AN INTRODUCTION TO COMPUTER SIMULATION IN APPLIED SCIENCE AN INTRODUCTION TO COMPUTER SIMULATION IN APPLIED SCIENCE EDITED BY FARID F. ABRAHAM IBM Scientific Center Palo Alto, California and Consulting A ssociate Professor Materials Science Department Stanford University Stanford, California AND WILLIAM A. TILLER Professor Materials Science Department Stanford University Stanford, California ~ PLENUM PRESS • NEW YORK-LONDON • 1972 Library of Congress Catalog Card Number 72-83047 ISBN-13: 978-1-4684-1976-4 e-ISBN-13: 978-1-4684-1974-0 DOl: 10.1007/978-1-4684-1974-0 © 1972 Plenum Press, New York Softcover reprint of the hardcover 1s t edition 1972 A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N. Y. 10011 United Kingdom edition published by Plenum Press, London A Division of Plenum Publishing Company, Ltd. Davis House (4th Floor), 8 Scrubs Lane, Harlesden, London NWI0 6SE, England All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher THIS BOOK IS DEDICATED BY FARID ABRAHAM TO HIS FATHER ANTONY F. ABRAHAM Preface This set of lectures is the outgrowth of a new course in the Department of Materials Science at Stanford University. It was taught collectively by the authors of the various sections and represents an attempt to increase the awareness of students in the materials area of computer simulation techniques and potentialities. The topics often ranged far afield from the materials area; however, the total package served the intended purpose of being an initiation into the world of computer simulation and, as such, made a useful first iteration to the intended purpose. The second iteration, which is in process, deals exclusively with the materials area. The course was designed to teach students a new way to wrestle with "systems" problems in the materials science work area that require the synthesis and interactions of several disciplines of knowledge. This course was a response to the realization that effective handling of real problems, which are essentially systems problems, is one of the most important at- tributes of a graduate materials scientist. About a third of the course was devoted to the student's selected problem, in the materials area, which he simulated using the digital computer. The set of lectures begins with Professor Tiller presenting the essential philosophy for dissociating real problems into a system of identifiable and interacting parts and rationalizing the vital importance of computer simula- tion in this resolution process. The importance of the computer simulation technique to the synthesis of knowledge within the student and to his acquiring the confidence to become a significant problem-solver is stressed. Finally, some examples are given to illustrate the dissociation of a problem area into a system of subroutines with the relevant parameters and variables identified for a particular level of modeling. Jacob Fromm presents the basic needs for computation of nonlinear fluid flows. Included are considerations of the governing equations and vii viii Preface their finite difference representation. The value of the linear stability analysis of the difference equations is emphasized. Finally, an outline of a working program is given along with listings and a test problem solution. A series of results of the numerical program are discussed with the object of demon- strating the versatility of the program and suggesting potential uses in related areas. Farid Abraham's presentation discusses a "simulation language" that is easy to use, is powerful in solving a large number of differential equations, and is able to solve the algebraic as well as the differential equations. This simulation language is entitled "The Systemj360 Continuous System Mod- eling Program (Sj360 CSMP)" and does not require the user to be a proficient computer programmer. Sj360-CSMP is illustrated by obtaining numerical solutions for some heat diffusion problems. George White discusses vapor deposition simulation programs devel- oped by use of Monte Carlo methods to describe the molecular processes of condensation, evaporation, and migration on lattices. The principal application is to systems that permit a comparison with Honig's theoretical ",ark, although the simulation methods are easily applied to a variety of other problems. The results of the simulations demonstrate that Honig's treatment is quite accurate and is a substantial improvement over previous treatments. This agreement also serves to build confidence in the use of Monte Carlo methods in simulating molecular dynamics for vapor deposi- tion studies. Robert Kortzeborn introduces computational theoretical chemistry via the solution of the Schrodinger equation for the hydrogen and helium atoms. The concept of integral poles and approximate techniques that arise in the two electron integrals are discussed. Molecular systems are then considered with emphasis on the theoretical model and its relationship to the real world. A brief explanation of the Hartree-Fock model is presented followed by a detailed research method for computing multicentered, two- electron