Multiple Physics, Multiple Approaches

When adding the physics for a multiphysics model, there are a number of ways in which you can handle the physics setup. In COMSOL Multiphysics, there are three different approaches:

1. Fully automatic
2. Manual with predefined couplings
3. Manual with user-defined couplings

Each approach is advantageous for different modeling scenarios and varies in terms of ease of implementation and amount of effort required from the user. The manual approach involves selecting the individual physics interface and then adding the multiphysics couplings. This is typically done in a sequential order and by conducting multiple standalone analyses to incrementally build up to the multiphysics problem. The user-defined couplings involve including user-defined expressions, equations, or functions to combine the physics in your model when there are no predefined multiphysics couplings available. In this post, we will focus on the fully automatic approach and address the two others in subsequent blog posts.

The Automatic Approach for Defining a Multiphysics Model

The fully automatic approach is a highly recommended way to add physics for a multiphysics model. By automatic, we mean to use the predefined multiphysics interfaces that are built in and readily available in COMSOL Multiphysics, as well as automatically preconfigured for the multiphysics interaction being simulated. These interfaces are available for several disciplines of engineering and type of application area, including electromagnetics, structural mechanics, acoustics, fluid flow, heat transfer, and chemical engineering.

Multiphysics interfaces make the modeling process simple. Their use automatically adds all of the necessary physics and couplings to your model at once. Additionally, they automatically include the optimal element discretization settings for the combination of physics in a model. In these ways and more, the automatic approach lets you more quickly dive into defining the physics for a multiphysics model without having to get bogged down with the small details or do all of the work from scratch.

For a complete list of the multiphysics interfaces available to you for any COMSOL products you hold a license for, you can view the Specification Chart. There are a number of ways you can browse the chart, as noted in a previous blog post. Simply select the application area your module(s) fall under or the modules themselves (or both) and scroll down to the Predefined Multiphysics Interfaces section to see the list of built-in interfaces.

The Specification Chart with the Acoustics application area selected.

Below, we highlight one example of the numerous multiphysics interfaces included in the software, as well as what it can be used for.

Multiphysics Modeling: A Plasma Example

Perhaps you work in the plasma sciences and need to build a multiphysics model observing several effects, such as in plasma arc welding applications. Modeling this complex process requires including and coupling electricity, magnetism, heat transfer, and fluid flow. In COMSOL Multiphysics, you can use the Combined Inductive/DC Discharge multiphysics interface to automate the entire physics setup, thus enabling you to define the multiphysics phenomena present in plasmas in a significantly easier way.

The Combined Inductive/DC Discharge multiphysics interface selected and added under the Plasma branch of the Physics Wizard, with the branch containing the multiphysics couplings expanded.

Modeling Resistive Heating and Thermal Expansion Using the Automatic Approach

Let’s use the thermal-electrical-mechanical (“tem”) version of the thermal microactuator tutorial model found in the Application Gallery on the COMSOL website, to demonstrate the automatic approach for adding the physics for a multiphysics model.

Note: For the sake of brevity, assume that all steps in the modeling workflow, apart from defining the physics, are complete. Thus, the geometry, materials, mesh, and study are already set up. Additionally, we will narrow our focus to defining the multiphysics aspects of the model and not go into detail regarding how to add and define specific boundary conditions for the constituent physics interfaces.

First, we define the physics for Joule heating and thermal expansion. A voltage is supplied to the bottom of the anchor on the middle arm. The conducted current is resisted as it flows through the actuator, resulting in electric heating, which raises the temperature of the device and causes it to deform. The base of all of the anchors are fixed, but the dimples enable the arms to translate in the xy-plane.

The thermal microactuator geometry with the parts labeled. The device is made of polysilicon.

To study the electrical-thermal-structural interaction that occurs, we can use the predefined Joule Heating and Thermal Expansion multiphysics interface. To include this interface in the simulation:

1. Open the Add Physics window
2. Expand the Structural Mechanics branch
3. Select and add the Joule Heating and Thermal Expansion interface to the model component

The Joule Heating and Thermal Expansion multiphysics interface being added to a model component.

Upon adding Joule heating and thermal expansion to a model, notice that several physics nodes have also been added to the model tree. These include the Electric Currents interface; Heat Transfer in Solids interface; Solid Mechanics interface; and a multiphysics node containing the couplings for electromagnetic heating, the temperature field, and thermal expansion.

A screenshot of the physics setup that is automatically added when using the automatic approach.

The electromagnetic heating multiphysics coupling passes the resistive heating induced from the electrical problem (the current conducted through the actuator) to the heat transfer problem. The temperature multiphysics coupling passes the temperature field from the thermal problem (the heating of the actuator) to the structural problem. Depending on the materials assigned to the geometry, it also passes the temperature back to the electrical problem so as to modify any material properties that are temperature dependent, such as the electrical conductivity. Lastly, the thermal expansion multiphysics coupling combines the thermal problem with the structural problem (the structural stresses and strains induced due to thermal changes) by including the temperature field as a thermal load.

This is why we always recommend using the automatic approach whenever possible. By adding the multiphysics effect you want to analyze, it instantly adds all of the appropriate individual physics interfaces, automatically couples them, and includes the modified settings that are optimal for computing the combination of physics included in your simulation. You don’t have to worry about recalling if you’ve included the correct multiphysics nodes or applied the appropriate settings from one physics interface to another, since it is all automated.

From this point forward, we can focus on defining the physics for each individual physics interface, such as adding any necessary boundary conditions or constraints and editing the settings.

To see how to build the complete model using this approach, see the thermal microactuator tutorial model in the Application Gallery.

Speed Up the Model Building Process

Take advantage of the multiphysics interfaces and options available for different physics disciplines that are available in the COMSOL® software, which, in tandem with the multiple setting modifications performed automatically, provide several distinct benefits.

Watch the related tutorial video to see a demonstration of the fully automatic approach. The video uses the Bracket – Thermal Analysis tutorial model as an example, so between this blog post and the video, you get to see two different multiphysics examples!

Stay tuned for future blog posts that will discuss the other two manual approaches, using predefined couplings or user-defined couplings, for setting up a multiphysics model…

Deploy Applications to Anyone, Anywhere with COMSOL Compiler™

COMSOL Compiler is a product for compiling your COMSOL Multiphysics applications into standalone executable files that can be distributed at your discretion. To run a standalone application, users do not need a COMSOL Multiphysics or COMSOL Server license, and COMSOL Runtime™ is included in the executable file. Standalone applications and digital twins can be compiled for the Windows® and Linux® operating systems as well as macOS.

An example of an application that has been compiled with COMSOL Compiler.

When you have a COMSOL Compiler license, a single Executable button is added to the Application Builder user interface. This Executable feature includes a few simple settings, such as:

• The output directory
• Which platforms to compile the application for
• A customized splash screen

For a product that is so simple to use, COMSOL Compiler offers immense benefits, positioning you as not only the software user but also the software creator.

New Product: The Composite Materials Module

The new Composite Materials Module is an add-on to the Structural Mechanics Module. This product brings you modeling tools for analyzing layered composite structures, which is useful for anyone who wants to simulate fiber-reinforced plastic, laminated plates, and sandwich panels, among other use cases.

A wind turbine blade made out of a laminated composite material. The model was created with the new Composite Materials Module.

The technology behind the module is based on the layerwise theory and equivalent single-layer theory for modeling composite shells. You can use this functionality to simulate first-ply failure, linear buckling, delamination, and fatigue analysis.

As with any new add-on product, the Composite Materials Module is multiphysics ready. You can include the effects of thermal expansion, Joule heating, fluid-structure interaction, and acoustic-structure interaction in your composite layer analyses.

Enhanced Usability in the COMSOL Multiphysics® Software

COMSOL Multiphysics version 5.4 includes several new features to optimize your modeling workflow, especially for users who like to keep a clean and organized model tree. For instance, node groups can be created for many nodes in the model tree, including selections, probes, materials, physics boundary conditions, and derived values. Besides grouping these model tree nodes, you can also keep your parameters more organized with multiple parameters nodes. Keeping your model organized is helpful — especially when working with larger models.

Another new feature that will help you navigate models that have large, complex geometries is the ability to add color themes for geometry selections. Color themes make it easier to distinguish between the different parts of a model in the Graphics window.

An example of a large 3D model where colored selections make it easier to navigate the different model parts.

New Physics-Specific Features in the Add-On Modules

There are many more new features and updates included in the add-on modules in the COMSOL product suite — too many to mention in this blog post, in fact! Stay tuned as we discuss more of the noteworthy features in version 5.4 in the upcoming weeks and months.

For now, head to the Release Highlights to get a full rundown of the new version.

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Minimizing Over Functions

In elementary calculus, we find optimal values for a function of single or multiple variables. We look for a single number or a finite set of numbers that minimize (maximize) a function . In variational calculus, we search for a function that minimizes (maximizes) a functional . In some sense, we can think of this as infinite dimensional optimization. Roughly speaking, a functional takes a function and returns a number. For example, a definite integral is a functional.

In engineering problems, these functionals commonly represent some kind of energy. For example, in elasticity theory, we can find equilibrium solutions by minimizing the total potential energy. This terminology is often carried over to other variational problems, such as variational image processing. We call the functional the “energy” — even when it doesn’t physically represent energy in the usual sense.

