When modeling thin fractures within a 3D porous matrix, you can efficiently describe their pressure field by modeling them as 2D objects via the *Fracture Flow* interface. Significant fracture flux calculation issues, however, may arise for systems of practical interest, such as hydraulic fractures contained within unconventional reservoirs. See how a hybrid approach overcomes such difficulties.

To model an actual fracture as a 2D object using the Subsurface Flow Module, you first have to solve for the pressure field (through a tangential form of Darcy’s law) within the internal surface representing the fracture’s lateral extent. You can then calculate, in principle, the corresponding fluid flux through the actual fracture cross section by multiplying the component normal to an edge of the velocity vector (delimiting the 2D fracture object) by the fracture’s thickness. This approach is much more computationally efficient, as a very thin but otherwise ample 3D object can now be described as a 2D object, one that only needs to be meshed as a surface.

Say you have a 2D fracture object with the following characteristics:

- It has an edge contained within the porous matrix.
- It is significantly more permeable than its surrounding formation.
- It features a very large aspect ratio between its lateral and cross-sectional dimensions.

In a system of interest such as this, significant fracture flux computational errors can occur. Let’s take a look at one such example.

Note: With the latest version of the COMSOL Multiphysics® software — version 5.2a — you can also model heat transfer in thin fractures. This is made possible via the

Heat Transfer in Fracturesinterface.

The system shown below features a 3D penny-shaped hydraulic fracture embedded in a reservoir block and connected to a horizontal well. The inlet for this simplified system consists of the two reservoir boundaries shown in green at the top and the back of the block. The only outlet is through the narrow boundary where the fracture disc connects to the wellbore. Both the inlets and the outlet are set as pressure boundary conditions, with the values of ΔP and 0, respectively. The geometry only considers one quarter of the actual system, as it takes advantage of existing symmetry.

*A 3D penny-shaped fracture (shown in blue) embedded in a reservoir block and hydraulically connected to a horizontal well (shown in red). The two reservoir inlet boundaries are highlighted in green.*

Note that the dimensions of the above system are not representative of cases of practical interest. The dimensions are scaled down to allow for adequate 3D meshing of the discoidal fracture, which has a radius of 7.62 m (25 ft) and a thickness of 1.27 cm. (Properly meshing 3D fractures with radii of hundreds of feet, as encountered in field applications, would be quite computationally expensive.) The wellbore radius is 12.7 cm (5 in), while the reservoir block’s dimensions are approximately 8 m x 15 m x 15 m (25 ft x 50 ft x 50 ft). The entire mesh consists of 2,246,298 tetrahedral elements, 657,720 of which are used for the discoidal fracture domain alone. The minimum and average element quality values of the latter are 0.148 and 0.700, respectively, while the average quality for the entire mesh is 0.673.

*Outlet boundaries for the 3D (shown in green) and 2D (shown in red) realizations of the actual hydraulic fracture of thickness d _{HF}.*

Darcy’s law is used to solve for the pressure field *p* in incompressible, single-phase, and stationary flow parametric studies for various values of the drawdown ΔP. The fluid is a light liquid hydrocarbon with a dynamic viscosity value of 0.26 cP. The permeability of the reservoir matrix is taken as 1 mD, while that of the (propped) hydraulic fracture is 45.6 Darcy.

The fracture flux calculation issue referenced above is depicted in the following figure. This figure shows the inlet and outlet flow rates as functions of the drawdown ΔP when the hydraulic fracture (HF) is described as either a 3D or 2D object. While the first three curves (for the inlet flow rates and the 3D outlet flow rate) overlap as expected, the outlet flow rate for the 2D case represents only a quarter of the inlet flow rate. The fluxes for the first three curves were calculated as integrals of the normal component of the fluid velocity vector over the respective inlet and outlet surfaces (). The outlet flow rate for the 2D fracture, meanwhile, was calculated as an integral of along the outlet edge, multiplied by the fracture thickness *d _{HF}*: .

The flux calculation issue remains regardless of the applied meshing and no matter how is probed, with its integrand expressed as (dl.nx*dl.u + dl.ny*dl.v + dl.nz*dl.w), (sys1.e_n1*dl.u + sys1.e_n2*dl.v + sys1.e_n3*dl.w), dl.bndflux/dl.rho, or as (root.nx*dl.u + root.ny*dl.v + root.nz*dl.w). The ‘dl.’ identifier stands for the applied interface (*Darcy’s Law* interface); {nx,ny,nz} are the Cartesian components of the unit vector normal to the edge ; and {u,v,w} are the Cartesian components of the fluid velocity vector .

*Inlet and outlet flow rates as functions of the drawdown ΔP when the actual HF is modeled as either a 3D or 2D object for the respective system.*

Notice that when the hydraulic fracture is described as a 2D object, the discoidal fracture (3D) domain is omitted from the model and is considered instead only through its inner lateral boundary. Otherwise, the geometry and mesh are identical between the 2D and 3D descriptions. This simplification greatly reduces the size of the system and thus represents one of the most attractive elements of the *Fracture Flow* interface: it enables the modeling of much larger fracture surfaces with proper meshing. As such, it would be quite useful if there was a way to work around the 2D fracture outlet flux issue.

A hybrid approach, which combines a 2D description of the fracture away from the wellbore with a 3D one in its immediate vicinity, makes this possible. The figure below shows the meshed geometry of the hybrid implementation. The 3D component of the fracture is represented by the blue domain, while the 2D component is represented by a red surface, which depicts the boundary toward the porous matrix of the actual fracture. Note that the 3D part of the actual fracture that corresponds to the 2D component is excluded from the model.

*Meshed geometry of the hybrid fracture implementation. The blue domain represents the 3D component of the fracture, while the red surface represents the 2D component. The latter is chosen as the inner lateral boundary of the actual fracture (toward the matrix).*

In the hybrid approach, the pressure field continues to be properly accounted for at any point within the actual fracture, while the flux through the outlet boundary is computed without the shortcoming of the 2D description. The following table compares relevant quantities for the 3D, 2D, and hybrid realizations of the hydraulic fracture. These computations were performed using a direct solver on a machine with an Intel® Core™ i74770 processor and 32 GB RAM.

Hydraulic fracture | Degrees of freedom | Memory (GB) | Time per iteration (s) | ||
---|---|---|---|---|---|

3D | 3,231,747 | 23.74 | 247.5 | 1 | 1.00026 |

2D | 2,354,490 | 15.98 | 153.5 | 0.99948 | 0.24992 |

Hybrid | 2,397,891 | 16.50 | 158.0 | 0.99941 | 0.99967 |

*Comparison of relevant quantities for the 3D, 2D, and hybrid realizations.*

The plot below shows, in logarithmic scale, the pressure profiles along a diagonal line within the *YZ*-plane containing the 2D fracture surface for a drawdown of 100 psi. This probe line is delimited by the outlet (wellbore) at the lower-right part of the surface and by the inlet of the reservoir block at the other end. The white line at the inset of the plot highlights the probe line. The surface color of the inset corresponds to the pressure value within the probed *YZ*-plane, and the guiding arrows help map important graphing points on it. The graph’s curves overlap for all three cases, indicating that the pressure field solution is practically identical among the three fracture descriptions: 3D, 2D, and hybrid.

*Pressure profiles along a face diagonal line within the *YZ*-plane.*

Flux calculation issues can occur with a solely 2D description of a fracture. As we’ve demonstrated here today, the proposed hybrid approach for describing an actual fracture provides a viable solution. As such, this technique can be applied to various systems of practical interest that feature a greater number of arbitrarily thin fractures.

Ionut Prodan is the principal of Boffin Solutions, LLC, a COMSOL Certified Consultant. Prior to his founding of Boffin Solutions, Ionut worked within upstream technology at Shell and Marathon Oil. He earned his doctorate in physics from Rice University, where he conducted research on the photoassociation of ultra-cold atoms and computational solid-state chemistry.

*Intel and Intel Core are trademarks of Intel Corporation or its subsidiaries in the U.S. and/or other countries.*

For many businesses, numerical modeling and simulation are valuable tools at various stages of the design workflow, from product development to optimization. Apps further extend the reach of these tools, hiding complex multiphysics models beneath easy-to-use interfaces. Here’s a look at one such example: a solid oxide fuel cell stack app.

Consider a device that can generate heat and power for your home as well as power the electric motor of your car. Now think about using such a device to produce electricity with small losses, without being limited by Carnot efficiency. The thought that should come to mind is a fuel cell.

Fuel cells, like batteries, are devices that electrochemically convert chemical energy present in the fuel into electrical energy. But unlike batteries, which have to be recharged, fuel cells are continuously fed with chemicals. Solid oxide fuel cells (SOFCs), in particular, are recognized as one of the most efficient and flexible types of fuel cells. This is due to their very small kinetic losses, despite the slightly less advantageous thermodynamics at higher temperatures. However, from an engineering point of view, these devices present more of a challenge due to their high operating temperatures (600°C to 1000°C).

*A schematic of an SOFC. Image in the public domain, via Wikimedia Commons.*

Because of these high temperatures and the overall complexity of the system, measuring critical parameters like temperature, composition, and current densities within the stack is quite difficult. However, addressing such elements is particularly important in optimizing the performance of the device and identifying any potential issues that occur at different operating conditions.

