Spiral slot antennas have several advantages, including:
In addition, the design of spiral slot antennas allows them to be conformally mounted on a variety of objects. This is useful in, for example, the defense industry, where spiral slot antennas can be mounted on military automobiles and aircraft and used for communication and surveillance purposes.
Example of a spiral antenna. Image by Bin im Garten — Own work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
While there are multiple types of spiral antennas, one of the most common is the Archimedean spiral antenna. Here, we discuss modeling this antenna with the COMSOL Multiphysics® software and addon RF Module.
Let’s start by examining the geometry of the spiral slot antenna, which consists of a twoarm Archimedean spiral slot. We use parametric curves to create the spiral pattern on a singlesided metal substrate. The parametric curves provide the freedom to draw an arbitrary line shape by using a mathematical formulation. The substrate is a perfect electric conductor (PEC) with a very high conductivity and negligible losses on the surface. At the center of the spiral slot is a lumped port, which is used to excite the antenna.
Spiral slot antenna geometry (left) and mesh (right).
The antenna and substrate are surrounded by an air region and perfectly matched layer (PML), the latter of which is depicted in gray in the left image above. The physicscontrolled mesh, which is shown in the right image above, is generated by default. Here, the maximum mesh size is set to 0.2 wavelengths, which is based on the maximum frequency defined in a Frequency Domain study step. The mesh is also automatically scaled by material properties such as the permittivity and permeability inside the dielectric substrate. The PML is swept with five elements along the radial direction.
The first plot looks at the electric field norm at the antenna’s top surface. This plot shows more intense electric fields along the slot than over the rest of the antenna surface, confirming that the fields are well confined to the slotted substrate.
Next, we examine a plot of the calculated Sparameters. From the results, we determine that over the studied frequency range, S_{11} is around 10 dB.
The logscaled electric field norm on the xyplane (left) and an Sparameter plot (right).
To perform a farfield analysis, we first create a 2D polar plot. This plot enables us to visualize the bidirectional radiation patterns of the antenna at different frequencies. We see that the shape of the radiation pattern remains similar for different frequencies.
Polar plot on the yzplane.
Finally, we examine the bidirectional farfield radiation pattern in 3D at a frequency of interest (3 GHz in this case). The results suggest that the direction of maximum radiation is along the zaxis. In addition, we see a symmetric pattern in the far field.
The 3D farfield radiation pattern of the antenna at 3 GHz (left) along with the antenna (right).
To get started with modeling spiral slot antennas, click the button below. Doing so will take you to the Application Gallery, where you can log into your COMSOL Access account and download the MPHfile and stepbystep instructions for the example.
Picture a micromirror as a single string on a guitar. The string is so light and thin that when you pluck it, the surrounding air dampens the string’s motion, bringing it to a standstill.
Because this damping effect is important to many MEMS devices, micromirrors have a wide variety of potential applications. For instance, these mirrors can be used to control optic elements, an ability that makes them useful in the microscopy and fiber optics fields. Micromirrors are found in scanners, headsup displays, medical imaging, and more. Additionally, MEMS systems sometimes use integrated scanning micromirror systems for consumer and telecommunications applications.
Closeup view of an HDTV micromirror chip. Image by yellowcloud — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
When developing a micromirror actuator system, engineers need to account for its dynamic vibrating behavior and damping, both of which greatly affect the operation of the device. Simulation provides a way to analyze these factors and accurately predict system performance in a timely and costefficient manner.
To perform an advanced MEMS analysis, you can combine features in the Structural Mechanics Module and Acoustics Module, two addon products to the COMSOL Multiphysics simulation platform. Let’s take a look at frequencydomain (timeharmonic) and transient analyses of a vibrating micromirror.
We model an idealized system that consists of a vibrating silicon micromirror — which is 0.5 by 0.5 mm with a thickness of 1 μm — surrounded by air. A key parameter in this model is the penetration depth; i.e., the thickness of the viscous and thermal boundary layers. In these layers, energy dissipates via viscous drag and thermal conduction. The thickness of the viscous and thermal layers is characterized by the following penetration depth scales:
where is the frequency, is the fluid density, is the dynamic viscosity, is the coefficient of thermal conduction, is the heat capacity at constant pressure, and is the nondimensional Prandtl number.
For air, when the system is excited at a frequency of 10 kHz (which is typical for this model), the viscous and thermal scales are 22 µm and 18 µm, respectively. These are comparable to the geometric scales, like the mirror thickness, meaning that thermal and viscous losses must be included. Moreover, in real systems, the mirrors may be located near surfaces or in close proximity to each other, creating narrow regions where the damping effects are accentuated.
The frequencydomain analysis provides insight into the frequency response of the system, including the location of the resonance frequencies, Qfactor of the resonance, and damping of the system.
The micromirror model geometry, showing the symmetry plane, fixed constraint, and torquing force components.
In this example, we use three separate interfaces:
By modeling the detailed thermoviscous acoustics and using the Thermoviscous Acoustics, Frequency Domain interface, we can explicitly include thermal and viscous damping while solving the full linearized NavierStokes, continuity, and energy equations. In doing so, we accomplish one of the main goals for this model: accurately calculating the damping experienced by the mirror.
To set up and combine the three interfaces, we use the AcousticsThermoviscous Acoustics Boundary and ThermoviscousAcousticsStructure Boundary multiphysics couplings. We then solve the model using a frequencydomain sweep and an eigenfrequency study. These analyses enable us to study the resonance frequency of the mirror under a torquing load in the frequency domain.
Let’s take a look at the displacement of the micromirror for a frequency of 10 kHz and when exposed to the torquing force. In this scenario, the displacement mainly occurs at the edges of the device. To view displacement in a different way, we also plot the response at the tip of the micromirror over a range of frequencies.
Micromirror displacement at 10 kHz for phase 0 (left) and the absolute value of the zcomponent of the displacement field at the micromirror tip (right).
Next, let’s view the acoustic temperature variations (left image below) and acoustic pressure distribution (right image below) in the micromirror for a frequency of 11 kHz. As we can see, the maximum and minimum temperature fluctuations occur opposite to one another and there is an antisymmetric pressure distribution. The temperature fluctuations are closely related to the pressure fluctuations through the equation of state. Note that the temperature fluctuations fall to zero at the surface of the mirror, where an isothermal condition is applied. The temperature gradient near the surface gives rise to the thermal losses.
Temperature fluctuation field within the thermoviscous acoustics domain (left) and the pressure isosurfaces (right).
The two animations below show a dynamic extension of the frequencydomain data using the timeharmonic nature of the solution. Both animations depict the mirror movement in a highly exaggerated manner, with the first one showing an instantaneous velocity magnitude in a cross section and the second showing the acoustic temperature fluctuations. These results indicate that there are highvelocity regions close to the edge of the micromirror. We determine the extent of this region into the air via the scale of the viscous boundary layer (viscous penetration depth). We can also identify the thermal boundary layer or penetration depth using the same method.
Animation of the timeharmonic variation in the local velocity.
Animation of the timeharmonic variation in the acoustic temperature fluctuations.
When the problem is formulated in the frequency domain, eigenmodes or eigenfrequencies can also be identified. From the eigenfrequency study (also performed in the model), we can determine the vibrating modes, shown in the animation below (only half the mirror is shown as symmetry applies). Our results show that the fundamental mode is around 10.5 kHz, with higher modes at 13.1 kHz and 39.5 kHz. The complex value of the eigenfrequency is related to the Qfactor of the resonance and thus the damping. (This relationship is discussed in detail in the Vibrating Micromirror model documentation.)
Animation of the first three vibrating modes of the micromirror.
As of version 5.3a of the COMSOL® software, a different take on this example solves for the transient behavior of the micromirror. Using the same geometry, we extend the frequencydomain analysis into a transient analysis. To achieve this, we swap the frequencydomain interfaces with their corresponding transient interfaces and adjust the settings of the transient solver. In the simulation, the micromirror is actuated for a short time and exhibits damped vibrations.
The resulting model includes some of the most advanced air and gas damping mechanisms that COMSOL Multiphysics has to offer. For instance, the Thermoviscous Acoustics, Transient interface generates the full details for the viscous and thermal damping of the micromirror from the surrounding air.
In addition, by coupling the transient perfectly matched layer capabilities of pressure acoustics to the thermoviscous acoustics domain, we can create efficient nonreflecting boundary conditions (NRBCs) for this model in the time domain.
Let’s start with the displacement results. The 3D results (left image below) visualize the displacement of the micromirror and the pressure distribution at a given time. We also generate a plot (right image below) to illustrate the damped vibrations caused by thermal and viscous losses. The green curve represents the undamped response of the micromirror when the surrounding air is not coupled to the mirror movement. The timedomain simulations make it possible to study transients of the system, like the decay time, and the response of the system to an anharmonic forcing.
Micromirror displacement and pressure distribution (left) and the transient evolution of the mirror displacement (right).
We can also examine the acoustic temperature variations surrounding the micromirror. The isothermal condition at the micromirror surface produces an acoustic thermal boundary layer. As with the frequencydomain example, the highest and lowest temperatures are located opposite to one another.