A soap film between rings.

Consider a soap film between two rings on the yz-plane whose centers are on the x-axis. We want to find the function that gives us the shape of the soap film when we revolve the graph of the function around the x-axis. This function minimizes the following functional

(1)

More generally, in variational calculus, we are looking for a function that minimizes

(2)

Most engineering problems deal with functionals that contain, at most, a first-order derivative. In the beginning of this series, we will focus on such problems in one spatial dimension. Later on, we will generalize to higher dimensions, higher-order derivatives, and several unknowns. Lastly, maximizing is the same as minimizing the negative, thus we will only talk about minimization in the sequel.

The functional we will be dealing with, unless specified otherwise, is

(3)

Solving Variational Problems

Say you find yourself blindfolded in a valley. How do you know when you’ve reached the bottom? (By the way, removing the blindfolds is not an option.) You might feel the ground around you with your hands and feet, and if every area that you test feels higher than where you stand, you are at the bottom of the valley (at least on a local depression). The same idea is used to check minima in both calculus and variational calculus. In calculus, you test neighboring points, whereas in variational calculus, you test neighboring functions.

The function minimizes the functional if and only if, for every admissible variation , it follows that

(4)

for a small number .

Not every variation is admissible, since every has to satisfy constraints on the solution. For example, for a soap film fixed to wires at the ends, every function we compare with the minimal function has to be fixed to the wires as well. Consequently, we only consider those variations with We will deal with constraints in detail in later blog posts.

A necessary condition for Eq. 4 is

(5)

assuming sufficient smoothness to allow differentiation.

In this series, we are not going to discuss problems with moving or open boundaries. In such cases, we can move the derivative inside the integral and apply the chain rule to obtain

(6)

Note that we vary only the dependent variable and its derivatives, not the spatial coordinate .

If you are interested in problems with moving boundaries or interfaces, check out these blog posts on using the level set and phase field methods for free surface problems and modeling free surfaces with moving mesh.

As shown in Eq. 1 for soap films, we have . Consequently, the variational problem for soap films is to find such that

(7)

Euler-Lagrange Equation

In classical variational calculus, we apply integration by parts to move the spatial derivative from the variation to the solution to obtain the Euler-Lagrange equation

(8)

and use ordinary differential equation (ODE) methods to find the solution.

In higher dimensions, the Euler-Lagrange equation becomes a partial differential equation (PDE).

In our case, we do not need to use the Euler-Lagrange equation, so we will not talk about it any further. The reason is that the finite element method works with the variational formulation. In COMSOL Multiphysics, for example, if you use the Coefficient Form PDE or General Form PDE interfaces to specify the Euler-Lagrange equation, the software internally formulates and solves the corresponding variational equation, so why waste the effort? As we will see later, the variational form also provides natural ways of thinking about sophisticated domain and boundary conditions.

Implementing a Variational Problem in COMSOL Multiphysics®

To specify a variational problem in COMSOL Multiphysics, we use the Weak Form PDE interface. How do we differentiate between the solution and the corresponding test function ? For the latter, we can use the test operator. For example, for the soap film problem, the integrand in the variational formulation has to be entered as sqrt(1+ux^2)*test(u) + u*ux/sqrt(1+ux^2)*test(ux) in the Weak Form PDE node, as shown below.

Specifying a variational problem.

We consider the simple constraint that the soap film is fixed on wire loops on the left and right sides. The radii of the left and right rings are 1 and 0.9, respectively, thus we know the primary variable at both ends. The Dirichlet Boundary Condition node is used to specify such boundary conditions. For numerical reasons, in this particular problem, we provide an initial value of 1 instead of the default 0 in Initial Values 1.

Using the Dirichlet Boundary Condition node to specify known boundary values.

If we compute the solution, we obtain the shape shown below.

Profile of a soap film hanging between two vertical circular wire rings.

Specifying a Simpler Symbolic Problem

In the above example, we carried out the partial differentiation of with respect to and manually. We can avoid unnecessary labor and potential errors by using the symbolic mathematics capabilities of COMSOL Multiphysics.

Using symbolic differentiation to reduce manual work.

Variational Solution Versus Direct Optimization

We can solve a functional minimization problem by direct optimization. In this approach, we do not need to derive the variational problem. On the downside, it requires more computational tools. In COMSOL Multiphysics, for example, direct optimization requires the Optimization Module. If you are interested in direct optimization, check out our blog post on solving the brachistochrone problem using the Optimization Module.

Coming Up Next…

Today, we showed you how to solve variational problems with simple constraints using the Weak Form PDE interface. This interface comes with every COMSOL Multiphysics installation. You can learn more about the weak form in this blog post.

In upcoming posts, we will show how to add more sophisticated constraints, such as point, distributed, and integral constraints. The series will conclude by generalizing to higher spatial dimensions, higher-order derivatives, and multiple fields. Stay tuned!

In the meantime, contact us to learn more about the equation-based modeling features of COMSOL Multiphysics via the button below.

View More Blog Posts in the Variational Problems and Constraints Series

• Part 2: Specifying Boundary Conditions and Constraints in Variational Problems
• Part 3: Methods for Dealing with Numerical Issues in Constraint Enforcement
• Part 4: Methods for Enforcing Inequality Constraints
• Part 5: Image Denoising and Other Multidimensional Variational Problems

The original version of this post was written by Alexandra Foley and published on July 15, 2013. It has since been revised with additional details, animations, and an updated version of the featured model.

The Doppler Effect, Explained

One of the most common ways we experience the Doppler effect in action is the change in pitch caused by either a moving sound source around a stationary observer or a moving observer around a stationary sound source. When the sound source is stationary, the sound that we hear is at the same pitch as the sound emitted from the sound source.

Sound waves propagating from a stationary sound source in a uniform flow (this corresponds to the source moving at constant speed).

When the sound source moves, the sound we perceive changes. Going back to the ambulance example, when an ambulance drives past us, the siren sounds different than it would if we were standing right next to it. The moving ambulance has a different pitch as it approaches, when it is closest to us, and as it passes us and drives away.

As the ambulance moves toward us, each successive sound wave is emitted from a closer position than that of the previous wave. Because of this change in position, each sound wave takes less time to reach us than the one before. The distance between wave crests (the wavelength) is thereby reduced, meaning that the perceived frequency of the wave increases and the sound is perceived to be of a higher pitch. Conversely, as a sound source moves away, waves are emitted from a source that is farther and farther away. This creates an increased wavelength, a decreased perceived frequency, and a lower pitch.

The situation is mirrored when we drive by the siren of an ambulance that is parked. In this instance, the observers (us) move toward the source (the siren) and the sound waves reach us from closer and closer positions as we move.

Visualizing Another Example of the Doppler Effect

Another example of the Doppler effect that is easy to visualize involves waves on a water surface. For instance, a bug rests on the surface of a puddle. When the bug is stationary, it moves its legs to stay afloat. These disturbances propagate outward from the bug in spherical waves.

When the bug starts moving across the water, the water flow around the bug changes. The waves appear closer together when we look at the bug swimming toward us (eek!) and farther apart as it swims away (phew!) The animation above shows the principle for waves (ripples) on water, which move much slower than the speed of sound. The slower speed is why, in this instance, the Doppler effect can be seen with the naked eye.

Simulating the Doppler Effect

By using the COMSOL Multiphysics® software and the add-on Acoustics Module, you can simulate the Doppler effect and measure the change in frequency for a source moving at a certain velocity. Let’s assume that the air surrounding the sound source (the ambulance, in this case) is moving with a velocity of V = 50 m/s in the negative z direction. We also assume that the observer of the sound is standing 1 m from the ambulance as it passes by. In the figure below, we can see the change in the pressure as the ambulance approaches and passes an observer.

In this plot, the distance of the ambulance from the observer is represented on the x-axis. The solid line represents the pressure perceived by the observer of an approaching ambulance and the dashed line shows the pressure as the ambulance gets farther away.

From this plot, we can see how the amplitude of the wave (or pressure) drops off at a faster rate when the ambulance is moving away from an observer compared to when it approaches. The change in the amplitude of the wave depicts how the siren becomes quieter as the ambulance moves away. The rate at which the sound level decreases as the ambulance recedes is much faster than the rate at which the sound becomes louder as the ambulance approaches (as shown in the graph above).

To look at this effect in a different way, we can visualize the sound pressure level around the sound source (remember, the source is effectively moving in the positive z direction).

The sound pressure level around the sound source is represented by colors and contour lines. You can see how the outermost contour runs from well inside the physical domain to the perfectly matched layer, showing that the sound is greater below the source than above it.

Other Examples of the Doppler Effect

The Doppler effect is apparent in many other phenomena. One common example is Doppler radar, in which a radar beam is fired at a moving target. The time it takes for the beam to bounce off the target and return to the transmitter can provide information about a target’s velocity. Doppler radar is used by police to identify people driving faster than the speed limit.

The Doppler effect is also used in the field of astronomy to determine the direction and rate at which a star, planet, or galaxy moves compared to Earth. By measuring the change in the color of electromagnetic waves — called redshift or blueshift — an astronomer can determine a celestial body’s radial velocity. If you notice a star that appears red, it is quite far from Earth — and a visible sign that the universe is expanding!

