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Chapter 4: COMSOL Models for Physical Systems

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CHAPTER

4

COMSOL MODELS FOR

PHYSICAL SYSTEMS

OVERVIEW

M

any engineering problems involve either one type of physics or, if they involve multiphysics, can be simplified to a dominant one.

For example, consider the laminar flow of a fluid at a high value of Reynolds number. Since a Reynolds number may be interpreted (see

Reference 4.1) as the ratio of inertia over viscous forces exerted on the fluid, then for a fluid with a high value Reynolds number we can neglect viscous forces and consider the fluid as an ideal frictionless one. When turbulence effects are present, flow becomes more complicated and equivalent turbulence-induced viscous stresses (so called Reynolds stress) should be included. Or in the case of a steel beam, for example, we can neglect the effects of deformation due to shear stresses at the cross-section of the beam and assume that the beam cross-section remains perpendicular to the axis of the beam about which it is bent. These types of approximations in engineering are very common and sometimes are matters of “engineering art” or technical judgment. It takes much experience to simplify a problem without losing the dominant physics and obtain results that are useful in practice and have applications.

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Chapter 1: Introduction

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CHAPTER

1

INTRODUCTION

E

ngineering practice foundations are mathematical models, p

­ hysics principles, and empirical results obtained from experiments for defining design criteria. When we mention physics in this book we mean overall science, which includes all disciplines such as chemistry, biology, etc. An engineer should know the Laws of Physics very well and use the relevant mathematical models and their solutions, either exact or numerical, in practice to design parts, systems, and complex machines that work and function with certain reliability for an assumed lifetime.

Knowing the exact behavior of a complete system or a system component under actual “real life” type loads is an extremely difficult task.

Hence engineers use a safety/design factor to overcome the probability of weaknesses and defects in materials used or extreme loads applied to a system or component under extreme conditions. For example, an engineer who designs a curved beam for a given load assumes that the material is free from defects and the load does not exceed the corresponding design limits. However, that beam might be subject to extreme dynamic loads due to vibration resulting from a strong earthquake, and subsequently fail. Designing for extreme cases is “possible” but is neither practical nor economical. Safety factors should be considered in accordance with design codes, which usually address the minimum requirements. To assist in this task, an engineer may use modeling tools to simulate the behavior of the curved beam such that the beam’s strength is sufficient to resist the realworld loads applied to it. The modeling task and application of software tools are becoming more common in modern engineering practices, as shown in

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Chapter 3: COMSOL—A Modeling Tool for Engineers

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24 • COMSOL for Engineers

Another major feature of COMSOL is the ability to solve any PDE/

ODE that users might have and which may not fit into classical governing equations (e.g., wave, heat, equilibrium). A new feature in version 4, called (0D), allows users to solve problems that do not have space as a relevant defined dimension, such as electric or thermal equivalent networks.

The most recent feature to become available allows COMSOL to be run directly through a CAD software package interface, such as SolidWorks and some Autodesk products.

The author’s experience with this package includes its ongoing improvement in features and especially the solvers, which makes it an efficient modeling tool for small- to medium-size problems (in terms of geometrical size) and with multiphysics involved. In COMSOL, users can see the governing equations for the type of physics they solve right on the interface—a feature that is very helpful for assigning right values to the variables and boundary conditions as well as knowing what type of equations you are solving using the finite element method. Building the geometry of a model is possible either by using CAD facilities available in COMSOL or using live communication modules such as, LiveLink. LiveLink modules are available for major CAD packages (e.g., Inventor, SolidWorks, SpaceClaim) as well as MATLAB and Excel. The post-processing features allow users to study and see the modeling results and analyze them using color-coded surface graphs and data line graphs, among others. The Reports feature is very useful, enabling users to generate a file using modeling results in common word processing formats.

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Chapter 2: Finite Element Method—A Summary

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CHAPTER

2

FINITE ELEMENT

METHOD—A SUMMARY

OVERVIEW

F

inite element method (FEM) is the dominant computational method in engineering and applied science fields. Other methods including finite-volume, finite-difference, boundary element, and collocation are also used in practice. To provide general readers with a background for applications of FEM, either directly or with application of a software tool, we discuss the FEM principles in summary in this chapter. We also refer readers who are interested in further reading on this subject to a selection of available textbooks and references.

As discussed in Chapter 1, modeling has an ancient history. However, since the mid-twentieth century a new definition of modeling has gradually emerged (see References 2.1 and 2.2). This definition is a direct consequence of the development of advanced computational methods as well as huge advances in digital computers in terms of their CPUs and graphics processing power. As a result, computer modeling is synonymous to

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