COMSOL Multiphysics ® V ERSION 3.5a MODELING GUIDE How to contact COMSOL: Benelux COMSOL BV Röntgenlaan 19 2719 DX Zoetermeer The Netherlands Phone: +31 (0) 79 363 4230 Fax: +31 (0) 79 361 4212 info@comsol.nl www.comsol.nl Denmark COMSOL A/S Diplomvej 376 2800 Kgs. Lyngby Phone: +45 88 70 82 00 Fax: +45 88 70 80 90 info@comsol.dk www.comsol.dk Finland COMSOL OY Arabianranta 6 FIN-00560 Helsinki Phone: +358 9 2510 400 Fax: +358 9 2510 4010 info@comsol.fi www.comsol.fi France COMSOL France WTC, 5 pl. Robert Schuman F-38000 Grenoble Phone: +33 (0)4 76 46 49 01 Fax: +33 (0)4 76 46 07 42 info@comsol.fr www.comsol.fr Germany COMSOL Multiphysics GmbH Berliner Str. 4 D-37073 Göttingen Phone: +49-551-99721-0 Fax: +49-551-99721-29 info@comsol.de www.comsol.de Italy COMSOL S.r.l. Via Vittorio Emanuele II, 22 25122 Brescia Phone: +39-030-3793800 Fax: +39-030-3793899 info.it@comsol.com www.it.comsol.com Norway COMSOL AS Søndre gate 7 NO-7485 Trondheim Phone: +47 73 84 24 00 Fax: +47 73 84 24 01 info@comsol.no www.comsol.no Sweden COMSOL AB Tegnérgatan 23 SE-111 40 Stockholm Phone: +46 8 412 95 00 Fax: +46 8 412 95 10 info@comsol.se www.comsol.se Switzerland FEMLAB GmbH Technoparkstrasse 1 CH-8005 Zürich Phone: +41 (0)44 445 2140 Fax: +41 (0)44 445 2141 info@femlab.ch www.femlab.ch United Kingdom COMSOL Ltd. UH Innovation Centre College Lane Hatfield Hertfordshire AL10 9AB Phone:+44-(0)-1707 636020 Fax: +44-(0)-1707 284746 info.uk@comsol.com www.uk.comsol.com United States COMSOL, Inc. 1 New England Executive Park Suite 350 Burlington, MA 01803 Phone: +1-781-273-3322 Fax: +1-781-273-6603 COMSOL, Inc. 10850 Wilshire Boulevard Suite 800 Los Angeles, CA 90024 Phone: +1-310-441-4800 Fax: +1-310-441-0868 COMSOL, Inc. 744 Cowper Street Palo Alto, CA 94301 Phone: +1-650-324-9935 Fax: +1-650-324-9936 info@comsol.com www.comsol.com For a complete list of international representatives, visit www.comsol.com/contact Company home page www.comsol.com COMSOL user forums www.comsol.com/support/forums COMSOL Multiphysics Modeling Guide © COPYRIGHT 1998–2008 by COMSOL AB. All rights reserved Patent pending The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from COMSOL AB. COMSOL, COMSOL Multiphysics, COMSOL Reaction Engineering Lab, and FEMLAB are registered trademarks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their respective holders. Version: November 2008 COMSOL 3.5a Part number: CM020003 CONTENTS | i CONTENTS Chapter 1: Introduction Overview of the COMSOL Multiphysics Application Modes 2 Application Modes in COMSOL Multiphysics . . . . . . . . . . . . 2 Selecting an Application Mode . . . . . . . . . . . . . . . . . . 5 Modeling Guidelines 7 Using Symmetries . . . . . . . . . . . . . . . . . . . . . . 7 Effective Memory Management . . . . . . . . . . . . . . . . . 8 Selecting an Element Type . . . . . . . . . . . . . . . . . . . 9 Analyzing Model Convergence and Accuracy . . . . . . . . . . . . 9 Achieving Convergence When Solving Nonlinear Equations. . . . . . . 10 Avoiding Strong Transients . . . . . . . . . . . . . . . . . . . 11 Typographical Conventions . . . . . . . . . . . . . . . . . . . 11 Chapter 2: Using the Physics Modes The Physics Modes 14 Defining the Physics for a Model . . . . . . . . . . . . . . . . . 14 The Physics Modes . . . . . . . . . . . . . . . . . . . . . . 15 Physics Mode Documentation . . . . . . . . . . . . . . . . . . 16 Chapter 3: Acoustics Fundamentals of Acoustics 20 What is Acoustics? . . . . . . . . . . . . . . . . . . . . . . 20 Five Standard Acoustics Problems . . . . . . . . . . . . . . . . 20 Mathematical Models for Acoustic Analysis . . . . . . . . . . . . . 21 The Acoustics Application Mode 23 Variables and Space Dimensions . . . . . . . . . . . . . . . . . 23 ii | CONTENTS PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . 23 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . . 24 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 25 Scalar Variables . . . . . . . . . . . . . . . . . . . . . . . 28 Application Mode Variables . . . . . . . . . . . . . . . . . . . 29 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2D Axisymmetric. . . . . . . . . . . . . . . . . . . . . . . 