Other applications that take advantage of the Doppler effect include meteorological forecasts, sonar, medical imaging, blood flow measurement, and satellite communication.

Next Steps

Click the button below to try simulating the Doppler effect. With a COMSOL Access account and valid software license, you will be able to download the MPH-file for the example featured in this blog post.

Additional Resources

• Learn about the man who discovered the Doppler effect in this blog post on Christian Doppler
• Explore the Acoustics Module

Using Equation-Based Modeling in Your Simulations

With equation-based modeling, part of the core functionality of COMSOL Multiphysics, you can create your own model definitions based on mathematical equations and directly input them into the software’s graphical user interface (GUI).

These abilities give you complete control over your model, so you can tailor it to your exact specifications and add complexity as needed. To provide this flexibility, COMSOL Multiphysics uses a built-in interpreter that interprets equations, expressions, and other mathematical descriptions before producing a model. In addition, you can use tools like the Physics Builder to create your own physics interfaces, or the Application Builder to create entire new user interfaces.

Example of entering a custom partial differential equation into the COMSOL Multiphysics GUI.

Using this functionality, you can work with:

• Partial differential equations (PDEs)
• Ordinary differential equations (ODEs)
• Algebraic equations
• Differential algebraic equations (DAEs)
• Arbitrary Lagrangian-Eulerian (ALE) methods
• Curvilinear coordinate computation
• Sensitivity analysis

There is no limit to how creative you can be when setting up and solving your models with equation-based modeling, which expands what you can achieve with simulation. To show this functionality in action, let’s take a look at three examples…

Example 1: The KdV Equations and Solitons

In 1895, the Korteweg-de Vries (KdV) equation was created as a means to model water waves. Since the equation doesn’t introduce dissipation, the waves travel seemingly forever. These waves are now called solitons, which are seen as single “humps” that can travel over long distances without altering their shape or speed.

Today, engineers use the KdV equation to understand light waves. As a result, one of the main modern applications of solitons is in optical fibers.

Simulating the KdV Equations with Equation-Based Modeling

To solve the KdV equation in COMSOL Multiphysics, users can add PDEs and ODEs into the software interface via mathematical expressions and coefficient matching. It’s also possible to easily define dependent variables and identify coefficients via the General Form PDE interface.

With this setup, users are able to model an initial pulse in an optical fiber and the resulting waves or solitons. According to the KdV equation, the speed of the pulse should determine both its amplitude and width, which can be observed via simulation. In addition, the simulation reveals that, just like with linear waves, solitons can collide and reappear while maintaining their shape. This counterintuitive finding would be challenging to observe without simulation.

If you want to learn more about this example, see the KdV equation model in the Application Gallery.

Simulation showing how solitons maintain an intact shape when colliding and reappearing.

Example 2: Electrical Signals in a Heart

Moving on to a medical example, let’s see how simulation can be used to understand the rhythmic patterns of contractions and dilations in a heart. The rhythmic contractions are triggered when the heart passes an ionic current through the muscle. During this process, ions flow through small pores that exist in an excitation (open) or rest (closed) state within the cellular membrane. As such, to gain a better understanding of heart patterns, the electrical activity in cardiac tissue needs to be examined.

Studying the electrical signals in a heart is not a simple process and involves modeling excitable media. To address this challenge, users can implement two sets of equations to describe various aspects of the electrical signal propagation. One such example is the Electrical Signals in a Heart model, provided through the courtesy of Dr. Christian Cherubini and Prof. Simonetta Filippi from the Campus Bio-Medico University of Rome in Italy. The equations used in this model, FitzHugh-Nagumo and complex Ginzburg-Landau, are included in the PDE interfaces available in COMSOL Multiphysics.

Using 2 Different PDEs to Analyze Electrical Signal Propagation in Cardiac Tissue

By using the FitzHugh-Nagumo equations to simulate excitable media, it is possible to create a simple physiological heart model with two variables: an activator (corresponding to the electric potential) and inhibitor (the voltage-dependent probability that the membrane’s pores are open and can transmit ionic current). Using these equations and various parameters, users can visualize a reentrant wave that moves around the tissue without damping, which results in a characteristic spiral pattern. In the context of electrical signals, this pattern could generate effects similar to those of arrhythmia, a condition that disturbs the normal pulse of a heart.

Solving the FitzHugh-Nagumo equations at times of 120 (left) and 500 (right) seconds.

The complex Ginzburg-Landau equations help to model some parts of the transition from periodic oscillatory behavior to a chaotic state. During this transition, the amplitude of oscillations gradually increases and the periodicity decreases. These equations are used to study the dynamics of spiral waves in excitable media. The results show the diffusing species and the characteristic spiral patterns, which increase in complexity over time.

Solving the complex Ginzburg-Landau equations at times of 45 (left) and 75 (right) seconds.

Using both sets of equations enables the visualization of complicated real-world phenomena.

Example 3: A Lorenz Attractor

Lastly, let’s take a look at the Lorenz equations, which were developed to serve as a simple mathematical model for atmospheric convection. When using certain parameter values and initial conditions, a system of ODEs (a Lorenz system) has chaotic solutions. One such solution is a Lorenz attractor, which looks like a figure eight or butterfly when plotted in the phase space.

Example of the typical shape of a Lorenz attractor.

Using ODEs to Model a Lorenz Attractor

To solve the Lorenz attractor model, the Lorenz equations — a system of three coupled ODEs that contain three degrees of freedom — need to be added into the software. This is a straightforward process when using the Global ODEs and DAEs interface to define the Lorenz system.

Next, users can view an initial solution close to the attractor and study the growth of a very small perturbation to this initial data. The results (seen in the left image below) visualize how the difference between the original and perturbed problems increases over time. In addition, the simulation demonstrates that with the chosen parameter values, the Lorenz system behaves like a Lorenz attractor, with results showing the butterfly shape that these attractors are known for.

The differences between the unperturbed and perturbed solutions over time (left) and the normal pattern for a Lorenz attractor (right).

Next Step

Watch a quick video introduction to COMSOL Multiphysics to learn more about the software’s key features. When you’re ready, request a software demonstration.

COMSOL Multiphysics® and COMSOL Server™ Version 5.3a Updates

Before getting into the physics-specific updates, let’s take a look at a couple of the features that can be employed by every COMSOL Multiphysics user. The copy-paste functionality has been extended to physics interfaces and even entire model components. For example, when working on a model that has multiple components running similar but varied analyses, instead of redefining several of the same physics boundary conditions over again, you can copy your already defined physics interface into a different component and make slight changes. This could become the most used feature of the 5.3a release (learn more about it on the COMSOL Desktop® page in the Release Highlights).

Demonstrating the copy-paste functionality.

A new color table has been added that may be the most interesting new feature. It’s called Cividis and was created by Ryan Renslow, Chris Anderton, and Jamie Nuñez of the Pacific Northwest National Laboratory. The color table was designed for people with color vision deficiency, optimized for those with red-green color blindness, but useful for everyone when showing scalar values. Find more info on the Cividis color table on the Postprocessing and Visualization Updates page in the Release Highlights. (We are also publishing an in-depth blog post about Cividis soon, so stay tuned!)

A loudspeaker model visualized with the new Cividis color table.

The release of COMSOL Server version 5.3a also provides the most comprehensive list of new features since its debut. Administrators can set up automatic login for repeat users, eliminating the need to sign in every time they want to run an application. App creators can edit the title, description, and thumbnail for an app directly from the COMSOL Server user interface. A new app has been added to validate cluster setup; to test settings for cluster and remote computing. Notifications can be sent to users to inform them of new or updated apps. All of these features and more can be found on the COMSOL Server Updates page in the Release Highlights.

Images from the Release, Highlighting Physics-Specific News

Next Steps

To view all of the news in detail, check out the Release Highlights pages for COMSOL Multiphysics and COMSOL Server version 5.3a.

Be sure to keep up with the blog as well; we will be covering new features and tutorials in-depth over the next few months.

Updated List of Blog Posts Covering 5.3a Topics in Detail

• A Simulation Color Table for Engineers with Color Vision Deficiency
• Calculating Thermodynamic Properties for Liquids and Gases
• How to Model Moisture Flow in COMSOL Multiphysics®
• How to Model Roller Bearings in COMSOL Multiphysics®
• Using the Inflow Boundary Condition in Nonisothermal Flow Simulations
• How to Use the Boundary Element Method in Acoustics Modeling
• Using the Algebraic Multigrid (AMG) Method for Large CFD Simulations

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How to Set Up RF, Microwave, and Millimeter-Wave Circuits in COMSOL Multiphysics®

Regardless of the characteristics of your device, whether it is resonating, radiating, or attenuating, the rule of thumb for your electromagnetics simulation is simple: You need to make the modeling process efficient, as discussed in a previous blog post. Even if you are confident in your design, it is better to start with a simple structure so you can ensure that the modeling process is correct for the given basic geometry before adding complex design elements.

When electromagnetic waves are not radiating but captured in a device, they are guided, confined, or attenuated through a structure. The physical phenomena inside passive microwave and millimeter-wave circuits can be addressed via electromagnetic (EM) simulation by solving Maxwell’s equations.

