Building the COMSOL^® Model

To get a better idea of the interference these sensors may introduce, a collaborative team from GEPEA–Oniris (Process Engineering–Environment–Agri–Food Joint Research Unit, National College of Veterinary Medicine, Food Science and Engineering, Nantes, France) and CTCPA (agri-food technical center in France) ran a study that evaluated these in-package microwave heating processes by simulating the effects of different sensor positions within a single-mode microwave applicator. The study focused on a lab-scale 915-MHz microwave heating process for 300 g of mashed potato in a polypropylene tray. The 3D model geometry of the microwave equipment was built in the COMSOL Multiphysics^® software, while the experimentally measured geometry of the polypropylene tray was imported to COMSOL^® from a custom-made STL file.

Figure 1. Description of the 915-MHz single-mode microwave apparatus.

The model the team built was robust, taking into account all of the geometrical details of the microwave apparatus. It included the complete design of a waveguide transition, an impedance matching element (i.e., an iris), a 915-MHz single-mode microwave applicator, and a sliding short circuit. The microwave input power source was modeled with a coaxial port at the microwave antenna. All surfaces of the waveguide, including the copper sliding short circuit, were considered as perfect electric conductors. The dielectric properties of the mashed potato were experimentally measured and assumed to correspond to a constant moisture content of 76% on a wet basis.

To avoid microwave interference with metallic parts, the wireless sensor was surrounded by an ogive-shaped 316-L stainless steel microwave shield and placed at the upper surface of the mashed potato sample. Additionally, a polypropylene film was heat-sealed at the top surface of the polypropylene tray to close the system. Each of these components came together to create a numerical model that helped the team derive accurate insights into the best in-package sensor position to limit interactions with the electromagnetic field.

Figure 2. Description of the polypropylene tray filled with a mashed potato sample. MW = microwave, PP = polypropylene, and PTFE = polytetrafluoroethylene.

Simulation-Based Insights

The team used the RF Module, an add-on to COMSOL Multiphysics^®, to predict the electromagnetic field in the microwave cavity relative to the microwave-absorbed power within the sample of mashed potato. For the solver settings, boundary mode analysis was used as a first step to compute the mode at port 2 (see Figure 1), which comes out of the waveguide. The second step was using a Frequency-Stationary study at 915 MHz.

Without a sensor inside the tray, the simulation revealed microwave-reflected power of 24% (S₁₁ = -6.16 dB), with the optimal sliding short circuit position at 790 mm. S₁₁, the input port reflection coefficient (i.e., the measure of the port’s mismatch), was then evaluated as a function of varying microwave shield positions following different z-axis rotations, from alpha = 0 to 180^°, with 30^° increments.

Figure 3. Local microwave-absorbed power within the mashed potato as a function of four sensor positions.

The results showed that the microwave shield position needed to be aligned to the microwave propagation direction (alpha = 0^°) to obtain a similar S₁₁ parameter to the food tray without the sensor (S₁₁ ≈ -6 dB, RF ≈ 25%). When rotated counterclockwise by 90^°, so the microwave shield position faced the same direction as the transverse electric plane, the S₁₁ parameter dropped to -7.8 dB. Furthermore, by changing the rotation angle of the microwave shield from 90 to 180^°, simulation results indicated S₁₁ variations from -7.8 to -4.7 dB. This rotation from 90 to 180° also resulted in the simulated microwave reflected power increasing by a factor of 2, with 17% microwave reflected power at 90° and 33% at the 180° rotation. These results suggest that the sensor positioning is crucial to take into account, as these reflected power variations can impact overall microwave power efficiency.

A portable vector network analyzer (VNA) was used in the experimental validation of the numerical model. The VNA was connected at port 1 to evaluate the microwave-reflected power from the load according to different operating conditions. The S₁₁ parameter was used to account for the microwave-reflected power at the antenna relative to the incident microwave power. Overall, the simulation results were in agreement with the experimental measurements performed with the VNA. Discrepancies between the experimental and simulation results can be attributed to slight variations in sensor positioning due to the difficulty of precisely replicating the uneven surface of the mashed potato.

Figure 4. Reflection factor following different rotation angles of the microwave shield.