With the capabilities of the COMSOL Multiphysics® software, it is possible to achieve precise estimates of such parameters. The software also helps with the interpretation of experimental results and enables users to virtually perform potentially destructive tests, which can otherwise be quite expensive and difficult to fully understand.

Modeling an SOFC stack requires combining a rather intricate geometry with large aspect ratios and multiple physics in a 3D problem. Each SOFC features two reactant flows: air and fuel. These two elements can have different compositions and several parameters that describe inlet and environmental conditions.

*Model geometry of an SOFC stack.*

With regards to the multiple physics that are involved, the model needs to solve:

- Chemistry for the air and fuel sides
- Flow for the air and fuel sides
- Transport of species for the air and fuel sides
- Heat transfer
- Electrical current

Designing this complex model and modifying its parameters requires knowledge about both technology and mathematical modeling. The combined expertise needed to run such calculations often translates into a heavy workload for the model developers, who have to dedicate a great share of their time to run calculations.

Now, thanks to the Application Builder in COMSOL Multiphysics®, simulation experts have the ability to design a user-friendly tool that can be deployed to other team members and throughout an organization to study such complex systems. This can greatly improve overall productivity by delivering simulation results more quickly to users, while giving simulation experts more time to focus on model development and, of course, to continue to provide assistance with simulation strategies along the way.

Like the initial model highlighted above, the app presented here was developed at Topsoe Fuel Cell, a leader in SOFC technology. Both the model and the corresponding app have served as an important development tool throughout the organization for advancing the design of SOFC stacks.

The app’s user interface (UI) features several tabs, as highlighted in the following series of screenshots. Under the *Input* tab, for example, users with little to no simulation expertise can easily modify several operating conditions for the SOFC design. In this case, such conditions range from temperatures and currents to flows and heat losses — parameters that can be updated based on the specific modeling needs.

*Parameters available for modification under the* Input *tab.*

Perhaps an app user does have knowledge of the solver and other special settings. If so, under the *Solver & Special Input* tab, these users can modify various elements related to the solver, including mass constraint, the oxidized species, and the different species in the solver on the fuel side. There is also the option to make adjustments to the settings that pertain to the area-specific resistance (ASR).

*The* Solver & Special Input *tab.*

After applying the appropriate changes to the settings, the next step is to compute the simulation. While doing so, app users can follow the progress of the simulation as they await the results. So what types of computations can users perform with this tool? For one, the app can be used to compute the temperature distribution in the SOFC stack, as shown below.

*Temperature variation in the SOFC stack.*

With the app, users can also analyze the current densities of the SOFC stack as well as the hydrogen mole fraction. You can see such simulation results highlighted in the following plots.

*Left: Plot showing the current density. Right: Plot showing the H _{2} mole fraction.*

As the design workflow progresses and users obtain a better understanding of their device, they may request to change or include new input parameters and outputs for analysis. With the flexibility and customization available in the Application Builder, it is easy for app designers to make these modifications and thus meet the specific needs of their end users.

Turning a complex model into an app is a powerful solution for bringing simulation capabilities to a larger number of people. Whether deploying apps to other engineers, system developers, or salespeople at your organization, these tools serve as a viable resource in helping to deliver reliable simulation results quickly to verify operating strategies or provide some insight into experimental results.

From a consultant-to-client point of view, a design workflow that includes an app based on a validated model creates a greater value for clients. Rather than receiving a report with only a few sensitivity analyses, they have the power to investigate the impact of various parameters on their design and therefore achieve faster and more reliable results.

Matteo Lualdi is the simulation manager at resolvent ApS, a COMSOL Certified Consultant. He received two master’s degrees in energy engineering from Politecnico di Milano and the Royal Institute of Technology in Stockholm within the Top Industrial Managers for Europe (T.I.M.E.) framework. Matteo later earned his PhD in chemical engineering from the Royal Institute of Technology. His experience prior to resolvent ApS includes working in the fuel cell and catalyst business, where he was responsible for system simulations and catalytic reactor sizing.

]]>MP3 players, smartphones, and tablets allow us to listen to our favorite music almost everywhere. While driving in a car, we should also enjoy the highest sound quality. Learn how to use simulation to reproduce sound in one of the most difficult environments — a vehicle — to design better automotive sound systems.

Automotive acoustics experts are confronted with the complex problem of finding the optimal position for speakers in vehicles because of their extremely complex architecture and confined space. A mix of different materials in a vehicle cabin, such as textiles, plastics, metals, glass, etc.; acoustic influences; and disturbing engine or road noise must be analyzed until any noise concerns are rectified and eliminated.

*A CAD model of a vehicle door with speaker. Image by Konzept-X GmbH.*

The use of modern simulation methods in the concept and design of a vehicle’s acoustic components and the evaluation of the systems in the early phase of product development are becoming increasingly important. This is partly thanks to the demand for shorter development cycles and the increasing expansion of vehicle model ranges.

Using the COMSOL Multiphysics® simulation software, experts are able to simulate complete acoustics on the basis of a vehicle’s CAD model. As we’ll discuss, a proper modeling strategy is important to bring automotive interior acoustics to a new level of performance.

To begin, we create a multiphysics model of a vehicle’s loudspeakers, of which electromagnetism, mechanics, acoustics, and sometimes even thermal transfer and fluid dynamics are taken into account. It is most important to analyze the different physical domains, as each domain interacts bidirectionally with each other.

To enable a realistic simulation, we also have to deal with the path-dependent dynamic effects and nonlinearities (including instabilities) in each domain. We use a unique mixture of 1D (lumped parameter), 2D, and 3D (finite element) models for this multiphysics simulation.

Lumped parameter models are based on 1D scalar equations to describe a physical variable. Fortunately, they can describe the system behavior of a speaker that interacts with an enclosure and the resulting sound radiation by including several physical scalar variables. For the mechanical and acoustic (airborne sound) area, however, there are substantial limitations when using only scalar values. For example, in the mechanical system, only piston-like movements can be described. Thus, the applicable frequency range is limited. At higher frequencies, the vibration behavior is no longer piston-shaped and we have to use a multidimensional modeling approach. In that case, the finite element method is recommended. This fact has led to the development of models based on matrix methods.

For this demonstration, it is necessary to use a lumped parameter model for voice coil electromagnetics, a finite element model for the mechanical domain, and a finite element model for the acoustic domain in order to develop a fully coupled electrical-mechanical-acoustic simulation model for loudspeakers. If we tie together all of the governing equations of the different physical domains — electrical, mechanical, and acoustic — we get a coupled system of equations describing the multiphysics of an electroacoustic transducer in the frequency domain, as shown below.

*The coupled multiphysics system for a loudspeaker. Image by Konzept-X GmbH.*

C^{ma} and C^{am} are coupling matrices connecting the mechanical and acoustic domains. These coupling matrices arise from the assumption of continuity of the velocity and pressure in the mechanical and acoustic domains in the direction normal to the coupling surface. D^{e} is an electromagnetic damping, which is derived from the back electromagnetic force.

Consider that the coupled model alone will not automatically lead to realistic simulations. We also need to accurately describe the material properties in the electrical and structural domain. Thus, a key aspect of the simulation is the material measurement procedures specifically designed to measure electrical and mechanical parameters and their frequency dependence.

In the following figure, we compare the measured and simulated frequency responses of the radiated sound pressure. The accuracy of the simulation, based on the previously presented theory, is within the manufacturing tolerances of the loudspeaker, and thus can be deemed a realistic simulation.

*Comparing measurements (represented by green and black) and simulation results (shown in pink). Image by Konzept-X GmbH.*

The most important challenge for a successful automotive acoustic simulation is the strong coupling of all physical domains involved. Strong coupling within that context means that each physical domain interacts bidirectionally with other domains.

*A CAD model of a loudspeaker (cut). Image by Konzept-X GmbH.*

While the motor system is typically treated as an axisymmetric device, and thus simplified 2D models can be applied for a majority of applications, we must account for the strong coupling to the structural domain (the loudspeaker’s vibration system) via the voice coil acting in the magnet’s air gap. For some motor structures, the variation of the flux field in the axial direction is also of crucial importance. Thus, finite element models for detailed motor design and optimization are commonly used.

At large excursions of the voice coil (when the loudspeaker is driven in the region of nominal power), a significant portion of the voice coil moves out of the main flux field, and thus less mechanical force is being induced. This nonlinear effect is essential and causes unwanted distortion in the radiated sound. Additionally, voice coil inductance is also dependent on voice coil excursion and current. This leads to the need for nonlinear models to predict the loudspeaker behavior at large signals.

For system- or subsystem-level simulations (without the goal of designing a motor), 1D lumped models (that include additional nonlinearities to predict large signal behavior) are efficient.

As mentioned before, the structural domain has to be modeled via finite elements to account for nonpistonic effects. The existence of circumferential bending waves can only be accounted for by 3D models, thus 2D models have to be used with care. An additional challenge is the thin-walled structures of cone and dust caps, as well as surround and spider caps. By simply using 3D solid elements, we could end up with a very large and unhandy model, thus we should use shell finite elements to model the vibration system.

At large signals (and thus large excursions), major nonlinear effects arise from the material behavior and change in geometry, leading to a change in the stiffness of the vibration system, thus generating distortion in sound radiation. Even if we had a superlinear material, its change in geometric stiffness would lead to distortion. On top of this, due to the change in geometric stiffness, instabilities may occur (snap-through and bifurcation), defining an additional source of heavy distortion.