An electromagnetic wave confined via a microstrip meander line connected to SMA receptacles.

To efficiently describe the passive circuit measurement in a real lab environment, it is necessary to pick the right physics features and boundary conditions. It is challenging to accurately reflect real-world conditions in a model, while also being both time and memory efficient.

Below, we summarize some real-world measurement and testing situations and the possible modeling features to choose in COMSOL Multiphysics:

When setting up a passive circuit model, you don’t need to have a lot of complicated boundary conditions in the beginning of your modeling process. You can build a circuit, especially for low frequencies, using just two features in COMSOL Multiphysics and the RF Module. Let’s go over this process using a microstrip line example.

The geometry of a microstrip line.

A microstrip line circuit is composed of five objects, each with a specific purpose:

1. Block that acts as a metallic enclosure filled with air
2. Block that acts as a substrate
3. Rectangle that acts as a printed metallic trace
4. Rectangle that acts as lumped port 1 for excitation
5. Rectangle that acts as lumped port 2 for termination

The materials include a dielectric substrate (user defined) and air, which encloses the entire domain.

You must then choose the correct physics features for the model:

• Perfect Electric Conductor boundary condition, which mimics metallic surfaces with a high conductivity
• Lumped Port boundary condition, which excites or terminates the circuit and measures S-parameters

The metallic part in the microstrip line circuit: top copper trace and ground plane (left) and a Lumped Port boundary condition at one end of the microstrip line (right).

The simulation may only take a few seconds to solve at the intended operating frequency. You will get the default S-parameter evaluation and electric field distribution plot for a single frequency. When the simulation is performed on multiple frequencies, the electric field distribution plot, S-parameter plot, and Smith plot are plotted by default. You can also evaluate port impedance as necessary.

The electric field distribution on the surface of the substrate is visualized with the mesh view of the microstrip line.

The RF Module lets you add other physical effects to your electromagnetics simulations. This means that you can study all of the physical phenomena and define the properties to suit your unique needs. When validating your device design, it is useful to take into account not only a single physical effect, but also multiple physics — and to know the underlying physics involved.

Find more information on how to capture the details of underlying physics in previous blog posts on modeling metallic objects in wave electromagnetics problems and using perfectly matched layers and scattering boundary conditions for wave electromagnetics problems.

Next, you can start to design your microwave and millimeter-wave circuit, whether it is a coupler, power divider, filter, or wideband device.

Developing Microwave and Millimeter-Wave Circuits with the RF Module

The Application Library for the RF Module offers RF, microwave, and millimeter-wave examples for a wide range of applications. The tutorial models include conventional devices, such as basic transmission lines, couplers, power dividers, filters, and coils, as well as multiphysics examples like microwave ovens, SAR calculation, tunable filters, and more. There is also a circulator example that uses ferrite material properties.

Traditional Filter, Coupler, and Power Divider Examples

Couplers, power dividers, and filters are fundamental devices in microwave engineering. They are good introductory examples for learning how to model microwave circuits in the COMSOL® software. It’s also relatively easy to validate your simulation results with well-known textbook examples.

Textbook examples of a branch line coupler, quadrature 90° hybrid (left) and Wilkinson power divider (right).

Filters are essential in microwave circuits to refine signals in RF and microwave systems.

A coupled line filter (left) and waveguide iris filter (right).

Passive devices are not limited to the conventional shape of circuits on PCBs. For instance, another device is made up of periodic structures that can provide a frequency response for a bandpass or bandstop, called a frequency-selective surface, split ring resonator. In the example below, the signals around the center frequency are the only ones that are able to pass through the layer of the periodic complementary split ring resonator.

A model of a frequency-selective surface, complementary split ring resonator. You can use periodic boundary conditions to model an infinite 2D array on one unit cell.

Adding More Physical Effects for Multiphysics Simulation

Due to heat expansion, external force, or the behavior of piezo materials, the structure of a circuit can be distorted. The surface can be warped nonuniformly, which can cause nonuniform reactance distribution that cannot be simply addressed via a geometric parametric sweep. By including the real-world physical effects in your design, you can accurately analyze complex devices, such as tunable filters controlled by piezoactuators, and realize true multiphysics simulations.

Animated multiphysics examples of a tunable cavity filter controlled by a piezoactuator (top) and a printed low-pass filter affected by external force (bottom).

To perform multiphysics simulations, you just need to combine other physics with your microwave circuits, such as heat transfer to simulate microwave heating or structural mechanics to see how structural deformation affects the electromagnetic behavior of your device. Though you are handling multiple physics, you are still working in the same environment and using the same workflow.

Accelerating EM Simulation Using Reduced-Order Modeling Techniques

Certain EM devices, such as bandpass-filter-type high-Q devices in the frequency domain, are computationally expensive to simulate. The RF Module offers two analysis types to help accelerate bandpass filter modeling: asymptotic waveform evaluation (AWE) and frequency-domain modal analysis.

The Evanescent Mode Cylindrical Cavity Filter tutorial model shows the usage of the AWE method. It is very useful when simulating a single resonant circuit with many frequency points.

These cascaded rectangular cavity filter (left) and CPW bandpass filter (right) examples in the Application Libraries present the advantage of using the frequency-domain modal method when analyzing bandpass-frequency responses of a passive circuit resulting from a combination of multiple resonances. Here, eigenfrequency analysis is key to capturing the resonance frequencies of an arbitrary shape of a device.

Through the AWE or frequency-domain modal method, you can get a very fine frequency response, which could result in a huge file that contains a tremendous size of data. Usually, designers are only interested in S-parameters for passive microwave circuits, so it is not required to save all of the data in the entire simulation domain, but only data associated with the boundaries for S-parameter calculation: lumped port and port boundaries. These boundary sizes are relatively small and the file size of a model running the model order reduction techniques can be reduced a lot by storing the solution only on relevant port boundaries.

Time-Domain Reflectometry in Circuits

The Electromagnetic Waves, Transient physics interface simulates electromagnetic wave propagation in the time domain, where the time-domain reflectometry (TDR) of a microwave circuit can be calculated. By performing a TDR analysis of your device, you can predict the quality of a signal transmitted through the circuit. The voltage distortion due to the coupling between transmission lines and the impedance mismatch from any discontinuity on a transmission line degrades the quality of a signal: essence of signal integrity (SI). Since there are more demands for high-speed interconnect devices handling a higher data rate in the market, there is a fast-growing interest in the signal integrity application area. The examples below illustrate the TDR for two cases: the unwanted coupling between two adjacent microstrip lines and the impedance mismatch from a metalized via, respectively. In both cases, the unwanted effects are manifested in the time evolution of the lumped port voltages.

A microstrip line crosstalk model (left) and its TDR at each port, showing that the higher data rate signal causes the stronger crosstalk on another signal path (right).

TDR analysis can be used to optimize impedance matching properties for the design of high-speed interconnects.

Rapid Prototyping and Analyzing Transmission Line Equations

Interest in the millimeter-wave frequency range is growing due to upcoming 5G mobile network applications that need to support higher data rate communication. Computationally cost-effective simulation helps quickly validate the concept of a prototype. When a waveguide is operating on its dominant mode, 2D modeling techniques help to reduce the simulation time drastically.

A diplexer is a device that combines or splits signals into two different frequency bands, widely used in mobile communication systems. This waveguide diplexer example simulates splitting properties using a simplified 2D geometry.

When the coupling between transmission lines is marginal, the Transmission Line physics interface will save time and resources, too. A simulation that usually takes several minutes to hours can solve in a few seconds with transmission line equations.

A traditional low-pass filter (left) and 4×4 Butler matrix (right).

Once the basic performance of a transmission line circuit is evaluated, the design can be extended to connect to a 3D model.

An 8×1 phased array antenna using an 8×8 Butler matrix at 30 GHz: a combination of fast transmission line analysis and 3D full-wave FEM simulation.

Final Thoughts on Modeling Microwave Circuits

Here, we have discussed different techniques for modeling microwave circuits, as well as examples of devices that can be designed with RF simulation. With this information and the featured examples, you can design passive circuits in COMSOL Multiphysics, while maintaining optimized computational efficiency.

Why Are Surfaces Special?

Surfaces are the locations for what chemists call heterogeneous reactions. These are reactions that involve more than one phase, such as in the catalytic reduction of nitrogen oxides at a solid surface in a catalytic converter. Because phases refer to different immiscible components of a system, heterogeneous processes can only take place at the interfaces where different phases meet.

Surfaces also act as sites for adsorption. This is a process in which a molecule in the adjacent gas or liquid phase becomes localized at the surface, either through intermolecular forces, like van der Waals forces (physisorption), or through a direct chemical bond (chemisorption). Adsorption can allow molecules from the gas and liquid phase to spend more time in close proximity (while they are stuck to the solid surface) than would be the case when they are in the free gas or solution. Also, chemisorption can lower the activation energy required to break chemical bonds in the adsorbed molecule, so that reactions between adsorbed chemical species can proceed through a different mechanism from the case in the free phase. These are two important reasons why solid surfaces can catalyze reactions.