An Important Discovery

This study led to an important discovery: Aligning the sensor position with the microwave propagation direction resulted in a similar microwave reflection coefficient as tests run without a sensor. However, a significant reduction of the microwave-reflected power is observed if the microwave shield is positioned within the transverse electric plane of the waveguide.

Testing showed that the model is also able to predict the high local microwave power densities at the interface between the sensor and the food sample, which could further elucidate the best position for a temperature and pressure (T-and-P) sensor, though this would require more experimental validation and accounting for potential temperature-dependent dielectric properties of the food, as well as their coupled impact on the electric field distribution.

Further Learning

For more information on this research, read the GEPEA-Oniris and CTCPA team’s full paper, which won a Best Paper award at the COMSOL Conference 2025 Amsterdam! The paper describes the team’s full modeling approach and results.

The vibration of structures is affected by the surrounding media. Air, one of these media, is light and compressible and thus can often be neglected when a structural engineer evaluates the vibration of a design. Simulations might be performed without a physical representation of the air — as if the structure were vibrating in a vacuum without any interference with a surrounding medium. Conversely, when a structure is submerged in water, with one thousand times the density of air and a higher speed of sound (stiffness), the water’s effect on the vibration characteristics is often noticeable.

Accounting for the Effects of the Surrounding Medium

Traditionally, this effect has been denoted “added mass”, derived from the following equation used to describe it.

In the equation, is the mass of the structure, and is the added mass. The value of the added mass has often been empirically derived by experiments where the eigenfrequencies of the structure have been measured in air and water, respectively. Eigenfrequencies, also known as natural or resonance frequencies, are the frequencies at which a structure naturally reinforces vibrations. The difference in eigenfrequencies is then used to calculate the added mass.

Unfortunately, this approach has led to the misunderstanding that the effect of the surrounding medium is an added mass that vibrates together with the structure, whereas actually, the surrounding media acts with a pressure on the surface of the moving object, and this pressure is highest when the surface speed is highest. Therefore, the pressure is not constant but changes in amplitude and direction depending on the direction of the moving surface.

The true influence of the surrounding fluid on the vibration response can be captured using COMSOL Multiphysics^® and a coupled acoustic–structure interaction (ASI) modeling approach. Similar multiphysics simulations have been described in the blog post “What is the Best Way to Analyze Fuel Tank Vibration” and also in our blog posts on muffler design, such as “Evaluating the Effect of Shell Thickness on Muffler Performance“.

Simulating the Surrounding Media

The focus of this blog post is highlighting the effect of the surrounding media and clearly describing an alternative to “added mass”. The physical example used in this blog is a Kaplan turbine blade in a water domain. The effect on its frequency response from a varied gap to the pipe wall is studied. The figure below shows the full Kaplan turbine geometry and the considered gap. Note that pressure equalization at the leading and trailing edge of the blade is always possible, irrespective of the varied gap size.

Kaplan turbine geometry with arrows showing the gap to the pipe wall.

Acoustic–Structure Interaction

The fundamental part of acoustic–structure interaction is the multiphysics coupling applied to the boundary between the fluid and the structural domain. The pressure degree of freedom (DOF) in the fluid is coupled to the three displacement degrees of freedom (DOFs) in the structure (in the x, y, and z directions), as described below.

The conditions for an external boundary are

in which is the structural acceleration (applied to the fluid), is the total acoustic pressure, is the density of the fluid (the subscript c indicates that this can be a complex-valued quantity used to model losses in the fluid ), is an optional acoustic dipole domain source term, is the surface normal, and is the (acoustic) load (force per unit area) on the structure.

The acoustic pressure is applied to the structural faces, and the acceleration of the surface is coupled to the acoustic domain. To enable the modeling of acoustic–structure interaction in COMSOL Multiphysics^®, both the Pressure Acoustics, Frequency Domain and Solid Mechanics interfaces need to be added to the model. Furthermore, the multiphysics coupling described above needs to be enabled and added (under the Multiphysics node) to all boundaries between structural and fluid volumes. Here is how it looks in the Model Builder tree:

Building a vibroacoustic simulation.