33 Example—Reactive Muffler 35 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 35 Model Definition . . . . . . . . . . . . . . . . . . . . . . . 35 Results and Discussion. . . . . . . . . . . . . . . . . . . . . 37 Reference . . . . . . . . . . . . . . . . . . . . . . . . . 38 Modeling Using the Graphical User Interface . . . . . . . . . . . . 38 Chapter 4: Diffusion The Diffusion Application Mode 46 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . 46 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . . 46 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 47 Application Mode Variables . . . . . . . . . . . . . . . . . . . 49 The Convection and Diffusion Application Mode 51 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . 51 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . . 52 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 54 Application Mode Variables . . . . . . . . . . . . . . . . . . . 56 References . . . . . . . . . . . . . . . . . . . . . . . . . 57 Example—Effective Diffusivity in Porous Materials 58 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 58 Model Definition . . . . . . . . . . . . . . . . . . . . . . . 59 Results and Discussion. . . . . . . . . . . . . . . . . . . . . 60 Modeling in COMSOL Multiphysics . . . . . . . . . . . . . . . . 63 CONTENTS | iii Modeling Using the Graphical User Interface . . . . . . . . . . . . 64 Chapter 5: Electromagnetics The Electromagnetics Application Modes 78 Fundamentals of Electromagnetics 79 Electromagnetic Force for Particle Tracing . . . . . . . . . . . . . 83 Electromagnetic Forces . . . . . . . . . . . . . . . . . . . . 84 The Conductive Media DC Application Mode 85 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . 85 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 86 Line Sources . . . . . . . . . . . . . . . . . . . . . . . . 89 Point Sources . . . . . . . . . . . . . . . . . . . . . . . . 89 Application Mode Variables . . . . . . . . . . . . . . . . . . . 89 The Electrostatics Application Mode 91 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . 91 Application Scalar Variables . . . . . . . . . . . . . . . . . . . 92 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 92 Line Sources . . . . . . . . . . . . . . . . . . . . . . . . 94 Point Sources . . . . . . . . . . . . . . . . . . . . . . . . 94 Application Mode Variables . . . . . . . . . . . . . . . . . . . 94 Magnetostatics Application Mode 97 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . 97 Application Mode Property . . . . . . . . . . . . . . . . . . . 98 Application Scalar Variable . . . . . . . . . . . . . . . . . . . 98 Boundary and Interface Conditions . . . . . . . . . . . . . . . . 98 Point Sources . . . . . . . . . . . . . . . . . . . . . . . 100 Application Mode Variables . . . . . . . . . . . . . . . . . . 100 AC Power Electromagnetics 107 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 107 Application Scalar Variables . . . . . . . . . . . . . . . . . . 108 iv | CONTENTS Boundary and Interface Conditions . . . . . . . . . . . . . . . 108 Point Sources . . . . . . . . . . . . . . . . . . . . . . . 110 Application Mode Variables . . . . . . . . . . . . . . . . . . 110 Example—Electric Sensor 121 Model Definition . . . . . . . . . . . . . . . . . . . . . . 121 Results and Discussion. . . . . . . . . . . . . . . . . . . . 121 Modeling Using the Graphical User Interface . . . . . . . . . . . 122 Example—A Permanent Magnet 126 Modeling Using the Graphical User Interface . . . . . . . . . . . 126 Alternative Modeling Approach . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . 131 Chapter 6: Fluid Mechanics The Navier-Stokes Application Mode 134 Variables and Space Dimension . . . . . . . . . . . . . . . . 134 PDE Formulation and Equations . . . . . . . . . . . . . . . . 134 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . 135 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 136 Point Settings . . . . . . . . . . . . . . . . . . . . . . . 144 Numerical Stability—Stabilization Techniques . . . . . . . . . . . 144 Corner Smoothing . . . . . . . . . . . . . . . . . . . . . 