Multiphysics Modeling Using COMSOL 5 and MATLAB E-Book

Mercury Learning and Information

• Herausgeber: Packt Publishing
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 453

Veröffentlichungsjahr: 2024

Beschreibung

This updated edition of the book explores COMSOL 5 and MATLAB, essential modeling tools for engineers and scientists. It includes five new models and covers systems from 0D to 3D, introducing numerical analysis techniques in COMSOL 5.6 and MATLAB. Using examples from electromagnetic, electronic, optical, thermal physics, and biomedical models, the book provides fundamental concepts and step-by-step instructions for building each model. Companion files include all models and related animations.
The course starts with modeling methodology and material properties, progressing through 0D electrical circuit interface, 1D, 2D, 2D axisymmetric, 2D simple and complex mixed mode, and 3D modeling. Advanced topics like Perfectly Matched Layer models and Bioheat models are also covered. Each chapter builds on the previous one, ensuring a comprehensive understanding of modeling techniques.
Understanding these concepts is crucial for developing and analyzing engineering, science, and biomedical systems. This book transitions readers from basic to advanced modeling, combining theoretical knowledge with practical skills. Companion files enhance the learning experience, making this an essential resource for mastering COMSOL 5 and MATLAB.

Leseprobe

MULTIPHYSICS MODELING

USING COMSOL® 5 AND MATLAB®

SECOND EDITION

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MULTIPHYSICS MODELING

USING COMSOL® 5 AND MATLAB®

SECOND EDITION

Roger W. Pryor, Ph.D.

COMSOL Certified Consultant

MERCURY LEARNING AND INFORMATION

Dulles, Virginia

Boston, Massachusetts

New Delhi

Copyright ©2022 by MERCURY LEARNING AND INFORMATION LLC. All rights reserved.

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Publisher: David Pallai

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Roger W. Pryor. Multiphysics Modeling Using COMSOL®5 and MATLAB®, 2/E

ISBN: 978-1-68392-589-7

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CONTENTS

Preface

Introduction

Chapter 1: Modeling Methodology Using COMSOL Multiphysics 5.x

Guidelines for New COMSOL Multiphysics 5.x Modelers

Hardware Considerations

Simple Model Setup Overview

Basic Problem Formulation and Implicit Assumptions

1D Window Heat Flow Models

1D 1 Pane Window Heat Flow Model

1D 2 Pane Window Heat Flow Model

1D 3 Pane Window Heat Flow Model

First Principles as Applied to Model Definition

Some Common Sources of Modeling Errors

References

Suggested Modeling Exercises

Chapter 2: Materials Properties Using COMSOL Multiphysics 5.x

Materials Properties Guidelines and Considerations

COMSOL Materials Properties Sources

Other Materials Properties Sources

Material Property Entry Techniques

Multipane Window Model

Set Boundary Conditions

References

Chapter 3: 0D Electrical Circuit Interface Modeling Using COMSOL Multiphysics 5.x

Guidelines for Electrical Circuit Interface Modeling in 5.x

Electrical / Electronic Circuit Considerations

Simple Electrical Circuit Interface Model Setup Overview

Basic Problem Formulation and Implicit Assumptions

0D Basic Circuit Models

0D Resistor-Capacitor Series Circuit Model

0D Inductor-Resistor Series Circuit Model

0D Series-Resistor Parallel-Inductor-Capacitor Circuit Model

0D Basic Circuit Models Analysis and Conclusions

First Principles as Applied to 0D Model Definition

References

Suggested Modeling Exercises

Chapter 4: 1D Modeling Using COMSOL Multiphysics 5.x

Guidelines for 1D Modeling in 5.x

1D Modeling Considerations

1D Basic Models

1D KdV Equation Model

1D Telegraph Equation Model

1D Spherically Symmetric Transport Model

1D Spherically Symmetric Transport Model Animation

1D Advanced Model

1D Silicon Inversion Layer Model: A Comparison of the Results obtained from using the Density-Gradient (DG) Theory and the Schrodinger-Poisson (SP) Theory Methodologies