As the problem is solved in the frequency domain, the model is assumed linear, and only small structural deformations can be modeled. Losses in the fluid can be accounted for in several ways using a fluid model or even more advanced boundary conditions such as the Thermoviscous Boundary Layer Impedance feature. Damping due to vorticity in a moving fluid, and even the effect of a moving fluid, requires solving a more advanced acrostic formulation such as the linearized Navier–Stokes equations, also available in COMSOL^®.

Example Using a Kaplan Turbine Blade

In the following, the influence of the inner radius of a pipe has on the vibration response on the general Kaplan turbine, is investigated. With the functionality in COMSOL Multiphysics^®, it is possible to apply periodic boundary conditions and only model one blade while capturing the effect of overlapping blades. The water volume is created by offsetting the circular sections at the top and bottom so that their connecting sides pass between the blades. On those sides, the periodic boundary condition is applied (shown as yellow in the figure).

Water domain, with periodic boundary conditions in yellow.

The turbine blade was modeled as linear elastic with Young’s modulus 210 GPa, Poisson’s ratio 0.3, and density 7850 kg/m³. The water was assigned the properties of speed of sound at 1500 m/s and density at 1000 kg/m³.

For the model with a 10 mm gap to the pipe wall, 310,000 quadratic tetrahedral elements made up the mesh with refinements around the blade. The gap had at least 5 elements for all cases. The entire mesh was continuous, and no contact exists in the model.

All outer faces were set to the Sound Hard Boundary (Wall) boundary condition, which sets the normal component of the acceleration/velocity to zero. The condition is simply the natural Neumann condition for the total pressure, i.e., the normal derivative of the pressure is zero on the boundary

For the eigenfrequency analysis, the hub mount face of the blade was fully constrained

When determining the response to excitation in the frequency domain (the frequency response function, or FRF), an axial rotation of rad/s² was applied instead of the fully constrained boundary condition. That gives an acceleration of 1 rad/s², regardless of the frequency.

Results of the Acoustic–Structure Interaction Analysis

First, it was determined that the height of the water volume did not influence the results when the vertical distance between the blade and water boundary exceeded 10 cm above and under the blade. Therefore, this size of the water domain will be used throughout the study.

Second, the influence of the surrounding fluid (water or air) on vibration properties can be determined with the ASI approach. In the table below, the first six eigenfrequencies in water, with a gap large enough to not influence the turbine blade, are compared to the first six eigenfrequencies in free air. The table shows that water has a large influence on the eigenfrequencies, even when the gap has no effect.

Finally, the influence of a varied gap size on eigenfrequencies can be determined. As the gap increases, eigenfrequencies do as well, up to a gap of about 150 mm. Here, eigenfrequencies converge to a constant value, regardless of a continued increase in gap size. The first six real eigenfrequencies for each gap size were derived and are presented below.

The first six real eigenfrequencies as functions of the gap.

The decreasing eigenfrequencies for smaller gaps can be explained by the increase in resistance for pressure equalization around the blade. It is not to be confused with damping from viscous effects, nor should it be simplified to an additional mass. Instead, we see how the water, modeled explicitly, affects the eigenfrequency of the system consisting of the turbine blade and water domain coupled together.

The corresponding mode shape for the fifth eigenfrequency for a gap of 149 mm is shown below, left. In COMSOL Multiphysics^®, it is possible to plot the entire structure — not just the blade that is used in the calculation — giving a better overall picture of the state of the structure, as seen below, right.

Mode shape 5 with a 149 mm gap, showing the blade used in the calculation (left) and the entire structure (right).

3D illustration of a Kaplan turbine propeller.

When the structure is instead excited by adding a frequency-independent rotation on the hub mount face of the blade, it results in the frequency-domain response presented in the figure below. Note that the frequency domain is swept in steps of 1 Hz and therefore the amplitude of the eigenfrequencies might vary depending on how close to the exact value is reached in this simulation. Regardless, peak response amplitude is not of the highest interest since it is most often desired to avoid these frequencies with some margin of safety in product development.

Acceleration response in the frequency domain.

