The bolometer was first invented in 1878 by Samuel P. Langley, an American astronomer, with the intended use of studying solar irradiance and solar radiation intensity at different wavelengths. Bolometers consist of a strip of conducting material (a conducting absorber) that is thermally insulated from the rest of the device and mounted on a dielectric material, which acts as a heat sink and electrical insulator.
When a bolometer is exposed to incident electromagnetic radiation, the conducting absorber’s temperature increases in comparison to the rest of the device. As this temperature change occurs, the absorber’s electrical conductivity lowers, altering the flow of the bias current. This change in voltage is detected by a voltmeter and because the change is related to the amount of incident electromagnetic radiation, the device can be used as a sensor.
A schematic of a bolometer geometry that shows how the device operates. Image by J. Thomas, J.S. Crompton, and K.C. Koppenhoefer and taken from their COMSOL Conference paper submission.
In recent years, the use of bolometers has extended to night vision cameras and astronomy as well as particle and nuclear physics. NASA, for instance, uses transition-edge sensor (TES) bolometers to sense minor changes that occur in temperature when photons are absorbed and converted into heat. These devices help generate detailed maps for astrophysics projects that illustrate the polarization of cosmic microwave backgrounds.
A bolometer array. Image by Geni. Licensed under CC BY-SA 4.0, via Wikimedia Commons.
Researchers at the Herschel Space Observatory are also utilizing bolometers. Their application relates to investigating distant galaxies and the beginning stages of star formation. Doing so involves using a device called the Spectral and Photometric Imaging Receiver (SPIRE), which consists of five arrays of bolometers. Bolometers are ideal in this case because they have the highest sensitivity for light in the far-infrared to millimeter range of any direct detector.
With their range of applications, bolometer designs need to be customized for different uses, while optimizing their overall sensitivity (a key design parameter). When it comes to measuring the design quality, it is favorable that bolometers are able to detect temperature changes of less than 0.0001°C.
AltaSim Technologies recognizes the importance of improving bolometer designs. With the help of COMSOL Multiphysics, they studied the impact of different design parameters on a bolometer’s sensitivity and the level of detection of incident electromagnetic radiation. Let’s see what they found…
For their studies, the researchers noted that bolometer functionality is tied to three main physical phenomena:
In order to analyze such phenomena, the team used multiphysics modeling to bidirectionally couple heat transfer with electric currents, solving the two elements simultaneously. Solving this coupled electrical and thermal problem involved using a segregated solver approach, using an iterative linear system solver for the current substep and a direct solver for the thermal substep.
As for radiation, testing realistic sunlight conditions was possible via a solar source position functionality available in the Heat Transfer Module. The researchers further applied a uniform normal heating term of 100 W/m^{2} as a smoothed ramp function that ramps from 0 to 100 W/m^{2} over 0.1 s. This is illustrated in the plot below. The model featured a fixed operating temperature of 25 K at the bottom boundary, with a convective heat flux applied to all other boundaries in order to simulate the small cooling effect caused by cooled air inside the bolometer.
A plot showing radiation heating as a function of time. Image by J. Thomas, J.S. Crompton, and K.C. Koppenhoefer and taken from their COMSOL Conference presentation.
Now, let’s turn our attention to the sensitivity study. While there are multiple factors affecting the sensitivity of a bolometer — strip spacing, strip aspect ratio, materials, and operating temperatures — the focus here was evaluating the impact of specific design parameters.
In their study, the researchers chose to model a copper material absorbing strip with temperature-dependent material properties. Why copper? As an absorbent material with a strong dependence of conductivity to temperature, copper enabled the team to achieve the greatest sensitivity in their chosen temperature range, 10-50 K (cryogenic temperatures).
Graph showing the electrical conductivity as a function of T. Here, we can see the most sensitive temperature range for a copper material. Image by J. Thomas, J.S. Crompton, and K.C. Koppenhoefer and taken from their COMSOL Conference paper submission.
One point of interest was the dimensions of the contact strip. For this analysis, the team utilized the loft feature in the Design Module for automatic parameterization of the strip spacing parameter. They also generated a free tetrahedral mesh using the default physics-controlled mesh in COMSOL Multiphysics.
Left: Parameterization of the strip spacing. Right: Tetrahedral mesh. Images by J. Thomas, J.S. Crompton, and K.C. Koppenhoefer and taken from their COMSOL Conference paper submission.
The simulation analyses generated results for the temperature, current flow, and voltage caused by incident radiation at 0.2 s. Sensitivity, represented by S in this study, was defined as the change in voltage divided by the amount of absorbed incident light wattage.
Temperature (left), current flow (middle), and change in voltage caused by incident radiation (right) at 0.2 s. Images by J. Thomas, J.S. Crompton, and K.C. Koppenhoefer and taken from their COMSOL Conference paper submission.
To judge how strip spacing affects sensitivity, the researchers performed a parametric sweep for different strip spacing values. In order to account for the voltage drop change for the different configurations of the serpentine length, the sensitivity was normalized by an initial voltage drop. As the graph below indicates, increasing the space between the mounting board and serpentine absorber led to improved bolometer sensitivity. These results provided valuable information to the team at AltaSim, allowing them to move forward in creating a sensitive bolometer design that can more accurately detect and measure the power of incident electromagnetic radiation.
A graph highlighting the effects of strip spacing on bolometer sensitivity. Image by J. Thomas, J.S. Crompton, and K.C. Koppenhoefer and taken from their COMSOL Conference paper submission.
The simulation study presented here can be used to create customized bolometer designs for specific applications and identify other methods for maximizing device sensitivity. For the research team at AltaSim, the computational model can further serve as a foundation for examining additional design parameters that affect the sensitivity of the device. For example, they could study materials that are specially designed to be sensitive to conductivity at room temperatures. Further points of focus in the team’s simulation research include analyzing the serpentine geometry, strip material selection, and bias current magnitude.
Nanobots that reduce ocean pollution, solar cells that operate in the rain, and devices that transmit wireless data up to ten times faster. What binds these developing technologies together, aside from their innovative nature, is the use of a revolutionary material that has been a recurring topic of discussion on the COMSOL Blog. Its name? Graphene.
Recognizing the advantages of this strong, lightweight material, more and more industries have begun to embrace graphene’s potential use for a variety of applications. Take biosensors, for instance. Because of its high electrical conductivity and large surface area, graphene is an optimal material selection for these devices. This can be attributed to the fact that a faster transfer of electrons prompts greater accuracy and selectivity in the detection of biomolecules.
Glucose monitoring is one application of biosensors. Image by David-i98 (talk) (Uploads) — Own work. Licensed under CC BY-CA 3.0, via Wikimedia Commons.
In an effort to help bring new sophistication and reliability to biosensors for therapeutic solutions and personalized medical applications, a team from the Polytechnic University of Bucharest designed and analyzed a 3D multilayered graphene biosensor with COMSOL Multiphysics. Let’s have a look at their research, which was presented at the COMSOL Conference 2015 Grenoble.
As mentioned above, the research presented here is centered on a multilayered biosensor design. While single-layer graphene is known to be much more reactive than multilayered graphene structures, the edge of the material is more reactive than its surface. With graphene being rather inert, a multilayered graphene structure is an ideal option for biosensors.
In addition to the design of the biosensor itself, it is also important to consider the interface with which it will interact. In this case, that interface is human skin. Because it is the part of the human body with the largest surface area and because its responses to external and internal stimuli vary, human skin is a good environment through which to collect physical and chemical data.
Before developing their biosensor models, the team considered the following interfaces:
This series of interfaces led to the development of two biosensor device models that could describe the influence of process variables as well as environmental stimuli. The first of these models is a single-layer graphene/graphene-oxide sensor device with two electrodes. The second model is a multilayered sensor device with four electrodes. For the latter case, the inclusion and exclusion of the graphene composite structure was studied to further differentiate between the graphene responses.