In addition, by calculating the acoustic velocity variations of the micromirror, we see that a noslip condition at the micromirror surface results in a viscous boundary layer.
Acoustic temperature variations (left) as well as acoustic velocity variations for the xcomponent (center) and zcomponent (right).
These examples demonstrate that we can analyze micromirrors using advanced modeling features available in the Acoustics Module in combination with the Structural Mechanics Module. For more details on modeling micromirrors, check out the tutorials below.
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To understand the huge variety of magnetic behaviors in different materials, classification is useful. The easiest classification system for magnetic materials is as follows:
Materials in categories 2 and 3 are referred to as ferromagnetic.
However, this classification is not simple, because the separation between soft irons and permanent magnets might not be that sharp and certain behaviors could be intermediate between the two categories. It is possible for a material to have some small magnetization in the absence of external sources (resembling a permanent magnet), but this magnetization increases by a significant amount because of an externally applied magnetic field (resembling a soft iron).
Moreover, a material can exhibit hysteretic behavior, which means that after the application and removal of an external load, the magnetization could be different from the initial one. The external load could not only be the magnetic field generated by an electric current but also a physical displacement (as shown in the video below).
When dealing with ferromagnetic materials, we may need to describe very different behaviors. In this blog post, we will analyze the options available in the COMSOL® software to do so.
The variety of magnetic behaviors is found (desirably or undesirably) in different systems, and being able to characterize the span of behaviors is important.
The AC/DC Module offers automatic support for including all typical magnetic behaviors via eight predefined constitutive relations, which are listed in the first column of the table below, and one option for writing your own codebased external material. Weakly magnetic materials are typically characterized by the first option, Relative permeability, which is the default in the COMSOL Multiphysics® software.
Dealing with a ferromagnetic material may require one of the other options. The first four options in the table below are tuned for soft irons and the next four are tuned for permanent magnets. Both groups are ordered in an increasing level of complexity of the constitutive relation with an increasing number of properties to describe the magnetization dynamics.
Constitutive Relations  Soft Iron, (Fully Timedependent)  Soft Iron (AC Feeding)  Permanent Magnets (Fully TimeDependent)  Required Information 

Relative Permeability  ✓  ✓  1 scalar (or tensor)  
Magnetic Losses  ✓  2 scalars (or tensors)  
BH Curve  ✓  1 function  
Effective BH Curve*  ✓  1 function  
Magnetization  ✓  1 vector  
Remanent Flux Density  ✓  1 scalar (or tensor) and 1 vector  
BH Nonlinear Permanent Magnet  ✓  Function and a direction  
Hysteresis JilesAtherton Model  ✓  ✓  5 scalars (or tensors)  
External Magnetic Material**  ✓  ✓  ✓  Externally compiled code 
Summary of the use of the laws to model soft or hard irons and the number of parameters to input. *For Effective BH Curve, the function can be automatically calculated from the standard BH curve with an example simulation app in the AC/DC Module Application Library. You can learn more about these capabilities in a previous blog post.) **External Magnetic Material is a suboption of the BH curve. More information for this condition can be found in a blog post on accessing external material models.
In the section below, there are eight plots illustrating the typical dynamics in the BH plane for the various constitutive relations explained in the table above. In BH plots, the yaxis shows the magnetic flux density B. When interpreting the magnetic flux, there is not much space for ambiguity, as it is directly measurable. The xaxis is a measure of the magnetic field H. For H, the interpretation may depend on details of the analyzed system (this will be explained in an example later on).
For now, we consider an ideal magnetic circuit where the material is a torus of length L wound uniformly by a coil of N turns carrying a current I. In this case, H = N*I/L. Depending on the application, such a design (or a different one, such as an Epstein frame) may be used from the manufacturer to display BH curves.
Here, we mention some examples of how to use these conditions in common magnetic materials, employed in some typical applications.
Notice that we did not mention the external magnetic material option from the first table. This is a suboption, found by selecting the BH Curve constitutive relation, that can be used for modeling even more general magnetic laws. A detailed example is explained in this previous blog post. This option is typically used for custommade hysteresis models that may include conditional logic.
All of the parameters and functions discussed in the tables above can be functions of all other parameters in the model. This is extremely important, as the function can be used to include multiphysics effects or to have extra freedom in handling material nonlinearities.
An example where a nonlinear dependency is added manually to the Relative Permeability case so that it can behave exactly as the BH Curve case is featured in a tutorial model on the topology optimization of a magnetic circuit. The example shows that transforming the case is as easy as writing in the field for relative permeability: murOfB(mf.normB). This is useful because the permeability is then set to: 1p^2+p^2*murOfB(mf.normB), so that the law is describing air in the region where p is 0, and a soft iron in the regions where p is 1. (In the model, p is a function varying according to a topological optimization. Notice that, as is explained in the model documentation, writing a function of normB could also require other actions in order to avoid convergence issues. In the model, the option “Split complex variables in real and imaginary parts” was activated.)
Induction heating is another useful application for setting permeability as a function. In these cases, the material is crossing the Curie temperature. This is commonly done by means of setting a permeability to a function of the form 1+f(T)*(mur(normB)1), where f(T) is a function that decreases from unity at low temperatures to 0 at the Curie temperature (remaining at 0 above that temperature). This method is needed to accurately model many induction heating processes of steels (e.g., hardening). More generally, many functional dependencies of BH parameters versus temperature can be retrieved from either literature or data sheets, and we can insert them using the same method.
Many parameters, even if referred to as “scalar” or “function” in the table, can be “tensors” or a set of functions filling a vector or a tensor. This is important because magnetism is intrinsically vectorial in nature. In fact, for all of the described behaviors in the first table, the AC/DC Module gives us the option to model completely anisotropic materials. Such an example is discussed in this tutorial model on vector hysteresis, where an anisotropic JilesAtherton material is used and reproduces published data.
The vectorial nature of fields is crucial for modeling moving magnetic machines. The animation below shows the magnetic flux density in a rotating machinery simulation where the outer domain is described according to a JilesAtherton hysteresis model. On the left, the hysteretic domain rotates, and on the right, the inner magnet rotates. Comparing any point in the left image with the corresponding point in the right image, all components of the vectors B and H follow the transformation that vectors must satisfy due to rotation. This results in the animation on the right being a rigid rotation of the animation on the left.
Magnetic flux for rotating machinery, including a hysteretic material showing that the vector nature is exact so that the local fields are identical both in the frame of the magnetic field source (left) and in that of the hysteretic domain (right).
Next, let’s consider an example that shows that different laws can be used for the same piece of ferromagnetic material when modeling its behavior during different processes. This is given that only a limited set of typically retrievable material information is available.
Here, we model a magnetic circuit, depicted in the figure below. The red part is a nonlinear soft iron piece characterized by no significant remanent flux and a hysteretic BH curve (from the Soft Iron material available in AC/DC Module Material Library: knee reaching 1.5[T] at about 5400[A/m]). The blue domains represent a coil that is wound around the soft iron core. The green domain is the piece of interest, which we analyze using different laws. This could be an AlNiCo component that is initially not magnetized.
Geometry of the magnetic circuit, featuring soft iron (red); a coil (blue); and an AlNiColike material (green). The AlNiCo bar is initially nonmagnetic, then magnetized because of an applied current to the coil and demagnetized because of the extraction from the magnetic circuit (arrow).
We can simulate the magnetic circuit in four different working conditions:
We may be tempted to tune the constitutive relation for the whole cycle. This is indeed possible, but usually requires specific independent measurements from the iron manufacturer. For example, it could be easy to know the value of H where the material can be considered as fully magnetized, the corresponding remanent flux, and the demagnetizing curve.
For the current example, let’s say that we know that saturation is reached for an externally applied magnetic field of 30[kA/m] and that the (uniaxial) demagnetization curve in the second quadrant of the BH plane is reported in the table below. The curve starts at the remanent flux Br at H = 0 and approaches B = 0 at the (negative) coercive field Hc. Notice that the data reported in the table exactly represents the material you will find in COMSOL Multiphysics by selecting Demagnetizable Nonlinear Permanent Magnet under the AC/DC Module Material Library.
If you wish to import your own data, take a look at the embedded example material. Notice that you will need to provide the coercive magnetic field Hc and a properly placed demagnetization curve. Given the original curve spans the second quadrant, the curve should be translated along the Haxis by abs(Hc). When doing so, the input BH curve will start at (0,0) and reach the remanent flux density Br at the value abs(Hc). Further instructions can be found in the AC/DC Module User’s Guide.
H [kA/m]  B [T] 

50 (coercive magnetic field, Hc)  0 
48  0.5 
47  0.7 
46  0.85 
44  0.96 
40  1.03 
35  1.08 
30  1.11 
20  1.155 
10  1.187 
0  1.2 (Remanent flux, Br) 
Data for the second quadrant BH curve of the Demagnetizable Nonlinear Permanent Magnet material available in the AC/DC Module Material Library.