The major challenge here is the strong coupling to the structural domain. In other words, the movement of the loudspeaker’s vibration system has an effect on the surrounding acoustic medium (air and sound waves are generated) and the surrounding air has vice versa an effect on the movement of the structural domain (typically called an *added mass and stiffness* effect).

*The multiphysics of a loudspeaker (CAD model). Image by Konzept-X GmbH.*

The multiphysics simulation of a loudspeaker is the very first step to achieving an outstanding sound performance. Also of particular importance is the mechanical and acoustic integration of the loudspeakers in the vehicle. For developers, the space requirements of speakers can be particularly challenging.

By using CAE-based simulation tools, like COMSOL Multiphysics, we can optimize the speaker integration in the early design phase. As a result, we can avoid errors such as spurious oscillations of enclosures and structural parts. We can instead take completely new ways of integration into account.

One example is the use of the cavities of the vehicle structure as resonance volume for the speakers, because loudspeakers require an enclosure as resonance volume in order to reproduce low frequencies. By limiting available space, these enclosures often have a complex geometry and an important influence on the sound. Consequently, in the next step, the previously designed multiphysics model of the speakers has to be extended and the enclosures have to be integrated and provided with mechanical and acoustic characters.

*An example multiphysics model of a loudspeaker and enclosure (vehicle door). Image by Konzept-X GmbH.*

*Sound pressure level inside the door cavity. Image by Konzept-X GmbH.*

*Sound pressure level inside the woofer enclosure. Image by Konzept-X GmbH.*

Then we integrate the car cabin, with its mechanical and acoustic characters, into the simulation model. The car cabin has an important impact on the perception of sound too. Due to the mix of different materials, such as leather, plastic, and textiles, a hybrid approach is used, as the coupled multiphysics model alone will not automatically lead to realistic simulations. The various materials with totally different behaviors and joining techniques require detailed descriptions of the material properties in the electrical and structural domain. Thus a key aspect here is material measurement procedures specifically designed to measure electrical and mechanical parameters.

In order to create an outstanding sound performance of the complete sound system in the vehicle, psychoacoustic characteristics and properties are of importance as well (psychoacoustics deals with the human perception of sound). Through careful virtual tuning, irregularities and resonances in frequency response are compensated. Extensive tests and measurements are carried out in further steps. Finally, the virtual audio system must undergo an analysis by human ear. These listening tests are necessary for a final evaluation of the audio performance, especially in terms of spatial reproduction.

Within the multiphysics simulation model of our vehicle audio system, the distribution of the sound can be determined, even in the furthest corner of the car. All factors that affect the sound in the interior, however minimal, are taken into account. By means of a special reproduction technique, it is also possible to make the virtual sound system audible on a computer (also known as auralization). In the past, the integration of loudspeaker systems was evaluated mainly with real prototypes at a late stage of development. This process took a lot of time and money and also bore a risk in the progress of the project.

Thanks to modern simulation technology, we can virtually analyze the optimal positions of loudspeakers and select adequate components in the early stage of development. COMSOL Multiphysics interface couplings are able to bring automotive audio acoustic simulations to a new level of predictive power.

Dr. A. J. Svobodnik is a thought leader, entrepreneur, engineer, and technical scientist. He has worked in audio and acoustic applications for over 25 years. Svobodnik is managing director and cofounder of Konzept-X GmbH, a COMSOL Certified Consultant specializing in providing consulting services and innovative technologies for automotive audio and technical acoustics, as well as the developer of multidisciplinary virtually optimized industrial design M-voiD® technology. Konzept-X was founded in 2011.

*M-voiD is a registered trademark of Alfred J. Svobodnik in the U.S. and other countries.*

The acoustic design of mufflers in the automotive industry has traditionally been performed by an iterative process where different alternatives are compared by experimental methods until a satisfactory design is found. Numerical simulation can drastically reduce a project’s time and expenses, while simultaneously increasing the performance of the muffler.

Sweden-based Scania has utilized the benefits of numerical simulation for a long time and is constantly improving their modeling knowledge and skill set. Recently, Lightness by Design began working with Scania to evaluate the benefits of using a multiphysics simulation approach where acoustics are coupled to shell structural mechanics in the muffler design process.

Together, we found that introducing structural mechanics into an acoustic model can have a large impact on the predicted muffler characteristics. The differences can also, to some degree, explain previous discrepancies between a simulation and experimental results, where structural mechanics obviously had always been present but never taken into account.

*The exhaust system for an automotive muffler.*

We’ll use the classic COMSOL Multiphysics model of an automotive muffler as an example to illustrate the effect of including structural mechanics in a normal pressure acoustic simulation. The model is available in the Application Gallery and is modified by including shell physics and a multiphysics coupling.

The model, shown below, is solved for the following boundary constraints: Plane wave propagation is defined on both the inlet and outlet surfaces, with an incident pressure of 1 Pa applied at the inlet. All walls, interior and exterior, are defined as sound hard barriers or shells made of steel (E = 210 GPa, ν = 0.3, ρ = 7800 kg/m^{3}, η = 0.02, and t = 1 mm) in the *Pressure Acoustics* node. We also define the shell physics and restrict all degrees of freedom to zero at the inlet and outlet edges. In the image below, you can see the surfaces defined as shells in darker blue.

*The Automotive Muffler model with inlet, outlet, and highlighted shell boundaries.*

The actual multiphysics coupling is defined at the interface between the shells and the acoustic domain. This couples the pressure variable, *p*, which is solved for in the *Pressure Acoustics* node, and the nodal displacements, (*u*,*v*,*w*), which are solved for in the *Shell* physics node, according to

\mathbf{-n}\cdot(-\frac{1}{\rho}\nabla p_\textrm{t} )=\mathbf{n}\cdot\frac{\partial^2 \mathbf{u}}{\partial t^2 },

where **n** is the surface normal vector, is the air density, is the Laplace operator, is the total acoustic pressure, **u** is the shell surface displacement vector, and *t* represents time.

The condition represents continuity in normal acceleration. The multiphysics coupling *Acoustic-Structure Boundary* is predefined and located under the *Multiphysics* node in the model tree. See the image below for the final setup in the model tree.

*The model tree with* Pressure Acoustics*,* Shell*, and* Multiphysics *nodes.*

A free tetrahedral mesh with eight second-order elements per wavelength at maximum frequency is used, yielding the maximum element size

E_{max}=\frac{c }{f_{max}\cdot n_{min}},

where *c* is the speed of sound in air (343 m/s), *f _{max}* is the highest frequency of interest (800 Hz), and

The transmission loss is finally calculated as

TL=10\cdot log_{10} (\frac{P_{in}}{P_{out}}),

where and are the acoustic power at the muffler inlet and outlet, respectively, calculated as

P_{in}=\int_{A_{in}} (\frac{p^2_{in}}{2\cdot \rho \cdot c}) dA, \qquad P_{out}=\int_{A_{out}}(\frac{p^2_{out}}{2\cdot \rho \cdot c}) dA,

where and are the inlet and outlet areas, respectively; and are the inlet and outlet acoustic pressure, respectively; is the density of air; and *c* is the speed of sound in air.

Then, we solve the model; first for the eigenfrequencies and later by carrying out a parametric sweep in the frequency domain (10 to 800 Hz in 2 Hz increments).

The effect on the predicted transmission loss for the muffler, modeled with pure pressure acoustics, is compared to the multiphysics model in the image below. Looking at the calculated transmission loss, you can see that the curves are quite similar with small discrepancies, except for the frequency region between approximately 600 to 670 Hz. The behavior of the transmission loss in this region is similar with and without coupled shell physics, but we can see a large amplitude difference.

*A transmission loss comparison between pure acoustics and multiphysics models.*

This amplitude difference can be explained by the interaction at the interface between the encapsulated air mass and the shell structure. The mechanism behind this structure can be further explored by eigenfrequency analysis.

In the graph below, the eigenfrequencies are plotted against frequency and interesting eigenfrequencies are identified in the transmission loss diagram. Two calculated cases are presented: pressure acoustic physics only, yielding the frequencies where standing waves occur in the air mass, and shell physics only, yielding the frequencies where the structure is easily excited into vibration. This chart gives the possibility to detect coinciding eigenfrequencies in the exciting load (standing waves in the air mass) and the encapsulating structure.

*Transmission loss and eigenfrequencies for both models.*

There are at least three regions where the predicted transmission loss differs between the pure acoustics and multiphysics models. These are around 170 and 340 Hz and the entire region between 570 and 670 Hz. The first two frequencies (170 and 340 Hz) are pure shell modes that can be excited by the pressure pulses without acoustic resonance occurring; i.e., no coinciding eigenfrequencies are needed.

*The shell mode shapes at eigenfrequencies of 172 Hz and 342 Hz.*

A group of closely spaced eigenfrequencies, both acoustic and shell modes, is found around 600 Hz and 634 Hz. This is right in the region where the largest differences in the transmission loss occur. Let’s have a look at what is happening at the multiphysics interface at 634 Hz, since this is where the difference in transmission loss is the greatest.