A multilayered graphene biosensor device. Image by E. Lacatus, G.C. Alecu, and A. Tudor and taken from their paper “Models for Simulation Based Selection of 3D Multilayered Graphene Biosensors“.
With their models in place, the researchers ran a series of simulations in COMSOL Multiphysics to analyze the biosensing capabilities of both model configurations. Such analyses included measuring the temperature distribution at the interface of the device as well as the electrical potential (shown in the following set of figures). The findings revealed that the graphene-based structure possessed a sensing ability regardless of the specific design.
Temperature distribution plots at the interface for the device with two electrodes (left) and the device with four electrodes (right). Image by E. Lacatus, G.C. Alecu, and A. Tudor and taken from their paper “Models for Simulation Based Selection of 3D Multilayered Graphene Biosensors“.
Electric potential plots on the interface for the device with two electrodes (left) and the device with four electrodes (right). Image by E. Lacatus, G.C. Alecu, and A. Tudor and taken from their paper “Models for Simulation Based Selection of 3D Multilayered Graphene Biosensors“.
After identifying these sensing capabilities, the team tested a number of different biosensor device structures as a means of determining the optimal response for the PVA hydrogel on the sheets of graphene and for the protein-functionalized graphene biosensors. Simulation proved to be a useful tool for analyzing such properties. The simulation results shown below, for instance, highlight the spatial distribution of flux energy on the graphene biosensor as well as the interface charge distributions for the device with four electrodes.
Simulations corresponding to the spatial distribution of flux energy on the graphene biosensor (left) and the interface charge distributions (right) for the device with four electrodes. Image by E. Lacatus, G.C. Alecu, and A. Tudor and taken from their paper “Models for Simulation Based Selection of 3D Multilayered Graphene Biosensors“.
Furthering the researchers’ studies were analyses designed to show how different environmental stimuli influence the biosensor when reaching its active surface. Using the Acoustics Module, for example, the researchers were able to define the response of the interface to variations in acoustic pressure. The results below illustrate the effects of the acoustic stimuli on the graphene sensing structure for the device with four electrodes.
Analyzing the impact of acoustic stimuli on the device with four electrodes. Image by E. Lacatus, G.C. Alecu, and A. Tudor and taken from their presentation “Models for Simulation Based Selection of 3D Multilayered Graphene Biosensors“.
With COMSOL Multiphysics, the researchers could successfully identify the relevant properties of the graphene biosensing structures, all while relating them to the complex interface of the human skin. This simulation research, combined with the models themselves, provided valuable design solutions in the development of graphene-based biosensors.
In a previous blog post, we discussed the modeling of objects translating inside of domains that are filled with a fluid, or just a vacuum. This initial approach introduced the use of the deformed mesh interfaces and the concept of quadrilateral (or triangular) deforming domains, where the deformation is defined via bilinear interpolation. Such a technique works well even for large deformations, as long as the regions around the moving object can be appropriately subdivided. This, however, is not always possible.
A solid object moving along a linear path inside of a complicated domain.
Consider the case shown above, where an object moves along a straight line path, defined by \mathbf x(t), through a domain with protrusions from the sides. In this situation, it would be quite difficult to implement the original approach. So what else can we do?
The solution involves four steps. They are:
We can begin by dividing our original model space into two different geometry objects, as depicted in the figure below. Here, the red domains represent the stationary domains and the blue domains represent the regions in which our object is linearly translating. The subdivision process takes place in the geometry sequence, which is then finalized with the Form Assembly operation.
For a description of this functionality and steps on how to use it, watch this video.
Subdividing the modeling space into different geometry objects.
The Form Assembly step will allow the finite element meshes in the blue domains to slide relative to the meshes in the red domains. This step will also automatically introduce identity pairs that can be used to maintain continuity of the fields for which we will be solving. Let’s take a look at a representative mesh that may be appropriate in this case.
The subdivided domains with a representative mesh.
In the figure above, note that the dark red domains contain meshes that will not move at all. The dark blue domains, meanwhile, have translations completely defined by our known function, \mathbf x(t). The light blue domains are the regions in which the mesh will deform. We can simply use the previously introduced method of bilinear interpolation in these domains. You should also note that a Mapped mesh is used for these two rectangular domains and that the distribution of the elements is adjusted, such that they are reasonably similar in size or smaller than the adjacent nondeforming elements.
After the object has been linearly translated, the mesh in the light blue domains deforms.
From the previous images, you can clearly see that the meshes no longer line up between the moving and stationary domains. While COMSOL Multiphysics version 5.0 has introduced greater accuracy in the handling of noncongruent meshes between domains, there are some things that you should be aware of when using this functionality.
As a result of the Form Assembly geometric finalization step, COMSOL Multiphysics will automatically determine the identity pairs — the mating faces at which the mesh can be misaligned. We simply need to tell each physics interface within our model to maintain continuity at these boundaries. This can be accomplished via the Pairs > Continuity boundary condition, which is available within the boundary conditions for all of the physics interfaces.
Once this feature is added and applied to all of the identity pairs, the software will apply additional conditions at these interfaces to ensure that the solution is as smooth as possible over the mesh discontinuity. Each identity pair has a so-called source side and a destination side. The mesh on the destination side should be finer in all configurations of the mesh.
Assembly meshes with noncongruent meshes at the boundaries can be used in combination with most physics interfaces. There are, however, a few important exceptions. Whenever you are solving an electromagnetics problem involving a curl operator on a vector field, such a technique cannot be used. Common physics interfaces that fall into this category include the 3D Electromagnetic Waves interfaces, the 3D Magnetic Fields interfaces, and the 3D Magnetic and Electric Fields interfaces. This still, of course, leaves us with a wide range of physics.
Let’s look at one case of computing the temperature fields around our object, with differing temperatures for the object and the surrounding domain’s outer walls. The contour plots of the temperature fields shown below verify that the solution is quite smooth over the boundary where the mesh is not continuous.
The temperature fields over time are smooth across the Continuity boundary condition applied to the identity pair.
At this point, you can probably already see how this same technique can be applied to a rotating object. We simply create a circular domain around our rotating object and use all of the same techniques we have discussed here. Of course, if the object is only rotating, we no longer need the deforming mesh — making things even a bit simpler for us.
The figures below show the same object from before, except now the object is rotating.
An object rotating about a point.
An assembly composed of stationary and rotating objects as well as the mesh.
The expressions for the prescribed deformation of the rotating domain, (X_{r}, Y_{r}), can be expressed in terms of the angular frequency, \omega; the undeformed geometry coordinates, (X_{g}, Y_{g}); the point about which the object is rotating, (X_{0}, Y_{0}); and time, t. This gives us the following expressions:
where the prescribed deformation in the Deformed Geometry interface is quite simply:
Seems easy, right? In fact, the technique outlined in this section is actually applied automatically in COMSOL Multiphysics when using the Rotating Machinery, Fluid Flow and the Rotating Machinery, Magnetic physics interfaces. This provides you with a behind-the-scenes look at what is going on within these interfaces!
We have now introduced methods for modeling the motion of solid objects inside fluid- or vacuum-filled domains. Although we have simply prescribed a displacement in all of these cases, the displacement of our solid objects could be computed and coupled to the field solutions in the surrounding regions. That is, however, a topic for another day.
Interested in learning more about using the deformed mesh interfaces for your modeling? Download the tutorial from our Application Gallery.
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Suppose we want to set up a COMSOL Multiphysics model of a solid object moving around inside of a larger domain filled with fluid such as air, or even just a vacuum. To start, let’s assume that we know what path the object will take over time. We won’t worry about which physics we need to solve the model, but we’ll assume that we want to solve for some fields in both the moving domain and the surrounding domain. Of course, we will need a finite element mesh in both of these regions, but this finite element mesh will need to change.
A solid object moves freely around inside of a larger domain along a known path.
For situations like this, there are two options: The Deformed Geometry interface and the Moving Mesh interface. These two interfaces actually work identically, but are meant to be used in different situations.