The trajectory of the horizontal component of magnetic flux density in the center of the AlNiCo component during the fourpart processes is depicted in the image below. The colors represent the following stages:
Horizontal component of magnetic flux density in the center of the AlNiCo during the fourstep process.
The video below shows the conditions that have been applied to the AlNiCo component and result in the trajectory of the graph above.
Notice that all of these simulations are very easy and robust parametric stationary simulations, which are eventually starting from the previous solution. With such a setup, developing the same model in 3D or with more complex geometries would have been just as easy. As we commented before, we have used previous solution data to connect behaviors in different regions. This is the origin of the small discontinuities that are shown in the curve of the graph above.
The model can be adapted to make the process exactly continuous, but distinguishing eventual extra parameters would likely require extra information and measurements. By following this procedure, we have shown that such measurements may not be that compulsory and that reasonable solutions can be found using the material data that is typically available.
When looking at the hysteresis loop of the graph above, we should make a note regarding the quantity that is read on the xaxis. During Steps 1 and 2 on the xaxis, it is natural to read a quantity that is proportional to the driving current. During Steps 3 and 4, there is no current in the coil and the fields depend on the component’s spatial displacement. So it is not easy to unambiguously decide the quantity to be used on the xaxis. In Step 3, we use a builtin variable, called axialH. In Step 4, we use a normalized displacement from the circuit. These differing definitions are important when reading a BH curve to be sure of the researcher’s intentions (i.e., which experimental apparatus was used) when creating graphs.
In this example, we have shown that we can alternate different constitutive relations in different studies and that, in the properties of these constitutive relations, we can write any expression depending on variables computed in previous solutions. Here, we did so in the most simple way in order to avoid overcomplicating the discussion. For a more advanced and rigorous 3D model of AlNiCo extraction and reinsertion in a magnetic circuit, you can check out this selfdemagnetization model. There, a local linear recoil model for magnet reinsertion is added.
In this blog post, we have analyzed the wide variety of options present in COMSOL Multiphysics and the AC/DC Module for modeling magnetic materials. We started with the basic principles of magnetism and the suite of conditions that are offered, pointing to actual materials and devices for which each available law is more appropriate. We have also commented on the functionality for multiphysics modeling and implementation of more advanced conditions.
That said, we only covered a margin of the important aspects that are taken into account when deciding a constitutive law. We invite you to refer to the additional resources below and to contact us for more information about the software.
Get an overview of the capabilities of the AC/DC Module for modeling magnetic materials and more. If you would like to see the software in action, you can request a demonstration from the product page.
Strain gauges measure how structures — both manmade and biological — react to an applied strain. These devices are common in the mechanical and civil engineering fields for monitoring the structural health of bridges, detecting soil pressure changes near oil drilling platforms, and testing aircraft components. Strain gauges can even be used to analyze bone structure in humans and animals.
A mechanical strain gauge used to measure the growth of cracks on the HudsonAthens Lighthouse in New York. Image by Roy Smith. Licensed under CC BYSA 2.5, via Wikimedia Commons.
In terms of performance, MEMSbased gauges have several advantages over standard foil strain gauges. For one, MEMS sensors have higher strain sensitivity than foil sensors, resulting in more accurate measurements. MEMSbased gauges also have higher fracture strength and can withstand high operating and bonding temperatures, which makes them more durable than their foil counterparts and expands their range of applications.
DETF strain gauges have their own set of distinct advantages:
To optimize the design of a new DETF sensor, researchers from the School of Engineering Technology at Purdue University used MEMS simulation.
The research team created a 3D geometry of a strain gauge with the MEMS Module, an addon to COMSOL Multiphysics. The model consists of a DETF — including the beams, base, and anchors — and an electrostatic comb drive. The researchers wanted to validate the simulation results, so they used the dimensions of an analytical model when setting up the model’s components.
The DETF strain gauge. Image courtesy A. Bardakas, H. Zhang, and D. LeonSalas.
The research team used several of the builtin capabilities of the MEMS Module when modeling the strain gauge. The ThinFilm Damping feature was used to compute the forces between solid surfaces and the surrounding air. This effect accounts for the main cause of damping in the DETF.
The team also used the MEMS Module to set up a prestressed frequency analysis. This is important for many MEMS devices, as it helps determine the initial frequency of the device and how the frequency shifts once a load is applied.
Using a parametric sweep, the researchers determined how a range of applied forces affect the gauge without having to manually change the value and recompute the model each time. This enabled them to optimize their design more efficiently. Based on the results of the parametric sweep, the team modified the geometry to ensure that the device could accommodate a range of forces.
The resonance frequency of the DETF gauge was computed via two methods. Using a frequency analysis, the researchers found a resonance frequency of 84.060 kHz for the model. This is slightly higher than the frequencies found using a fundamental mode analysis (83.263 and 83.271 kHz). This difference is likely because a denser mesh was used for the mode analysis.
Resonance frequency of the DETF strain gauge for different mode shapes. Image courtesy A. Bardakas, H. Zhang, and D. LeonSalas
The team used the model to optimize the DETF strain gauge design by balancing its sensitivity and structural integrity. Next, they plan to use a twomask silicononinsulator process to fabricate the design. In addition, the researchers plan to investigate strain loss in the device via further analyses and experiments.
Born in 1707 in Basel, Switzerland, Leonhard Euler (pronounced “oiler”) was a prolific mathematician who published more than 800 articles during his lifetime. He studied under the famous Johann Bernoulli and received his master’s degree in philosophy from the University of Basel. Before moving to St. Petersburg, Russia, to work at the university, Euler submitted his first paper to the Paris Academy of Sciences, coming in second place at only 19 years old.
A portrait of Leonhard Euler. Image in the public domain, via Wikimedia Commons.
Euler quickly rose through the academic ranks and in 1733 succeeded Bernoulli as the chair of mathematics in St. Petersburg. Euler moved to Berlin in 1741 at the invitation of King Frederick II. In his 25 years there, he wrote around 380 articles and the first volume of his seminal book Introductio in Analysin Infinitorum, which formally defined functions for the first time; introduced the notation; popularized the and notation; and established the critical formula .
JosephLouis Lagrange (pronounced “luhgronj”) was born Giuseppe Lodovico Lagrangia in Turin. Today, this city is the capital of the region of Piedmont in Italy, but when Lagrange was born in 1736, it was ruled by the Duke of Savoy as part of the Kingdom of Sardinia. Lagrange developed an interest in mathematics and, after working independently on novel topics, began corresponding with Euler, whom he succeeded when Euler left Berlin.
A portrait of JosephLouis Lagrange. Image in the public domain, via Wikimedia Commons.
In Berlin, Lagrange developed most of the mathematics for which he is famous today. He played an important role in the development of variational calculus and came up with the Lagrangian approach to mechanics. Although Lagrangian mechanics makes the same predictions as Newton’s laws of motion, the Lagrangian functional introduced by Lagrange allows the classical mechanics of many problems to be described in a mathematically more straightforward and insightful manner than in Newtonian mechanics. Lagrange also developed the method of Lagrange multipliers, which allows constraints on systems of equations to be introduced easily in a variational approach.
The mathematical formulations of Euler and Lagrange are fundamental to the finite element method, which is used to solve equations in COMSOL Multiphysics.
In the Eulerian method, the dynamics of a system are considered from the viewpoint of an observer measuring the system’s evolution with respect to a fixed system of coordinates. This coordinate system is called the spatial frame in COMSOL Multiphysics. It could be understood to correspond to the laboratory frame in physical analysis, since the system of coordinates is oriented according to a fixed set of axes without any reference to the orientation of the components of the physical system itself.
The figure below illustrates a thin plate of material whose structural mechanics are modeled in a 2D plane. The plate is fixed to a rigid wall at the lefthand side and is deformed under its own weight, as gravity acts downward. With the results plotted in the spatial frame, we see the deformation of the object, as we would expect to observe in the laboratory.
A thin plate fixed to the gray block at the left deforms under its own weight, as viewed in the spatial (lab) frame. The deflection at the tip is about 5 mm for the given mechanical properties.
Formulating physical equations seems very natural in the Eulerian method. Indeed, this is the common formulation for problems such as electromagnetics and fluid physics, in which the field variables are expressed as functions of the fixed coordinates in the spatial frame.
For mechanical problems, though, the Lagrangian method offers a helpful alternative. In the Lagrangian method, the mechanical equations are written with reference to small individual volumes of the material, which will move within an object as it displaces or deforms dynamically. To put it another way, the object itself always appears undeformed from the point of view of the Lagrangian coordinate system, since the latter stays attached to the deforming object and moves with it, but external forces in the surroundings appear to change their orientation from the deforming object’s perspective. The corresponding coordinate system, which moves along with the deforming object, is called the material frame in COMSOL Multiphysics.
A point within the object, as measured in the spatial frame, is displaced from the position of the same point as expressed in the material frame by the mechanical displacement of that point. In the image below, we focus our view on the tip of the deforming plate in the example above and animate its deformation as the density of the object increases so that the weight increases too. As you can see, the material frame coordinate system (red grid and arrows) deforms together with the object, as the object’s dimensions in the spatial frame change. This means that anisotropic material properties — such as mechanical properties of composite materials — can be expressed conveniently in the material frame.