*The acoustic standing wave and shell mode shape at eigenfrequencies at 634 Hz.*

We are looking at an eigenmode where the shell is easily excited into vibration, which corresponds very well with the pressure mode when excited at 634 Hz. This causes high acceleration levels at the shell end plates. This interaction effect is most likely causing the difference in transmission loss.

Normally, an increased transmission loss is beneficial for a muffler, but if it comes at the price of a vibrating structure, it could be questionable both from a fatigue and shell noise standpoint. Whether the shell vibrations at the excited eigenmodes pose a problem or not is mostly dependent on the amplitude of the vibrations. If there are large pressure amplitudes, they can give rise to both noticeable shell noise as well as structural strains with increased risk for fatigue problems.

The endplate modes at 170 and 340 Hz could be easily removed if necessary by increasing the endplate thickness with only a small associated weight penalty. These two modes are excited by the incident pressure, since no standing wave in the air mass exists at these frequencies. Hence, the severity of these modes can be estimated by comparing the predicted operating frequencies and pressure amplitudes of the muffler with the resonating shell frequencies.

The third example mode, at 634 Hz, where the largest difference in transmission loss occurs, is more complex to examine using simulation only. This study would require some experimental evaluation where the amplitudes of the system can be measured. This is mostly because of the broad frequency range (570 to 670 Hz) and the combination of both acoustic and shell modes. The acoustic modes change the pressure distribution and can locally increase the acoustic pressure. Hence, noticeable vibrations and possible fatigue problems could arise.

When designing automotive mufflers, numerical simulation with COMSOL Multiphysics results in better products, while simultaneously reducing the risk and cost of the project and the end product’s time-to-market. With the implementation of a multiphysics coupling for acoustics and structural mechanics, new possibilities for increasing the efficiency of the design phase and the reliability of the simulation results are opening up.

As seen with market leader Scania, keeping the leading edge requires continuously honing skills and utilizing advances in simulation technology. This organization keeps developing their knowledge with the help of their work with COMSOL Multiphysics and COMSOL Certified Consultant Lightness by Design.

Download one of the following example models or applications to see how you can use multiphysics simulation to optimize your own automotive muffler design:

- Automotive Muffler
- Absorptive Muffler
- Eigenmodes in a Muffler
- Absorptive Muffler Designer
- Absorptive Muffler with Shells

Linus Fagerberg from Lightness by Design is an experienced consultant working with simulation-supported product development. He holds a PhD from KTH Royal Institute of Technology and is specialized in the structural mechanics of composites, stability, and optimization. Linus believes that numerical simulation is a great tool to consistently deliver high-quality products, improve performance, and mitigate risks. Lightness by Design is a COMSOL Certified Consultant based in Stockholm, Sweden.

]]>Phononic crystals are rather unique materials that can be engineered with a particular band gap. As the demand for these materials continues to grow, so does the interest in simulating them, specifically to optimize their band gaps. COMSOL Multiphysics, as we’ll show you here, can be used to perform such studies.

A *phononic crystal* is an artificially manufactured structure, or material, with periodic constitutive or geometric properties that are designed to influence the characteristics of mechanical wave propagation. When engineering these crystals, it is possible to isolate vibration within a certain frequency range. Vibration within this selected frequency range, referred to as the *band gap*, is attenuated by a mechanism of wave interferences within the periodic system. Such behavior is similar to that of a more widely known nanostructure that is used in semiconductor applications: a *photonic crystal*.

Optimizing the band gap of a phononic crystal can be challenging. We at Veryst Engineering have found COMSOL Multiphysics to be a valuable tool in helping to address such difficulties.

When it comes to creating a band gap in a periodic structure, one way to do so is to use a unit cell composed of a stiff inner core and a softer outer matrix material. This configuration is shown in the figure below.

*A schematic of a unit cell. The cell is composed of a stiff core material and a softer outer matrix material.*

Evaluating the frequency response of a phononic crystal simply requires an analysis of the periodic unit cell, with Bloch periodic boundary conditions spanning a range of wave vectors. It is sufficient to span a relatively small range of wave vectors covering the edges of the so-called *irreducible Brillouin zone* (IBZ). For rectangular 2D structures, the IBZ (shown below) spans from Γ to X to M and then back to Γ.

*The irreducible Brillouin zone for 2D square periodic structures.*

The Bloch boundary conditions (known as the Floquet boundary conditions in 1D), which constrain the boundary displacements of the periodic structure, are as follows:

u_{destination} = exp[-i\pmb{k}_{F} \cdot (r_{destination} - r_{source})] u_{source}

where **k**_{F} is the wave vector.

The source and destination are applied once to the left and right edges of the unit cell and once to the top and bottom edges. This type of boundary condition is available in COMSOL Multiphysics. Due to the nature of the boundary conditions, a complex eigensolver is needed. The system of equations, however, is Hermitian. As such, the resulting eigenvalues are real, assuming that no damping is incorporated into the model. The COMSOL software makes this step rather easy, as it automatically handles the calculation.

We set up our eigensolver analysis as a parametric sweep involving one parameter, *k*, which varies from 0 to 3. Here, 0 to 1 defines a wave number spanning the Γ-X edge, 1 to 2 defines a wave number spanning the X-M edge, and 2 to 3 defines a wave number spanning the diagonal M-Γ edge of the IBZ. For each parameter, we solve for the lowest natural frequencies. We then plot the wave propagation frequencies at each value of *k*. A band gap appears in the plot as a region in which no wave propagation branches exist. Aside from very complex unit cell models, completing the analysis takes just a few minutes. We can therefore conclude that this approach is an efficient technique for optimization if you are targeting a certain band gap location or if you want to maximize band gap width.

To illustrate such an application, we model the periodic structure shown above, with a unit cell size of 1 cm × 1 cm and a core material size of 4 mm × 4 mm. The matrix material features a modulus of 2 GPa and a density of 1000 kg/m^{3}. The core material, meanwhile, has a modulus of 200 GPa and a density of 8000 kg/m^{3}. The figure below shows no wave propagation frequencies in the range of 60 to 72 kHz.

*The frequency band diagram for selected unit cell parameters.*

To demonstrate the use of the band gap concept for vibration isolation, we simulate a structure consisting of 11 x 11 cells from the periodic structure analyzed above. These cells are subjected to an excitation frequency of 67.5 kHz (in the band gap).

*The structure used to illustrate vibration isolation for an applied frequency in the band gap.*

The animation below highlights the response of the cells. From the results, we can gather how effective the periodic structure is at isolating the rest of the structure from the applied vibrations. The vibration isolation is still practically efficient, even if fewer periodic cells are used.

*An animation of the vibration response at 67.5 kHz.*

Note that at frequencies outside of the band gap, the periodic structure does not isolate the vibrations. These responses are depicted in the figures below.

*The vibration response at frequencies outside of the band gap. Left: 27 kHz. Right: 88 kHz.*

To learn more about the 2D band gap model presented here, head over to the COMSOL Exchange, where it is available for download.

- P. Deymier (Editor),
*Acoustic Metamaterials and Phononic Crystals*, Springer, 2013. - M. Hussein, M. Leamy, and M. Ruzzene,
*Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook*, Appl. Mech. Rev 66(4), 2014.

Nagi Elabbasi, PhD, is a managing engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is the modeling and simulation of multiphysics systems. He has extensive experience in the finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, failure analysis, and material testing and modeling, and is a COMSOL Certified Consultant.

]]>When manufacturing electronic devices, leading companies often look to simulation in order to thermally characterize their products. At BE CAE & Test, we have found a more efficient way of answering such requests: designing apps that are tailored to our customers’ needs. Our surface-mount device app, presented here, is just one testament to what apps can achieve.

An innovative concept came about some time ago at COMSOL. What if a standard finite element model could be turned into an intuitive simulation tool that was accessible by a broad audience? The “from model to app” idea came to fruition with the release of the Application Builder.

*With the Application Builder, you can turn your model into an easy-to-use app.*

Apps mark a revolutionary page in the history of simulation. With these flexible and user-friendly tools, people of varying levels of expertise can access, exploit, and share simulation power. As such, simulation apps can create more business opportunities with customers. Beyond simply providing them with a technical report, you are also supplying them with an interactive tool.

We at BE CAE & Test strongly believe that apps represent the natural evolution of numerical simulation applied to real-world physics. Trusting this concept, our goal has been to promote the use of apps, which has included performing demonstrations for customers.

When proposing the idea of a simulation app to customers, one of the first questions they often ask is: “What is a COMSOL app?”

In our opinion, an app can be defined as a customized user interface (UI) that allows users to carry out parametrical simulations without building models. For “parametric” analyses, you have the flexibility to select a wide variety of parameters to include in your app’s design, which app users can then easily manage and modify through the interface. Such parameters can range from geometries and materials to model assumptions and individual functional working conditions (i.e., source terms).

While apps are designed to hide the complexity of the underlying model, they still provide you with the functionality to fully exploit all of the visualization features available in COMSOL Multiphysics, making it easy to communicate results.

In order to explain more tangible contents of apps, we want to share an app with you today that we presented at the COMSOL Conference 2015 Grenoble. The app was built for demonstrative purposes, created according to our experience and expertise in using the COMSOL Multiphysics software.