The actual use of these two interfaces is identical, but choosing between them depends on which other physics you want to solve, as the interfaces handle each type of physics differently. Although we won’t cover how to choose between these two interfaces in this blog post, it is worth reading the sections “Deformed Mesh Fundamentals” and “Handling Frames in Heat Transfer” in the COMSOL Multiphysics Reference Manual as a starting point.
It is also worth mentioning that the Solid Mechanics interface cannot be combined with the Moving Mesh interface. The Solid Mechanics interface already computes the domain deformation via the balance of momentum. Other physics, such as heat transfer in solids, are solved on this deformed shape. On the other hand, it is reasonable to combine the Deformed Geometry interface with the Solid Mechanics interface if you want to study the change in stresses due to material removal, or if you want to perform a parametric sweep over a dimension without parameterizing the geometry, as described in this previous blog post.
Here, we look at the conceptual case of an object moving around inside of a larger domain with stationary boundaries, as shown in the figure above. The path of the object over time is known. We will look at how to set up the Deformed Geometry interface for this problem. But first, we need to take a quick look at which equations will be solved in COMSOL Multiphysics.
Our case of an object moving around inside of a domain is actually a boundary value problem. All boundaries have known displacements, and these boundary displacements can be used to define the deformation of the mesh within the interior of both domains.
There are four types of approaches for computing the deformation of the mesh within each domain: Laplace, Winslow, Hyperelastic, and Yeoh smoothing types. Here, we will address only the simplest case, referred to as a Laplace smoothing, and demonstrate how this approach is sufficient for most cases. The Laplace smoothing approach solves the following partial differential equation within the domain:
where lowercase (x,y,z) are the deformed positions of the mesh and uppercase (X,Y,Z) are the original, undeformed positions.
Since the displacements at all boundaries are known, this is a well-posed problem, and theoretically, the solution to this equation will give us the deformation of the mesh. However, in practice, we may run into cases where the computed deformation field is not very useful. This is illustrated in the figure below, which shows the original mesh on the original domain and the deformed mesh as the part is moved along the diagonal. Observe the highlighted region and note that the mesh gets highly distorted around the moving part edges, especially at sharp corners. This high distortion prevents the model from solving the above equation past a certain amount of deformation.
Original and deformed mesh. The region where the mesh gets highly distorted is highlighted.
In the above image, the deformation of the blue domain is completely described by its boundaries and can be prescribed. On the other hand, the deformation within the red region requires solving the above partial differential equation, and this leads to difficulties. What we want is an approach that allows us to model greater deformations while minimizing the mesh deformation.
If you have a mathematical background, you will recognize the above governing equation as Laplace’s equation and you might even know the solutions to it for a few simple cases. One of the simpler cases is the solution to Laplace’s equation on a Cartesian domain with Dirichlet boundary conditions that vary linearly along each boundary and continuously around the perimeter. For this case, the solution within the domain is equal to bilinear interpolation between the boundary conditions given at the four corners. As it turns out, you can use bilinear interpolation to find the solution to Laplace’s equation for any convex four-sided domain with straight boundaries.
The first thing that we have to do is subdivide our complicated deforming domain into convex four-sided domains with straight boundaries. One such possible subdivision is shown below.
Subdividing the domain so that the deforming region (red) is composed of four-sided convex domains.
The deforming domain is divided into convex quadrilateral domains. In fact, we could have also divided it into triangular domains since that would simply be a special case of a quadrilateral with two vertices at the same location — a so-called degenerate domain. We would only need to decompose the domain into triangles if it were not possible to split the domains into quadrilaterals.
Now that we have these additional boundaries, we need to completely define all of the boundary conditions for the deformation within the domain. The boundaries adjacent to the deforming domain are known and there is no deformation at the outside boundaries. But what about the boundaries connecting these? We have a straight line connecting two points where the deformation is known, so we could just apply linear interpolation along these lines to specify the deformation there as well.
And how can we easily compute this linear interpolation? As you might have already guessed, we can simply solve Laplace’s equation along these connecting lines!
A very general way of doing this is by adding a Coefficient Form Boundary PDE interface to our model to solve for two variables that describe the displacement along each of these four boundaries. This interface allows you to specify the coefficients of a partial differential equation to set up Laplace’s equation along a boundary. We know the displacements at the points on either end of the boundary, which gives us a fully defined and solvable boundary value problem for the displacements along the boundaries.
These new help variables completely define the deforming domains. The results are shown below and demonstrate that larger deformations of the mesh are possible. Of course, we still cannot move the object such that it collides with the boundary. That would imply that the topology of the domain would change; also, the elements cannot have zero area. We can, however, make the deformed domain very small and thin.
The undeformed and deformed mesh after adding the help variables for the Deformed Geometry along the interior boundaries.
You are probably thinking that the mesh shown above appears rather distorted, but keep in mind that all of these distorted elements still have straight-sided edges, which is good. In practice, you will often find that you can get good results even from what appear to be highly distorted elements.
However, we can observe that there are now very many small, distorted elements in one region and larger, stretched elements in other parts of our moving domain. The last piece of the puzzle is to use Automatic Remeshing, which will stop a transient simulation based on a mesh quality metric and remesh the current deformed shape.
The deformed geometry immediately before and after the Automatic Remeshing step.
We can see from the above images that Automatic Remeshing leads to a lower element count in the compressed region and adds elements in the stretched region, such that the elements are reasonably uniform. This total number of elements in the mesh stays about the same. There is also an added computational burden due to the remeshing, so this step is only warranted if the element distortion adversely affects the accuracy of the results.
We have looked at a case where we know exactly how our solid object will move around in our fluid domain. But what if there is an unknown deformation of the solid, such as due to some applied loads that are computed during the solution? A classic example of such a situation is a fluid-structure interaction analysis, where the deformation of the solid is due to the surrounding fluid flow.
In such situations, we can use the Integration Component Coupling operator, which makes the deformation at one point of a deforming solid structure available everywhere within the model space. The deformation of one or more points can then be used to control the deformation of the mesh. A good example of this technique is available in the Micropump Mechanism tutorial. The technique is visualized below.
When the actual deformation is unknown, an integration component coupling at a helper point can be used to control a helper line that defines the mesh deformation.
We can see in the image above that the modeling domain is actually not divided into convex quadrilaterals, and that the helper line is allowed to slide along the top boundary of the modeling domain. So this modeling approach is a little bit less strict, yet still allows the mesh to deform significantly. Hopefully it is clear that there is no single best approach to every situation. You may want to investigate a combination of techniques for your particular case.
We have described how to use the deformed mesh interfaces efficiently by decomposing the deforming domain into quadrilateral domains and introducing help variables along the boundaries. This approach makes the problem easier for the COMSOL Multiphysics software to solve. The addition of Automatic Remeshing is helpful if there is significant deformation. The approach outlined here can also be applied to 3D geometries. An example that uses both 2D and 3D cases is available here.
So far, we have only looked at translations of objects inside of relatively simple domains where it is easy to set up deforming domains. When we need to consider geometries that cannot easily be subdivided and cases with rotations of objects, we need to use a different approach. We will cover that topic in an upcoming blog post on this subject, so stay tuned!
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Based on their operation, magnetic bearings can be classified into one of two groups: active magnetic bearings (AMBs) and passive magnetic bearings (PMBs). AMBs function through an attractive force between a ferromagnetic material and electromagnets (coil and core). PMBs function through a repulsive force between the permanent magnets (PMs) and/or a conducting surface and PMs.
Active magnetic bearings consist of a stationary part — the stator — that contains the electromagnets and the position sensors, and a rotating part — the rotor — that moves with the shaft. Under normal operating conditions, the rotor is ideally centered with an equal gap around the stator. However, in perturbed situations, the rotor’s position is controlled using a closed-loop feedback system. The change in rotor position is measured by the sensor, which is then passed to the digital controller. After processing the data, the controller sends the signals to the power amplifier. The amplifier readjusts the currents in the electromagnets to push the rotor back to its original position. To readjust the rotor, it is important for designers to know the magnetic force at various offset rotor positions as well as the corresponding currents.