Zoomedin view of the tip of a thin plate deforming under its own weight, as its density is increased. The red grid denotes the material frame coordinates, tied to the object, as viewed in the spatial (lab) frame. The red and green arrows show the x and ycoordinate orientations of the material frame, as viewed in the spatial frame.
In the limit of very small strains for this type of mechanical problem, the spatial and material frames are nearly coincident, because the mechanical displacement is small compared to the object’s size. In this case, it is common to use the “engineering strain” to define the elastic stressstrain relation for the object, and the resulting stressstrain equations are linear. As the mechanical displacement increases, though, the linear approximation used to evaluate the engineering strain is increasingly inaccurate — the exact GreenLagrange strain is required. In COMSOL Multiphysics, the term “geometric nonlinearity” means that the GreenLagrange strain is used.
For further details on the mathematics, see my colleague Henrik Sönnerlind’s blog post on geometric nonlinearity.
Geometric nonlinearity is handled in COMSOL Multiphysics by allowing the spatial frame to be separated from the material frame, according to a frame transformation due to the computed mechanical displacement. It remains convenient to access the material frame to express properties such as anisotropic mechanical material properties, since these properties will usually remain aligned with the material frame coordinates, even as the object deforms.
By contrast, external forces such as gravity have a fixed orientation in the spatial frame. From the perspective of the material frame, external forces like gravity change direction as the object deforms. The image below shows the tip of the thin plate as above, but here, the displacement magnitude is plotted with colors. Arrows are used to illustrate the force due to gravity, as expressed in the material frame coordinates. Since the material frame coordinates remain fixed with respect to the object, the dimensions of the object appear not to change. However, the displacement magnitude increases with the object’s weight and the gravity force increasingly changes direction with respect to the deformed material in conditions of greater deformation.
Zoomedin view of the tip of a thin plate deforming under its own weight as its density increases. The plot is in the material frame as used for the Lagrangian formulation, so the deformation is not apparent, although displacement increases. The red arrows indicate the apparent direction of gravity (which is constant in the spatial frame) as perceived from the material frame of reference within the deforming object.
Neither the Lagrangian nor Eulerian formulation is more “physical” or “correct” than the other. They are simply different mathematical approaches to describing the same phenomena and equations. Through coordinate transformation, we can always transform the physical equations for any phenomenon from the material frame to the spatial frame or vice versa. From the perspective of interpretation and implementation, though, each approach has certain advantages and common applications. Some of these are summarized in the table below:
Strengths  Common Applications  

Eulerian Method 


Lagrangian Method 


What about multiphysics problems, such as fluidstructure interaction (FSI) or geometrically nonlinear electromechanics? In these cases, one physical equation might be formulated most naturally with the Eulerian method, while another might be better expressed with the Lagrangian method. This is where the ALE method comes in. This method solves the equations on a third coordinate system, which is not required to match either the spatial frame or the material frame coordinate systems.
The third coordinate system is called the mesh frame in COMSOL Multiphysics. There is one mathematical mapping between the spatial frame and the underlying mesh frame, and one between the material frame and the underlying mesh frame, so at all points in time, the equations formulated in the spatial and material frames can be transformed into the mesh frame to be solved.
In domains representing solids in a model, mechanical displacement is predicted using structural mechanics equations in the Lagrangian formulation. Here, the relation of the spatial and material frames is given by the mechanical displacement, as above. The ALE method adds more equations to allow the apparent positions and shapes of mesh elements in neighboring domains to displace in the spatial frame. That is in order to account for how mechanical deformation can change the shape of the boundaries of any domain where the physics are described in the Eulerian formulation. These additional equations are called a Moving Mesh or Deformed Geometry in COMSOL Multiphysics.
At boundaries between Lagrangian and Eulerian domains, a boundary condition for these additional equations requires that the displacement of the spatial frame (as defined through the moving mesh) for the Eulerian domain must match the mechanical displacement of the spatial frame away from the material frame in the Lagrangian domain. Even where no mechanical equations are solved, such that no Lagrangian method is used, the ALE method can still be used to express moving boundaries due to deposition or loss of material.
If you find the ALE method quite mathematical, that’s OK! It’s a difficult concept to follow in the abstract. To better understand the way the ALE method works, let’s take a look at an example within COMSOL Multiphysics.
The ALE method plays an important role in modeling FSI. In COMSOL Multiphysics, this method enables the automated bidirectional coupling of fluid flow and structural deformation, a capability demonstrated in our Micropump Mechanism tutorial model.
At the heart of this micropump mechanism are two cantilevers, which perform the same function as valves in conventional pumping devices. These cantilevers are flexible enough that the fluid flow causes them to deform. As fluid is alternately pumped into or out of the channel at the top, the force of the fluid flow causes the two cantilevers to deform so that fluid flows out to the right or in from the left.
The micropump mechanism. Pumping fluid into or out of the top tube produces opposite reactions in the two cantilevers, pushing fluid in or out of the chamber. Even though there is no timeaveraged net flow into the upper tube, there is a timeaveraged net movement of fluid from left to right.
The cantilevers deform enough that there is an appreciable change in the position of the boundary where the fluid and solid meet: a geometrically nonlinear case. The selfconsistent handling of the fluid’s pressure on the solid and the solid’s force on the fluid, together with the deformation of the mesh, are handled automatically by the FluidStructure Interaction interface. The interface employs the ALE method to account for the change in shape in the solid and fluid regions.
For solids, the mechanical equations with geometric nonlinearity define the displacement of the spatial frame with respect to the material frame. In the fluid equations, it’s necessary to deform the mesh on which the equations are solved in order to express the displacement of the solid boundaries in the spatial frame where the fluid equations are formulated. The deformation at the boundaries is controlled by the mechanical displacement from the solution to the structural problem. Within the fluid, though, the exact position or orientation of mesh nodes isn’t important, as the equations are formulated in the fixed spatial frame. Instead, the deformation of the mesh is smoothed in order to ensure that the numerical problem remains stable with highquality mesh elements.
To explain the ALE method for the FSI problem, we could paraphrase a common explanation for general relativity: forces due to fluid flow (Eulerian) tell the structure how to deform in the material frame (Lagrangian), while the structural deformation (Lagrangian) tells the mesh how to move in the spatial frame (Eulerian).
Top: The micropump’s operation, including pressure, flow, and cantilever deformation, as plotted in the spatial frame. Bottom: Mesh deformations calculated by the ALE method.
As of COMSOL Multiphysics version 5.3a, the Moving Mesh feature to define mesh deformation in this type of problem is located under Component > Definitions. This allows consistency in the definition of material and spatial frames between all physics included in a model, even if several physics interfaces are included. The screen capture below shows where these settings are located in the COMSOL Multiphysics Model Builder tree.
Screen capture showing Moving Mesh features under Component > Definitions, and physical coupling between two physics interfaces through Multiphysics > FluidStructure Interaction.
Turning to an electrochemical problem, the Copper Deposition in a Trench tutorial model shows that the ALE method can be vital for simulating electrodeposition problems. In this model, copper is deposited onto a circuit board that has a small “trench”. The deposited copper layer becomes thick compared to the overall size of the trench, so the size and orientation of the copper surface change appreciably as deposition proceeds. Since the rate of copper deposition at different points on this surface is nonuniform, the shape and movement of the boundary cannot be neglected.
A schematic of the physical problem being solved in the electrodeposition model.
To calculate the rate of deposition at a given point on the copper electrodeelectrolyte interface, we need the concentration of the species and the electrolyte potential of the solution adjacent to that point. As the deposition progresses and the boundary moves, the shape of the electrolyte volume has to change continuously. Similarly, the concentration and potential distributions on the altered shape must be recalculated.
The coupling of the deposition rate to the boundary motion rate and the calculation of the changing shape are accomplished with the ALE method and fully automated multiphysics couplings with the Tertiary Current Distribution and Deformed Geometry interfaces. Here, the Deformed Geometry displaces the copper surface in the spatial frame at a rate proportional to the local current density for electrodeposition, as computed from the electrochemical interface.
With this model, we can accurately account for the deposition process in order to optimize its parameters. We can also experiment with different applied potentials and deposition surface geometries to improve the uniformity of the deposition, which produces a more efficient process and a higherquality end product.
Animations showing the evolution of the deposition process in time. It is clear that the deposition happens unevenly, resulting in a pinching of the trench opening at its top.
Thermal ablation, discussed in this previous blog post, involves a very high temperature applied to an object, causing the surface to melt and vaporize. Examples of thermal ablation include the removal of material by lasers — such as in the etching process, laser drilling, or laser eye surgery — and a spacecraft’s heat shield as it reenters the atmosphere.
Animation showing the effect of thermal ablation on a material.
Since we expect that an object’s shape will change when some of its material is removed, deforming meshes are clearly a key part of thermal ablation simulation. What we need to know is how the shape of the object will change. This depends on how we balance the applied heat with heat lost to ablation and heat dissipation throughout the structure by mechanisms such as conduction.