Kinetics of Surface Reactions

The rate of a homogeneous reaction can be measured according to the number of moles of material that react per unit volume and per unit time; this sets the reaction rate in units mol m^-3 s^-1. By contrast, the rate of a heterogeneous reaction depends on the unit area that is available for a reaction, so the reaction rate is expressed in units mol m^-2 s^-1. These are units of molar flux. When setting up a chemical model in COMSOL Multiphysics, a direct way to specify a heterogeneous reaction is to add a Flux boundary condition.

We can imagine a flux of the reactant “into” the surface and, correspondingly, a flux of products “out of” the surface. The figure below illustrates the flux of a reactant measured into a catalytic surface.

Flux of the solution-phase reactant RBr at a series of catalytic surfaces in a microreactor. Since the reaction consumes the RBr reactant, the inward flux to the solution is negative.

The rate of a surface reaction is expressed according to a rate law as a function of reactant and product concentrations as well as other local properties, such as temperature or pressure. Many of the same rate laws encountered for homogeneous reactions can also apply to heterogeneous reactions. If you’d like to review some different kinetic rate laws, take a look at this introductory blog post on chemical kinetics.

As an example of a rate law for a heterogeneous reaction, a first-order reaction consuming species A might set the following inward flux of A:

Here, N_A is the flux (mol m^-2 s^-1) and c_A is the concentration of the reactant located at the surface, but measured in the adjacent phase (mol m^-3).

Hence, the first-order rate constant has units m s^-1, representing an effective “velocity” of the chemical species into the boundary due to its reaction. It is common to find rate constants for surface processes, including those in electrochemistry, represented in units of velocity.

The Hydrocarbon Dehalogenation in a Tortuous Microreactor tutorial model is a good example of how surface processes are modeled. Here, two competing surface reactions are both active on specific catalytic surfaces represented by a Flux boundary condition in the Transport of Diluted Species interface. The respective rates of these processes are controlled by Arrhenius kinetics with different activation energies. Compared with the hydrogenation reaction, the competing dimerization reaction has a higher activation energy, hence it is accelerated more rapidly with an increasing temperature. The temperature dependence of the product ratios could be predicted with this model and compared with experimental data.

The image below shows a section of the Model Builder with the Flux boundary condition. As you can see, this condition applies specifically to the catalytic surfaces where the reactions take place. The reaction rates are defined using the kinetic mechanism specified in the coupled Chemistry interface. See the following section for more details on how this interface is used for surface reactions.

The Model Builder, showing the settings for a Flux boundary condition that defines the flux of three solute species at catalytic surfaces due to a heterogeneous reaction.

Adsorption and Transport at Surfaces

In an adsorption process, the flux of a chemical reactant into the surface is not balanced by a flux of a chemical reactant out of the surface. Instead, the surface concentration (mol m^-2) of the adsorbed reactant changes. Consider the adsorption of a chemical species from the gas phase. We can write this as an equation of the form: A(g) A(ads).

The conservation of mass can be expressed by writing:

This is a transport equation on the surface. The quantity N_ads is the flux of the adsorbed species tangent to the surface due to processes such as surface diffusion. In many cases, this surface diffusion is approximately zero. On the right-hand side, the flux acts as a reaction term to apply a source or sink of mass for the adsorbed species.

This equation is implemented in the Surface Reactions interface in COMSOL Multiphysics. Typically, it can be coupled to the Transport of Diluted Species or Transport of Concentrated Species interface for the mass transport in the adjacent gas or liquid phase. To see an example of the use of this interface, take a look at this blog post on modeling protein adsorption.

In this example, a section of an ion-exchange column is simulated to predict the rate of uptake of two protein species from a flow of an aqueous solution onto an active surface. The spherical ion-exchange beads are explicitly included in the geometry and the boundaries at their surface are used as the location for the adsorption reaction. The model includes four interfaces:

1. Chemistry interface for the chemical mechanism
2. Transport of Diluted Species interface for the concentrations of dissolved species in the flow (including heterogeneous reactions)
3. Surface Reactions interface for the surface concentrations of adsorbed species on the ion-exchange bead surfaces
4. Laminar Flow interface to predict the velocity field in the flow of water, hence the contribution of convection to transport of dissolved species

A typical plot is shown below, illustrating the adsorption extent after 30 seconds of exposure to the flow.

The surface concentration of an adsorbed protein in an ion-exchange column after 30 seconds of exposure to a flow of water containing the dissolved protein. Concentrations are higher toward the upper (inflow) face of the ion-exchange column, where the catalytic surfaces are more accessible to the dissolved protein in the flow.

Note that in this example, the surface diffusivity of the adsorbed species is set to zero. Therefore, the rate of change of the adsorbed species concentration at any point depends only on the local adsorption and desorption fluxes. The dynamic equilibrium between the kinetic adsorption and desorption processes is often described using a mathematical relation called an adsorption isotherm (one example is the Langmuir isotherm). We’ll discuss the choice and implementation of isotherms in more detail in an upcoming blog post in this series.

Defining Surface Reactions in the Reaction Engineering and Chemistry Interfaces

When using the Reaction Engineering interface to describe the behavior of a perfectly mixed reactor, surface reactions can be included by adding any surface-bound species to the reaction mechanism. Surface-bound species are indicated by the suffix (ads) at the end of a species name. Any reaction involving a surface species is understood to take place at a surface and thus has a reaction rate measured in mol m^-2 s^-1.

The option Generate Space-Dependent Model from a Reaction Engineering interface creates a Chemistry interface that stores the same mechanism as well as a Transport of Diluted Species interface or another chemical species transport interface for the bulk phase. The concentrations of adsorbed species are not traced by default, but you can add an extra Surface Reactions interface to do so.

There are two options for treating surface reactions. If they are assumed to apply at boundaries that will be explicitly resolved in the geometry of the space-dependent model, a Flux condition is added to the chemical species transport interface to describe the heterogeneous processes.

Fast reactions at surfaces can also be described with the help of a Surface Equilibrium Reaction feature. This constrains the concentrations of reactants and products at an interface to maintain a certain ratio, according to the equilibrium constant of a reaction. The screenshot below shows the Model Builder settings in the protein adsorption example.

The Model Builder, showing the settings for a surface reaction within the specification of a kinetic mechanism through the Chemistry interface. The reaction is assumed to be very fast so that equilibrium according to a specified equilibrium constant is maintained.

Concluding Thoughts on Modeling Surface Reactions

In this blog post, we have introduced some basic kinetic theory for surface reactions as well as the conservation of mass between concentrations of species localized at a surface and concentrations of the same chemical species in an adjacent bulk phase (gas or liquid). The Flux boundary condition and Surface Reactions interface in COMSOL Multiphysics are used to add these reactions to chemical models. So far, we have only considered cases where the boundary has a simple shape so that it can be included in the geometry directly in order to define the location of surface reactions.

Stay tuned for the next blog post in this series, where we will discuss the theory of how reactions at the surfaces of porous media with complicated surface geometries can be approximated using homogenization.

Further Resources

• Learn more about modeling chemical reactions on the COMSOL Blog:
  • How to Model Heat and Moisture Transport in Porous Media with COMSOL®
  • Sensing the Bio in Biosensor Design with a Simulation App
  • Modeling Approaches in Heterogeneous Catalysis

EEDF and Plasma Modeling

While it’s important to model plasma accurately (to match experimental results, for example), we also want the model to be as simple as possible. To keep the model simple, we typically choose a Maxwellian or Druyvesteyn EEDF and a constant electron mobility. The transport properties can then be computed automatically using the Einstein relation. However, at low-pressure discharges, the EEDF often deviates significantly from a Maxwellian EEDF. At higher pressures, high accelerating fields might also result in a non-Maxwellian distribution.

For a more accurate EEDF shape, we can solve the Boltzmann equation (typically, the two-term approximation). The COMSOL Multiphysics® software even provides a dedicated interface for this purpose — appropriately named the Boltzmann Equation, Two-Term Approximation interface.

Comparison of Maxwell, Druyvesteyn, and Boltzmann distribution functions for a typical argon discharge at a mean electron energy of 5 eV.

We then run into a catch-22. To compute the EEDF with the Boltzmann equation, we must have the electron density. This is the result of the plasma model, which we solve using the EEDF and transport properties.

How can we get out of this loop? Well, one possibility is to use an iterative process. The first step is to guess the mean electron density, using it to calculate the EEDF with the Boltzmann Equation, Two-Term Approximation interface. Next, we compute the plasma model with the EEDF to get a new mean electron density. We must then recompute the EEDF, the plasma model, and so on.

Performing this procedure one step at a time can be annoying. A better option is to use the Application Builder to design an app that automates this process. Let’s explore one such example from the Application Gallery.

Using an App to Model Plasma with a Non-Maxwellian EEDF

To use the app, we only need an initial guess for the electron density and the mole fraction of excited argon atoms. The values for the applied voltage, reactor length, gas temperature, and other variables can be adapted. We can also add a quick circuit at the electrode on the right side. Next, we select our desired type of EEDF (Boltzmann, Maxwellian, Druyvesteyn, or generalized).