The embedded model is a very simple model for conducting a thermal analysis in solids. As the initial figure below shows, the electronic surface-mount device (SMD) is comprised mainly of a copper frame, a lead-free solder layer (also called a solder die), and a silicon die equipped with a front metal that is connected to device pins by several ribbons. In the second figure, the applied source term and boundary conditions are presented graphically.

*Base layout of the system for our model to app demonstration.*

*Source terms and boundary conditions applied in the embedded model.*

When designing the app, we leveraged a number of functionalities within the Application Builder: the *Action Button, Input Field, Combo Box, Graphic*, and *Data Display*.

In our app, we labeled the action buttons as the following: *Build, Mesh, Run, Reset data, Report, Layout*, and *Contact us*. What, you might ask, is the purpose of each of these buttons? Let’s take a look:

*Build*: Show the actual geometry in the graphics window after applying the solder thickness to the related input field*Mesh*: Generate the tetrahedral elements mesh*Run*: Launch the simulation*Reset data*: Return to the default settings*Report*: Create a COMSOL report*Layout*: Show the graphical contents presented in the above figure of the base layout*Contact us*: Display the contact information for BE CAE & Test

Users can manually introduce geometrical and functional parameters within the model via the app’s input fields. With the flexible nature of simulation apps, the parameters that are included in the app can be tailored to specific design needs. In this case, app users can modify the thickness of the solder layer and the dissipated thermal power.

Within our app, we implemented two combo boxes in order to choose the constitutive material for the solder layer and die. As the figure below illustrates, the material choice was added to the embedded model via a global materials definition and a material link. Each material is indexed with a string in which the global variables are defined, and an initial value and a choice list are both associated with the global variables. The value of the global variable, which app users can modify, is controlled by an interactive combo box. When the value is changed, the local method implemented in the Application Builder permits changes to the materials in the underlying model.

*The global materials and local link enable users to change constitutive materials.*

Further, we conceived data display objects to show the outputs of interest for a typical thermal-electronic device application, that is, the maximum junction temperature, T_{jc}(MAX), and the junction-to-case thermal resistance, R_{th-jc}, of the SMD. Meeting this goal involved using a standard probe that returned the output for the maximum value of the temperature evaluated over all domains. We also defined a second probe, assuming a global variable value for the junction-to-case thermal resistance. The value of the second probe is dependent on the value of the first probe, as the following figure indicates.

*The global variable and probes are used to allow evaluation of the junction-to-case thermal resistance.*

The app appears graphically, as it does in the earlier figure of the base layout. While running the app, the main graphics window changes to correlate with the different steps. Depending on the specific step, the window shows the actual geometry of the SMD, the finalized mesh, and the computed thermal distribution.

Now the app is ready to be run. When doing so, app users can evaluate many important thermal elements within the electronic device’s design:

- Thermal distribution within the device
- The maximum temperature over the whole device
- The junction-to-case thermal resistance as a function of solder thickness and dissipated power
- The constitutive material of the solder and the die

From our own experience with building and sharing simulation apps, we have found that apps offer an innovative way of interacting with customers. Rather than simply sending simulation results to them, you can provide customers with a flexible tool that they can use to investigate the problem on their own — all while ensuring accuracy in their results. This not only allows them to obtain results more quickly, but it also gives simulation experts more time to work on adding complexity to the model behind the app versus running simulations.

“What happens in my system if…” is a question that the team at BE CAE & Test typically hears when first meeting with customers. Traditionally, our reply was something like this: “Let us build a reference model for your system and then carry out parametrical simulations. We will be able to give you useful predictions, which are an essential advantage to virtual prototyping.”

Now, with the use of simulation apps, we have a much simpler reply: “Let us provide you with a COMSOL app, and you will be able to check yourself.”

Giuseppe Petrone is a cofounder and the sole administrator of BE CAE & Test, a COMSOL Certified Consultant. He received his master’s degree in mechanical engineering from the University of Catania in Italy and later earned his PhD in energetic and process engineering from the Université of Paris-Est in France. Before starting at BE CAE & Test, Petrone devoted his time to academic research ventures, which included exploiting numerical methods in fluid dynamics and thermal analyses. He has been a user of COMSOL Multiphysics since 2005.

]]>In a webinar highlighting electrochemical recycling processes, SIMTEC presented a computational approach for predicting current distributions in a molten salt electro-refiner. The three main types of current distribution (primary, secondary, and tertiary) were treated, with a particular emphasis on the first two types. Using COMSOL Multiphysics, we implemented primary and secondary current distributions in an electrolysis cell.

A common pyrometallurgical route for the recovery of many metals and rare earths is high-temperature molten salt electrolysis. This process involves a molten electrolyte made of a molten salt in which the metal to be recovered, most commonly present in its oxide form, is dissolved. When a current is applied between the cathode and anode, the metal is deposited as a solid or a liquid at the cathode, while some gas (generally a carbon oxide, such as CO or CO_{2}) evolves at the carbon-based anode, as depicted in the figure below.

*Principle of a molten salt electrolysis cell for the refining of rare earths.*

Numerical simulation can be advantageously employed to predict the main cell features (e.g., the reaction rate distribution on the electrodes, the cell voltage, or the electrolyte temperature) in order to optimize the design and operational conditions of the process. Even though high-temperature electrolysis involves many physical phenomena that are complex and strongly coupled to one another, the reaction rate distribution on the electrodes can be approximated with a rather simple electrostatic approach that involves the calculation of the current densities associated with the reactions.

The current flow in an electrochemical device can be described by three different types of distributions: primary, secondary, and tertiary. The type of distribution depends on the reaction kinetics and whether the cell is limited by the transport of active species.

The primary current distribution assumes that the reaction kinetics are so fast that no reaction overpotential is associated with the activation of the reactions. The cell can be considered a simple resistive charge conductor. The only equation to solve is the conservation of the current density in all domains. An example of the equation and possible boundary conditions to implement in order to compute the primary current are represented schematically in the following image. In this example, the cathode potential is arbitrarily set to zero, while a current is prescribed at the anode.

*An example of equations and boundary conditions of a primary current model of an electrolysis cell.*

The secondary current distribution accounts for the activation overpotentials of the reaction(s) at the electrode/electrolyte interfaces, which can be required when the electrochemical reactions are significantly slow. A kinetic relation is implemented as a boundary condition, which relates the electrode overpotential to the Faradaic current density. An ordinary kinetics expression is the so-called *Butler-Volmer equation*, which includes exponential activation terms. A schematic of the secondary current approach is illustrated below, with an example of a classical Butler-Volmer relation expressed with the exchange current density *i _{0}*.

*An example of equations and boundary conditions of a secondary current model of an electrolysis cell.*

The design of the molten salt electro-refiner considered here is represented in the following figure. It is made up of several anode rods mounted in a circle around a central cathode rod. The electrodes are immersed in a molten salt-based electrolyte containing the metal (a rare earth, such as neodymium) to be recovered at the cathode. Symmetrical considerations allow the implementation of 1/32^{nd} of the geometry without the loss of information, which significantly diminishes the degrees of freedom within the system.

*The geometry of the molten salt electro-refiner considered for simulation and an elementary unit obtained taking symmetry planes into consideration.*

The primary and secondary current models are implemented in COMSOL Multiphysics using the *Electric Currents* physics interface of the AC/DC Module. Predefined physics interfaces are available in the electrochemistry-based COMSOL Multiphysics modules, such as the Electrodeposition Module, that provide easy definitions of terms such as the Butler-Volmer equations and enable the simulation of current distributions in electrochemical devices. However, we chose to use the *Electric Currents* physics interface found in the core COMSOL Multiphysics product and introduce the electrochemical reaction kinetics as expressions in the source term. Three electric currents physics are used, one for each domain (anode, cathode, and electrolyte). For both models, a *Ground* condition (V = 0) and a *Normal Current Density* condition are set on top of the cathode and on top of the anode, respectively.

In the case of the primary current model, the *Electric Potential* condition is set at both electrode surfaces, with the value defined in the primary current model example above. For the secondary current model, the *Normal Current Density* condition is used on these boundaries, selecting “Inward Current Density” with values defined by the Butler-Volmer relations in the previous secondary current model example. The conductivity of the electrodes and that of the electrolyte are set as constant. The kinetics parameters associated with the secondary current description are *i _{0}* = 1 A.cm

The domains are meshed using tetrahedral elements. There is no need for a special level of refinement. However, it is important to ensure that there are enough elements on top of the anode, where the current is injected, as well as on the electrode edges, where current peaks are expected (edge effects).

The current densities simulated at the electrode surfaces for a total prescribed current of 1000 Amps are shown for both models in the next image.

*Current densities simulated at the electrode surface for the primary and secondary current models.*

In both current approaches, it is observed that the gas evolution reactions on the anodes are mainly concentrated on the inner parts facing the cathode. The deposition rate is quite uniform along the cathode height, with a significant current peak taking place around the bottom edge. The main difference between the primary and secondary descriptions is the uniformity of the current associated with the anodic reaction.

In the case of a primary distribution, the current density ranges from 0 to nearly 8 A/cm². In the secondary distribution, the anodic current density range is twice as small, with a maximum value of 3.63 A/cm^{2}. This is due to the homogenization effect induced by the activation overpotentials, which are accounted for in the secondary model. The difference between the primary and secondary currents is much more attenuated at the cathode, where a similar level of heterogeneity is observed due to the high value of *i _{0}* considered in the secondary model for the cathodic reaction.