To obtain this information, we can predict the temperature profile as a function of space and time by solving the heat transfer equations using the Heat Transfer interface. Because the mass and shape of the object are changing, the Heat Transfer interface is coupled to a Deformed Geometry interface, using the ALE method to displace the boundary according to the rate of ablation. The Heat Transfer equations predict the temperature distribution in the object as its shape evolves.
By performing these steps, we can attain accurate calculations for the thermal ablation process. Moreover, we can determine the final shape of the object after ablation is complete. This might enable us to check whether a laser weld will fall within acceptable tolerances or whether a spacecraft will survive an emergency landing.
The contributions of Leonhard Euler and JosephLouis Lagrange in the field of mathematics have paved the way for simulating a variety of systems involving multiphysics applications. The combination of their individual methods has led to the development of the ALE method, which can be used to predict physical behavior when objects deform or displace. By properly accounting for these movements, you can set up highly accurate models. Remember to thank Euler and Lagrange as you investigate these and other models that exploit the ALE method!
The ALE method is one of many builtin physics capabilities in the COMSOL Multiphysics® software. See more of them:
In the search for planets outside our solar system, astronomers rely on a wide variety of tools, one of which is the échelle spectrograph. This device consists of mirrors, gratings, and lenses that separate polychromatic light into a highly dispersed spectrum. Échelle spectrographs boast a high resolving power and can capture spectra over a large wavelength. Thus, 2D échelle spectral formats enable very large spectral grasps. The visible light instruments can even cover the full visible spectrum in a single exposure.
An example of a 2D échelle spectral format. Here, the primary (or échelle) dispersion moves from left to right and the cross dispersion moves from bottom to top.
Échelle spectrographs can be used for precision Doppler velocimetry, the highresolution analysis of stellar atmospheres, and highthroughput Raman spectroscopy. For instance, the échelle spectrograph HARPS helps astronomers to locate new exoplanets — such as the Earthlike Ross 128 b — by detecting tiny wobbles in the motion of the stars, which betray the presence of an orbiting planet.
Moving forward, a successor of HARPS with increased precision called ESPRESSO may further enhance our ability to search for new planets. Other teams, such as those from NASA and those working on the Giant Magellan Telescope (GMT), are also helping us move toward the next generation of instruments for exoplanet discovery.
The HARPS spectrograph (left) and an artistic interpretation of Ross 128 b (right). Images by ESO (left) and ESO/M. Kornmesser (right). Licensed under CC BYSA 2.0, via the ESO copyright guidelines.
To evaluate the design of échelle spectrographs and aid in their continued advancement, we can model these devices using COMSOL Multiphysics and the addon Ray Optics Module.
In this example, a “white pupil” version of an échelle spectrograph is focused with a Petzval lens. The model includes two grating components:
This design can generate 2D spot arrays for various échelle orders and wavelengths.
A depiction of a white pupil échelle spectrograph and a magnified view of a Petzval lens system, which shows only the axial rays for the different wavelengths.
To streamline the process of building the complex model geometry, we use readymade parts available in the Part Library. We further simplify the model creation process by building the spectrograph geometry separately. To do this, we split the spectrograph parameters (which can be given a numeric value or expression) into two categories:
Since the model geometry is fully parameterized, we can adjust all of the main physical and geometric properties with the Parameters node.
The full geometry sequence for the échelle spectrograph model.
We then perform a parametric sweep across multiple orders, each with several wavelengths. Here, we run a ray trace over 5 échelle orders with 11 wavelengths per order, but this analysis can be extended to include as many wavelengths and orders as desired.
First, let’s look at the 2D and 3D ray diagrams in which we trace rays as they travel through the spectrograph. In these plots, we only trace the marginal rays from the entrance slit to verify the spectrograph geometry and clear aperture parameters.
2D (left) and 3D (right) ray diagrams for the model of the white pupil échelle spectrograph.
Next, we generate an échelle diagram that plots the location of every wavelength in the parametric sweep on the image plane. As seen in the plot below, the échelle dispersion runs from left to right and the cross dispersion moves from bottom to top. These are the same directions that we saw in the 2D échelle spectral format example located at the beginning of this blog post.
Échelle diagram of the white pupil échelle spectrograph.
Want to try this example of a white pupil échelle spectrograph? Find the model documentation and download the related MPHfiles by clicking the following button.
In electromagnetic simulations, the wavelength always needs be resolved by the mesh in order to find an accurate solution of Maxwell’s equations. This requirement makes it difficult to simulate models that are large compared to the wavelength. There are several methods for stationary wave optics problems that can handle large models. These methods include the socalled diffraction formulas, such as the Fraunhofer, FresnelKirchhoff, and RayleighSommerfeld diffraction formula and the beam propagation method (BPM), such as paraxial BPM and the angular spectrum method (Ref. 1).
Most of these methods use certain approximations to the Helmholtz equation. These methods can handle large models because they are based on the propagation method that solves for the field in a plane from a known field in another plane. So you don’t have to mesh the entire domain, you just need a 2D mesh for the desired plane.
Compared to these methods, the Electromagnetic Waves, Beam Envelopes interface in COMSOL Multiphysics (which we will refer to as the Beam Envelopes interface for the rest of the blog post) solves for the exact solution of the Helmholtz equation in a domain. It can handle large models; i.e., the meshing requirement can be significantly relaxed if a certain restriction is satisfied.
A beam envelopes simulation for a lens with a millimeterrange focal length for a 1um wavelength beam.
We discuss the Beam Envelopes interface in more detail below.
Let’s take a look at the math that the Beam Envelopes interface computes “under the hood”. If you add this interface to a model and click the Physics Interface node and change Type of phase specification to User defined, you’ll see the following in the Equation section:
Here, is the dependent variable that the interface solves for, called the envelope function.
In the phasor representation of a field, corresponds to the amplitude and to the phase, i.e.,
The first equation, the governing equation for the Beam Envelopes interface, can be derived by substituting the second definition of the electric field into the Helmholtz equation. If we know , the only unknown is and we can solve for it. The phase, , needs to be given a priori in order to solve the problem.
With the second equation, we assume a form such that the fast oscillation part, the phase, can be factored out from the field. If that’s true, the envelope is “slowly varying”, so we don’t need to resolve the wavelength. Instead, we only need to resolve the slow wave of the envelope. Because of this process, simulating largescale wave optics problems is possible on personal computers.
A common question is: “When do you want the envelope rather than the field itself?” Lens simulation is one example. Sometimes you may need the intensity rather than the complex electric field. Actually, the square of the norm of the envelope gives the intensity. In such cases, it suffices to get the envelope function.
The math behind the beam envelope method introduces more questions:
To answer these questions, we need to do a little more math.
Let’s take the simplest test case: a plane wave, , where for wavelength = 1 um, it propagates in a rectangular domain of 20 um length. (We intentionally use a short domain for illustrative purposes.)
The outofplane wave enters from the left boundary and transmits the right boundary without reflection. This can be simulated in the Beam Envelopes interface by adding a Matched boundary condition with excitation on the left and without excitation on the right, while adding a Perfect Magnetic Conductor boundary condition on the top and bottom (meaning we don’t care about the y direction).
The correct setting for the phase specification is shown in the figure below.
We have the answer , knowing that the correct phase function is or the wave vector is a priori. Substituting the phase function in the second equation, we inversely get , the constant function.
How many mesh elements do we need to resolve a constant function? Only one! (See this previous blog post on highfrequency modeling.)
The following results show the envelope function and the norm of , ewbe.normE
, which is equal to . Here, we can see that we get the correct envelope function if we give the exact phase function, constant one, for any number of meshes, as expected. For confirmation purposes, the phase of , arg(E1z)
, is also plotted. It is zero, also as expected.
The envelope function (red), the electric field norm (blue), and the phase of the envelope function (green) for the correct phase function k_{0}x, computed for different mesh sizes.
Now, let’s see what happens if our guess for the phase function is a little bit off — say, instead of the exact . What kind of solutions do we get? Let’s take a look:
The envelope function (red), the electric field norm (blue), and the phase of the envelope function (green) for the wrong phase function, 0.95 k_{0}x, computed for different mesh sizes.
What we see here for the envelope function is the socalled beating. It’s obvious that everything depends on the mesh size. To understand what’s going on, we need a pencil, paper, and patience.
We knew the answer was , but we had “intentionally” given an incorrect estimate in the COMSOL® software. Substituting the wrong phase function in the second equation, we get . This results in , which is no longer constant one. This is a wave with a wavelength of = 20 um, which is called the beat wavelength.
Let’s take a look at the plot above for six mesh elements. We get exactly what is expected (red line), i.e., . The plot automatically takes the real part, showing . The plots for the lower resolutions still show an approximate solution of the envelope function. This is as expected for finite element simulations: coarser mesh gives more approximate results.
This shows that if we make a wrong guess for the phase function, we get a wrong (beatconvoluted) envelope function. Because of the wrong guess, the envelope function is added a phase of the beating (green line), which is .