A schematic depicting the components of AMBs. Image via Wikimedia Commons.
AMBs are advantageous due to their active control of the rotor position, but this means a higher cost for the electronic circuit as well as greater operational costs. The operational costs can, of course, be reduced by optimizing the design of the electromagnets so that less power is needed to run them. COMSOL Multiphysics is a helpful tool in this optimization process.
The most appropriate way to simulate AMBs, including the stator and the rotor, is with the Rotating Machinery, Magnetic interface in the AC/DC Module. The modeling process is very similar to that of an electric generator or motor, as illustrated in our Generator in 2D tutorial. For general guidelines, you can refer to our previous blog post “How to Model Rotating Machinery in 3D“.
Using the Magnetic Fields interface, you can model PMs as well as conducting coils (single-turn or multi-turn coils). However, you cannot model the induced currents due to rotations. If the induced currents can be neglected, you can set up the model in the stationary or frequency domain and add a parametric sweep study for various rotor positions, computing the magnetic force or torque.
Passive magnetic bearings use permanent magnets and do not require sensor and control circuitry or input power. The constant air gap is maintained by the magnetic repulsion force between the opposite poles of PMs, as demonstrated here, or by the electrodynamic suspension (EDS) between the PMs and the rotating conducting disc or shaft, as illustrated here. The geometry and the simulation results for the passive magnetic bearing using PMs is shown below.
Left: Geometry of an axial magnetic bearing using permanent magnets. The magnetization direction of the permanent magnets is depicted by black arrows. Right: The results plot showing the magnetic flux density (arrow plot) and the surface plot of the magnetic flux density norm.
When rotating in a magnetic field produced by a PM, an electrically conducting rotor induces eddy currents on the conducting rotor. These eddy currents, in turn, generate a magnetic field that opposes the magnetic fields by the magnets and induces a repulsive force between the rotating conductor and the stationary PMs. The displacement of the rotor is always balanced by this repulsive magnetic force. Thus, the rotor is rotating at the center with a uniform gap.
Electrodynamic bearings can be further classified as radial electrodynamic bearings and axial electrodynamic bearings. This is based upon whether the magnetic flux is parallel to the rotor axis or perpendicular to the rotor axis, respectively.
A radial electrodynamic bearing consists of a conducting cylinder affixed to the rotating shaft. The components of the PMs are stacked in between the iron rings, such that the radially inward or outward magnetic flux — with reference to the shaft axis — is created in the air gap between the stator and the rotor. The Electrodynamic Bearing 3D tutorial available in our Application Gallery is solved using the Magnetic and Electric Fields interface. The magnetic forces are computed for various offset positions.
Left: The 3D geometry of a radial electrodynamic bearing. Right: Radial electrodynamic bearing depicting the magnetic flux density in the stator (iron and magnets) and the eddy currents (grayscale) in the conducting rotor for the x-axis offset position of 1.5 mm.
The cross-sectional cut configuration for an axial electrodynamic bearing is shown below. The conducting disc is attached to the rotor and the magnetic material (iron yoke) is used to guide the magnetic fields of the PMs, such that the magnetic flux lines are parallel to the rotor axis. This is where the term axial electrodynamic bearing comes from.
In this design, the magnetic flux path is very effective with a relatively small air gap. The full tutorial is available to download from our Application Gallery.
Left: Cross-sectional cut of an axial electrodynamic bearing. Right: An axial electrodynamic bearing showing the magnetic flux density in the stator and the eddy currents in the conducting rotor. An arrow plot of the magnetic flux density in the stator and the eddy currents in the rotor are also shown.
Both electrodynamic bearing examples discussed above are modeled in COMSOL Multiphysics using the Magnetic and Electric Fields interface. In both cases, the Velocity (Lorentz term) feature is used to prescribe the rotational velocity. By using this approach, you do not need to use the Moving Mesh interface to account for the rotation of the rotor.
Note that the Velocity (Lorentz Term) feature can only be used when the moving domain does not contain magnetic sources like currents or magnetization (fixed or induced) that move along with the material, and the moving domains are invariant in the direction of motion. For instance, the Velocity (Lorentz Term) feature can be used to model conductive (not magnetic) homogeneous spinning disks. Some examples include magnetic brakes, an electrodynamic bearing, a homopolar generator, magnets over a moving infinite homogeneous plane (such as a falling magnet through a copper tube or maglev trains), a flow of homogeneous conducting fluid past a magnet (such as liquid metal pumps or hall generators/thrusters).
Note: The domain assigned to the Velocity (Lorentz Term) feature cannot contain currents or permanent magnets. Additionally, the rotating domain cannot be permeable or saturable materials (i.e., iron) as they would contain induced magnetization. The moving magnetic sources induced in the magnetic material would, in turn, induce an electric field in the surrounding region, which is not modeled by the feature.
In COMSOL Multiphysics, there are two available methods for calculating electromagnetic forces and torques. The most general method is the Maxwell stress tensor method, which is used by the Force Calculation feature in the Magnetic Fields interface; the Magnetic Fields, No Currents interface; the Magnetic and Electric Fields interface; and the Rotating Machinery, Magnetic interface.
For instance, by adding this feature, the spatial component of the magnetic forces (mf.Forcex_0, mf.Forcey_0, mf.Forcez_0
) and the axial torque ( mf.Tax_0
) in the Magnetic Fields interface are available for postprocessing. The Force Calculation feature simply integrates the Maxwell stresses, evaluated just outside of the selected domain (or domains) and over the entire outer boundary of the domain selection, which should be a group of domains moving together (a single mechanical body). Since this method is based on surface integration, the computed force is sensitive to mesh size. It is important when using this method to always perform a mesh refinement study to correctly compute the force or torque.
Calculation force will not be correct if the domain applying the Force Calculation feature is touching an external boundary, a periodic boundary, and an identity pair. Furthermore, to compute the force on a magnet attached to a ferromagnetic surface, contact boundaries must have a thin low permeability gap (thin low permittivity gap for the electrostatic equivalent) assigned, as the Maxwell stresses should be evaluated in air rather than in the ferromagnetic (dielectric) material.
The second method — the Lorentz force method — works only in special cases for computing the magnetic force on nonmagnetic, current-carrying domains. The Lorentz force is defined as \mathbf{F = J \times B}, where J is the current density and B is the magnetic flux density. The Lorentz force is very accurate for force calculations in electrically conducting domains as it is evaluated on volume rather than boundaries. Therefore, whenever possible, the Lorentz force method is preferred over the Maxwell stress tensor method.
In addition to those built-in methods referenced above, the magnetic force and torque can also be calculated using the virtual work method, or the principle of virtual displacement. In this technique, the force is calculated by studying the effect of a small displacement on electromagnetic energy. The virtual work method can be employed by using the features for deformed mesh and sensitivity analysis in COMSOL Multiphysics, as demonstrated here.
In solid mechanics, stiffness is the rigidity of an object — the extent to which the object resists deformation in response to an applied force. Similarly, in magnetic bearing applications, the parameter is defined as the magnetic stiffness, and it is the negative of the derivatives of the total magnetic force with respect to position. If the magnetic force is F_z, the magnetic stiffness with respect to position z is given by:
A tutorial illustrating this method for computing the magnetic stiffness in an axial magnetic bearing is available here. This example, however, is limited to 2D axisymmetry. Therefore, the magnetic stiffness in the x- and y-directions cannot be evaluated. To evaluate the magnetic stiffness in all directions, you would need to model the problem in 3D. Here, we will create a 3D version of the same axial magnetic bearing discussed above and determine the stiffness k_x. You can download this example tutorial from our Application Gallery.