147 Application Mode Variables . . . . . . . . . . . . . . . . . . 148 Solver Settings . . . . . . . . . . . . . . . . . . . . . . . 149 Khan and Richardson Force for Particle Tracing . . . . . . . . . . 152 References . . . . . . . . . . . . . . . . . . . . . . . . 153 Example—Steady Incompressible Flow 154 Introduction . . . . . . . . . . . . . . . . . . . . . . . 154 Model Definition . . . . . . . . . . . . . . . . . . . . . . 154 Results and Discussion. . . . . . . . . . . . . . . . . . . . 155 References . . . . . . . . . . . . . . . . . . . . . . . . 158 Modeling Using the Graphical User Interface . . . . . . . . . . . 159 CONTENTS | v Chapter 7: Heat Transfer Heat Transfer Fundamentals 168 What Is Heat Transfer? . . . . . . . . . . . . . . . . . . . 168 The Heat Equation . . . . . . . . . . . . . . . . . . . . . 169 The Conduction Application Mode 171 Variables and Space Dimensions . . . . . . . . . . . . . . . . 171 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 171 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . 172 Boundary Condition Types . . . . . . . . . . . . . . . . . . 173 Boundary Settings . . . . . . . . . . . . . . . . . . . . . 176 Application Mode Variables . . . . . . . . . . . . . . . . . . 177 The Convection and Conduction Application Mode 179 Variables and Space Dimensions . . . . . . . . . . . . . . . . 179 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 179 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . 180 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 184 Application Mode Variables . . . . . . . . . . . . . . . . . . 185 References . . . . . . . . . . . . . . . . . . . . . . . . 186 Examples of Heat Transfer Models 188 1D Steady-State Heat Transfer with Radiation . . . . . . . . . . . 188 Model Definition . . . . . . . . . . . . . . . . . . . . . . 188 Results. . . . . . . . . . . . . . . . . . . . . . . . . . 189 Modeling Using the Graphical User Interface . . . . . . . . . . . 189 2D Steady-State Heat Transfer with Convection . . . . . . . . . . 192 Model Definition . . . . . . . . . . . . . . . . . . . . . . 192 Results. . . . . . . . . . . . . . . . . . . . . . . . . . 193 Modeling Using the Graphical User Interface . . . . . . . . . . . 194 2D Axisymmetric Transient Heat Transfer . . . . . . . . . . . . 196 Model Definition . . . . . . . . . . . . . . . . . . . . . . 196 Results. . . . . . . . . . . . . . . . . . . . . . . . . . 197 Modeling Using the Graphical User Interface . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . 202 vi | CONTENTS Chapter 8: Structural Mechanics The Structural Mechanics Application Modes 204 Theory Background 205 Strain-Displacement Relationship . . . . . . . . . . . . . . . . 205 Stress-Strain Relationship. . . . . . . . . . . . . . . . . . . 205 Implementation . . . . . . . . . . . . . . . . . . . . . . 207 Application Mode Descriptions 210 Application Mode Properties . . . . . . . . . . . . . . . . . 210 Material Properties . . . . . . . . . . . . . . . . . . . . . 212 Constraints . . . . . . . . . . . . . . . . . . . . . . . . 213 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Damping . . . . . . . . . . . . . . . . . . . . . . . . . 215 The Solid, Stress-Strain Application Mode 216 Variables and Space Dimensions . . . . . . . . . . . . . . . . 216 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 216 Application Mode Variables . . . . . . . . . . . . . . . . . . 217 The Plane Stress Application Mode 220 Material . . . . . . . . . . . . . . . . . . . . . . . . . 220 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 220 Application Mode Variables . . . . . . . . . . . . . . . . . . 221 The Plane Strain Application Mode 225 Material Properties . . . . . . . . . . . . . . . . . . . . . 225 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 226 Application Mode Variables . . . . . . . . . . . . . . . . . . 226 The Axial Symmetry, Stress-Strain Application Mode 230 PDE Formulation . . . . . . . . . . . . . . . . . . . . . . 231 Application Mode Variables . . . . . . . . . . . . . . . . . . 232 Examples of Structural Mechanics Models 235 Tapered Membrane End Load . . . . . . . . . . . . . . . . . 