The simulated cell voltages can be decomposed into several contributions. In the primary current model, the cell voltage (8 V) is the sum of the thermodynamic voltage E^{0} (1.7 V) and the Ohmic drops occurring across the cell (6.3 V). In the secondary model, there is an additional activation term of 2.1 V due to the electrode kinetics, which brings the cell voltage up to 10 V.

The secondary current model is then expected to be more accurate than the primary model in terms of current and voltage, especially for such processes involving kinetically slow gas evolution reactions at the anode.

To bring the current model further, a tertiary distribution can be implemented to take into account the effect of the active species concentration in the electrode vicinity. However, such a model is more complicated as it requires an appropriate description of the species transport as well as the electrolyte convective motions in the cell. While an electrochemistry-based module, such as the Electrodeposition Module, has specific physics interfaces for defining and simulating tertiary current distributions, other physics sometimes need to be considered within the simulation. In our case, gas evolution at the anode as well as natural convection occurring through thermal gradients can affect the electrolyte concentration near the anode.

An example of bubbly flow simulated in the multi-anode cell with the *Laminar Bubbly Flow* interface is given in the following figure. It is shown that the main electrolyte movement occurs around the anodes where gas bubbles are released and rise, while slower zones are also apparent, for example, at the bottom of the cathode.

*An example of a bubbly flow simulation in the electro-refiner.*

In this short presentation, we reported on two current models developed with COMSOL Multiphysics for simulating the electrochemical reaction rate distribution in a typical molten salt electrolysis cell. The primary current model considers the reactions to be infinitely fast, while the secondary current model includes kinetics relations at the electrode/electrolyte interface. The results differ in terms of current uniformity (more uniform distributions for the secondary distribution) as well as cell voltage (significant contribution of the activation overpotentials in the secondary model). For such cells, a secondary approach is clearly better. While a tertiary current distribution is not presented, an example of the convective flow due to gas bubble formation at the anode is provided and can be used to solve for the tertiary currents.

Alexandre Oury received a master’s degree in materials science and engineering from Joseph Fourier University (Grenoble). He later earned his PhD in electrochemistry and process engineering from the National Polytechnic Institute of Grenoble, with a research topic related to battery storage systems. He currently works at SIMTEC as a modeling engineer, developing numerical models that are mostly applied to electrochemical systems and electrolysis cells.

]]>Induction heating has become an important process in many applications, from cooking meals to manufacturing. It is valued for its precision and efficiency along with being a non-contact form of heating. In this guest post, I will describe how to build an induction furnace model in COMSOL Multiphysics and demonstrate how it can enhance your design.

The physical principles that govern the process of induction heating are quite simple: An alternating current flows in a solenoid (coil), which generates a transient magnetic field. Following Maxwell’s equations, this magnetic field induces electric currents (eddy currents) in nearby conductor materials. If the application is a furnace and due to the Joule effect, heat is generated and the melting point of the charge (metal) can be reached. By adjusting the current parameters, the molten metal can be maintained as a liquid or its solidification can be precisely controlled.

*Induction heating. (In the public domain, via Wikimedia Commons).*

When building the model, we begin by describing the geometry and the associated materials. As is often the case for such industrial applications, an axisymmetric assumption can be considered. The chosen geometry (shown in the figure below) is composed of the classical components of an induction furnace: the crucible that contains the charge (metal), a thermal screen that controls the heat radiation, and a water-cooled coil in which the electrical power is applied.

*The model’s geometry.*

By using the *Induction Heating* multiphysics interface, two physic interfaces — *Magnetic Field* and *Heat Transfer in Solids* — are automatically added to the component. The multiphysics couplings add the electromagnetic power dissipation as a heat source, while the electromagnetic material properties can depend on the temperature. A strong coupling is then ensured by applying the preselected study step, which can be the *Frequency-Stationary* or the *Frequency-Transient* study. In these cases, Ampere’s law is solved for each time step for a given frequency, and then the thermal problem is solved for a transient or stationary state.

Through considering the axisymmetric assumption, only the component of the magnetic vector potential that is perpendicular to the geometry plane () is non-zero. In order to apply boundary conditions, we can assume that a state of magnetic insulation is apparent quite a “far” distance from the furnace. It is important to ensure that this state of insulation is far enough away to guarantee that it does not affect the solution. An efficient technique is to use the *Infinite Elements* domain, available in the *Definition* item of the component. This method allows you to limit the size of the problem by applying a coordinate scaling to a layer of virtual domains surrounding the physical region of interest.

*The* Infinite Elements* domain.*

To apply the electromagnetic source, different methods are available. The selected method depends on the type of geometry and how well the electric properties are known. In our case, the geometry of the coil is truly represented (by four turns), and a *Single-Turn Coil* condition is thus added to these copper surfaces.

Concerning our knowledge of the coil excitation, we consider a case where the coil power is known. To apply this quantity to the entire coil, the *Coil Group* mode has to be activated to ensure that the voltage used to compute the global coil power is the sum of the voltages of all the turns. By using this kind of excitation, the problem becomes nonlinear, and COMSOL Multiphysics automatically adds the related equations to compute the correct power (see here).

The heat equation is solved for only the solid parts by neglecting the effect of the surrounding air. Indeed, heat is essentially transferred by radiation in this problem. Therefore, the *Surface-to-Surface Radiation* boundary condition is added to the *Heat Transfer in Solids* physics interface by selecting the external boundaries of each component.

*Surface-to-surface radiation boundaries.*

A circulation of water is classically used in industrial furnaces for cooling the coil. The design of the coil enables this channel flow by using a circular hollow section (see the geometry of the coil in the next figure). A convective volume loss term is then added to each turn by considering the mass flow rate and the heat capacity of water , the inlet temperature of water , and the internal radius of the coil :

Q_{loss} = \frac{\dot m C_p (T_{in} -T)} {2 \pi r * \pi r_{int}^{2}}

For each computation, an important parameter has to be quantified: the skin depth, as most of the electric current will flow through this skin depth. This parameter is dependent on the vacuum permeability, , the relative permeability of the material, , the electrical conductivity, , and the frequency by the following formula:

\delta = \sqrt\frac {1} {\pi \cdot \mu_R \cdot \mu_0 \cdot \sigma \cdot f}

The higher the frequency, the thinner the skin depth. Consequently, by modulating the current frequency, the location of the heat source can be precisely controlled. Numerically, it means that for each conductor material, the mesh has to be sufficiently fine to ensure precision. Convention requires that at least four elements cover this area. This can be easily achieved by using the *Boundary Layers* mesh type, as illustrated here:

*The* Boundary Layers* mesh type for the external coil boundaries. The figure also shows the inner tube where cooling water flows.*

The model can now be solved by specifying the frequency in the study step. In our case, a frequency of 1000 Hz is used and a stationary solution is obtained in less than a minute with a laptop.

The magnitude (norm) of the resulting current density is plotted in the following figure, together with the magnetic field fluxlines. We can see that the maximum current density is located in the coil domains. The distribution of the current density is not uniform throughout the coil section and the current tends to flow in the inner part of the turns. In the charge (metal), the magnetic field fluxlines are highly deformed and an eddy current flowing in the opposite direction is induced.

*Global and local plots of the current density norm.*

As it flows into the resistive charge, this current dissipates energy in the material. The resulting temperature in each part of the furnace is highlighted below. We can observe that, even if the current is very intense in the coil, the temperature is close to ambient thanks to the water cooling system. On the contrary, the temperature in the charge is high and close to the melting point of the material due to eddy currents and the Joule effect. The other parts of the furnace are heated by radiation.

*A model illustrating the temperature distribution in the furnace.*

The furnace geometry can now be customized according to different design constraints. The coil characteristics (frequency, power, type of geometry, number of turns, etc.) and the geometry of all the parts can now be optimized to reduce the energy consumption and ensure a controlled melting of the material.

To take things a step further and understand how the melted metal behaves in the furnace, hydrodynamics equations can easily be added to the model. By considering a homogeneous temperature of the melted pool, surface tension and buoyancy effects can be neglected, with only the Lorentz forces remaining. An additional source term is then added to the fluid momentum equation, with representing the current density and representing the magnetic flux density:

F^{Lorentz} = j \times B

Both vectors are complex entities and were previously obtained for a given frequency. The time-averaged Lorentz force contribution, given in COMSOL Multiphysics by the “mf.FLtzavr” and “mf.FLtzavz” parameters, must be used. Through neglecting the effect of the fluid on the magnetic field, the hydraulic problem can then be solved in an uncoupled way.

The image below shows the melted metal behavior at a stationary state. Two typical recirculation zones are generated in the fluid. Stirring can be controlled by adapting the frequency or the power of the current. This can have both positive and negative effects. On the one hand, it is a way to improve the homogeneity of the bath. On the other hand, stirring may lead to a rapid erosion of the refractory walls. As for the heating phase, and depending on the design constraints, parametric studies can now be computed to improve the process.

*Velocity vectors in the melted pool.*

Vincent Bruyere received his PhD in mechanical engineering from the National Institute of Applied Science (Lyon), with a research topic related to lubricated contacts. Following a post-doctoral position at the Atomic Energy and Alternative Energies Commission (CEA), Bruyere now works at SIMTEC as a modeling engineer. He develops numerical models applied predominantly to fluid dynamics as well as to thermal and electromagnetic applications.