This approach primarily involves the use of the Magnetic Fields, Deformed Geometry, and Sensitivity interfaces. As in the 2D model, the Magnetic Fields interface is used. The magnets are modeled using the Ampère’s Law feature with constitutive relation set to a remanent flux density of 1[T]. The Force Calculation feature is added to the inner magnets only, with the geometry parameterized such that the inner magnets are a function of the parameter dX
(an offset in the x-axis). This parameter will later be used for a parametric sweep as well as to specify the prescribed mesh displacement in the Deformed Geometry interface. Only a quarter of the geometry will be used here to compute the magnetic stiffness in the x-direction.
In this configuration, note that only the x-axis forces are correctly computed. Due to the symmetry, the force in the y- and z-directions should be zero. However, due to the fact that only a quarter of the geometry is modeled, the computed force will be quite large. The stiffness in the y-direction can be computed similarly by analyzing a quarter model cut symmetric to the yz- and xy-planes.
Left: The quarter 3D model of an axial magnetic bearing for stiffness calculation. Right: The magnetic flux density norm and the magnetic flux density arrow plot in a half model. The results are plotted using a Mirror 3D data set at the xy-plane.
The Deformed Geometry interface is solved everywhere except in the infinite element domain region. To solve for this region, begin by adding a Free Deformation node on the air region around the magnets. Similarly, you can add a prescribed deformation of dX
on the domains of the inner magnets along the x-axis. Finally, add two Prescribed Mesh Displacement nodes for the inner magnet boundaries and the symmetry cut plane boundaries, as illustrated below.
Settings for prescribed deformation on the inner magnets’ domains.
Settings for prescribed mesh displacement on the inner magnets’ boundaries.
Settings for prescribed mesh displacement on the symmetry cut boundaries.
In the Sensitivity interface, add the Global Objective feature and specify the total force in the x-direction (i.e., 4*mf.Forcex_0
) in the objective expression under the Global Objective settings. Here, mf.Forcex_0
is the x-component of the Maxwell stress tensor force that is being calculated by the Force Calculation feature in the Magnetic Fields interface. Similarly, add the global control variable dX
, as shown below.
Settings for the global objective (left) and the global control variables (right).
Since the Sensitivity (and Optimization) study cannot be combined with a Parametric Sweep study node, the model must be solved in two separate studies. The first study will include the Sensitivity solver and the Stationary solver. The second study will then have a Parametric Sweep study but will use Study 1 as a study reference. Perform the parametric sweep over a parameter dX
using the interval of range(0,1.5/20,1.5), with the settings shown below.
Left: Settings for the Stationary study for sensitivity analysis. Right: Settings for the Parametric Sweep study using Study 1 as a reference.
The magnetic force in the x-axis and the magnetic stiffness in the x-direction are plotted as 1D global plots.
Settings for the magnetic force plot (left) and the magnetic stiffness (right).
Left: The electromagnetic force x-component as a function of offset in the x-axis. Right: The magnetic stiffness k_x as a function of offset in the x-axis.
Here, we have discussed several types of magnetic bearings that can be modeled in COMSOL Multiphysics using interfaces available in the AC/DC Module. Simulation provides a simplified approach to addressing design parameters within these types of bearings, helping to optimize their performance as well as their lifespan.
In the next blog post in this series, we will focus on how COMSOL Multiphysics can be used to model magnetic gears. Stay tuned!
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The entire COMSOL® software product suite is built on top of the general-purpose software platform, COMSOL Multiphysics. This platform contains all of the core preprocessing, meshing, solving, and postprocessing capabilities, as well as several physics interfaces. (See our Specification Chart for complete details about what is available in each product.)
With COMSOL Multiphysics®, you can import 2D DXF™ files and 3D STL and 3D VRML files. You can use the 2D DXF™ file format to import profiles and extrude, revolve, or sweep them along a path to create 3D geometry.
The STL and VRML formats are best suited for simple shapes; complex CAD data does not transfer reliably since these formats lack the sophistication of modern CAD file formats. To work with STL files containing 3D scans, we recommend that you import those as a mesh and use the built-in functionality to convert the imported mesh to geometry. Depending on the complexity and quality of the 3D scan, the resulting geometry can then be combined with other geometric objects that are either imported or created in COMSOL Multiphysics.
Also part of the core COMSOL Multiphysics capabilities, the Virtual Operations approximate the CAD data for meshing purposes and are useful for cleaning up all imported CAD data, or even COMSOL native geometry.
The LiveLink™ products allow you to work with the data directly from your CAD program. Supported CAD packages include SOLIDWORKS® software, Inventor® software, Autodesk® AutoCAD® software, PTC® Creo® Parametric™ software, PTC® Pro/ENGINEER® software, Solid Edge® software, and the building information modeling (BIM) software Autodesk® Revit® Architecture software. Both LiveLink™ for SOLIDWORKS® and LiveLink™ for Inventor® offer the One Window interface, which directly embeds the COMSOL® modeling environment within the CAD software user interface. The list of version compatibility with these products is maintained here.
When using these LiveLink™ tools, you must have both COMSOL Multiphysics and the CAD program installed and running on the computer you are using. The CAD data as well as materials definitions and other selections will be bidirectionally synchronized between your CAD package and COMSOL Multiphysics, with full associativity. You can read more about that here. This means that any modifications that you make within the CAD package will be available within the COMSOL environment, and you can use COMSOL Multiphysics to change any of the dimensions within your CAD file. The functionality of each of these modules is described here.
Since the data is transferred with associativity, as you change the dimensions in your CAD program to reshape the part, the COMSOL software will track these changes and appropriately re-map all of the boundary conditions and other geometry- and selection-based settings. To see a demonstration of this, please watch the relevant videos in our Video Gallery. You will find this functionality useful when you want to perform parametric sweeps over the dimensions in your CAD file or perform dimensional optimization using the COMSOL Optimization Module.
In addition to synchronizing CAD data between a CAD software and COMSOL, the LiveLink™ products also include support for file import of the full range of CAD file formats supported by the CAD Import Module. If you are solving problems where you actually want to model the volume inside of the CAD domain (such as for fluid flow models), you can also use the Cap Faces command to create enclosed volumes based upon an existing geometry, as described here. You will also be able to perform repair and geometric clean-up (defeaturing) operations on your CAD data and write out the resultant geometry, or any geometry you create in COMSOL Multiphysics®, to the Parasolid® software or ACIS® software file format.
The LiveLink™ products are the best option for you if you can have your CAD software and COMSOL software installed on the same computer and you want to take advantage of the benefits offered by the included integration. However, if you are working with CAD data that is coming from someone else and don’t have their CAD software on your computer, then you may want to use the CAD Import Module or the Design Module instead.
The CAD Import Module and the Design Module will allow you to import a wide range of CAD file formats. You can find the complete list of formats and versions that are importable here.
If you are planning to make many design iterations, then the relative drawback of both the CAD Import Module and the Design Module compared to the LiveLink™ products is that the data import is one-way and there is no associativity that is maintained between the CAD data and the COMSOL model. That is, if you make a modification to the CAD file and have to re-import the geometry, the physics features and other geometry-based settings in the COMSOL model may not reflect these changes. You will need to manually check all settings and re-apply them to the modified geometry. Additionally, the dimensional data in the original CAD file is not accessible, so you will not be able to perform parametric sweeps or optimization.
It is possible to work around this limitation as described in the “Parameterizing the Dimensions of Imported CAD Files” blog post, but this technique is usually only practical for simpler geometries.
The Design Module provides additional functionality for creating geometry. It includes all of the capabilities of the CAD Import Module, but also provides extra geometric modeling tools. The Parasolid® software Kernel functionality is used to provide 3D Fillet, 3D Chamfer, Loft, Midsurface, and Thicken operations. You can learn more about these operations in this introduction to the Design Module.