First Principles as Applied to 1D Model Definition

References

Suggested Modeling Exercises

Chapter 5: 2D Modeling Using COMSOL Multiphysics 5.x

Guidelines for 2D Modeling in 5.x

2D Modeling Considerations

2D Basic Models

2D Electrochemical Polishing Model

2D Hall Effect Model

First Principles as Applied to 2D Model Definition

References

Suggested Modeling Exercises

Chapter 6: 2D Axisymmetric Modeling Using COMSOL Multiphysics 5.x

Guidelines for 2D Axisymmetric Modeling in 5.x

2D Axisymmetric Modeling Considerations

2D Axisymmetric Heat Conduction in a Cylinder Model

2D Axisymmetric Basic Models

2D Axisymmetric Cylinder Conduction Model

2D Axisymmetric Transient Heat Transfer Model

First Principles as Applied to 2D Axisymmetric Model Definition

References

Suggested Modeling Exercises

Chapter 7: 2D Simple and Advanced Mixed Mode Modeling Using COMSOL Multiphysics 5.x

Guidelines for 2D Simple Mixed Mode Modeling in 5.x

2D Simple Mixed Mode Modeling Considerations

2D Simple Mixed Mode Models

2D Electric Impedance Sensor Model

2D Metal Layer on a Dielectric Block Model

Heat Transfer 2 (ht2) Interface

Heat Transfer in Solids (ht) Interface

First Principles as Applied to 2D Simple Mixed Mode Model Definition

References

Suggested Modeling Exercises

Chapter 8: 2D Complex Mixed Mode Modeling Using COMSOL Multiphysics 5.x

Guidelines for 2D Complex Mixed Mode Modeling in 5.x

2D Complex Mixed Mode Modeling Considerations

2D Complex Mixed Mode Models Using the RF Module

Finding the Impedance of a Two (2) Wire, Parallel-Wire, Air-Dielectric, Transmission Line

2D Finding the Impedance of a Two (2) Wire, Parallel-Wire, Air-Dielectric, Transmission Line Model Summary and Conclusions

2D Finding the Impedance of a Concentric, Two (2) Wire, Transmission Line (Coaxial Cable)

2D Finding the Impedance of a Concentric, Two (2) Wire

2D Axisymmetric Transient Modeling of a Coaxial Cable

First Principles as Applied to 2D Complex Mixed Mode Model Definition

References

Suggested Modeling Exercises

Chapter 9: 3D Modeling Using COMSOL Multiphysics 5.x

Guidelines for 3D Modeling in 5.x

3D Modeling Considerations

3D Models

3D Spiral Coil Microinductor Model

3D Linear Microresistor Beam Model

Multiphysics Thermal Linear Elastic 1 (te1)

Heat Transfer in Solids (ht)

First Principles as Applied to 3D Model Definition

References

Suggested Modeling Exercises

Chapter 10: Perfectly Matched Layer Models Using COMSOL Multiphysics 5.x

Guidelines for Perfectly Matched Layer (PML) Modeling in 5.x

Perfectly Matched Layer (PML) Modeling Guidelines and Coordinate Considerations

Perfectly Matched Layer Models

Building the 2D Concave Metallic Mirror PML Model

Building the 2D Energy Concentrator PML Model

First Principles as Applied to PML Model Definition

References

Suggested Modeling Exercises

Chapter 11: Bioheat Models Using COMSOL Multiphysics 5.x

Guidelines for Bioheat Modeling in 5.x

Bioheat Modeling Considerations

Bioheat Transfer Models

2D Axisymmetric Microwave Cancer Therapy Model

First Principles as Applied to Bioheat Model Definition

References

Suggested Modeling Exercises

Appendix: A Brief Introduction to LiveLinkTM for MATLAB® Using COMSOL Multiphysics 5.x

Guidelines for LiveLink Exploration through Modeling in 5.x

Getting Started using LiveLink for MATLAB with COMSOL Multiphysics 5.x on a Windows® 10 platform

First Principles as Applied to Scripting and GUI Model Definition

References

Suggested Modeling Exercises

Index

PREFACE

The purpose of this second edition is to update the model building ­instructions to comply with COMSOL® Version 5.6, MATLAB® R2021a, and later. The MATLAB material contained in the Appendix is revised and updated to facilitate LIVELINK® interaction with COMSOL 5.x. This second edition also introduces five new hands-on model building and problem-solving techniques. In this new book, scientists, engineers, biophysicists, and other readers interested in exploring the behavior of potential physical device structures built on a computer (a virtual prototype), can develop exploratory models, before going to the workshop or laboratory and attempting to physically create the whatever-it-is (a real prototype).

The models presented herein are built within the context of the currently well-established laws of the physical world (applied physics) and are explored in light of widely applied, well-known, First Principle Analysis techniques. As with any other method of problem solution (mathematically computed answer), the information obtained (derived) through the use of such modeled solutions, as these computer simulations (virtual prototypes), can ultimately be only as accurate as the materials properties values and the fundamental assumptions employed to build (create) these simulations.