]]>We’ll look at how we can make use of COMSOL Multiphysics® simulation software to determine the operating read range of a passive RFID tag powered by a reader’s interrogating field. Additionally, we will look at how we can maximize this operating range by optimizing the tag’s antenna design.

Radio-frequency identification (RFID) involves the wireless use of radio-frequency electromagnetic fields to transfer information and identify and track objects with the use of an RFID transponder, or tag, attached to the object in question. These tags are now frequently seen in everyday products, produce, payment cards, and even livestock.

A reader is used to interrogate the tag through electromagnetic fields as illustrated in figure (i) below and information is transmitted back from the tag. With their ever increase in use, there is a rising need to reduce their energy requirements and the size of the tags, while maintaining or maximizing the read range, i.e., the distance at which the tag can be detected.

*Illustration of (i) RFID system and (ii) equivalent circuit of RFID tag.*

RFID tags essentially consist of an antenna and a chip that have complex input impedances as illustrated in figure (ii) above. The chips are typically located at the terminals of the antenna and the voltage (V_{a}), developed at the antenna terminals from the reader’s interrogating field, powers the chip.

In order to maximize the tag’s read range, one simply needs to ideally match the impedance of the RFID tag antenna to the chip utilized in the tag (references listed in the “Further Reading” section) and ensure that for a particular reader, a minimum threshold power (*P _{th}*) is achieved at this distance to activate the chip at the required operating frequencies.

Fortunately, I’m not going to go into all the theory here. (If you’d like the full details, check out my paper “Impedance Matching of Tag Antenna to Maximise RFID Read Ranges & Optimising a Tag Antenna Design for a Particular Application”, to be released soon via the COMSOL Conference 2014 resource.) However, one can obtain an equation for the Power Transmission Coefficient, τ, which describes the impedance match. Here, as τ tends to unity, the better the impedance match between the chip and antenna, as follows:

(1)

\quad \tau = {\frac{4R{_c}R{_a}}{|Z{_c}+Z{_a}|^2}}

Here, *R _{c}* and

(2)

\quad r = {\frac{\lambda}{4\pi}} \sqrt{\frac{P{_r}G{_r}G{_a}\tau}{P{_t}{_h}}}

Here, λ is the wavelength, *P _{r}* is the power transmitted by the reader,

With COMSOL Multiphysics® and the RF Module, one is able to develop an analysis model for the RFID tag, including substrate, antenna and chip geometry, and material properties. In addition, we can input the reader system details such as power transmitted, *P _{r}*; the reader antenna gain,

In our numerical model, we’re able to perform the electromagnetic field and frequency domain analysis of the combined chip and antenna design to determine the antenna’s complex impedance, *Z _{a}*; gain,

Furthermore, with the use of the Optimization Module, one can optimize the antenna’s geometric design to maximize the read range. The figure below illustrates the basic features of the RFID tag model, including air domain, perfectly matched layer (PML) regions, tag substrate, and antenna and chip geometries.

*COMSOL Multiphysics® model of an RFID tag, including substrate, antenna, and chip.*

In order to have confidence in the analysis results from any numerical model, it is important to validate the model. This can be a very costly and time-consuming exercise. Due to budget and time constraints, we’ll compare COMSOL Multiphysics numerical results to physical test data obtained from literature.

In this case, we’ll make use of physical test data presented by Rao et al. (2005), which provides enough physical test data, including read range, *r*, and power transmission coefficients, τ, at various frequencies for validation purposes. It should be noted that Rao et al. only provided a single chip impedance value for the frequency range. In addition, the geometric and material details of the antenna and tag design were extracted from available images and text.

We set up and ran a frequency sweep of the equivalent tag. Then, we compared the results of the read range and power transmission coefficient with the physical test data provided by Rao et al. and illustrated our findings graphically, as such:

*Comparisons of (i) read range and (ii) power transmission coefficient obtained from model vs. physical test data from Rao et al.*

As you can see in the figure above, the model trends observed follow those of the physical data, but the COMSOL Multiphysics model peaks are found to be at slightly higher frequencies than those presented by Rao et al. As expected, there will be a slight difference between the numerical and physical test data due to the limited chip impedance and material data provided in the literature. Moreover, small errors in the extraction of the antenna geometric dimensions may have occurred.

We did, however, consider the small percentage deviations from the physical test data acceptable under these circumstances. We also deemed the modeling method able to correctly predict the observed read range.

Now that we’ve developed a COMSOL Multiphysics model and compared it to physical test data from literature, we’re confident enough to use it to predict the read range of the tag for different chip or antenna designs, in combination with various reader and reader antenna systems. If we’re not completely satisfied with the read range, we can then optimize the design to maximize the read range.

In this example, we’re going to make use of a specific chip, reader, and reader antenna from known suppliers to find the read range of an example tag antenna design. The example design was required to have a maximum 75×45 millimeter footprint area and was based on the “Murata-A3” inlay antenna design for a durable tag. The figure below illustrates the example tag antenna design and compares it to the “Murata-A3” (95×15 mm) antenna.

*Example tag antenna design (71.2×15 mm) and comparison to Murata-A3 (95×15 mm) inlay antenna design for a durable tag.*

The specific chip, reader, and reader antenna from known suppliers:

- Murata MAGICSTRAP® component (Murata Manufacturing Co., Ltd., Japan)
- Chip frequency: 866.5 MHz

- OBID i-scan® LRU1002 UHF Long Range Reader (FEIG Electronic GmbH, Germany)
- Reader Power: 1W (mid-range value)

- OBID i-scan® UHF Reader Antenna (FEIG Electronic GmbH, Germany)
- Reader Antenna: ID ISC.ANT.U.270/270
- Reader Antenna Gain: 9 dBi
- Chip Impedance: 15-45 j ω

- Tag Substrate Material: FR4 (250 mm thick)

When running this model, we obtained 0.303 and 1.59 meters for the power transmission coefficient, τ, and the read range, respectively. The read range was deemed a little low compared to the required 2 meters for the application at hand. That’s when we turned to the Optimization Module to find an optimal antenna design that would give a read range greater than 2 meters.

In order to get the maximum read range, one can simplify the analysis work and optimize for the maximum power transmission coefficient, τ, for the tag design and then determine the read range based on Equation (2) in combination with the reader system. The geometric variables involved in the antenna optimization process included 34 length and width parameters as illustrated below.

*Schematic illustrating the tag antenna design and geometric variables on one side only.*

In addition to the constraint of keeping the antenna design within a 75×45 mm footprint area, manufacturing tolerance constraints obtained from a subcontractor were implemented, as well as some constraint on the possible lengths and widths.

We looked at two gradient-free optimization methods in the work; namely, the bound optimization by quadratic approximation (BOBYQA) method and the Monte Carlo method. These were chosen because the objective function does not need to be differentiable with respect to the control variables, and the definition of the problem and the geometric relations and constraints will be discontinuous, making traditional *hill climbing* optimization methods unsuitable.

Overall, it took a total of 42 hours and 23 minutes of simulation runtime to find the optimized antenna design, using both the BOBYQA and the Monte Carlo methods in series, with the use of a PC with two E5649 (2.53 GHz) Xeon® processors and 32 GB RAM.

The final objective value found was 0.675, a vast improvement on the initial 0.303. This gave a read range of 2.38 meters when used in combination with the OBID i-scan® LRU1002 UHF Long Range Reader and the OBID i-scan® UHF Reader Antenna, 0.38 meters greater than our minimum requirement of 2 meters.

The geometric characteristics of the optimized tag antenna design are detailed in the figure below. As you may see, this optimized antenna design is vastly different from the initial design, where the final solution fills a large percentage of the space available and has a dramatically different design scheme.

*Optimized RFID tag antenna design.*

In addition to this, by changing the reader power settings and the type of reader antenna used, one can also assess different reader system specifications. So, for example, by boosting the reader to 2 W and using the larger 600/270 OBID i-scan® UHF Reader Antenna, the read range can be increased to 4.23 meters.

With this optimized design, one can also assess the tag’s response across a frequency range for various regional requirements. For example, the Industrial, Scientific and Medical (ISM) Radio Band requirements for Europe are 865-868 MHz, while those of the USA are 902-928 MHz.

So, how will the same tag design respond in the USA? We can easily test this in our COMSOL Multiphysics model. The results of power transmission coefficient, τ, and read range, *r*, are presented graphically in the figure below for the tag design from 800 Hz to 1000 Hz.

*Frequency response of optimized antenna design.*

As we can see from the figure, for the USA, the tag’s minimum read range is 0.73 meters at 928 MHz. Thus, we know that this design is not going to fare well in the U.S. and we need to look at optimizing the design for the U.S. as well as for the European market

In the end, we find that COMSOL Multiphysics software can not only help us find the read range of a passive RFID tag design, but it can also assist in the design of better antennas, which ideally match the chip and maximize read range for specific requirements and regions worldwide.