When we click the Compute all button, the app automatically alternates between calculating the plasma model and the EEDF model. Each time, the computed mean electron density of the plasma model is used as an input parameter to determine a new EEDF. This continues until the deviation in mean electron density before and after calculating the plasma model falls below a user-defined value. Alternatively, we can solve only the Boltzmann or plasma model by clicking either the Compute Boltzmann only button or the Compute Plasma only button, respectively.

As for our results, we get the plasma species densities, like electron and ion density, as well as the electron temperature. The EEDF and electron transport properties for different mean electron energies are also shown. Using the EEDF at specified position tab under Boltzmann Analysis, we can examine the EEDF at each point in the reactor geometry as well as compare the computed EEDF with a Maxwellian EEDF. In addition, the app evaluates mean values for species densities and mean electron energy.

The Underlying Plasma and EEDF Models

Let’s take a look at the underlying models. To model plasma with a non-Maxwellian EEDF, we need two models:

1. A plasma model
2. A model to compute the EEDF

The plasma model used in the app is a simple 1D glow discharge model, similar to the one in the Application Gallery. The left electrode of the plasma reactor is grounded, while a voltage is applied to the right. We set the value of this voltage (as well as other parameters like the reactor length, gas pressure, and temperature) in the app.

The model tree for the Plasma interface.

We also use a simple 1D model to compute the EEDF. Here, the x-axis does not represent a space coordinate, but rather the electron energy. We then perform a parametric sweep over the mean electron energies. As an example, this model corresponds with the Argon Boltzmann Analysis tutorial.

The model tree for the Boltzmann Equation, Two-Term Approximation interface.

It is important to note that this app uses the mean electron density in the reactor to compute the EEDF. So, the EEDF only depends on the average properties of the discharge. In reality, the EEDF should be computed at every point in the reactor geometry. This would be tremendously expensive to compute and is thus avoided in this example.

Learn More About Plasma Modeling Using COMSOL Multiphysics®

• Browse these blog posts related to plasma modeling:
  • The Boltzmann Equation, Two-Term Approximation Interface
  • Electron Energy Distribution Function
  • Microwave Plasmas
  • Thermodynamic Equilibrium of Plasmas
  ]]> https://www.comsol.de/blogs/introduction-to-plasma-modeling-with-non-maxwellian-eedfs/feed/ 3 https://www.comsol.de/blogs/introduction-to-modeling-stress-linearization-in-comsol-multiphysics/ https://www.comsol.de/blogs/introduction-to-modeling-stress-linearization-in-comsol-multiphysics/#comments Tue, 16 May 2017 08:02:44 +0000 http://com.staging.comsol.com/blogs?p=217941 In some applications, it is necessary to approximate a general 3D stress state by a set of linearized stresses through a cross section of a thin structure. This is important for applications like the analysis of pressure vessels, fatigue analysis of welds, and determination of reinforcement requirements in concrete. In this blog post, we discuss why such an approach is useful as well as how to compute linearized stresses in the Structural Mechanics Module for COMSOL Multiphysics® version 5.3.

  Defining Membrane and Bending Stresses

  When performing a structural analysis with plate or shell elements, there is an underlying assumption that the variation of the in-plane stresses through the thickness is linear. In a local coordinate system, where z is oriented along the normal to the shell surface, it is thus possible to write

  \sigma_{ij}(x,y,z) = \sigma_{ij,m}(x,y)+\frac{2z}{d}\sigma_{ij,b}(x,y)

  where d is the thickness. The indices i and j can be either x or y. In this decomposition, is called the membrane stress and (or ) is called the bending stress.

  
  The decomposition of a linear stress distribution into a membrane stress and bending stress.

  For the other stress components, shell theory implies that

  \sigma_{zz} \approx 0

  and

  \sigma_{iz} \propto \left( 1- \left( \frac{2z}{d} \right)^2 \right).

  Unless the shell is thick, the transverse shear stresses, σ_iz, are significantly smaller than the in-plane stresses.

  Why Membrane Stresses Are More Dangerous

  A membrane stress has the same value all through the section. If the material is assumed to be elastoplastic with no hardening, then all points reach the failure stress at the same time. The load that causes incipient plasticity is thus also the failure load.

  
  The stress-strain curve for an elastoplastic material with no hardening. The variable σ_y is the yield stress.

  Now, consider pure bending with a uniaxial stress state, as in a beam. As long as the material is elastic, the stress distribution is linear through the section, with the value being zero at the midsurface. As the load increases, the stress in the outermost fibers reaches the yield limit. However, the rest of the section is still elastic. It is thus possible to further increase the load without a complete failure.

  
  The stress distribution at incipient yielding (left), partly through yielding (middle), and collapse (right).

  The bending moment at failure is 1.5 times the bending moment at initial yield. Thus, if the allowed stress only takes the maximum stress into account, the risk of collapse is larger for a membrane state than it is for a bending state.

  If we consider a state of mixed bending and tension, it is possible to compute the combinations of moment, M, and axial force, N, which cause failure.

  M=\frac{\sigma_y}{4} \left( d^2 -\left(\frac{N}{\sigma_y}\right)^2 \right)

  
  The stress state at collapse for combined tension and bending.

  The membrane and bending stresses are, for an elastic case, related to the moment and axial force through

  \sigma_m = \frac{N}{d}

  and

  \sigma_b =\frac{6 M}{d^2}.

  By writing the moment and axial force in terms of membrane and bending stresses, we arrive at the following interaction formula:

  \frac{\sigma_b}{\sigma_y} = \frac{3}{2} \left( 1 -\left(\frac{\sigma_m}{\sigma_y}\right)^2 \right).

  When the Detailed Stress Distribution Does Not Matter

  In a full 3D case, the stress distribution differs significantly from linear in the vicinity of geometric discontinuities. This is where the concept of stress linearization becomes important. The sum of the membrane and bending stress provides a linear approximation to the true stress distribution, having the property that the resultant force and moment are preserved.

  
  The linearization of a stress tensor component from a 3D solution.

  In the graph above, the maximum computed stress is 305 MPa. If the stress state is uniaxial — and the yield stress of the material is 350 MPa — this means that 87% of the load giving initial yield has been reached. However, the linearized stress predicts only 64% of the yield stress. The membrane stress contributes 32% of the yield stress.

  If we want to compute a safety factor against collapse, the actual stress distribution does not matter. At failure, the stress everywhere is equal to the yield stress, either in tension or in compression. The relation between tensile and compressive stresses is uniquely determined by force and moment equilibrium.

  In the figure below, we can see an example of how the stress is distributed along a stress linearization line as the load is increased in an elastoplastic analysis. The yield stress is first reached when the load parameter rises slightly above 0.38. When the load parameter reaches 0.76, a collapse ensues.

  
  The stress distribution over a cross section as the external load is increased. The load parameter value is the ratio between the membrane stress and yield stress.

  In this example, the values have been chosen so that σ_m = σ_b. Using the interaction formula above, this means that collapse should occur when

  \sigma_m = \sigma_b = \frac{\sqrt{10} -1}{3} \sigma_y = 0.72 \, \sigma_y.

  This value matches the final parameter value of 0.76 rather well. The difference can be explained by the fact that a small plastic hardening is used in the model to stabilize the analysis.

  The conclusion is that for determining safety conditions within plastic collapse, the linearized stress is the relevant parameter, since it is proportional to the axial force and bending moment. Using the true peak stress gives an overly conservative design. The safety factor, which is implicit in the bending collapse, must also be taken into account.

  If the structure is subjected to cyclic loading, the peak stresses are of utmost importance, as they determine the risk of fatigue crack initiation at the surface.

  The ASME Pressure Vessel Code

  The concept of stress linearization is an important part of the qualification of pressure vessels, as described in ASME Boiler & Pressure Vessel Code, Section III, Division 1, Subsection NB. Here, we are required to classify stresses as either primary or secondary.

  A primary stress is a stress that is required for maintaining force and moment equilibrium. Secondary stresses are caused by other effects. Typically, secondary stresses are local effects caused by either geometric discontinuities or displacement-controlled loading. Secondary stresses do not lead to a collapse when they exceed elastic limits, since they are just redistributed.

  During the analysis, the stress is studied along a number of lines through the section, referred to as stress classification lines (SCLs). The choice of SCL is not unique, so here we must use our engineering judgment to find the critical locations.

  Although not fully correct (but conservative), the linearized stresses are sometimes viewed as equivalent to the primary stresses. Without going into detail, the basic requirements of the code are:

  • The stress intensity (Tresca equivalent stress) from the primary membrane stresses should not exceed two-thirds of the yield stress. This gives a safety factor of 1.5 against plastic collapse when only membrane stresses are present.
  • The stress intensity from the sum of membrane and bending stress should not exceed the yield stress. If there are only bending stresses, the safety factor against collapse is again 1.5. This is because incipient yield is not equal to the full collapse of the section in this case.
  • Secondary stresses are allowed to reach twice the yield limit.
  • There are similar requirements, but with higher safety factors against reaching the ultimate stress.

  Interestingly enough, this means that if the membrane stress is at the limit allowed by the first criterion, it is still allowed to add a certain amount of bending stress. The discussion above tells us why: The bending stress reduces the stress over part of the section.