The primary advantage of the combination of computer simulation (virtual prototyping) and First Principles Analysis to explore artifacts (device structures) is that the modeler can try as many different approaches to the solution of the same underlying problem as are needed in order to get it right (or close thereto) before fabrication of the device components and the assembled device (real prototype) in the workshop or laboratory for the first time.

ACKNOWLEDGMENTS

I would like to thank David Pallai of Mercury Learning and Information for his ongoing encouragement. I would also like to thank the many staff members of COMSOL, Inc. for their help and encouragement.

I would especially like to thank Beverly E. Pryor, my wife, for her patience and encouragement during the creation of the manuscript of this book. Any residual errors in this work are mine and mine alone.

Roger W. Pryor, Ph.D.October 2021     

INTRODUCTION

COMSOL Multiphysics software is a powerful, Partial Differential Equation (PDE) solution engine. The basic COMSOL Multiphysics 5.x software has over twenty-five (25) add-on modules that expand the capabilities of the basic software into a broad collection of application areas: AC/DC, acoustics, batteries and fuel cells, CFD, chemical reaction engineering, electrodeposition, geomechanics, heat transfer, MEMS, microfluidics, plasma, RF, structural mechanics and subsurface flow, to name a few. The COMSOL Multiphysics software also has other supporting software, such as the Optimization Module, the Material Library Module, the CAD Import Module and LiveLink™ interfaces for several engineering software programs.

In this book, scientists, engineers, and others interested in exploring the behavior of different physical device structures through computer modeling are introduced to the techniques of hands-on building and solving models through the direct application of the basic COMSOL Multiphysics software, along with some samples using the AC/DC, heat transfer, rf, semiconductor, and structural mechanics modules. The next to the last technical chapter explores the use of Perfectly Matched Layers in the RF Module. The final technical chapter explores the use of the Bioheat Equation in the Heat Transfer and RF Modules.

The models presented herein are built within the context of the physical world (applied physics) and are presented in light of First Principle Analysis techniques. The demonstration models emphasize the fundamental concept that the information derived from the modeling solutions through the use of these computer simulations is only as good as the materials coefficients and the fundamental assumptions employed in building the models.

The combination of computer simulation and First Principle Analysis gives the modeler the opportunity to try a variety of approaches to the solution of the same problem as needed in order to get the design right or nearly right in the workshop or laboratory before the first device components are fabricated and tested. The modeler can also use the physical device test results to modify the model parameters and arrive at an improved solution more rapidly than by simply using the cut and try methodology.

CHAPTER TOPICS

The eleven (11) technical chapters in this book demonstrate to the reader the hands-on technique of model building and solving. The COMSOL concepts and techniques used in these chapters are shown in Figure Int.1. The COMSOL modules employed in the various models in specific chapters are shown in Figure Int.2, and the physics concepts and techniques employed in the various models in specific chapters are shown in Figure Int.3.

Figure Int.1 COMSOL Concepts and Techniques

Figure Int.2 COMSOL Modules Employed

Figure Int.3 Physics Concepts

The information in these three figures link the overall presentation of this book to the underlying modeling, mathematical and physical concepts. In this book, in contrast to other books with which the reader may be familiar, key ancillary information is, in most cases, contained in the notes.

NOTE

Please be sure to read, carefully consider, and apply, as needed, each note.

CHAPTER 1 MODELING METHODOLOGY

Chapter 1 provides an overview of the modeling process by discussing the fundamental considerations involved: the hardware (computer platform), the coordinate systems (physics), the implicit assumptions (lower dimensionality considerations), and First Principles Analysis (physics). Three relatively simple 1D models are presented that build and solve, for comparison, single-, double- and triple-pane thermal insulation window structures. Comments are also included on common sources of modeling errors.

CHAPTER 2 MATERIALS PROPERTIES

Chapter 2 discusses various sources of materials properties data, including the COMSOL Material Library, basic and expanded module, as well as print and Internet sources. A multi-pane thermal insulation window structure model demonstrates three techniques for entering material properties: user-defined direct entry, user-defined parameters, and material definitions. Also included are instructions for building a user-defined material library for storage within COMSOL 5.x.

CHAPTER 3 0D ELECTRICAL CIRCUIT INTERFACE

COMSOL 5.x uses zero-dimensional models to provide for the modeling of electrical circuitry. The models in this chapter illustrate techniques for modeling various basic circuits: a resistor-capacitor series circuit, an inductor-resistor series circuit, and a series resistor, parallel inductor-capacitor circuit. Considerations for the proper setup of the circuits are discussed along with the basics of problem formulation and the implicit assumptions built into COMSOL 5.x relative to electrical circuit modeling.