- Hsieh et al.,
*Key Factors Affecting the Performance of RFID Tag Antennas, Current Trends and Challenges in RFID*, Chapter 8, 151-170, Prof. Cornel Turcu (Ed.), InTech (2011). - N. D. Reynolds, “Long Range Ultra-High Frequency (UHF) Radio-frequency Identification (RFID) Antenna Design“, MSc Thesis, Purdue University (2005).
- Serkan Basat et al., “Design and Modeling of Embedded 13.56 MHz RFID Antennas“, Antennas and Propagation Society International Symposium, IEEE (2005).
- Rao et al., “Impedance Matching Concepts in RFID Transponder Design“, Fourth IEEE Workshop on Automatic Identification Advanced Technologies (2005)
- Yeoman et al. “The Use of Finite Element Methods & Genetic Algorithms in Search of an Optimal Fabric Reinforced Porous Graft System“, Annals of Biomedical Engineering, 37 (2009).

Mark Yeoman is the founder of Continuum Blue. He has a degree in engineering and a PhD in computational modeling and applied mathematics. Yeoman’s postgraduate studies focused on the development of cardiovascular implants, using simulation techniques and genetic algorithms for Medtronic Inc. Before starting Continuum Blue, he was a lecturer in applied dynamics and engineering. With over 15 years of experience in the field of multiphysics modeling, he has worked and continues to work in a variety of industries, including oil and gas, aerospace, automotive, chemical, and biomedical.

*OBID i-scan is a registered trademark of FEIG ELECTRONIC GmbH.*

*MAGICSTRAP is a registered trademark of Murata Manufacturing Co., Ltd.*

Using COMSOL Multiphysics, we implemented a wear model and validated it by simulating a pin-on-disc wear test. We then used the model to predict wear in an automotive disc brake problem. The results we found showed good agreement with published wear data.

*Wear* is the process of the gradual removal of material from solid surfaces that are subjected to sliding contact. It is a complex phenomenon that is relevant to many problems involving frictional contact, such as mechanical brakes, seals, metal forming, and orthopedic implants. The rate of wear depends on the properties of the contacting materials and operating conditions.

Archard’s law is a simple but widely used wear law that relates the volume of material removed due to wear to the normal contact force , sliding distance , material hardness , and a material-related constant

W=\frac{KF_N L_T}{H}

In our work, we considered a modified version of Archard’s law:

\.{w}=k(H,T)p_N V_T

This modified law relates the wear depth at any point to the normal contact pressure , magnitude of sliding velocity , and a constant that is a function of the material and temperature. The wear constant may be computed from experimental wear data, which is typically in the form of weight loss for a specific contact pressure and velocity.

Wear equations are not directly available in finite element analysis (FEA) codes, although their implementation in COMSOL Multiphysics is straightforward. We incorporated the wear equations within our simulations by defining boundary ordinary differential equations (ODEs) on the destination contact surfaces with the wear depth as the independent variable. The wear depth is then used as an offset between contacting surfaces (e.g., brake pad and disc) within the contact formulation in COMSOL Multiphysics. In particular, contact is enforced when the penetration between the contact surfaces is equal to the wear depth , as shown in image below.

*Modification of contact gap calculation: is the wear depth, is the gap, and is the contact pressure.*

This wear algorithm is very efficient since it does not involve altering the nodal locations to account for material loss due to wear. It is only suitable, however, for cases where the wear depth is significantly less than the width of the contact surface.

You can enhance this wear algorithm by including more sophisticated effects, such as anisotropic wear behavior, dependence on the mean and deviatoric stresses in the solid (not just the contact pressure), threshold pressure/stress below which no wear occurs, and more. The assumption of small wear depth must still hold for this modeling approach to be accurate.

We validated the new, contact-offset-based wear model implementation by simulating a pin-on-disc wear test. Only a small section of the disk is modeled, as shown below.

*Pin-on-disc wear test model.*

The disc in this model is much stiffer than the pin and all the wear is assumed to occur in the pin. A force is applied to the pin, resulting in a circular, Hertzian-type contact pressure distribution. A constant tangential velocity is then applied to the disc. The graph below shows how the wear depth varies radially along the pin at four time instances. The total volume loss, calculated as the integral of wear depth over the contact surface, was similar to the value calculated using Archard’s law.

*Wear depth vs. radial distance in the pin-on-disc model.*

We also used the model to predict wear in an automotive disc brake problem, which is similar to the Heat Generation in a Disc Brake model that can be downloaded from the COMSOL Model Gallery. We developed a 3D thermal-structural disc brake model involving representative brake disc/rotor and brake pads.

*Disc brake model used in the COMSOL Multiphysics wear simulation.*

The structural and thermal processes are coupled through frictional heat generation, thermal expansion, and thermal contact. Both physics fields are also coupled to the wear depth evolution boundary ODE. We used a fully-coupled direct solver that converged rapidly, keeping solution times similar for problems with and without wear.

The results for both the pin-on-disc validation example and the disc brake problem were in good agreement with published wear data. In the disc brake example, the model captured the non-uniform wear rate that is typically observed on brake pads; it was higher near the outer radius and leading edge, as shown below.

*Typical brake pad wear depth profile.*

We will present more of our results, including contact pressure and wear contours, at both the Cambridge and Boston stops of the COMSOL Conference 2014.

Nagi Elabbasi, PhD, is a Managing Engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is modeling and simulation of multiphysics systems. He has extensive experience in finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, material testing and modeling, and failure analysis, and is a member of the COMSOL Certified Consultant program.

]]>Modular orthopedic devices, common in replacement joints, allow surgeons to tailor the size, material, and design of an implant directly to a patient’s needs. This flexibility and customization is counterbalanced, however, by a need for the implant components to fit together correctly. With parts that are not ideally matched, micro-motions and stresses on mismatched surfaces can cause fretting fatigue and corrosion. Researchers at Continuum Blue Ltd. have assessed changes to femoral implant designs to quantify and prevent this damage.

Take a few steps and see how your hips rotate. You’ll find that your body weight is continuously shifting between the left and right sides, while your legs bend, swing, and then straighten out with each step. Thus, a good modular hip replacement system will need to be able to freely allow for the natural motions of the human body — walking, running, or going up and down stairs. In addition to this, it has to be durable enough to take the continually changing, and sometimes excessive, loads placed on it during these movements, while being comprised of lightweight materials that fit and interact well with the body.

Modular implants often include stems, heads, cups, or entire joint systems. A range of materials from steel and titanium alloys to polymers and ceramics offer the surgeon many options depending on the needs of the patient. However, material and geometric selections affect the amount of wear and tear that will occur over time, so certain combinations of components are better than others. With so many different factors at play, it is not surprising that these assemblies require tight tolerances and the right material combinations to function properly and last a lifetime.

*Virtual implantation of hip replacement in resected patient femur.*

Studying how a modular combination of parts will behave under dynamic loads and stresses is a crucial part of the design and decision-making process. In order to understand the available combinations better and aid medical professionals in decisions, engineers at Continuum Blue have modeled three combinations of modular femur stem and head implants to investigate the *fretting fatigue*; the fatigue wear caused by the repeated relative sliding motion of one surface on another.

The femur head contains an angled channel for the neck of a femur stem, which in turn must be tapered correctly to fit the channel. The engineers studied three different geometric configurations using different materials for the head and stem to determine which of the three was best for minimizing fretting fatigue.

*Different stem and head configurations with an ideal fit, positive mismatch, and negative mismatch.*

Using kinematic load data from Bergmann et al. and based on averages from four patient sets, Continuum Blue created a COMSOL Multiphysics simulation to analyze the cyclic loading on a femur head. They used their model to determine the loading at different points during a walking gait cycle, knowing that the load would change at different locations in the rotation, and validated their results against the kinematic data.

*Simulation results showing the dynamic loads and stresses during the gait cycle.*

Material fatigue can be determined by studying the mean stress and stress amplitude that occur during the cyclic loading of the joint. Like the loading in the femur head shown earlier, the stresses in the femur stem will change over the course of a gait cycle. With regular leg movements, the stresses observed will take on an oscillation that reflects the repeated motion of the person walking.

*SN curves for the titanium stem and cobalt chromium head used in the study.*

Continuum Blue assessed the three configurations with two different materials: a cobalt chromium alloy for the head and a titanium alloy for the stem of the modular implant. For each material domain, they calculated the stresses observed over a single gait cycle and related these to both the SN curves of the material and the micro-motions of the contact surfaces. This allowed them to predict the number of cycles the device could undergo before fretting fatigue became an issue.

*Areas where fretting fatigue occurs over gait cycle for each configuration.*

Their results showed a surprising fact: the “ideal” fit, where the femur head channel is exactly aligned to the sides of the femur stem, was *not* found to be the best configuration for minimizing fretting fatigue. Rather, the configuration with a slight positive misalignment turned out to be a better choice, exhibiting lower stresses and overall fretting fatigue.

Through their simulation, Continuum Blue was able to predict the stress, contact pressure, and areas most susceptible to fretting fatigue at different points in a gait cycle. There are many other factors that will be accounted for in future research, such as the sensitivity of the implant to varying degrees of misalignment; additional designs and geometric changes; different materials; and the effects of surface finishes, coatings, or roughness that may impact the results. However, their modeling work offers a unique promise for evaluating the lifetime of a modular implant device. It was validated as an accurate way to predict the wear and tear that will occur for these three configurations of the implant. If you ever need a joint replacement analysis — you’ll know who to call.

- COMSOL Conference 2012 presentation: “Fretting Wear and Fatigue Analysis of a Modular Implant for Total Hip Replacement“