The CAD Import Module is recommended only if you are certain that you will never be using COMSOL Multiphysics in conjunction with any of the CAD packages for which there is a LiveLink™ product and if you do not want to create any complex CAD geometries within the COMSOL environment. The Design Module is recommended over the CAD Import Module since it provides all of the same functionality, but will also allow you to create more complex CAD geometries within COMSOL Multiphysics. These geometries can then be exported to the Parasolid® software or ACIS® software file formats. Both modules include the full suite of defeaturing operations as well as the Cap Faces operation.
In addition to the products mentioned here, there is also the File Import for CATIA® V5 Module, which can import CATIA® V5 software files and is an add-on to any of the LiveLink™ products for CAD packages, the CAD Import Module, or the Design Module.
The ECAD Import Module is used for the import of ECAD data, which are files that are typically meant to describe the layout of an integrated circuit (IC), micro-electro-mechanical systems (MEMS) device, or printed circuit board (PCB) and thus contain planar layouts, and in some cases thickness and elevation information.
While the data transfer with this module is without associativity, you can take advantage of selections created by the import functionality to preserve model settings after importing a changed file. Also, the layered structure of the generated geometric objects makes the use of coordinate-based selections for model settings especially suitable to automate model set-up with imported ECAD files. Look out for a future blog post on how to do this.
We recommend the LiveLink™ products if you have your CAD software and COMSOL simulation software installed on the same computer. The Design Module or the CAD Import Module can be used if you only want to import files, and the Design Module is preferred since it has enhanced functionality. The add-on File Import for CATIA® V5 Module is only needed for that specific file type. Finally, to incorporate geometry from ECAD layout files into your simulations, you will need the ECAD Import Module.
If you have any other questions about how best to interact with your CAD data, please contact us.
ACIS is a registered trademark of Spatial Corporation.
Autodesk, the Autodesk logo, AutoCAD, DXF, Inventor, and Revit are registered trademarks or trademarks of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries.
CATIA is a registered trademark of Dassault Systèmes or its subsidiaries in the US and/or other countries.
Parasolid and Solid Edge are trademarks or registered trademarks of Siemens Product Lifecycle Management Software Inc. or its subsidiaries in the United States and in other countries.
PTC, Creo, Parametric, and Pro/ENGINEER are trademarks or registered trademarks of PTC Inc. or its subsidiaries in the U.S. and in other countries.
SOLIDWORKS is a registered trademark of Dassault Systèmes SolidWorks Corp.
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In December, I came across an article from the BBC drawing the public’s attention to computer-aided design in architecture. The article cites a number of essential contributions of modeling, which include the structural analysis of Jørn Utzon’s radical design for the Sydney Opera House and a fluid-structure (facade pressure) study of the Burj Khalifa in Dubai. Additionally, fluid flow, microclimate, and acoustic studies are also referenced.
The Sydney Opera House.
From this article, it is clear that architecture benefits from software outside the traditional fields of drawing tools and structural analysis. The way that modern architects use modeling software is underlined by the breadth of interests discussed in the “Building Design Simulation” user presentation session at the COMSOL Conference 2014 Cambridge. Some of this research includes mass transport and flow in the determination of cleanroom quality, acoustic development of soundproof windows, thermal analysis towards “Energy Zero Building”, and so on.
One of the poster prize winners at the COMSOL Conference 2014 Cambridge was Carmelo Galante of Newtecnic (London, UK). He presented his work using COMSOL Multiphysics to study the heat transfer performance of a facade, specifically a double-glazed window in which the air cavity also contains “solar control devices”. These are essentially blinds that can reflect a proportion of the incident solar radiation. You can read the details of his analysis here.
Galante’s study used the Heat Transfer Module to investigate the combined influence of all three modes of heat transfer: conduction, convection, and radiation. The cavity is mechanically ventilated so fluid flow can arise due to forced convection as well as natural convection. On the basis of a “U-value” that quantifies thermal performance, the design was optimized by changing the number and position of forced flow inlets and outlets. By combining fluid and thermal physics with radiation in a single modeling environment, the interplay of the different physical effects can be evaluated.
In the Model Gallery, you can find an assortment of other examples from the world of architecture. Many architects will be interested in thermal analysis, as in the conference work mentioned above. Additionally, in the Heat Transfer Module, there are several examples related to the building and construction industry. One of them considers the thermal performance of a window, illustrating the effect of different glazing configurations.
Below, you can see the material configuration of an ISO standard sliding window, together with the temperature profile from a COMSOL Multiphysics analysis. The predicted heat loss of 0.65 W/(m.K) is quite large due to the high thermal conductivity of the aluminum structure.
The geometry of the sliding window.
Temperature distribution in the sliding window.
Structural stability analysis is still important, too. In the Geomechanics Module, there is a model of a concrete beam in which a mechanical model is used to assess the additional structural rigidity due to steel “rebars”. Rather than assuming that the concrete is just a linear elastic material, the empirical Ottosen plasticity model is used to describe the concrete.
My colleague Bridget recently wrote about a larger-scale study of facade reflectivity and coupled thermal effects: the formation of a caustic surface due to the concave glass facade of the Vdara® hotel in Las Vegas. Here, COMSOL Multiphysics was used to investigate a harmful consequence of the building design “after-the-fact” using the ray tracing functionality of the new Ray Optics Module.
Just as ray tracing allows us to study light paths and reflection at the building scale, ray acoustics can be used for acoustic analysis in the limit where the acoustic cavity is much larger than the wavelength of the sound. This is typically the case for large rooms, such as theaters and concert halls, where a clear understanding of the acoustic properties is essential. In the Model Gallery for the Acoustics Module, you can find an example of a small concert hall with the pressure and energy impulse responses computed.
With the release of COMSOL Multiphysics version 5.0, we introduced a new product that is designed to integrate architectural designs with multiphysics simulations: LiveLink™ for Revit®. This integration tool and interface transfers design elements from Autodesk® Revit®, a leading software within architectural design, to the COMSOL Multiphysics software.
Using LiveLink™ for Revit®, you can take geometries for particular rooms and architectural elements within a building and then synchronize them with your COMSOL Multiphysics model. You can add additional physical effects to the model as needed, just as if the geometry were built in COMSOL Multiphysics. By choosing which details you would like to include, you can ensure an accurate simulation.
With multiphysics modeling for architecture active in the news and the scientific community, we encourage you to learn more about these different architectural models and download them from our Model Gallery:
Autodesk and Revit are registered trademarks or trademarks of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries.
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Heat transfer computation often needs to include surface-to-surface radiation to reflect reality with accuracy. The numerical tools used to simulate surface-to-surface radiation differ significantly from those used for conduction or convection. Whereas the latter are based on local discretization of partial differential equations (PDEs), surface-to-surface radiation relies on non-local quantities — the view factors between diffuse surfaces that emit and receive radiation.
When surface-to-surface radiation is activated, the heat interface creates a set of operators that are evaluated like the irradiation variables in surface-to-surface radiation. Thanks to these operators, it is possible to retrieve the irradiation variable values and to compute the geometrical view factor in a given geometry.
In this blog post, I’ll explain how to compute geometrical view factors in a simple 3D geometry where analytical values of the view factor are available.
The new operators in COMSOL Multiphysics version 5.0 offer full access to all of the information used to generate surface-to-surface radiation equations. This is true for even the more advanced configurations, such as radiation on both sides of a shell and multiple spectral bands with different opacity properties.
Let’s start with the simplest case where we assume that surfaces behave like gray surfaces. In this case, we don’t need to distinguish between the spectral bands. We have two operators, one for each face (up or down) of the surfaces. They are as follows:
radopu(expr_up,expr_down)
radopd(expr_up,expr_down)
These two operators are designed to be evaluated on a boundary where the surface-to-surface radiation is active. Assuming that the heat interface tag is ht
, ht.radopu(ht.Ju,ht.Jd)
returns the mutual surface irradiation, ht.Gm_u
, that is received at the evaluation point on the upside of the boundary.