CHAPTER 4 1D MODELING

The first part of Chapter 4 models the 1D KdV Equation. The KdV Equation is a powerful tool that is used to model soliton wave propagation in diverse media (e.g., physical waves in liquids, electromagnetic waves in transparent media, etc.). It is easily and simply modeled with a 1D PDE mode model.

The second part of Chapter 4 models the 1D Telegraph Equation. The Telegraph Equation is a powerful tool that is used to model wave propagation in diverse transmission lines. The Telegraph Equation can be used to thoroughly characterize the propagation conditions of coaxial lines, twin pair lines, microstrip lines, etc. The Telegraph Equation is easily and simply modeled with a 1D PDE mode model.

The third part of Chapter 4 is a 1D Spherically Symmetric Transport model that illustrates the technique of simplifying models with spherical components from 3D to 1D by assuming that they are essentially symmetrical.

The fourth part of Chapter 4 is an Advanced 1D Silicon Inversion Layer Model using DG and SP Theory Methodologies. It is the purpose of this model to demonstrate the modeling techniques needed to reproduce the calculated inversion layer electron density below the gate oxide, as a function of depth curves.

CHAPTER 5 2D MODELING

The first half of Chapter 5 models the surface smoothing process by using a 2D Electrochemical Polishing Model. This model is a powerful tool that can be used for diverse surface smoothing projects (e.g., microscope samples, precision metal parts, medical equipment and tools, large and small metal drums, thin analytical samples, vacuum chambers, etc.).

The second half of Chapter 5 models Hall Effect magnetic sensors. The 2D Hall Effect Model is a powerful tool that can be used to model such sensors when used for sensing fluid flow, rotating and/or linear motion, proximity, current, pressure, orientation, etc.

CHAPTER 6 2D AXISYMMETRIC MODELING

Modeling a 3D device that is symmetrical on one axis by treating it as a 2D Axisymmetric object simplifies the model for quicker first approximation results.

The first half of Chapter 6 discusses a 2D Axisymmetric Heat Conduction in a Cylinder Model and demonstrates the use of contour plotting of the solver results to show non-linear temperature distribution in the cylinder.

The second half of Chapter 6 models transient heat transfer in a niobium sphere immersed in a medium of constant temperature by using a 2D Axisymmetric model.

CHAPTER 7 2D SIMPLE AND ADVANCED MIXED-MODE MODELING

In this chapter, simple and advanced mixed-mode 2D models are presented. Such 2D models are typically more conceptually complex than the models that were presented in earlier chapters of this text. 2D simple and advanced mixed-mode models have proven to be very valuable to the science and engineering communities, both in the past and currently, as first-cut evaluations of potential systemic physical behavior under the influence of mixed external stimuli. The 2D mixed-mode model responses and other such ancillary information can be gathered and screened early in a project for a first-cut evaluation. That initial information can potentially be used later as guidance in building higher-dimensionality (3D) field-based (electrical, magnetic, etc.) models.

The first part of Chapter 7 uses a 2D Electrical Impedance Sensor Model to demonstrate this technique. The concept of electrical impedance, as used in alternating current (AC) theory, is an expansion on the basic concept of resistance as illustrated by Ohm’s Law, in direct current (DC) theory.

The second part of Chapter 7 uses a 2D Axisymmetric Metal Layer on a Dielectric Block Model to demonstrate more aspects of the technique. The modeler was introduced to Fick’s laws for the diffusion (mass transport) of a first item (e.g., a gas, a liquid, etc.) through a second item (e.g., another gas, liquid, etc.). In the case of this model, the diffusing item is heat.

In the first part of Chapter 7 implemented above, a copper layer was approximated by the Thin Conductive Layer function in the Heat Transfer in Solids Interface. In this, Advanced Model, the third study implemented in this chapter, the copper layer will be a geometrical copper layer. The modeler will now be able to compare the results of the earlier models, to the results of this Advanced model calculation, on nominally the same modeling problem, employing diamond as the substrate. The modeler can now determine the relative trade-offs required when he chooses different materials to model one methodology or another.

These models are examples of the Good First Approximation type of models because they demonstrate the significant power of relatively simple physical principles, such as Ohm’s Law, Joule’s Laws, and Fick’s Laws, when applied in the COMSOL Multiphysics Modeling environment. The equations can, of course, be modified by the addition of new terms, insulating materials, heat loss through convection, etc.