integrals via transformation techniques. This method is illustrated with the appropriate mathematics and FORTRAN code. Finally, Harwood Kolsky describes the physical phenomena occurring in the atmosphere and the problems of modeling them for computer analysis. The numerical methods commonly used in general circulation models are described briefly, and the relative advantages are discussed. An analysis of the computer requirements for global weather calculations is developed, and the need is pointed out for very fast computers capable of executing the equivalent of hundreds of millions of instructions per second. Preface ix We are indebted to Jacob Fromm, Harwood Kolsky, Robert Kort- zeborn, and George White for giving us a timely review of the present state of the art of digital simulation of applied science problems and for unfailing cooperation in making this manuscript possible. To Ms. Barbara Merrill, we express a special thanks for typing and organizing the manuscript. FARID F. ABRAHAM WILLIAM A. TILLER Contents Chapter I. Rationale for Computer Simulation in Materials Science W. A. Tiller I. Introduction . . . . I I. Patterns of Science. . . . . . 2 II I. The Student and His Research 9 IV. Examples of Systems Events in the Crystallization Area 12 A. An Overview of the Scientific Subroutines . . 12 B. Technological Understanding of Ingot Defects 17 C. Solute Distribution in Pulled Crystals . . . . 20 Chapter 2. Lectures on Large-Scale Finite Difference Computation of Incompressible Fluid Flows . . . . . . . . . .. 23 Jacob E. Fromm L The Differential Equations . . . . . . . . . 23 A. Introduction . . .. ....... 23 B. Numerical Solution of Laplace's Equation 24 C. The Inclusion of a Source Term and the Potential Solution 26 D. The Time-Dependent'Vorticity Field Due to Diffusion and Convection. . . . . . . . . . . . 28 E. The Dynamic Equations and Scaling . 30 F. Suggested Reading . . . . . . . . . 32 II. Stability Analysis of the Difference Equations 32 A. Introduction . . . . . . . . . . . . 32 B. Stability Analysis of the Heat Conduction Equation 32 C. Stability Analysis of Laplace's Equation . . . . . . 34 D. Stability Analysis of the Nonlinear Convection Equation 36 E. Suggested Reading .. ............. 39 xi xii Contents III. Applications of the Numerical Program for Incompressible Flow. . . . . . . . . . . . . . . . . 40 A. Introduction . . . . . . . . . . . 40 B. The Differential Equations of Fluid Flow 40 C. The Difference Equations . . 42 D. Karman Vortex Street Flows 46 E. The Benard Problem . . 49 F. References . . . . . . . 51 IV. Description of the Numerical Program for Incompressible Flow. . . . . . . . . . . . . . . . 52 A. Discussion of the Block Diagram . 52 B. Discussion of the Program Listings 54 C. Suggested Reading . . . . . . . 58 Appendix: Computer Listings of the Hydrodynamic Programs 59 Chapter 3. Computer Simulation of Diffusion Problems Using the Continuous System Modeling Program Language 71 Farid F. Abraham I. Introduction . 71 II. System/360 Continuous System Modeling Program (S/360 CSMP) . 74 A. Types of Statements 74 B. Elements of a Statement 75 C. Important Features of S/360 CSMP 76 D. The S/360 CSMP Library of Functions. 76 E. The S/360 CSMP Library of Data and Control Statements 78 F. Integration Methods 79 G. The MACRO Function 82 H. The Structure of the Model 82 I. Advantages of S/360 CSNI;P 83 J. Sample Problem 84 III. Heat Transfer in an Insulated Bar 90 A. Finite Differencing the Heat Equation 90 B. Finite Difference Approach in the Modeling 92 C. Fourier Solution 93 D. The S/360-CSMP Solution. 96 IV. The Freezing of a Liquid. 97 A. Finite Differencing the Governing Equations 98 B. The S/360-CSMP Solution. 103 Contents xiii Chapter 4. Computer Simulation of Vapor Deposition on Two- Dimensional Lattices . . . . . . . 107 George M. White I. Basic Concepts of Physical Processes 107 A. Introduction 107 B. The Honig Model 108 C. The Simulated Processes. 109 D. The Rate Equations 109 II. The Computer Simulation Model 112 A. Boundary Conditions 112 B. Evaporation, Migration, and Nearest Neighbor Effects 113 C. Initial Conditions . 115 III. Random Numbers and Simulation Strategy 116 A. Monte Carlo Methods 116 B. Generation of Random Numbers. 116 C. Gaussian Distribution Generated by Random Numbers II7 D. Use of Random Numbers to Select Dynamic Processes- Simulation Strategy . 120 IV. Real and Simulated Time. 121 V. The VDS Programs 122 VI. The Computer Simulation Resul,ts 124 References . 128 Appendix: Fortran Code 131 Chapter 5. Introduction to Computational Theoretical Chemistry. 139 Robert N. Kortzeborn I. Basic Concepts of Computational Theoretical Chemistry 139 II. The Nature of the Problem . 141 A. The Hydrogen Atom (Ground Electronic State) 143 B. The Helium Atom (Ground Electronic State) 145 III. Real Molecular Systems 149 A. Introduction 149 B. The Method of Hartree and Hartree-Fock 151 C. Multicenter Integrals 154 IV. The Calculation of Quantum-Mechanical Two-Electron Multi- center Integrals via Transformation Theory 156 A. Introduction 156 B. General Theory . 156