. Pasquetti, A. Caruso, L.C. Wrobel 2.1 Introduction Boundary Integral Equation Space and Time Discretization Evaluation of the Coefficients of Matrices HI, H2, Gl and G2 " Boundary Conditions Initial Conditions Treatment of Heat Sources Applications 50 References 60 Chapter 3 - Solving Linear Heat Conduction Problems by the Multiple Reciprocity Method A.J. Nowak 3.1 Introduction Fundamentals of the Multiple Reciprocity Method Heat Conduction with Heat Sources Linear Transient Problems Numerical Examples 77 Acknowledgements 82 References 82

8 Chapter 4 - Solving Nonlinear Heat Transfer Problems Using the Boundary Element Method R. Bialecki 4.1 Introduction Applying BEM to Nonlinear Problems. General Remarks Nonlinear Boundary Conditions Nonlinear Material (Nonlinear Differential Operator) Nonlinear Source Term Moving Boundaries Conclusions 112 Acknowledgements 113 References 114 Chapter 5 - Coupled Conduction-Convection Problems L.C. Wrobel, D.B. DeFigueiredo 5.1 Introduction BEM Formulation for Steady-State Problems BEM Formulation for Transient Problems BEM Formulation for Variable Velocity Fields Conclusions 142 Acknowledgements 142 References 142 Chapter 6 - Solving Coupled Problems Involving Conduction, Convection and Thermal Radiation A.J. Nowak 6.1 Introduction Coupled Thermal Problems with Non-Participating Medium Coupled Thermal Problems with Participating Medium Concluding Remarks 168 Acknowledgement 169 References 169 Chapter 7 - Advanced Thermoelastic Analysis V. Sladek, J. Sladek 7.1 Introduction Governing Equations Fundamental Solutions Integral Representations of the Temperature and the Displacement Fields. Boundary Integral Equations Integral Representations of the Temperature Gradients and Stresses Stress Tensor and Temperature Gradient on Boundary Numerical Solution 196

9 7.8 Stationary Problems in Media with Temperature Dependent Young's Modulus and Coefficient of Thermal Expansion 215 Appendix A 223 Appendix B 227 Appendix C 228 Appendix D 229 References 232 Chapter 8 - Integral Equation Analyses of Natural Convection Problems in Fluid Flow N. Tosaka, N. Fukushima 8.1 Introduction '" Natural Convection Problems Steady Analysis Unsteady Analysis Numerical Examples Conclusions 256 Acknowledgements 266 References 266 Chapter 9 - Improperly Posed Problems in Heat Transfer D.B. Ingham 9.1 Introduction Formulation Non-Linear Formulation Existence of Solution of Problem I Mathematical Models for the Solution of Problem I Mathematical Model for the Solution of Problem II Solutions of Some Test Examples for Problem I Solution of Some Test Examples for Problem II Conclusions 290 Acknowledgements 293 References 293

10 PREFACE Heat transfer problems in industry are usually of a very complex nature, simultaneously involving different transfer modes such as conduction, convection, radiation and others. Because of that, very few problems can be solved analytically and one generally has to resort to numerical analysis. The boundary element method is a numerical technique which has been receiving growing attention for solving heat transfer problems because of its unique ability to confine the discretization process to the boundaries of the problem region. This allows major reductions in the data preparation and computer effort necessary to solve complex industrial problems. The purpose of this book is to present efficient algorithms used in conjunction with the boundary element method for the solution of steady and transient, linear and nonlinear heat transfer problems. It also aims to reflect research being carried out by several active groups around the world, and its chapters have accordingly been written by scientists working in renowned centres of excellence. The first three chapters all deal with transient heat conduction using alternative boundary element formulations which require boundary discretization only. Chapter 1 presents the dual reciprocity technique which is attracting considerable interest because of its ability to transform domain integrals, resulting from effects such as internal heat generation, into equivalent boundary integrals. The technique is general and is applied in this chapter to steady and transient, linear and nonlinear problems. A more traditional approach using time-dependent fundamental solutions is described in chapter 2. Also included is a discussion on the treatment of some types of initial conditions and internal loadings by equivalent boundary integrals, and an efficient convolution-type time-marching scheme. The multiple reciprocity method is described in chapter 3. This method may be seen as an extension of Galerkin-vector techniques for non-harmonic loads, and can also be applied to transient problems. Chapter 4 deals with nonlinear heat transfer problems. The types of nonlinearity discussed include those of material, boundary conditions, heat sources and moving boundaries. Several practical examples of application are presented, and areas pointed out where further research is still necessary. Boundary element solutions to the convection-diffusion equation are the subject of chapter 5. The fundamental solution to the steady-state equation with constant coefficients is employed, and features such as transient effects and variable parameters are accounted for by using dual reciprocity approximations. Coupled problems are also discussed in chapter 6 which deals with heat transfer involving conduction, convection and radiation in enclosures. The formulation developed can be implemented into standard boundary element codes, and is equivalent to introducing a new fundamental solution. Special consideration is given to the resulting set of nonlinear equations which is solved by an efficient pre-elimination technique employing the Gauss-Jordan algorithm. Chapter 7, on thermoelasticity, starts with a brief classification of thermoelastic problems; next, it defines the fundamental solution for the Laplace transforms in general coupled thermoelasticity, and the time-dependent fundamental solutions when

11 these are available. A pure boundary formulation is then given for both the Laplace transform and the time-dependent fields, and boundary integral equations written in an advanced regularized form without any singular integral. Finally, a BEM formulation for solution of stationary problems in media with temperature-dependent Young's modulus and coefficient of thermal expansion is presented. Natural convection in fluid flow is the subject of chapter 8. This chapter is a collection of recent results obtained by the authors using an integral equation method based on boundary-domain discretization for solving two-dimensional thermal convection problems. The formulation uses the primitive variables, i.e. velocity and pressure, and constructs fundamental solution tensors for the differential operators corresponding to a linearized set of equations. Approximate solution procedures of the nonlinear system of integral equations are derived based on Newton-Raphson techniques. The last chapter deals with inverse heat conduction problems. Three different mathematical models, namely direct, least squares and minimum energy methods, are presented for two Laplace-type problems. It is found that the minimum energy method always gives a good, stable approximation to the solution, whereas the direct and least squares methods do not. We are indebted to all the authors for their contribution, patience and continuous support during the production stages of this book. Special thanks are due to Ms. Christine Seward for the excellent work in the preparation of the final manuscript. Luiz C. Wrobel Carlos A. Brebbia February 1992