Using functionality in the COMSOL Multiphysics^® software, this example study has shown how modeling acoustic–structure interaction can be used to evaluate the frequency response for a structure submerged in fluid. This is an important step in the product development process for all products where the vibration response is a design criterion in the same way for underwater products as for products in air, where frequency analysis has been utilized for a long time.

In this example, it was made clear how the resistance of the flow of the fluid influences the frequency response of the structure in addition to the difference of submerging in water instead of in air. The eigenfrequency decreased by 57%, from 165 Hz to 71 Hz, by accounting for the surrounding water and minimizing the gap (to 3 mm).

Dynamic Loads on the Blade

A useful feature in COMSOL^® is the Rotating Frame feature, which simplifies rotor dynamics by keeping your mesh stationary. By directly applying centrifugal, Coriolis, and Euler forces as body loads, it accurately simulates high-speed spin effects such as stress stiffening. This feature was not used in this example due to the low rotational speeds for this general application, but it is a powerful tool if the application requires it.

About the Guest Author

Tobias Jonsson is a structural mechanics specialist and consultant at Lightness by Design in Stockholm, Sweden. With an MSc from KTH Royal Institute of Technology, Tobias uses numerical simulation to help companies with simulation-driven product development. As part of a COMSOL Certified Consultancy, he is dedicated to integrating advanced simulation into the core of the product development process.

The consultants at Lightness by Design have contributed to the field of acoustic–structure interaction by coauthoring several research papers published in scientific journals. Furthermore, they have used their expertise in acoustic–structure interaction approaches to simulation and vibration design to help multiple clients in the product development process.

The Resonator Essentials: Physics, Admittance, Damping, and Electromechanical Coupling

Before we dive into the circuit formalism, I will set the scene from a physics standpoint. A standard bulk acoustic wave (BAW) or surface acoustic wave (SAW) resonator uses the piezoeffect to trap acoustic energy at natural frequencies inside a microscale cavity. To analyze this behavior from the electrical point of view, we track terminal admittance, , where its:

• Real part, conductance, , represents energy transfer
• Imaginary part, susceptance, , tracks energy storage

This electromechanical interaction yields two primary frequencies. At the series resonance frequency , standing acoustic waves constructively interfere. Internal mechanical impedance drops to a minimum, causing a sharp peak in electrical conductance . The antiresonance or parallel resonance frequency is located slightly higher. Here, the piezoelectrically generated charge from mechanical strain cancels out the electrostatic plate charge on the electrodes. The device draws minimal net current, driving the total admittance magnitude to a minimum.

The gap between and defines the effective electromechanical coupling coefficient , which is a measure of the energy conversion efficiency: . A larger frequency split provides a higher , directly translating to a wider achievable filter bandwidth. The exact definition of the effective electromechanical coupling coefficient depends on the standard or approximation you follow. The formula may deviate a bit between BAW and SAW applications or if your conditions satisfy the small-coupling approximation.

In an ideal lossless resonator, the conductance peak at would approach infinity, and the admittance valley at would drop to absolute zero. Real-world piezoelectric devices, however, experience energy dissipation due to internal material damping, both mechanical and dielectric, as well as other mechanisms, such as anchor leakage into the substrate. This total dissipation dictates the device’s Q factor, . On the admittance curve, higher material damping directly suppresses and flattens the response. It rounds off the sharp resonance peaks and valleys, lowering the maximum conductance at and raising the minimum admittance floor at . For RF filter designers, minimizing this damping to achieve a high Q factor is essential to ensure low insertion loss and steep filter skirts.

A convenient admittance magnitude |Y| curve sketch with the series resonance and parallel antiresonance points mapped out.

Representing a Piezoelectric Resonator in a FEM Manner

If you were to create a representative FEM model of a piezoelectric resonator, it would be straightforward in COMSOL Multiphysics^® and reveal the powerful multiphysics capabilities of the software. The steps are as follows: Start with a Piezoelectricity interface that couples an Electrostatics interface with a Solid Mechanics interface. For Solid Mechanics, select the Piezoelectric Material material model and specify the relevant lossy behavior through its subnodes, such as Mechanical Damping and Dielectric Loss.