Note that ht.Ju
and ht.Jd
define the radiosity on the up- and downsides of the boundaries, respectively. Similarly, ht.radopd(ht.Ju,ht.Jd)
returns the mutual surface irradiation, ht.Gm_d
, on the boundary downside.
When multiple spectral bands are considered, a given boundary can be opaque for a spectral band and transparent for another band. Hence, one pair of operators is needed per spectral band. They work exactly like the operator for gray surface and are named as follows:
ht.radopB1u(expr_up,expr_down)
and ht.radopB1d(expr_up,expr_down)
ht.radopB2u(expr_up,expr_down)
and ht.radopB2d(expr_up,expr_down)
ht.radopB3u(expr_up,expr_down)
and ht.radopB3d(expr_up,expr_down)
Let’s consider two diffuse gray surfaces: S1 and S2. We’ll assume that radiation occurs only on the upside of these surfaces. From a thermal perspective, the view factor between S1 and S2, F_{S1-S2}, is the ratio between the diffuse energy leaving S1 and intercepted by S2 and the total diffuse energy leaving S1.
Using the operators described above, we have
(1)
Note that for clarity, the ht.
prefix has been removed.
Assuming that radiosity is the same value on all surfaces, the above definition can be simplified and no longer depends on J. In that case, F_{S1-S2} depends only on the geometrical configuration and no longer on the thermal configuration. Let’s call this a geometrical view factor to distinguish it from the view factor based on thermal radiation.
We now have
(2)
where S1 represents either the surface name or its area and I_{\textrm{S1}} is the function indicator of the surface S1, which returns 1 when it is evaluated on S1 and 0 elsewhere.
In order to get used to the new operators and check their accuracy, I chose a simple configuration. The geometry consists of two concentric spheres of radius R_{int} and R_{ext} (with R_{int} < R_{ext}), as shown below:
The radiation occurs between the external side of the small sphere and the internal side of the large sphere. The geometrical view factors are:
Here, S_{int} and S_{ext} represent the interior and exterior sphere, respectively.
To compute the geometrical view factor in the COMSOL Multiphysics simulation software, we need to add a heat interface with surface-to-surface radiation activated, then draw the geometry and build the mesh.
Then, we don’t really need to run a heat transfer simulation since we are interested in the geometrical view factor. To have access to the radopu
and radopd
operators is enough to get initial values.
Before doing that, though, we’d better prepare a few tools that will help us with the postprocessing.
In the geometrical view factor expression, we have used the surface indicators I_{\textrm{S1}} and I_{\textrm{S2}}. These are defined as Variables in COMSOL Multiphysics and use the Geometric Entity Selection, so that it is 0 everywhere except on the corresponding surface where it is 1. Let’s name them ext
and int
.
Screenshots of the Geometric Entity Selection settings for the surface indicators.
Next, we define the integration operators intop_ext
and intop_int
. They will make it easy to compute surface integrals; for example, the surface of S_{ext} can be evaluated as intop_ext(1)
.
Screenshots of the settings for the integration operators.
We have seen that radiation may occur on the upside, downside, or on both sides of the boundaries. The radiation operators are designed to be able to distinguish the radiation coming from each side. Therefore, we need to check the sides on this model.
We can do this easily via the Diffuse Surface feature, where the radiation direction can be set to “Negative normal direction” (downside) or “Positive normal direction” (upside). Using this option prompts arrows to be automatically displayed to show the direction of radiation leaving the surface. In our example, the radiation occurs on the downside for S_{ext} and the upside for S_{int}.
With all these tools available to us, evaluating the view factor using an expression similar to (1) is direct. For example,
(3)
is evaluated in COMSOL Multiphysics syntax with intop_ext(comp1.ht.radopd(0,ext))/intop_ext(1)
.
Note that the use of radopd
is due to the fact that the radiation occurs on the downside of S_{ext}. The first argument of radopd
is 0 for the same reason.
(4)
is evaluated with intop_ext(comp1.ht.radopd(int,0))/intop_int(1)
, where radopd
is due to the fact that the radiation occurs on the downside of S_{ext}. But this time, the second argument of radopd
is 0 because radiation occurs on the upside of S_{int}.
I’ve gathered all the results in a table:
Analytical value | Computed value | Error | |
---|---|---|---|
F_{S_{\textrm{int}}-S_{\textrm{int}}} | 0 | 0 | 0 |
F_{S_{\textrm{int}}-S_{\textrm{ext}}} | 1 | 0.998 | 2e^{-3} |
F_{S_{\textrm{ext}}-S_{\textrm{ext}}} | 0.91 | 0.9102 | 2e^{-4} |
F_{S_{\textrm{ext}}-S_{\textrm{int}}} | 0.09 | 0.09 | 1e^{-6} |
Thanks to the radiation operators, we were able to retrieve the geometrical view factor.
With COMSOL Multiphysics 5.0, it is possible to compute geometrical view factors between diffuse surfaces, thanks to dedicated operators. This offers a solution for the requests we received since the surface-to-surface features have been released. But, these operators can do much more. They are flexible enough to provide all terms of the surface-to-surface radiation equation. They may also be used to formulate equations for other quantities.
Let’s look at a model of a bent copper wire. You have applied a voltage of 5 mV between its ends, and the model computes the resulting total current running through it. With the default Normal mesh, you get a current of 490 A. You decide to try a few other mesh settings in order to determine the accuracy of this current. The results vary only by a few mA and there is markedly less variation between the finer mesh cases. You judge that the result is mesh convergent and sufficiently accurate for your needs.
Wire model set-up (left) and computed electric potential distribution in mV (right).
Next, you would like to evaluate the maximum current density in the wire. It is important that this value isn’t too high. If it is, the resulting heat production risks wearing out the protective jacket and can even be a potential fire hazard.
The normal mesh gives you a maximum current density of 6.2 A/mm^{2}. It is clear that this value appears at the inside corner of the bend, so this time you take care to refine the mesh within that area. The graph below shows the (quite alarming) results: Even if you refine the mesh by a factor of 100, the maximum current density is still growing and there is no indication that it will ever stop.
Singular current density at the inside corner (left) and calculated maximum current density vs. local mesh density (right).
What you have run into is a geometrically singular result. The current density is proportional to the electric field, which in turn is (the negative of) the gradient of the electric potential. While the electric potential itself remains smooth and well-defined at a sharp corner, its gradient is in theory infinite. Numerically, it will tend towards infinity as you refine the mesh. In reality, there are of course no perfectly sharp edges. However, the sharper the edge, the greater the local current density.
If you give the edge a finite radius of curvature, you can limit the current density in the model to simulate reality far better.
Here’s how you do it:
With the fillet in place, the model now returns a smooth current distribution, with a convergent maximum value of 5.1 A/mm^{2}.
Smooth current density distribution (left) and calculated maximum current density vs. mesh density on the fillet (right).
If you would have been interested only in the total current, voltage distribution, or the current density some distance away from the singularity, you would have received good results even without the fillets. This, importantly, also goes for lumped parameters such as resistances and impedances. In an electromagnetic heating model, even a small thermal conductivity is sufficient to give a smooth temperature distribution and a convergent maximum temperature despite a locally singular current density.
Still, if you want to ensure that you get your local fields and currents right, keeping it smooth is a good idea. In general, edges and corners in 3D, as well as corners in 2D, may give you singular electric or magnetic fields. This holds true if they both demark the outer limits of your model or if they separate materials with different properties. If the fields are singular, so are most of the variables that depend explicitly and locally on them.
The next image shows the Maxwell stress tensor on the surface of an iron bar next to a horseshoe magnet. As the stress tensor is proportional to the magnetic field squared, it becomes singular if the bar has sharp corners and edges.
Magnetic flux density around a horseshoe magnet and Maxwell stress tensor distribution on a nearby iron rod. The close-ups show the local flux density on the iron bar without (left) and with (right) fillets.