CHAPTER 8 2D COMPLEX MIXED MODE MODELING

In this chapter, three new primary analysis concepts are introduced to the modeler: 2D electromagnetic impedance calculation for a planar, two-wire geometry (side by side), for a concentric two-wire geometry (coaxial cable), and a 2D Axisymmetric model of the transient behavior of a Concentric 2 Wire Geometry (Coaxial Cable).

CHAPTER 9 3D MODELING

In this chapter, the modeler is introduced to three new modeling concepts: the Terminal boundary condition lumped parameters and coupled thermal, electrical and structural multiphysics analysis. The Terminal boundary condition and the lumped parameter concepts are employed in the solution of the 3D Spiral Coil Microinductor Model. The fully coupled multiphysics solution is employed in the 3D Linear Microresistor Beam Model.

The lumped parameter (lumped element) modeling approach approximates a spatially distributed collection of diverse physical elements by a collection of topologically (series and/or parallel) connected discrete elements. This technique is commonly employed for first approximation models in electrical, electronic, mechanical, heat transfer, acoustic and other physical systems.

CHAPTER 10 PERFECTLY MATCHED LAYER MODELS

One of the fundamental difficulties underlying electromagnetic wave equation calculations is dealing with a propagating wave after the wave interacts with a boundary (reflection). If the boundary of a model domain is terminated in an abrupt fashion, unwanted reflections will typically be incorporated into the solution, potentially creating undesired and possibly erroneous model solution values. Fortunately, for the modeler of today, there is a methodology that works sufficiently well that it essentially eliminates reflection problems at the domain boundary. That methodology is the Perfectly Matched Layer.

Chapter 10 includes two models, the 2D Concave Metallic Mirror PML Model and the 2D Energy Concentrator PML Model, to demonstrate the use of the Perfectly Matched Layer methodology.

CHAPTER 11 BIOHEAT MODELS

The Bioheat Equation plays an important role in the development and analysis of new therapeutic medical techniques (e.g., killing of tumors). If the postulated method raises the local temperature of the tumor cells, without excessively raising the temperature of the normal cells, then the proposed method will probably be successful. The results (estimated time values) from the model calculations will significantly reduce the effort needed to determine an accurate experimental value. The guiding principle needs to be that tumor cells die at elevated temperatures. The literature cites temperatures that range from 42 °C (315.15 K) to 60 °C (333.15 K).

The first half of Chapter 11 models the Bioheat Equation as applied with a photonic heat source (laser).

The second half of Chapter 11 models the Bioheat Equation as applied with a microwave heat source.

CHAPTER 1

MODELING METHODOLOGY USING COMSOL MULTIPHYSICS 5.X

In This Chapter

●Guidelines for New COMSOL Multiphysics 5.x Modelers

–Hardware Considerations

–Simple Model Setup Overview

–Basic Problem Formulation and Implicit Assumptions

●1D Window Heat Flow Models

–1D 1 Pane Window Heat Flow Model

–1D 2 Pane Window Heat Flow Model

–1D 3 Pane Window Heat Flow Model

●First Principles as Applied to Model Definition

●Some Common Sources of Modeling Errors

●References

●Suggested Modeling Exercises

GUIDELINES FOR NEW COMSOL MULTIPHYSICS 5.x MODELERS

NOTE

First, for purposes of clarity in this book, when the term “COMSOL Multiphysics 5.x software” or “5.x” occurs, that term means: COMSOL Multiphysics software version 5.6 or later.

Second, the user of this text should be sure to READ ALL NOTES in this book, as the Notes contain information that is not presented elsewhere and should facilitate comprehension and, hopefully, minimize modeling errors for modelers unfamiliar with 5.x.

When building models, new or otherwise, it is VERY IMPORTANT to SAVE EARLY and OFTEN.

Hardware Considerations

There are two basic rules for selecting hardware that will support successful modeling with 5.x. The first rule is that the modeler should be sure to determine the minimum system requirements their version of 5.x requires before borrowing or buying a computer to run his new modeling software. This book introduces the use of 5.x for modeling within the Microsoft Windows® and the Macintosh OS X® operating systems environments.

COMSOL Multiphysics 5.x software supports shared memory parallelism under both the Microsoft Windows and the Macintosh OS X operating systems. Neither distributed memory nor cluster computing will be covered in this book.

NOTE

The number of platform cores is equal to the number of coprocessors designed into the computer (e.g., 1, 2, 4, 8, 12, etc.).

Shared memory parallelism means that multiple cores in the same computer share the same memory array. Distributed memory parallelism means multiple computers with multiple cores are connected in a cluster with the problem divided into multiple parallel computational sub-segments {1.1} distributed over the computational units of the cluster.