  As noted above, the detailed stress state is not important when it comes to static failure, as the stress distribution in the collapse state is fully determined by the force and moment equilibrium. In the figure below, the collapse interaction curve is compared with the stress limits imposed by the code.

  
  The fundamental ASME criteria for primary stresses. The stresses are normalized by the yield stress.

  It should be noted that because pressure vessels often operate at elevated temperatures, room temperature values of allowed stresses might not be sufficient.

  The requirement on the secondary stresses is set to avoid cyclic plastic deformation upon repeated loading–unloading cycles. The purpose is to avoid plastic strains accumulating in each load cycle, which can lead to a fast failure due to low-cycle fatigue.

  Other Applications of Stress Linearization

  Some rules for qualifying structural elements are based on the stresses being “hand calculated” or the result of a shell or plate analysis. When we do a full 3D analysis, the effect can be that we get results that are “too good”. The effects of local stress concentrations are already taken into account by providing low allowable nominal stresses. Because of this, we might end up in a situation where using the accurate results of a full 3D analysis leads to a highly conservative design. In this case, stress linearization can provide a useful tool for converting the 3D stress state back into a set of nominal stresses.

  For instance, this situation can occur when analyzing welds. Typically, the local geometry at the weld is not even well defined (unless it is a very high-quality weld that has been ground smooth). Thus, the actual local stress is not even meaningful to compute, so we must resort to methods based on nominal stresses.

  
  A weld in a pipe used for district heating. Image by Björn Appel, Benutername Warden. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

  Applying a Stress Linearization in COMSOL Multiphysics®

  A stress linearization does not affect the analysis as such; it is a type of result presentation. The variables to be used are set up in the Solid Mechanics interface. We add a line for stress linearization either under Variables in the context menu for the Solid Mechanics interface or under Global on the Physics tab in the ribbon.

  
  Adding a Stress Linearization node from the context menu.

  
  Adding a Stress Linearization node from the ribbon.

  Depending on whether the component is in 3D or not, the definition of the stress linearization line comes in two different flavors. In either case, we select an edge (or set of edges) that forms a straight line through the thickness of the component that we are evaluating. In 3D, we must also define the axis orientation of the local coordinate system in which the stresses along the SCL are represented.

  
  The settings for stress linearization in 3D.

  The stress tensor components along an SCL are represented in a local coordinate system, where 1 is the direction along the line. The 2 direction is perpendicular to the line and has the following orientations:

  • In the plane of a 2D model
  • In the hoop direction of a 2D axially symmetric model
  • Along a user-specified direction for a 3D model

  For the last bullet point, note that the Second Axis Orientation section of the Stress Linearization node provides several options for entering the orientation.

  If we have defined the SCLs prior to the analysis, then one edge data set is generated for each SCL. At the same time, a default plot called Stress Linearization is added.

  
  The default data sets and graph plot group.

  The stress linearization plot contains three graphs along the selected SCL:

  1. Actual stress
  2. Membrane stress
  3. The sum of the membrane and bending stress

  
  An example of a default stress linearization plot.

  In the stress linearization plot, we can change to another SCL by selecting the corresponding edge data set. In the default plot, the 22 stress tensor component is displayed. Of course, we can change to other components. Usually, 33 and 23 are the most important.

  If we add Stress Linearization nodes after running the analysis, we must click on the Update Solution button to make the newly created variables accessible for result presentation. No default plots or data sets are automatically generated in this case.

  Graphing along the SCLs is important for understanding the stress state at different locations, but at the end of the day, it is the stress intensity that is important. The maximum stress intensity for each SCL can be presented by adding a Global Evaluation node. When computing the stress intensity for the bending stress plus the membrane stress, the bending part of the out-of-plane stress components (which are supposedly small) is ignored. This approach is customary in this type of analysis.

  
  The result quantities for stress linearization when selecting data for a Global Evaluation node.

  In addition to the stress intensities, the peak stress tensor at the two ends of the SCL is available. We can also directly access the section forces and moments corresponding to the linearized stresses.

  Final Remarks on Performing a Stress Linearization in the Structural Mechanics Module

  As of version 5.3 of COMSOL Multiphysics® and the Structural Mechanics Module, the functionality for stress linearization provides us with a set of built-in tools for converting a 3D stress state to one of pure bending and tension. This makes it much easier to produce results that comply with various design codes.

  Get the Prestressed Bolts in a Tube Connection Tutorial

  Further Resources

  • Learn more about structural analysis in these blog posts:
    • Obtaining Material Data for Structural Mechanics from Measurements
    • Designing an App to Analyze Stress in a Pressure Vessel
  • Watch this video for an introduction to modeling structural mechanics
  ]]> https://www.comsol.de/blogs/introduction-to-modeling-stress-linearization-in-comsol-multiphysics/feed/ 0 https://www.comsol.de/blogs/introduction-to-multiscale-modeling-in-high-frequency-electromagnetics/ https://www.comsol.de/blogs/introduction-to-multiscale-modeling-in-high-frequency-electromagnetics/#comments Wed, 11 Jan 2017 15:58:34 +0000 http://com.staging.comsol.com/blogs?p=200191 This post begins a comprehensive blog series where we will look at several approaches to multiscale modeling in high-frequency electromagnetics. Today, we will introduce the supporting theory and definitions that we will need. In subsequent posts, you will learn how to implement multiscale modeling of high-frequency electromagnetics for different scenarios in the COMSOL Multiphysics® software. Let’s get started…

  Practical Scope: Antennas and Wireless Communication

  Multiscale modeling is a challenging issue in modern simulation that occurs when there are vastly different scales in the same model. For example, your cellphone is approximately 15 cm, yet it receives GPS information from satellites 20,000 km away. Handling both of these lengths in the same simulation is not always straightforward. Similar issues show up in applications such as weather simulations, chemistry, and many other areas.

  While multiscale modeling can be a general topic, we will focus our attention on the practical example of antennas and wireless communication. When we wirelessly transmit data via antennas, we can break the operation down into three main stages:

  1. An antenna converts a local signal into free space radiation.
  2. The radiation propagates away from the antenna over relatively long distances.
  3. The radiation is detected by another antenna and converted into a received signal.

  
  Modern communications require long-distance wireless data transfer via antennas.

  The two length scales that we will consider for this process are the wavelength of the radiation and the distance between the antennas. To use a specific example, FM radio has a wavelength of approximately three meters. When you listen to the radio in your car, you are often ten km or more away from the radio tower. Because many antennas, such as dipole antennas, are similar in size to a wavelength, we will not consider this to be another distinct length scale. As a result, we have one length scale for the emitting antenna, a different length scale for the signal propagation from source to destination, and then the original length scale again for the receiving antenna.

  Let’s go over some of the most important equations, terms, and considerations when working with multiple scales in the same high-frequency electromagnetics model.

  The Friis Transmission Equation

  The Friis transmission equation calculates the received power for line-of-sight communication between two antennas separated by a lossless medium. The equation is

  P_r = p(1-|\Gamma_t|^2)(1-|\Gamma_r|^2)G_t\left(\theta_t,\phi_t\right)G_r\left(\theta_r,\phi_r\right)\left(\frac{\lambda}{4\pi r}\right)^2P_t

  where the subscripts r and t discriminate between the transmission antenna and the receiving antenna, G is the antenna gain, P is the power, is the reflection coefficient for impedance mismatch between antenna and transmission line, p is the polarization mismatch factor, λ is the wavelength, r is the distance between the antennas and is associated with the so-called free-space path loss, and and are the angular spherical coordinates for the two antennas.

  Note that we have explicitly included two impedance mismatch terms, and so:

  • P_t refers to the power provided to a transmission line attached to an emitting antenna
  • P_r refers to the power received from a transmission line attached to a receiving antenna

  The Friis transmission equation is derived in many texts, so we will not do so again here.

  
  A visualization of the gain for a transmitting and receiving antenna. When using the Friis transmission equation, we require the orientation of each antenna for correct gain specification. The distance between the antennas is r.

  Spherical Coordinates

  Let’s now discuss spherical coordinates , since they are incredibly useful for antenna radiation and we will use them repeatedly. Starting from the Cartesian coordinates (x, y, z), we can easily express these as follows.

  \begin{align}
  r& = sqrt(x^2 + y^2 + z^2)\\
  \theta& = acos(z/r)\\
  \phi& = atan2(y,x)
  \end{align}

  For convenience, we have used the actual COMSOL Multiphysics commands — sqrt(), acos(), and atan2(,) — instead of their mathematical symbols. In our simulation setup, we will also make use of the Cartesian components of the spherical unit vector .

  \begin{align}
  \hat{\theta_x}& = cos(\theta)cos(\phi)\\
  \hat{\theta_y}& = cos(\theta)sin(\phi)\\
  \hat{\theta_z}& = -sin(\theta)
  \end{align}

  Similar assignments can be made for the Cartesian components of and , but is the most important for our purposes. This will be discussed later in this blog series when we cover ray optics.

  
  A given point shown in both Cartesian (x, y, z) and spherical coordinates. The unit vectors for the spherical coordinates are also included. Note that the spherical unit vectors are functions of location.

  The Poynting Vector and Radiation Intensity

  We are generally interested in the radiated power from antennas. The power flux in W/m² is represented by the complex Poynting vector .