235 CONTENTS | vii Modeling in COMSOL Multiphysics . . . . . . . . . . . . . . . 236 Results. . . . . . . . . . . . . . . . . . . . . . . . . . 236 Modeling Using the Graphical User Interface . . . . . . . . . . . 237 Tapered Cantilever Gravity Load . . . . . . . . . . . . . . . . 240 Modeling in COMSOL Multiphysics . . . . . . . . . . . . . . . 240 Results. . . . . . . . . . . . . . . . . . . . . . . . . . 241 Modeling Using the Graphical User Interface . . . . . . . . . . . 241 Reference . . . . . . . . . . . . . . . . . . . . . . . . 244 Chapter 9: PDE Modes for Equation-Based Modeling The PDE Modes 246 Using the PDE Modes 247 Starting a Model Using a PDE Mode. . . . . . . . . . . . . . . 247 Using the Coefficient Form PDEs. . . . . . . . . . . . . . . . 247 Using the General Form PDE . . . . . . . . . . . . . . . . . 257 Solving Time-Dependent Problems . . . . . . . . . . . . . . . 264 Solving Eigenvalue Problems. . . . . . . . . . . . . . . . . . 266 Interpreting PDE Coefficients . . . . . . . . . . . . . . . . . 267 Classical PDEs . . . . . . . . . . . . . . . . . . . . . . . 268 Equation Variables . . . . . . . . . . . . . . . . . . . . . 268 Boundary-Coupled Equation Variables . . . . . . . . . . . . . . 270 Ideal and Non-Ideal Constraints . . . . . . . . . . . . . . . . 273 Variables for PDEs in Weak Form . . . . . . . . . . . . . . . 274 Application Mode Variables . . . . . . . . . . . . . . . . . . 277 PDE Coefficients and Boundary Conditions with Time Derivatives . . . 279 Implementing a Point Source . . . . . . . . . . . . . . . . . 279 Modeling Using the Graphical User Interface . . . . . . . . . . . 281 Using Equation Contributions on Interior Mesh Boundaries 285 Specifying Equation Contributions . . . . . . . . . . . . . . . 285 Specifying Interior Mesh Boundary Expressions. . . . . . . . . . . 286 viii | CONTENTS Chapter 10: Sensitivity Analysis The Sensitivity Analysis Application Mode 288 Introduction . . . . . . . . . . . . . . . . . . . . . . . 288 Sensitivity Variable Selection . . . . . . . . . . . . . . . . . 288 Scalar Objective Functions . . . . . . . . . . . . . . . . . . 290 Model Navigator . . . . . . . . . . . . . . . . . . . . . . 294 Application Mode Properties . . . . . . . . . . . . . . . . . 294 Subdomain, Boundary, and Edge Settings . . . . . . . . . . . . . 294 Point Settings . . . . . . . . . . . . . . . . . . . . . . . 296 Scalar Settings . . . . . . . . . . . . . . . . . . . . . . . 297 Example—Predicting the Effect of a Geometrical Change 299 Introduction . . . . . . . . . . . . . . . . . . . . . . . 299 Model Definition . . . . . . . . . . . . . . . . . . . . . . 299 Results and Discussion. . . . . . . . . . . . . . . . . . . . 302 Modeling Using the Graphical User Interface . . . . . . . . . . . 305 Chapter 11: Optimization The Optimization Application Mode 310 Introduction . . . . . . . . . . . . . . . . . . . . . . . 310 Optimization Problem Formulation . . . . . . . . . . . . . . . 310 Model Navigator . . . . . . . . . . . . . . . . . . . . . . 312 Application Mode Properties . . . . . . . . . . . . . . . . . 313 Subdomain Settings . . . . . . . . . . . . . . . . . . . . . 313 Boundary Settings . . . . . . . . . . . . . . . . . . . . . 317 Edge Settings (3D Only) . . . . . . . . . . . . . . . . . . . 318 Point Settings . . . . . . . . . . . . . . . . . . . . . . . 318 Scalar Settings . . . . . . . . . . . . . . . . . . . . . . . 319 Solver Settings . . . . . . . . . . . . . . . . . . . . . . . 321 Example—Minimizing the Flow Velocity in a Microchannel 326 Introduction . . . . . . . . . . . . . . . . . . . . . . . 326 Model Definition . . . . . . . . . . . . . . . . . . . . . . 