The second rule of successful modeling is the modeler should run 5.x on the best platform with the highest processor speed and the most memory obtainable; the bigger, the faster, the better. It is the general rule that the speed of model processing increases in proportion to instruction size (32 bit, 64 bit), the core speed, the number of platform cores, and to the amount of usable, available memory.

The platform this author uses is an Apple Mac Pro®, running Mac OS X version 10.15.x. That platform has 12-2.7 GHz cores and 64 GB of RAM and is configured for 64-bit processing and runs at the 64-bit rate when using 5.x. If the new modeler desires a different 64-bit operating system than Macintosh OS X, then they will need to choose a LINUX® platform, using UNIX® or a PC with a 64-bit Microsoft Windows operating system, as specified in the COMSOL Operating System Requirements (https://www.comsol.com/system-requirements).

NOTE

The 2.7 GHz specification is the operating speed of each of the cores and the 64 GB is the total shared Random Access Memory (RAM). 64-bit refers to the width of a processor instruction.

Currently available Macintosh Pro hardware can now be obtained with up to 28 cores and 768 GB of shared ECC {1.2}.

Once the best available processor is obtained, within the constraints of your budget, install your copy of 5.x, following the installer instructions.

After installation, as a matter of caution, the modeler should restart the processor and, once stable, start the COMSOL Multiphysics 5.x software. 5.x will then present the modeler with a configurable Graphical User Interface (GUI). For computer users not familiar with the GUI concept, information is presented primarily in the form of pictures with supplemental text, not exclusively as text. See Figure 1.1a (Mac) and Figure 1.1b (PC) to observe the details of the 5.x GUI interface.

FIGURE 1.1(a) Default COMSOL desktop display on a Mac immediately after startup.

FIGURE 1.1(b) Default COMSOL desktop display on a PC immediately after startup.

Simple Model Setup Overview

Figures 1.1a&b show the default COMSOL Desktop Display immediately after startup, for both the Mac and the PC. The startup screen for the Mac and the PC are the same in 5.x on the two platforms. COMSOL Multiphysics Version 5.x displays two (2) buttons. The two buttons on the Mac and the PC are the same. Those buttons are the “Model Wizard” button and the “Blank Model” button.

In 5.x, after the Model Wizard button has been selected, a row of buttons is displayed that allows the modeler to select the desired geometry (3D, 2D Axisymmetric, 2D, 1D Axisymmetric, 1D, and 0D) appropriate to the model under development. In the case of the 3D coordinate system, the coordinates that are pictorially shown in the Graphics window are based on the Right-Hand-Rule.

NOTE

The Right-Hand-Rule is derived, as it states, from your right hand. Look at your right hand, point the thumb up, point your index (first) finger away from your body, at a right angle (90 degrees) to your thumb and point your middle (second) finger, at a right angle to the thumb, tangent to your body. Your thumb represents the Z-axis, your index (first) finger represents the X-axis, and your middle (second) finger represents the Y-axis.

In a 3D coordinate system that obeys the Right-Hand-Rule, X rotates into Y and generates Z. Z is the direction in which a Right-Handed Screw would advance (move into the material into which it is being screwed).

NOTE

The modeler should note that the default COMSOL Desktop for 5.x, built by using the Model Wizard button, displays four active windows: Model Builder, Settings, Graphics, and Messages. The Desktop Display can, of course, be modified as needed to show more or fewer (see COMSOL Desktop Environment {1.3}). In this book, the Desktop Display will be used in the default configuration, except when modifications are required.

The most singularly important currently active window for the modeler, at this point, is the Model Wizard window. Before anything else can happen relative to creating a new model, 5.x requires that the modeler choose a new coordinate system.

The Model Wizard window of the Desktop Display shows six coordinate system (Space Dimension) choices available for selection: 3D, 2D Axisymmetric, 2D, 1D Axisymmetric, 1D, and 0D. See Figures 1.2a&b to observe the interface details.

FIGURE 1.2(a) Default model wizard window for model coordinate system (space dimension) selection for a Mac.

Figure 1.2a shows the Model Wizard window for model coordinate system (Space Dimension) selection for a Mac.

FIGURE 1.2(b) Default model wizard window for model coordinate system (space dimension) selection for a PC.

Figure 1.2b shows the Model Wizard window for model coordinate system (Space Dimension) selection for a PC.

Geometric coordinate systems in COMSOL range from the extremely complex to the nominally very simple. The default geometries are the standard Cartesian geometries 3D, 2D, and 1D, as shown in Figures 1.3, 1.4, and 1.5.