What about the norm of ? Look at the blue line in the plots above. It looks like the COMSOL Multiphysics software generated a correct solution for ewbe.normE
, which is constant one. Let’s calculate: Substituting both the wrong (analytical) phase function and the wrong (beatconvoluted) envelope function in the second equation, we get , which is the correct fast field!
If we take a norm of , we get a correct solution, constant one. This is what we wanted. Note that we can’t display itself because the domain can be too large, but we can find analytically and display the norm of with a coarse mesh.
This is not a trick. Instead, we see that if the phase function is off, the envelope function will also be off, since it becomes beatconvoluted. However, the norm of the electric field can still be correct. Therefore, it is important that the beatconvoluted envelope function be correctly computed in order to get the correct electric field. The above plots clearly show that. The sixelement mesh case gives the completely correct electric field norm because it fully resolves the beatconvoluted envelope function. The other meshes give an approximate solution to the beatconvoluted envelope function depending on the mesh size. They also do so for the field norm. This is a general consequence that holds true for arbitrary cases.
No matter what phase function we use in COMSOL Multiphysics, we are okay as long as we correctly solve the first equation for and as long as the phase function is continuous over the domain. When there are multiple materials in a domain, the continuity of the phase function is also critical to the solution accuracy. We may discuss this in a future blog post, but it is also mentioned in this previous blog post on highfrequency modeling.
So far, we have discussed a scalar wave number. More generally, the phase function is specified by the wave vector. When the wave vector is not guessed correctly, it will have vectorvalued consequences. Suppose we have the same plane wave from the first example, but we make a wrong guess for the phase, i.e., instead of . In this case, the wave number is correct but the wave vector is off. This time, the beating takes place in 2D.
Let’s start by performing the same calculations as the 1D example. We have and the envelope function is now calculated to be , which is a tilted wave propagating to direction , with the beat wave number and the beat wavelength .
The following plots are the results for θ = 15° for a domain of 3.8637 um x 29.348 um for different max mesh sizes. The same boundary conditions are given as the previous 1D example case. The only difference is that the incident wave on the left boundary is . (Note that we have to give the corresponding wrong boundary condition because our phase guess is wrong.)
In the result for the finest mesh (rightmost), we can confirm that is computed just like we analyzed in the above calculation and the norm of is computed to be constant one. These results are consistent with the 1D example case.
The electric field norm (top) and the envelope function (bottom) for the wrong phase function , computed for different mesh sizes. The color range represents the values from 1 to 1.
The ultimate goal here is to simulate an electromagnetic beam through optical lenses in a millimeterscale domain with the Beam Envelopes interface. How can we achieve this? We already discussed how to compute the right solution. The following example is a simulation for a hardapertured flat top incident beam on a planoconvex lens with a radius of curvature of 500 um and a refractive index of 1.5 (approximately 1 mm focal length).
Here, we use , which is not accurate at all. In the region before the lens, there is a reflection, which creates an interference. In the lens, there are multiple reflections. After the lens, the phase is spherical so that the beam focuses into a spot. So this phase function is far different from what is happening around the lens. Still, we have a clue. If we plot , we see the beating.
Plot of . The inset shows the finest beat wavelength inside the lens.
As can be seen in the plot, a prominent beating occurs in the lens (see the inset). Actually, the finest beat wavelength is in front of the lens. To prove this, we can perform the same calculations as in the previous examples. The finest beat wavelength is due to the interference between the incident beam and reflected beam, but we can ignore this because it doesn’t contribute to the forward propagation. We can see that the mesh doesn’t resolve the beating before the lens, but let’s ignore this for now.
The beat wavelength in the lens is for the backward beam and for the forward beam for n = 1.5, which we can also prove in the same way as the previous examples. Again, we ignore the backward beam. In the plot, what’s visible is the beating for the forward beam. The backward beam is only a fraction (approximately 4% for n = 1.5 of the incident beam, so it’s not visible). The following figure shows the mesh resolving the beat inside the lens with 10 mesh elements.
The beat wavelength inside the lens. The mesh resolves the beat with 10 mesh elements.
Other than the beating for the propagating beam in the lens, the beating in the subsequent air domain is pretty large, so we can use a coarse mesh here. This may not hold for faster lenses, which have a more rapid quadratic phase and can have a very short beat wavelength. In this example, we must use a finer mesh only in the lens domain to resolve the fastest beating.
The computed field norm is shown at the top of this blog post. To verify the result, we can compute the field at the lens exit surface by using the Frequency Domain interface, and then using the Fresnel diffraction formula to calculate the field at the focus. The result for the field norm agrees very well.
Comparison between the Beam Envelopes interface and Fresnel diffraction formula. The mesh resolves the beat inside the lens with 10 mesh elements.
The following comparison shows the mesh size dependence. We get a pretty good result with our standard recommendation, , which is equal to . This makes it easier to mesh the lens domain.
Mesh size dependence on the field norm at the focus.
As of version 5.3a of the COMSOL® software, the Fresnel Lens tutorial model includes a computation with the Beam Envelopes interface. Fresnel lenses are typically extremely thin (wavelength order). Even if there is diffraction in and around the lens surface discontinuities, the fine mesh around the lens part does not significantly impact the total number of mesh elements.
In this blog post, we discuss what the Beam Envelopes interface does “under the hood” and how we can get accurate solutions for wave optics problems. Even if we get beating, the beat wavelength can be much longer than the wavelength, which makes it possible to simulate large optical systems.
Although it seems tedious to check the mesh size to resolve beating, this is not extra work that is only required for the Beam Envelopes interface. When you use the finite element method, you always need to check the mesh size dependence for accurately computed solutions.
Try it yourself: Download the file for the millimeterrange focal length lens by clicking the button below.
The beginning is a very good place to start, as most would say. Part 1 of the tutorial series is where you meet the model — a threecore leadsheathed crosslinked polyethylene, highvoltage alternating current (XLPE HVAC) submarine cable. You’ll also get details on what to expect in the other five parts of the series.
A submarine cable similar to the one modeled throughout this series. Image by Z22 — Own work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
The overview of the fundamentals of electromagnetism and numerical modeling is helpful if you are new to the electromagnetics field, simulation, or both. Feel free to skip ahead if these topics are old hat to you, but if not, this primer covers subjects such as:
The cross section (left) and mesh (right) for a model of a typical leadsheathed XLPE HVAC submarine cable with three cores. The geometry has been parameterized to allow for quick modification; any cable with the same basic structure can be investigated with ease.
The second tutorial focuses on modeling the cable’s capacitive properties and validates an important assumption: An analytical approach is sufficient for the analysis of capacitance and charging effects. This will be useful throughout the series.
This tutorial is included for beginners, but the results also support the other parts of the series, as it demonstrates the significance of the material properties and cable length. In the cross section of the cable model, the large contrast in material properties enables you to consider the XLPE as a perfect insulator and lead and copper materials as perfect conductors. These results correspond to the analytical approximations.
Left: The electric potential distribution after 10 km of cable for singlepoint bonding (at phase φ = 0). Right: The inplane displacement current density norm in the insulators (primarily the XLPE).
In terms of cable length, you will see that the analytical approximations are sufficient for a 10km cable. This stays true even under the worst possible nominal conditions, which occur when singlepoint bonding is applied and all voltageinducing effects are inphase.
Part 3 of the series builds on the previous tutorial, which showed that you may neglect the capacitive coupling between phases and consider one isolated phase. This reduces the model to an axisymmetric problem. In order to cover the full 10 kilometers of cable, we use a scaled 2D axisymmetric geometry in the model.
Left: The 2D axisymmetric geometry of an isolated phase with three separate bonding sections and a different scale for transverse and longitudinal directions. Right: The norm of the resulting charging current accumulated along the cable (for cross bonding).
The charging currents that leak into the screen build up along the cable and reach a maximum at the ground point, or intersection. The Bonding Capacitive tutorial analyzes the current buildup for different bonding types as well as the corresponding losses. The results are as follows:
Bonding Type  Total Accumulated Charging Current at Ground Point/Intersection  Corresponding Losses per Screen 

SinglePoint Bonding  55 A  1.5 kW 
Solid Bonding  28 A  0.38 kW 
Cross Bonding  10.7 A  85 W 
This part of the series builds on the previous two tutorials, which show that there is a weak coupling between the inductive and capacitive parts of the cable. The relatively small losses caused by inplane displacement and eddy currents justify approximating the cable using a 2D inductive model with outofplane currents only.
Animation of the instantaneous magnetic flux density norm in the cable’s cross section, for solid bonding and with armor twisting included.
Animation of the current density induced in the cable’s armor and screens, for solid bonding and with armor twisting included.
This model focuses on the importance of wire twist with respect to both phase conductors and armor, and investigates the corresponding losses. For instance, when armor twist is applied to the cable, the armor currents are suppressed and the total losses decrease by ~11%.
In addition to this, we demonstrate two different ways of modeling the central conductors. The first example assumes the central conductors to consist of solid copper, resulting in a typical skin and proximity effect. The other shows a perfectly stranded Litz wire approach, resulting in a homogenized current distribution.
The simulation results found in this tutorial are validated using actual product data sheets following the official international standards. The comparison shows a good match, especially for the inductance.