To run a Frequency Domain study, you might want to drive your device via a Terminal condition, added in the Electrostatics interface. Among other benefits, this feature automatically derives the terminal admittance, making it readily available for plotting and further results evaluation. The results evaluation includes extracting the mentioned resonance frequencies and Q factors. Alternatively, you may opt for an Eigenfrequency analysis, which also helps retrieve resonance frequencies and Q factors.

Bridging Finite Elements and Lumped Parameters

The beauty of a lumped-parameter circuit model lies in its simplicity, as solving a network of a few resistors, inductors, and capacitors takes fractions of a second but still provides you with the sought admittance curve. The electromechanical equivalent framework, which is central to this blog post, traces its lineage back to Stephen Butterworth’s filter designs and Walter Guyton Van Dyke’s early-20^th-century work mapping the behavior of quartz crystals. The “modified” modern version adds parasitic elements to fit the stringent demands of high-frequency MEMS resonators.

Modern RF MEMS resonators demand an engineering synergy between fast circuit models and modern numerical solvers. A circuit model is only as good as its input parameters. You cannot simply guess the equivalent capacitance of a complex, layered SAW interdigital finger topology or the motional resistance of a novel piezoelectric thin film. By using COMSOL Multiphysics^® to simulate the true geometric and material complexities of your device, you can extract the exact lumped parameters needed to populate your mBVD circuit. Altogether, the FEM provides the ground-truth physics, while the circuit representation allows for the system-level speed.

Note: The classic mBVD model configuration to be discussed here is chiefly designed for one-port resonators featuring a single signal terminal and a ground terminal. If your design boils down to a multiport configuration or the like, you will need to expand this framework into a more elaborate one.

Similarly, for the sake of clarity, we will focus on one particular resonance. You will need to add an extra parallel motional branch for every resonance mode you need to consider.

Breaking Down the mBVD Circuit

The conventional mBVD model captures a standard piezolelectric resonator behavior by splitting the electrical behavior after the initial series routing into two parallel paths: a static branch representing the physical dielectric layer stackup and a motional branch representing the electromechanical acoustic resonance. A circuit schematic is provided below.

A schematic of the one-port mBVD circuit topology showing the series resistance, , connected to the parallel combination of the motional and static branches. The motional branch includes , , and , which form a series contour, and the static branch includes and , which are connected in parallel.

The table below can be used to consider each element, how the elements impact the electrical admittance response, and how to cleanly extract their values from your FEM simulation.

Circuit Element	Physical Meaning	Admittance Impact	How to Extract It
Series resistance,	This element represents the purely ohmic resistance of the metal electrodes, busbars, and interconnect wires leading up to the active area of the resonator.	It limits the maximum peak of the admittance magnitude at series resonance .	can be calculated directly from the conductivity and geometry of the metal routing in your model, say via an auxiliary Electric Currents setup. You typically don’t need to include it explicitly in your main Piezoelectricity simulation.
Static capacitance,	represents the electrical plate capacitance, which is formed by the overlapping electrodes and the intervening piezoelectric material. is independent of mechanical movement.	It establishes the baseline “floor” of the imaginary admittance curve and determines the frequency separation between your series resonance and parallel antiresonance .	Run a frequency-domain simulation at an arbitrary point, , well below , where the device is mechanically “dead” and the motional branch acts as an open circuit. Evaluate the imaginary part of the admittance to isolate the static capacitance as
Static resistance,	represents the bulk dielectric material losses and electrical leakage paths through the piezoelectric layer.	It introduces a clean, flat baseline conductance to the real part of the admittance curve at low frequencies.	Assuming that your model goes with a nonzero dielectric loss, in the same low-frequency regime used to find , i.e., well below acoustic resonance, measure the real part of the admittance to directly isolate the flat baseline dielectric leakage as . For a case of small dielectric loss, is supposed to get a huge value.
Motional capacitance,	This element represents the mechanical elasticity or compliance of the resonator structure, which is intimately tied to the effective electromechanical coupling coefficient of the piezoelectric crystal.	It dictates the overall strength of the resonance. A higher coupling coefficient increases . Subsequently, it widens the window between and , which is a vital feature for wide-bandwidth RF filters.	can be pulled instantly using your extracted value and the resonance frequency ratio via .
Motional inductance,	This element represents the effective acoustic mass or inertia of the vibrating crystal lattice or thin-film membrane.	Together with , it determines the exact frequency location of the series resonance peak. This calculation follows the classical formula for resonance frequency .	Once you know and have identified from your simulation data, you can define the motional inductance as .
Motional resistance,	This resistance accounts for the total mechanical and acoustic energy dissipation in the device.	It dictates the height and sharpness of the admittance peak at resonance. A lower value signifies lower loss, leading to a much higher Q factor.	In the Frequency Domain sweep, the peak admittance at is dominated by the motional branch. The peak conductance simplifies to . Alternatively, you can pull the device’s mechanical Q factor, , from the Eigenfrequency study and use the following relation: .