FIGURE 1.3 3D Cartesian coordinate geometry.

Figure 1.3 shows a Right-Hand 3D Cartesian Coordinate Geometry Axes configuration.

FIGURE 1.4 2D Cartesian coordinate geometry.

Figure 1.4 shows a 2D Cartesian Coordinate Geometry Axes configuration.

FIGURE 1.5 1D Cartesian coordinate geometry.

Figure 1.5 shows a 1D Cartesian Coordinate Geometry Axis configuration.

NOTE

Comprehension of the 3D, 2D, 1D Cartesian Coordinate Systems is relatively easy for the modeler, assuming that the modeler has some previous engineering or science training. These coordinate systems are simply an orthogonal combination of rectilinearly incremented axes comprising 1, 2, or 3 measurement dimensions.

The 2D and 1D Axisymmetric Geometries are somewhat more complex. Both the 2D and 1D Axisymmetric Geometries comprise cylindrical coordinate systems. The primary benefit derived through the use of the axisymmetric coordinate systems is that they allow the modeler to reduce the effective model dimensionality and the calculational complexity of a problem, for example, 3D → 2D, through the assumption of axial symmetry and rotationally uniform boundary conditions.

Due to the complex nature of the underlying assumptions in the 1D Axisymmetric Coordinate System and its applications, the modeler is referred to the COMSOL literature {1.4} for further details.

FIGURE 1.6 2D axisymmetric coordinate geometry (space dimension).

Figure 1.6 shows a 2D Axisymmetric coordinate geometry axis (Space Dimension) configuration.

The last Space Dimension choice available is 0D. In the case of 0D, the underlying equations in the multiphysics model have either no relational dependence on geometrical factors or react in a homogeneous and isotropic manner (effectively geometrically relationally independent).

The 0D Space Dimension will be used in this book only to explore electrical and/or electronic circuit model behavior through the use of SPICE calculations.

Select A Space Dimension:

For this first demonstration of the Desktop Display, Select (Click) the 1D selection button. See Figure 1.7.

NOTE

The term Select (Click) means for the modeler to place the display screen computer cursor in proximity to or superimposed on the item to be selected and then tap (momentarily depress) the physical mouse activation key (button).

The selection of the 1D button will set the dimensionality to 1D and cause the Model Wizard display to switch to the Select Physics page. See Figure 1.8.

FIGURE 1.7 1D Button.

Figure 1.7 shows the dimensionality buttons (1D button)

Figure 1.8 shows the Select Physics page. See Figure 1.8.

FIGURE 1.8 Select physics page.

Figure 1.8 shows the Select Physics page.

Once a Physics Interface is clicked, the Desktop Display changes to show a description of the Physics Interface and the Add Physics window, as shown in Figure 1.9a. In the case of Figure 1.9a, all of the licensed COMSOL Multiphysics Physics Interfaces are displayed,. If the modeler has not licensed all of the available modules, only the licensed modules are displayed.

FIGURE 1.9(a) Desktop display with the description of the physics interface and the add physics window.

Click the Add button below the Select Physics window.

Figure 1.9b shows the Desktop Display with the Added Physics in the Add Physics window.

FIGURE 1.9(b) Desktop display with the selected physics interface in the add physics window.

NOTE

A Physics Interface is the set of relevant equations, typical initial conditions, independent variable(s), dependent variable(s), default settings, boundary conditions, etc., that are appropriate for the solution of a particular problem type for a given branch or sub-branch of physics (e.g., acoustics, electromagnetics, heat transfer, etc.).

As is indicated by the name Multiphysics, several different branches or sub-branches of physics can be applied, as needed, simultaneously or serially to achieve the solution of a given model. Diverse models that demonstrate the application of the Multiphysics concept through the incorporation of multiple different Physics Interfaces in a given model will be explored in detail as this book progresses.

The modeler has now been shown fundamentally how to select a set of Space Coordinates and how to access Physics Interfaces.

If the modeler is using a Macintosh, Select > COMSOL Multiphysics menu > Quit COMSOL Multiphysics.

If the modeler is using a PC, Select > File > Exit.

Basic Problem Formulation and Implicit Assumptions

A first-cut problem solution is the equivalent of a back-of-the-envelope or on-a-napkin solution. Problem solutions of that type are more easily formulated, more quickly built, and typically provide a first estimate of whether the final solution to the full problem will be deemed to be (should possibly be) within reasonable time constraints or budgetary bounds. Creating first-cut solutions will often allow the 5.x modeler to easily decide whether or not it is worth the additional effort and cost needed to build a fully implemented higher dimensionality model.