The objective of Part 5 is to further examine the different bonding types that were suggested in Part 3 (and 4): singlepoint, solid, and cross bonding. (Cross bonding is especially of interest for terrestrial cable systems.) As opposed to Part 3, this part focuses on inductive effects.
You will learn how to individually consider three different cable sections by coupling three separate magnetic fields physics interfaces to a circuit. The resulting model allows for investigating debalanced cables and cables with dissimilar section lengths.
In addition to this, the tutorial demonstrates the effects of using a simplified geometry. Simplification is an overarching theme in this tutorial series: It is often justified to use a much simpler geometry than you think. It isn’t the quantity of details, but the quality that optimizes a model.
In the final installment of the series, electromagnetic heating and temperaturedependent conductivity are added to the cable model. Building on Part 4, you’ll learn how to set up a twoway coupling between the electromagnetic field and heat transfer part by implementing a frequencystationary study.
Left: An example of a preset resistance curve R_{ac} (T). Right: The resulting temperature distribution when using a temperaturedependent conductivity such that R_{ac} (T) is matched.
Results show the effect of temperature on losses for the cable’s phases and screens. When electromagnetic heating is added (without temperaturedependent conductivity) the cable heats up, but the electromagnetic properties are still identical to those reported in Part 4. When adding linearized resistivity to the phases specifically, phase losses increase but not the screen losses. The temperature reaches a maximum. If linearized resistivity is applied to the screens as well, the temperature lowers and losses decrease for both the phases and the screens.
In this case still, the material properties are provided and the numerical model determines the corresponding AC resistance. However, for thermal cable models, it’s common practice to use the temperaturedependent AC resistance as an input (as provided by the IEC 60287 series of standards). The final part of the tutorial demonstrates how to use any temperaturedependent resistance curve as an input and let the model determine the corresponding material properties.
Check out the Cable Tutorial Series if you’re looking for a selfpaced electromagnetics modeling resource, whether you want to examine each section in detail or skip ahead depending on what interests you.
You can access the materials, which include stepbystep PDF instructions and MPHfile downloads, via the button below:
Model documentation is available with a COMSOL Access account. To download the MPHfiles, you also need a software license.
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In today’s society, we’re able to constantly share information using smartphones, computers, and wearable technology. This information is usually transmitted via wired and wireless networks. While these networks are the backbone of our current communication methods, they don’t reach everywhere on the planet.
To deliver highbandwidth data throughout the world, including remote areas without access to a network, one potential option is to use artificial satellites to form a suborbital highdatarate communications network called IoS.
Illustration of a satellite in outer space.
This satellitebased network could send information between Earth and space and distribute it globally. By working in conjunction with existing networks, IoS technology can even increase our data transfer capabilities. This is a key benefit due to the consistent increase in the amount of devices that can share data (such as IoT devices) as well as the amount of data they are sharing, causing a strain on existing communications networks.
In recent years, technological advancements have facilitated the development of IoS technology. For instance, satellite deployment costs have dropped and there have been advancements in phased array technology and microsatellites.
The Tactical Satellite4 (TacSat4) is one example of an improved satellite that can send realtime data collected in space to Earth (left). Animation showing how the antenna on this satellite opens in an “umbrellalike” fashion (right). Left image in the public domain in the United States, via Wikimedia Commons. Right animation is by NASA/JPLCaltech and is in the public domain via Wikimedia Commons.
However, before IoS technology can be fully realized, there are a few more hurdles to overcome.
One step toward a commercially viable IoS is to improve the design of antennas, such as dish and inflatable satellite antennas, by minimizing their size and weight, while ensuring that they retain their ability to send signals into orbit. To achieve this feat, engineers must ensure that their antenna designs maintain a specific radiation pattern that allows for accurate and longdistance satellite communication (SatCom).
Two types of antennas that can be used for IoS are dish antennas (left) and inflatable satellite communications antennas (right). While dish antennas work well for this application, they have a limited portability. In cases where antennas need to be more easily deployed and moved, IoS technology can rely on inflatable satellite antennas. Left image by Julie Missbutterflies — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons. Right image by The Official CTBTO Photostream — Own work. Licensed under CC BY 2.0, via Wikimedia Commons.
To understand what type of radiation pattern is needed for IoS applications, picture yourself in a kitchen far from the depths of outer space. You are handed a ball of dough and tasked with making it as long as possible. You work the dough into a long, thin “noodle” that can cover the longest distance before breaking apart.
Similarly, to send signals into space, we need to generate a radiation pattern that can be visualized as a thin and sharp noodle or needlelike shape. The ideal radiation pattern uses a minimal amount of electromagnetic energy to efficiently communicate with satellites in orbit.
Animation showing an example of dish reflector excitation by a circular horn antenna and its response for SatCom applications (created with the COMSOL Multiphysics® software and addon RF Module). The generated farfield radiation pattern looks like a needle.
Simulating dish or parabolic reflector antennas is notoriously difficult and computationally expensive due to their large size. In addition, dish antennas, inflatable satellite antennas, and devices like the aforementioned TacSat4 satellite are are all axisymmetric. To efficiently model these antennas, you can use 2D axisymmetric modeling, a technique available in the RF Module, an addon product to the COMSOL Multiphysics software. As compared to a full 3D analysis, using 2D axisymmetry is very fast and greatly reduces the size of your model.
The tutorial we discuss here uses 2D axisymmetry to model a parabolic reflector antenna with a radius larger than 20 wavelengths. In this example, you can use the 2D axisymmetric formulation of the electromagnetic wave equation to simulate the antenna. Both the axial feed circular horn and parabolic reflector antenna are solids of revolution.
A 3D model of a parabolic reflector antenna.
This technique enables you to easily predict the electric field norm of the antenna as well as the direction and relative magnitude of power flow. It’s also possible to evaluate the antenna’s 3D farfield radiation pattern. As you can see below, these results show that the antenna generates a very sharp and highgain radiation pattern, like the sharp “needle” shape that is optimal for IoS technology.
The electric field norm and power flow.
The very sharp farfield radiation pattern.
A plot with the dB scale to characterize sidelobes.
Simulations like these can help engineers improve antenna designs to create devices that advance IoS technology, connecting us all to the future.
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Ion funnels consist of a stack of ring electrodes that have decreasing inner diameters. Due to a combination of RF and DC potentials as well as the presence of a background gas, these devices can focus ion beams by confining ions radially and moving them toward the narrow end of the funnel. In doing so, the funnel can transport ions between the ion source and mass filter with minimal ion losses.
Simulation of an ion funnel.
Ion funnels can be used to inject ions into quadrupole mass filters and ion mobility spectrometers, enabling them to separate and analyze mixtures of ionized gases. These devices have a wide variety of applications, such as:
Of course, before ion funnels can be put to use, we need to gain insight into their design and functionality.
In this example, we analyze the focusing effect of an ion funnel that combines RF and DC potentials. The model contains a set of insulated ringshaped electrodes that are exposed to an RF potential and have adjacent electrodes out of phase. In addition, there is a neutral argon buffer gas within the funnel. To model the interaction of the ions and neutral background gas, we use the Collisions node with an Elastic subnode and the Monte Carlo collision setting.
The RF potential radially confines the ions, and a DC bias directs them toward the increasingly narrow electrodes. The superposition of these two fields enables the funnel to focus the ions, sending them through the funnel and counteracting the thermal dispersion and Coulombic repulsion effects.
To create this model, we use three different interfaces in the COMSOL Multiphysics® software:
The simulation results for the ion funnel show that the positive ions are successfully moved from the wider end of the funnel to the narrow end via the gradual DC bias. To keep the ions inside the funnel, the AC voltage is kept out of phase between the adjacent electrodes. As seen below, this results in a very large electric potential gradient near the electrodes.
The combined electric potential of the electrodynamic ion funnel when time = 0.
Using this model, we also investigate the ion trajectories in the funnel. These trajectories show that the ions are confined to the increasingly small area. Due to this confinement, the ions can be efficiently transported to another device, such as a mass filter.
Positive ion trajectories in the electrodynamic ion funnel.
Moving on, let’s take a closer look at the ions located at the narrow end of the funnel. While the ions are released along the positive xaxis, they become uniformly distributed around the zaxis when they reach the end of the funnel.
The x and ycoordinates of the ions at the narrow end of the funnel. In this plot, blue indicates particles still in the funnel and red indicates particles that have exited the funnel. Note that these results may differ from the former two plots because the Collisions node uses random numbers to decide if a collision takes place at every time step.
Want to try this ion funnel example? Access the model documentation and associated MPHfiles via the button below.
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The beam of light that Erasmus Bartholinus observed traveling straight through the crystal is called an ordinary ray. The other light beam, which bends while traveling through the crystal, is an extraordinary ray. Anisotropic materials, such as the crystal from the stone and bench experiment described above, are found in applications ranging from detecting harmful gases to beam splitting for photonic integrated circuits.
Ordinary and extraordinary rays traveling through an anisotropic crystal.