326 [...] .. . Weak Constraints 35 0 Specifying Weak Constraints 35 1 Weak Constraints in Assemblies 35 2 Limitations of Weak Constraints 35 3 Example—Coupling Variables and Boundary Constraints 35 4 Boundary Constraints in a Heat Transfer Model 35 4 No Weak Constraints 35 8 Weak Ideal Constraints 35 8 Weak Non-Ideal Constraints .. . can visualize any variable or expression of variables by working with the Plot Parameters, Cross-Section Plot Parameters, and Domain Plot Parameters dialog boxes You can also have COMSOL Multiphysics compute integrated values with the help of the Subdomain Integration and Boundary Integration dialog boxes Each application mode provides a special set of application-specific variables called application .. . Standard Acoustics Problems Five standard problems or scenarios occur frequently when analyzing acoustics: • The radiation problem A vibrating structure (a speaker, for example) radiates sound into the surrounding space A far-away boundary condition is necessary to model the unbounded domain • The scattering problem—An incident wave impinges on a body and creates a scattered wave A far-away radiation .. . defined Many variables that are available on subdomains are also available on boundaries, edges, and points, but they then take the average value of the values in the subdomains around the boundary, edge, or point (for the subdomains in which the variable is present) In addition to the letters above, indicating the domains where the application mode variable is valid, a V indicates that it is a vector-valued .. . the acoustic analysis provides a load (the sound pressure) to the structural analysis, and the structural analysis provides accelerations to the acoustic analysis • The transmission problem—The incident sound wave propagates into a body, which can have different acoustic properties Pressure and acceleration are continuous on the boundary 20 | CHAPTER 3: ACOUSTICS Mathematical Models for Acoustic Analysis .. . there are variations in the radial (r) and vertical (z) direction only and not in the angular (θ) direction You can then solve a 2D problem in the rz-plane instead of the full 3D model, which can save considerable memory and computation time Many COMSOL Multiphysics application modes are available in axisymmetric versions • Symmetry and Antisymmetry Planes or Lines are common in both 2D and 3D models .. . USING A SINGLE APPLICATION MODE Most of the physics application modes contain stationary, eigenvalue, and dynamic (time-dependent) analysis types As already mentioned, these application modes provide modeling interfaces where you can create models using material properties, boundary conditions, and initial conditions Each of the application modes comes with a template that automatically supplies the appropriate .. . temperature gradient are Tx1, Ty1, Tz1 Note: All application mode variables include a suffix indicating which application mode they belong to Table 2-1 on page 15 lists the default suffix for the physics modes For example, the default suffix added to application mode variable names for an Incompressible Navier-Stokes application mode is _ns Note: The default space coordinate names are x, y, z for Cartesian .. . the application mode The default values are physical constants such as the permittivity of vacuum or arbitrary values, for example, the frequency 50 Hz for the AC Power Electromagnetics application mode Not every application mode has application scalar variables SUBDOMAIN SETTINGS The Subdomain Settings section lists the subdomain properties in the application mode such as material properties and sources .. . vector-valued variable In some cases the table also lists the solution form for which a variable is available: • Coefficient form: c • General form: g • Weak form: w The Name column in the table of application mode variables lists the variables available for use, whereas the Expression column lists the implementation of the application mode variables in terms of other variables The italic i and j in the variable . + 1 -3 1 0-4 4 1-4 800 Fax: + 1 -3 1 0-4 4 1-0 868 COMSOL, Inc. 744 Cowper Street Palo Alto, CA 9 430 1 Phone: + 1-6 5 0 -3 2 4-9 9 35 Fax: + 1-6 5 0 -3 2 4-9 936 info @comsol. com www .comsol. com For a complete list of international. England Executive Park Suite 35 0 Burlington, MA 018 03 Phone: + 1-7 8 1-2 7 3- 3 32 2 Fax: + 1-7 8 1-2 7 3- 6 6 03 COMSOL, Inc. 10 850 Wilshire Boulevard Suite 800 Los Angeles, CA 90024 Phone: + 1 -3 1 0-4 4 1-4 800. www .comsol. de Italy COMSOL S.r.l. Via Vittorio Emanuele II, 22 251 22 Brescia Phone: +3 9-0 3 0 -3 7 938 00 Fax: +3 9-0 3 0 -3 7 938 99 info.it @comsol. com www.it .comsol. com Norway COMSOL AS Søndre gate