A modeler can generate a first-cut problem solution as a reasonable first estimate, by choosing initially to use a lower dimensionality coordinate space than 3D (e.g., 1D, 2D Axisymmetric, etc.). By choosing a low-dimensionality Space Dimension, a modeler can significantly reduce the ultimate time needed to achieve a detailed final solution for the chosen prototype model. However, both new modelers and experienced modelers must be especially careful to fully understand the underlying (implicit) assumptions, unspecified conditions, and default values that are automatically incorporated into the model by simply selecting a lower dimensionality Space Dimension.

Reality, as we currently understand it, comes in four basic dimensions, three space dimensions (X-Y-Z), and one time dimension (t) (e.g., X-Y-Z-t, r-ϕ-θ-t, etc.). Relativistic effects can typically be neglected in most cases, except where high velocity or ultra-high accuracy is involved. Models with relativistic effects will not be covered in this book.

NOTE

Relativistic effects typically only become a concern for bodies in motion with a velocity approaching that of the speed of light (~3.0 × 108 m/s) or for ultra-high resolution time calculations at somewhat lower velocities.

The types of calculations presented within this book are typically for steady-state models, quasi-static models, and for relatively low-velocity transient model solutions.

NOTE

In a steady-state model, the controlling parameters are defined as numerical constants and the model is allowed to converge to reach the final equilibrium state defined by the specified constants.

In the quasi-static methodology, a model solution to a problem is found by finding an initial steady-state solution, not the final steady-state solution. Then, the modeling constants are incrementally modified. That incremental change in the constants moves the modeling solution toward the desired final steady-state solution.

In a transient solution model, all of the appropriate variables in the model are a function of time. The model solution builds from a set of suitable initial conditions, through a set of incremental intermediate solutions, to a final solution.

1D WINDOW HEAT FLOW MODELS

Consider, for example, a brief comparison between a relatively simple 1D heat flow model and the identical problem as a 3D model. The following models are those of a single pane, a dual pane, or a triple pane window mounted in the wall of a building on a typical winter’s day. The basic question to be answered is: Why use a dual pane window rather than a single pane window or a triple pane window?

1D 1 Pane Window Heat Flow Model

1D 1 Pane Window Model Analysis and Conclusions

TABLE 1.2 1 Pane Window Calculation Results.

NameCoordinate Location (x)TemperatureInterior Surface0.0e−3[m]≈35.926 °FMidpoint2.5e−3[m]≈35.000 °FExterior Surface5.0e−3[m]≈34.074 °FInterior ΔT0.0e−3[m]≈34.074 °FExterior ΔT5.0e−3[m]≈34.074 °FWindow ΔT0.0−5.0e−3[m]≈1.852 °FRoom ΔT0.0 to −∞ [m]≈34.074 °FExterior ΔT5.0e−3 to ∞ [m]≈34.074 °F

The modeler should notice that this calculation shows that for a 1 Pane Window, the temperature drop across the Pane is approximately 2 degrees Fahrenheit. It also shows that the Interior Surface will be perceptibly cold to the touch. The 1 Pane Window will, due to the temperature difference between the Interior Window Pane surface and the room air, cause condensation of the water vapor in the room on the surface.

1D 2 Pane Window Heat Flow Model

Now that the modeler has had an initial experience in model development, it is time to consider, for example, how the addition of a second Pane in series with the first Pane will alter the heat flow through the window, given the same environmental conditions. That comparison is relatively simple to implement in a 1D heat flow model. The following model is similar to the first model, but sufficiently different that the modeler should build it with care.

Run 5.x > Click the Model Wizard button > Select 1D in the Model Wizard.

Click the Heat Transfer twistie.

Select > Heat Transfer in Solids (ht) in the Add Physics window.

Click > Add button.

Click > Study (right pointing arrow).

Select General Studies > Stationary.

Click > Done button.

FIGURE 1.41 Initial 5.X model build for the 1D 2 pane window heat transfer.

Figure 1.41 shows the initial 5.x model build for the 1D 2 Pane Window Heat Transfer Desktop Display.

Save the Model as > MM2E5X_1D_W2Pane.mph. See Figure 1.42.

FIGURE 1.42 Saved initial model build MM2E5X_1D_W2Pane.mph.

Figure 1.42 shows the saved initial model build for MM2E5X_1D_W2Pane.mph.

Now that the model has been saved, the modeler needs to enter the constant values needed for this model.

In the Model Builder window of the Desktop Display,

Click > Global Definitions > Parameters 1.