In a physical context, when an unpolarized electromagnetic beam of light propagates through an anisotropic dielectric material, it polarizes the dielectric domain, leading to a distribution of charges known as electric dipoles. This phenomenon leads to induced fields within the anisotropic dielectric material, wherein two kinds of waves experience two different refractive indices (ordinary and extraordinary).
The ordinary wave is polarized perpendicular to the principal plane and the extraordinary wave is polarized parallel to the principle plane, where the principal plane is spanned by the optic axis and the two propagation directions in the crystal. Because of this behavior, the waves propagate with different velocities and trajectories.
In a previous blog post, we discussed silicon and how its derivative, silicon dioxide, is used extensively in photonic integrated chips due to its compatibility with the CMOS fabrication technique. Bulk silicon, which has an isotropic property, is used to develop prototypes for photonic integrated chips. However, due to unique optical properties such as splitting beams and polarizationbased optical effects, anisotropy comes into play at a later stage.
Anisotropy in silicon photonics occurs unintentionally due to the annealing process while fabricating the waveguide. The difference in thermal expansion between the core and cladding causes geometry mismatch due to stress optical effects, which results in effects such as mode splitting and pulse broadening. Anisotropy could also be intentionally introduced by varying the porosity of silicon dioxide. This enables researchers to work with a range of effective refractive indices from silicon dioxide (n ~1.44) to air (n ~1), giving them the edge to perform very sensitive optical sensor applications.
To perform qualitative analyses of anisotropic media, researchers investigate how optical energy propagates within planar waveguides (also known as modes of propagation). In planar waveguides, we define modes using and terminology (Ref. 2), where x and y depict the direction of polarization and p and q depict the number of maxima in the x and ycoordinates.
Picture it this way: You are walking on an “landscape” (as shown below). The “winds” (polarization) are along ±x direction, and you encounter two distinct peaks when traveling from the x to +x direction. When you move from the y to +y direction, you observe both of the peaks simultaneously.
Mode analysis of the planar waveguide. Top row, left to right: and . Middle row, left to right: and . Bottom row, left to right: and . The arrow plot represents the electric field; contour and surface plot represent outofplane power flow (red is high and blue is low magnitude).
Before launching a beam of light through a waveguide using a laser source, it is important to know which optical modes could persist within a specified core/cladding dimension of the waveguide. Performing a mode analysis using a full vectorial finite element tool, such as the COMSOL Multiphysics® software, could be very helpful to qualitatively and quantitatively analyze the optical modes and dispersion curve respectively.
Performing a modal analysis on any isotropic material requires the definition of a single complex value, while in the case of an anisotropic material, a full tensor relative permittivity approach is required. The electric permittivity essentially relates the electric field with the material property. Here, tensor refers to a 3by3 matrix that has both diagonal (_{xx}, _{yy}, _{zz}) and offdiagonal (_{xy}, _{xz}, _{yx}, _{yz}, _{zx}, _{zy}) terms as shown below.
However, for all materials, you can find a coordinate system in which you only have nonzero diagonal elements in the permittivity tensor, whereas the offdiagonal elements are all zero. The three coordinate axes in this rotated coordinate system are the principal axes of the material and, correspondingly, the three values for the diagonal elements in the permittivity tensor are called the principal permittivities of the material.
There are basically two kinds of anisotropic crystal: uniaxial and biaxial crystal. With a suitable choice of coordinate system, where only the diagonal elements of the permittivity tensor are nonzero, in terms of optical properties, uniaxial crystal considers only the diagonal terms, that is _{xx} = _{yy} = (n_{o})^{2}, _{zz} = (n_{e})^{2}, where n_{o} and n_{e} are the ordinary and extraordinary refractive indices. However, when , it is known as a biaxial crystal.
To put this argument into a modeling perspective, we can extend the buried rib waveguide example from this blog post on silicon photonics design. We can perform a modal analysis on the 2D cross section of the waveguide with the square core and cladding length of 4 um and 20 um, respectively (shown below). The operating wavelength for all the cases is considered as 1.55[um].
Schematic of 3D buried rib optical waveguide where the mode analysis was performed at the inlet 2D cross section. The intensity plot and arrow plot representing the mode and polarization of Efields respectively.
Core of the rib waveguide depicting the optic axis (red) along the xaxis and the principal axis (blue).
In the classic case of a uniaxial material, we assume the optic axis (i.e., caxis) is along the principal xaxis (as shown above) and consider the diagonal relative permittivity _{yy} and _{zz} terms (which are orthogonal to the caxis) as the square of ordinary refractive index (~1.5199^{2} ~ 2.31). The _{xx} component element that is along the caxis is considered as the square of extraordinary refractive index (~1.4799^{2} ~ 2.19) (as per Ref. 3). In addition, the offdiagonal terms are considered zero (as shown below) and the cladding has an isotropic relative permittivity (~1.4318^{2}). The optical modes derived are the 6 modes shown above. Note the difference in the refractive indices: “n_{xx} – n_{yy}” is known as birefringence, where n_{xx} = and n_{yy} = .
Relative permittivity tensor with diagonal elements.
By evaluating the optical modes, we can visually comprehend the behavior of the optical waveguide. However, the dispersion curves could also be handy for performing quantitative analyses. A dispersion curve represents the variation of the effective refractive index with respect to the length of the waveguide or the operating frequency.
A modal analysis is performed while parametrically sweeping the length of the waveguide from 0.5 um to 4 um to derive the dispersion curve for the anisotropic core, as shown in the figure below. We assume the earlier case stated, with diagonal anisotropy terms of the core (i.e., _{xx} = 2.19, _{yy} = _{zz} = 2.31 and all of the diagonal elements are zero). The results are compared with Koshiba et al. (Ref. 3).
Dispersion curve with transverse anisotropic core.
When the optic axis (i.e., caxis) lies in XY plane and makes an angle of with the xaxis, the diagonal components _{xx}, _{yy}, _{zz} and offdiagonal components _{xy} and _{yz} are nonzero, while the rest of the components are zero. The full relative permittivity tensor could be evaluated by using the rotation matrix [R] as shown below, where the rotation matrix [R] is specifically for rotating the caxis in the XY plane. _{xx} is the square of the extraordinary refractive index (~2.19), because the caxis lies along the principal xaxis, while _{yy} and _{zz} are the square of the ordinary refractive index (~2.31). The offdiagonal elements _{xy} and _{yz} are derived from the multiplication of the matrices as stated below.
The caxis lying in the XY plane and making an angle of with the xaxis.
The relative permittivity tensor ε is treated along with a rotation matrix, rotating the caxis in the XY plane with angle .
Finally, the modal analysis of the waveguide with offdiagonal anisotropic core and isotropic cladding, where the optic axis makes angles of 0, 15, 30, and 45 degrees with respect to the principal xaxis, as shown below. Here, it could be observed that the direction of the inplane magnetic field changes according to the change in the angle of the optic axis. The dispersion curve could also be plotted by parametrically sweeping the length of the core and cladding from 0.5 um to 4 um, while considering the angle as 45°. The dispersion curve tends to be similar to the dispersion curve of the diagonal anisotropy, as discussed above.
Mode analysis, including offdiagonal terms, for θ = 0° (topleft), θ = 15° (topright), θ = 30° (bottomleft), and θ = 45° (bottomright). The figure represents the magnetic field lines within the core for different rotation angles.
Finally, when considering the longitudinal anisotropy where the optic axis (i.e., caxis) lies in the YZ plane and makes an angle of with the yaxis, the diagonal components _{xx}, _{yy}, _{zz} and the offdiagonal components _{yz} and _{zy} are nonzero, while the rest of the components are zero. The relative permittivity tensor could be evaluated by using the rotation matrix [R] as shown below, where the rotation matrix [R] is specifically for rotating the caxis in the YZ plane. _{yy} is the square of the extraordinary refractive index (~2.19), because the caxis lies along the principal yaxis, while _{xx}, _{zz} is the square of the ordinary refractive index (~2.31). The offdiagonal elements _{yz} and _{zy} are derived from the multiplication of the matrices as stated below.
The caxis lying in the YZ plane and making an angle of with the xaxis.
The relative permittivity tensor ε is treated along with a rotation matrix, rotating in the YZ plane with angle .
A modal analysis is then performed where the length of the waveguide is parametrically swept from 0.5 um to 4 um to derive the dispersion curve for the longitudinal anisotropic core, as shown in the figure below. In this case, = 45° (i.e., the caxis lies in the YZ plane and makes 45° with the yaxis) (Ref. 3).
Dispersion curve with longitudinal anisotropic core.
In this blog post, we performed qualitative analyses (modes of propagation) and quantitative analyses (dispersion curves) of the anisotropic optical waveguide using modal analysis in COMSOL Multiphysics. Diagonal anisotropy as well as offdiagonal transverse and longitudinal anisotropy were considered to derive their dispersion relationships. These types of analyses give us more flexibility when carrying out optimization of material and geometric parameters to help us gain an indepth and intuitive understanding of the physics of anisotropic materials.
A simple tutorial model to help you started would be the StepIndex Fiber, which involves mode analysis over a 2D cross section of the 3D optical fiber.