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LICENSE, DISCLAIMER OF LIABILITY, AND LIMITED WARRANTY By purchasing or using this book (the “Work”), you agree that this license grants permission to use the contents contained herein, but does not give you the right of ownership to any of the textual content in the book or ownership to any of the information or products contained in it. This license does not permit uploading of the Work onto the Internet or on a network (of any kind) without the written consent of the Publisher. Duplication or dissemination of any text, code, simulations, images, etc. contained herein is limited to and subject to licensing terms for the respective products, and permission must be obtained from the Publisher or the owner of the content, etc., in order to reproduce or network any portion of the textual material (in any media) that is contained in the Work. MERCURY LEARNING AND INFORMATION (“MLI” or “the Publisher”) and anyone involved in the creation, writing, or production of the companion disc, accompanying algorithms, code, or computer programs (“the software”), and any accompanying Web site or software of the Work, cannot and do not warrant the performance or results that might be obtained by using the contents of the Work. The author, developers, and the Publisher have used their best efforts to insure the accuracy and functionality of the textual material and/or programs contained in this package; we, however, make no warranty of any kind, express or implied, regarding the performance of these contents or programs. The Work is sold “as is” without warranty (except for defective materials used in manufacturing the book or due to faulty workmanship). The author, developers, and the publisher of any accompanying content, and anyone involved in the composition, production, and manufacturing of this work will not be liable for damages of any kind arising out of the use of (or the inability to use) the algorithms, source code, computer programs, or textual material contained in this publication. This includes, but is not limited to, loss of revenue or profit, or other incidental, physical, or consequential damages arising out of the use of this Work. The sole remedy in the event of a claim of any kind is expressly limited to replacement of the book, and only at the discretion of the Publisher. The use of “implied warranty” and certain “exclusions” vary from state to state, and might not apply to the purchaser of this product.

COMSOL® FOR Engineers Mehrzad Tabatabaian MERCURY LEARNING AND INFORMATION Dulles, Virginia Boston, Massachusetts New Delhi

Copyright ©2014 by Mercury Learning and Information. All rights reserved. This publication, portions of it, or any accompanying software may not be reproduced in any way, stored in a retrieval system of any type, or transmitted by any means, media, electronic display or mechanical display, including, but not limited to, photocopy, recording, Internet postings, or scanning, without prior permission in writing from the publisher. Publisher: David Pallai Mercury Learning and Information 22841 Quicksilver Drive Dulles, VA 20166 info@merclearning.com www.merclearning.com 1-800-758-3756 This book is printed on acid-free paper. M. Tabatabaian. COMSOL® for Engineers. ISBN: 978-1-938549-53-3 The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others. Library of Congress Control Number: 2013958000 141516321 Our titles are available for adoption, license, or bulk purchase by institutions, corporations, etc. For additional information, please contact the Customer Service Dept. at 1-800-758-3756 (toll free). The sole obligation of Mercury Learning and information to the purchaser is to replace the disc, based on defective materials or faulty workmanship, but not based on the operation or functionality of the product.

To the memories of my father, whose support and guidance are still felt to this day. To my family, for their support and encouragement.

CONTENTS Preface Chapter 1: Introduction Chapter 2: Finite Element Method—A Summary Overview FEM Formulation Matrix Approach Example 2.1: Analysis of a 2D Truss General Procedure for Global Matrix Assembly Example 2.2: Global Matrix for Triangular Elements Weighted Residual Approach Galerkin Method Shape Functions Convergence and Stability Example 2.3: Heat Transfer in a Slender Steel Bar Exercise Problems References xi 1 5 5 8 9 9 13 14 15 15 16 17 18 21 21

viii • Contents Chapter 3: COMSOL—A Modeling Tool for Engineers Overview COMSOL Interface COMSOL Modules COMSOL Model Library and Tutorials General Guidelines for Building a Model Chapter 4: COMSOL Models for Physical Systems Overview Section 4.1: Static and Dynamic Analysis of Structures Example 4.1: Stress Analysis for a Thin Plate Under Stationary Loads Example 4.2: Dynamic Analysis for a Thin Plate: Eigenvalues and Modal Shapes Example 4.3: Parametric Study for a Bracket Assembly: 3D Stress Analysis Example 4.4: Buckling of a Column with Triangular Cross-section: Linearized Buckling Analysis Example 4.5: Static and Dynamic Analysis for a 2D Bridge-support Truss Example 4.6: Static and Dynamic Analysis for a 3D Truss Tower Section 4.2: Dynamic Analysis and Models of Internal Flows: Laminar and Turbulent 23 23 24 32 33 34 37 37 38 38 49 53 67 76 88 95 Example 4.7: Axisymmetric Flow in a Nozzle: 96 Simplified Water-jet Example 4.8: Swirl Flow Around a Rotating Disk: Laminar Flow 105 Example 4.9: Swirl Flow Around a Rotating Disk: Turbulent Flow 114 Example 4.10: Flow in a U-shape Pipe with Square Cross-sectional Area: Laminar Flow Example 4.11: Double-driven Cavity Flow: Moving Boundary Conditions 118 129

Contents • ix Example 4.12: Water Hammer Model: Transient Flow Analysis Example 4.13: Static Fluid Mixer Model Section 4.3: Modeling of Steady and Unsteady Heat Transfer in Media Example 4.14: Heat Transfer in a Multilayer Sphere Example 4.15: Heat Transfer in a Hexagonal Fin Example 4.16: Transient Heat Transfer Through a Nonprismatic Fin with Convective Cooling Example 4.17: Heat Conduction Through a Multilayer Wall with Contact Resistance Section 4.4: Modeling of Electrical Circuits Example 4.18: Modeling an RC Electrical Circuit Example 4.19: Modeling an RLC Electrical Circuit Section 4.5: Modeling Complex and Multiphysics Problems Example 4.20: Stress Analysis for an Orthotropic Thin Plate Example 4.21: Thermal Stress Analysis and Transient Response of a Bracket Example 4.22: Static Fluid Mixer with Flexible Baffles Example 4.23: Double Pendulum: Multibody Dynamics Example 4.24: Multiphysics Model for Thermoelectric Modules Example 4.25: Acoustic Pressure Wave Propagation in an Automotive Muffler Exercise Problems References Suggested Further Readings Trademark References Index 142 150 159 159 165 173 180 185 185 188 193 194 197 205 214 219 228 238 243 244 244 245

PREFACE This book is written for engineers, engineering students, and other practitioners in engineering fields. The main objective of the material is to introduce and help readers to use COMSOL as an engineering tool for modeling by solving examples that either directly could be used or could become a guide for modeling similar or more complicated problems. It would be exhaustive to include all features available in COMSOL in a single book; our objective is to provide a collection of examples and model- ing guidelines through which readers could build their own models. Readers are assumed to know or at least be familiar with the princi- ples of numerical modeling and finite element method (FEM). We took a flexible-level approach for presenting the materials along with using practi- cal examples. The mathematical fundamentals, engineering principles, and design criteria are presented as integral parts of examples. At the end of each chapter we have added references that contain more in-depth physics, technical information, and data; these are referred to throughout the book and used in the examples. This approach allows readers to learn the materi- als at their desired level of complexity. COMSOL for Engineers could be used as a textbook complementing another text that provides background training in engineering computa- tions and methods, such as FEM. Examples provided in this book should be considered as “lessons” for which background physics could be explained in more detail. Exercise problems, or their variations, could be used for homework assignments. We start each chapter with an overview, background physics, and mathematical models to set the foundation. We then present the relevant modeling techniques and materials through several examples. The examples

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