  Many antenna texts also use radiation intensity, which is defined as the power radiated per solid angle and measured in W/steradian. Mathematically speaking, this is . For clarity, we have included two conventions here, as it is common to use in electrical engineering, while physicists will generally be more familiar with . We can then calculate the radiated power by integrating this quantity over all angles.

  Gain and Directivity

  Gain and directivity are similar in that they both quantify the radiated power in a given direction. The difference is that gain relates this radiated power to the input power, whereas directivity relates this to the overall radiated power. Put more simply, gain accounts for dielectric and conductive losses and directivity does not. Mathematically, this reads as and for gain and directivity, respectively. P_in is the power accepted by the antenna and P_rad is the total radiated power. While both quantities can be of interest, gain tends to be the more practical of these two as it accounts for material loss in the antenna. Because of its prevalence and usefulness, we also include the definition of gain (in a given direction) from “IEEE Standard Definitions of Terms for Antennas”, which is: “The ratio of the radiation intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically.”

  IEEE also includes three notes about gain in their definition:

  1. “Gain does not include losses arising from impedance and polarization mismatches.”
  2. “The radiation intensity corresponding to the isotropically radiated power is equal to the power accepted by the antenna divided by 4π.”
  3. “If an antenna is without dissipative loss, then in any given direction, its gain is equal to its directivity.”

  Gain, Realized Gain, and Impedance Mismatch

  In practice, an actual antenna will be connected to a transmission line. Because the antenna and the transmission line may not have the same impedance, there can be a loss factor due to impedance mismatch. The realized gain is simply the gain when accounting for impedance mismatch. Mathematically, this is , where is the reflection coefficient from transmission line theory, Z_c is the characteristic impedance of the transmission line, and Z is the impedance of the antenna.

  When using a lumped port with a characteristic impedance in COMSOL Multiphysics, the far-field gain that is calculated corresponds to the IEEE realized gain. This is important to mention explicitly, since various definitions of gain have changed over the last few decades. Starting with COMSOL Multiphysics version 5.3, which will be released in 2017, the variable names in the COMSOL software will be changed to match the IEEE definitions.

  
  The realized gain and electric field from a Vivaldi antenna, simulated using COMSOL Multiphysics and the RF Module. You can find the Vivaldi Antenna tutorial model in the Application Gallery.

  Receiving Antennas, Lorentz Reciprocity, and Received Power

  The terms we have discussed so far have referred to antennas emitting radiation, but they are also generally applicable to receiving antennas. The reason we have put more emphasis on emission thus far is because antennas generally obey reciprocity (the Lorentz reciprocity theorem is a fixture in most antenna textbooks). Reciprocity means that an antenna’s gain in a specific direction is the same regardless of whether it is emitting in that direction or receiving a signal from that direction. Practically speaking, you can calculate the gain in any direction from a single simulation of an emitting antenna, which is easier than simulating the inverse process for each desired direction.

  When we talk about receiving antennas, we are often interested in calculating the received power for an incoming signal. This can be done by multiplying the effective area, , of the antenna by the incident power flux and accounting for impedance mismatch in the line, yielding . As you may expect, this bears a striking similarity to several terms of the Friis transmission equation.

  Emitter Example: A Perfect Electric Dipole

  Today, we will talk about one type of emitter: the perfect electric point dipole. Depending on the literature, you may have seen this referred to as a perfect, ideal, or infinitesimal dipole. This emitter is a common representation of radiation for electrically small antennas. The solution for the field is

  \overrightarrow{E} =\frac{1}{4\pi\epsilon_0}\{k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}+[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}\}

  where is the dipole moment of the radiation source (not to be confused with the polarization mismatch) and k is the wave vector for the medium.

  
  One breakdown of the various regions for the electromagnetic field generated from an electrically small antenna.

  In this equation, there are three factors of 1/rⁿ. The 1/r² and 1/r³ terms will be more significant near the source, while the 1/r term will dominate at large distances. While the electromagnetic field will be continuous, it is common to refer to different regions of the field based on the distance from the source. One such distribution for an electrically small antenna is shown above, although there are other conventions that refer to the magnitude of kr.

  Later, we will see how to calculate the fields at any distance from a given source, but the most important region for antenna communications is the far field or radiation zone, which is the region farthest away from the source. In this region, the fields take the form of spherical waves, , a fact that we will take advantage of.

  We will now split up the E-field equation above into two terms. For simplicity, we will call the 1/r term the far field (FF) and the 1/r² and 1/r³ terms the near field (NF).

  \begin{align}
  \overrightarrow{E}& = \overrightarrow{E}_{FF} + \overrightarrow{E}_{NF}\\
  \overrightarrow{E}_{FF}& = \frac{1}{4\pi\epsilon_0}k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}\\
  \overrightarrow{E}_{NF}& = \frac{1}{4\pi\epsilon_0}[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}
  \end{align}

  As mentioned before, we can calculate the radiated power in watts by integrating over all angles. Note that only the far-field term will contribute to this integral, which is a primary reason why the far field is of practical interest to antenna engineers. The total power radiated from a point dipole is , where Z₀ is the impedance of free space and c is the speed of light. The maximum gain is 1.5 and is isotropic in the plane normal to the dipole moment (e.g., the xy-plane for a dipole in ).

  A note on units: The equations above are given with the traditional definition of the dipole moment in Coulomb*meters (Cm). In antenna and engineering texts, it is common to specify an infinitesimal current dipole in Ampere*meters (Am). COMSOL Multiphysics follows the engineering convention. The two definitions are related by a time derivative, so for a COMSOL software implementation, the dipole moment should be multiplied by a factor of to obtain the infinitesimal current dipole.

  Receiver Example: A Half-Wavelength Dipole

  We will use a perfectly conducting half-wavelength dipole as our receiving antenna.

   

  A visual representation of radiation incident on a half-wavelength dipole antenna.

  Many texts cover an infinitely thin wire, which has an impedance of and a directivity of . It is worth mentioning that the antenna impedance will change from these values for an antenna of finite radius. The receiving antenna we use here has a length of 0.47 λ and a length-to-diameter ratio of 100. With these values, we simulate an impedance of , which is close to the infinitely thin value and also agrees reasonably well with experimental values. Regrettably, there is no theoretical value to compare to this number, but this highlights the need for numerical simulation in antenna design.

  The comparison between the directivity of the infinitely thin dipole and our simulated dipole antenna is shown below. Because the antenna is lossless, this is equivalent to the antenna gain. You can download the dipole antenna model here.

  
  A comparison of the directivity for two half-wavelength antennas (oriented in z) as a function of theta. The COMSOL Multiphysics® simulation is of a finite radius cylinder and the theory is for an infinitely thin antenna.

  Computing the Received Power

  We can now use the Friis transmission equation to calculate the power that is emitted from a perfect point dipole and received by a half-wave dipole antenna. To use this equation, we simply need to know the gain and impedance mismatch (or realized gain), wavelength, distance between the antennas, and input power. Since we are using a point electric dipole, we have a dipole moment instead of input power and impedance mismatch. We can account for this by removing the impedance mismatch term and replacing the input power by the radiated power of the perfect electric dipole from above — effectively saying that power in equals power out.

  P_r = p(1-| \Gamma_r|^2) G_t \left(\theta_t,\phi_t\right) G_r \left(\theta_r,\phi_r\right) \left(\frac{\lambda}{4\pi r}\right)^2 P_{rad}

  If we assume that our emitter and detector are both located in the xy-plane, are polarization matched, and are separated by 1000 λ, as well as that the dipole moment of the emitter is 1 Am in , the Friis equation yields a received power of 380 μW. We will simulate this value in part 3 of this series for verification of our simulation technique. We can then use our simulation to confidently extract results and introduce complexity that the Friis equation cannot account for.

  Concluding Thoughts

  In this blog post, we have introduced the idea of multiscale modeling and discussed all of the relevant terms, definitions, and theory that we will need moving forward. For those of you with a strong background in electromagnetics and antenna design, this has likely been a quick review. If the concepts presented here are new to you, we strongly recommend further reading in a book on classical electromagnetics or antenna theory.

  In the following blog posts, we will focus primarily on practical implementation of multiscale modeling in COMSOL Multiphysics and we will repeatedly refer to concepts discussed today.

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  Coming Up Next…

  Stay tuned for more installments in our multiscale modeling blog series:

  • In part 2, we will simulate the emission from a point electric dipole using the Electromagnetic Waves, Frequency Domain interface. We will discuss the Far-Field Domain node, which calculates the far-field radiation from a source, and show how the Electromagnetic Waves, Frequency Domain interface can be coupled to the Electromagnetic Waves, Beam Envelopes interface to simulate fields in the intermediate zone.
  • In part 3, we will simulate a point dipole radiating to a half-wavelength dipole antenna an arbitrary distance away. For verification, we will calculate the power received by the half-wavelength dipole antenna and verify our results using the Friis transmission equation.
  • In part 4, we will couple our emitting source, the point electric dipole, to a ray optics simulation using the Geometrical Optics interface.
  • In part 5, we will couple the two antennas using the Geometrical Optics interface. We will again verify our results and discuss how this more general method can account for inhomogeneous media and multipath transmission.
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