Outside of electromagnetics, many other solution-variable gradients (heat fluxes, stresses, strains, etc.) exhibit similar singularities and will become smooth if you fillet your edges and corners. You can read more about singularities in Walter’s blog post on the same topic.
Last but not least: Don’t cut corners! It may be tempting to keep the total mesh count down by using the simpler Chamfer operation instead of applying fillets. This, however, gives you a straight section instead of the smooth bend and literally replaces each singularity you remove with at least two additional ones.
Fillets (and chamfers) in 2D are included in all installations of COMSOL Multiphysics version 5.0. In 3D, you need the new Design Module.
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The Loft operation comes in handy when Extrude or Sweep won’t do the job because they do not support multiple cross sections. You can create solid or surface objects from cross sections defined by a face or a profile curve. The profile can be closed or open. For start and end profiles, you can specify the direction of the loft in relation to the profile or adjacent faces. To further control the shape of the loft, you can include guide curves that connect the profiles.
The highlighted regions of this bracket geometry are generated using the loft command by joining the two cross-sectional faces. The loft shape is controlled by guide curves and a setting that keeps the loft direction perpendicular to the profile faces.
The Fillet and Chamfer operations do not need much introduction; they do exactly what we expect them to do — now in 3D. You can create fillets and chamfers with constant radii on edges on 3D solids and surfaces. The fillet or chamfer can automatically propagate to edges that are tangent to the selected edges, which makes the selection of the input entities quite easy.
The fillets on this turbine stator geometry are created using the Design Module. On the enlarged detail (bottom of the image), the fillet propagates to edges tangent to the selected edge (marked with a red arrow). The tool automatically handles the overflow of the fillet into the adjacent fillet in the front.
Setting up a finite element model on the midsurface of a solid object is an analysis technique that is used quite often for parts with a very thin geometry (e.g., something made out of sheet metal). The Design Module enables the calculation of the midsurface of a solid object. The inverse of the Midsurface operation, called thicken, allows you to add material to a surface to convert it to a solid. You can add the thickness in the default normal direction to the surface or specify a direction. The Thicken operation can be symmetric, where an equal amount of volume is added on each side of the surface. It can also be asymmetric, which allows you to specify the thickness on each side.
The geometry of this imported speaker cone is reduced from a solid to a surface object by the Midsurface tool.
All of the newly added geometric tools are fully integrated into the parametric geometry modeling framework in the COMSOL Multiphysics® simulation software. You can use parameter-based expressions in the settings for the operations and thereby control the geometry from the parametric solver or an optimization solver (requires the Optimization Module).
The Design Module also includes all the functionality of the CAD Import Module, such as the file import of widely used CAD file formats and tools to repair and defeature imported geometry. To share your designs, you can export to the Parasolid® software and ACIS® software formats. Read more about the functionality of the Design Module here.
Now that we have introduced you to the Design Module, we will continue with additional blog posts focusing on specific aspects. Stay tuned.
In the meantime, check out the product page for the Design Module.
ACIS is a registered trademark of Spatial Corporation.
Parasolid is a registered trademark of Siemens Product Lifecycle Management Software Inc. or its subsidiaries in the United States and in other countries.
When we use the term CAD geometry, we are referring to a set of data structures that provide a very precise method for describing the shapes of parts. This method is called boundary representation, or B-rep. A B-rep model for solids consists of topological entities (faces, edges, and vertices) and their geometrical representation (surfaces, curves, and points). A face is a bounded portion of a surface, an edge is a bounded segment of a curve, and a vertex lies at a point.
In the B-rep data structures, surfaces are often represented by Non-Uniform Rational B-Splines, or NURBS. The B-rep model of a part is used as the basis for other operations, such as generating tooling paths in Computer Aided Manufacturing software, creating Rapid Prototyping files, and — most importantly — for your COMSOL Multiphysics modeling, generating the finite element mesh.
Your first choice in terms of element type will usually be the tetrahedral mesh for 3D models or a triangular mesh in 2D models. Any 3D geometry can be meshed with tetrahedral (“tet”) elements and any 2D geometry can be meshed with triangles. Additionally, these are the only elements that support Adaptive Mesh Refinement.
For the rest of this blog post, we will focus on the 3D case, since it is the most computationally challenging. At a very conceptual level, the COMSOL tetrahedral meshing algorithm first applies a mesh on all of the surfaces of an object. This mesh is then used to “seed” the volume mesh from which tetrahedral elements “grow” elements inwards. As these tetrahedral elements intersect, their sizes are adjusted with the objective of keeping the elements as isotropic (similar edge lengths and included angles) as possible and to have reasonably gradual transitions between smaller and larger elements.
An issue that you can run into with this algorithm is that the meshing is done based upon the underlying topological entities. There is no way for the meshing algorithm to insert larger elements if the underlying entities are small. As we saw in the previous blog post “Working with Imported CAD Designs,” we can use the CAD repair and defeaturing tools to simplify the geometry.
However, when these algorithms attempt to remove topological entities, they often need to modify the underlying NURBS surfaces and are therefore somewhat limited. An alternative in COMSOL Multiphysics software is to use Virtual Operations, which can keep the existing geometrical representations as a basis for constructing a new alternative topological structure purely for the purposes of meshing and defining the physics.
Let us take a look at the virtual operations and see what you can do with them through a series of examples. The first ten options in the Virtual Operations menu actually only represent five unique capabilities, but they can be used in different ways.
The Virtual Operations menu.
Let’s look at a quick example for each of these five.
The below image demonstrates the Ignore Vertices feature (top) and the Form Composite Edges feature (bottom), which result in the same geometry.
Below is a demonstration of the Ignore Edges feature (top) and the Form Composite Faces feature (bottom), which result in the same geometry.
The following image demonstrates that the Ignore Faces feature (top) can be used to ignore any faces that lie between two adjacent domains, resulting in a single domain. The Form Composite Domains feature (bottom) will also combine multiple domains into a single domain.
As shown next, the Collapse Edges feature (top) and the Merge Vertices feature (bottom) will result in the same geometry. The Merge Vertices feature gives the additional option of choosing which vertex to remove and which one to keep.
The Collapse Faces command (top) and the Merge Edges command (bottom) stand out, since they have been designed to work even in those cases where the faces are not continuous. A useful application for these commands is to get rid of slivers resulting from the union of components that are slightly misaligned or do not fit for other reasons.
Lastly, the Mesh Control Points, Edges, Faces, and Domains features will hide points, edges, faces, or domains during the set-up of the physics; however, these geometric entities will still be present during the meshing step. By using these operations, you can gain greater control over the meshing process by designating geometric entities for the control of the mesh size and distribution. The physics set-up is kept simple by excluding the control entities. A typical area of application is in CFD simulations, where regions of steep gradients in a volume need a high mesh density.
It appears that we have a lot of options here, and you may wonder which of these features you should be using. In practice, the Form Composite Faces can usually be your first choice. Almost all of the issues that you will typically run into, with the exception of forming composite domains, can be handled with this feature.
Let’s look at a case from the COMSOL Multiphysics Model Library: the stresses and strains in a wrench. This is a structural model of a combination wrench. The provided CAD geometry has some relatively complex sculpted surfaces and fillets and blends, which result in small faces in some parts of the model. These small faces force the tet mesher to use smaller elements, but we can see that Virtual Operations can be utilized to avoid this.
A detailed view of a CAD file shows that small faces result in a fine mesh. Using the Virtual Operations allows larger elements in these regions.
We can use the Form Composite Faces feature to abstract whole sets of faces. You can simply select all of the faces and then deselect those faces that you do not want to abstract. This is acceptable and recommended if you know you do not need high fidelity of the mesh in certain regions where there are many small faces.
Virtual Operations can be used to combine sets of surfaces and significantly simplify some parts of the geometry.
We have now seen why you would want to use these Virtual Operations and the many ways in which they can be used. If you want to see a step-by-step guide for using these features to simplify your geometry, please see the Model Library example on using Virtual Operations on a Wheel Rim Geometry.
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