Here we opted for the mBVD configuration featuring the static branch with and connected in parallel. The alternate convention with these elements connected in series also exists, and, if chosen, it will affect their extraction logic and interpretation. Two approaches are interchangeable in the sense that once the lumped parameters are obtained attentively, the outputs are to be identical.

Extracting the Parameters

Provided that we are now equipped with both theory and extraction logic, let’s extract the full set of lumped parameters and construct the mBVD circuit within COMSOL Multiphysics^®.

Note that this blog post acts as the primary resource for this model. If you’re interested in finding step-by-step instructions for leveraging similar models and plotting admittance curves, check out the example models we have available for the MEMS Module.

Here, we opt for a 3D layered one-port thin-film bulk acoustic resonator (FBAR), shown below. We also assume that is known beforehand and for further comparison, just add its contribution manually to the device’s admittance.

To accurately extract resonance frequencies and the peak conductance from the Frequency Domain study, you might want to refine the frequency step and use the Graph Markers for a Global admittance plot. Do not forget to assign the Dielectric Loss if you consider . Finally, it makes sense to run a separate study for a frequency several orders smaller than for the purpose of and extraction.

The UI in COMSOL Multiphysics^® showing a 3D FBAR model, with its admittance calculated via the Frequency Domain study.

Once all the parameters are calculated, you can bring these values directly into the built-in Electrical Circuit interface within the same model file to verify your work. Connect your newly defined lumped mBVD circuit to a voltage source and overlay its impedance response directly onto your original FEM data. Following the modern and handy AI trends, do not hesitate to ask the optional Chatbot window in COMSOL Multiphysics^® to provide you with COMSOL API code for adding the circuit with all the needed settings, as was done here.

The extracted mBVD parameters for the Electrical Circuit interface and the plot depicting good correlation between the full-fledged FEM model and the circuit representation.

The result is a great match around the operational bandwidth. From any practical standpoint, a minor discrepancy can be explained by the fact that the FEM model captures contributions from high-order modes present in the system. Once your mBVD circuit block matches the physics-based model, you are no longer constrained by the physical geometry. You can extend the circuit network right inside COMSOL Multiphysics^® by adding external tuning capacitors, inductors, or transmission lines, or by cascading multiple identical blocks to form complex ladder filters. Altogether, you end up with a verified, lightning-fast lumped model that seamlessly bridges microscale multiphysics with macroscopic system reality, enabling you to run massive sweeps in seconds.

Next Steps

Let’s wrap up by sharing the model featured in this blog post as well as related examples:

• Check out the model discussed in this blog post: Extracting mBVD Circuit Parameters from a 3D BAW Resonator
• To learn how to approach the same task from the Eigenfrequency side, see: Frequency-Domain Analysis of a SAW Unit Cell
• If you need a perfect match and/or a way to compensate for missing data (e.g., damping ratios), see this example, which makes use of the Parameter Estimation feature: Thin-Film BAW Resonator with Equivalent Circuit
• Check out tutorial models that leverage the MEMS Module: Application Gallery

The Agentic Loop

Until recently, most AI-assisted workflows focused on conversational interfaces: Users asked the AI tools modeling questions and received generated code or guidance while remaining in control of each individual step. The Chatbot window in COMSOL Multiphysics^® follows this paradigm, enabling engineers to ask modeling questions, generate and debug COMSOL API code, and streamline repetitive tasks, all within the software.

The next step in an AI-assisted workflow is moving from conversation to action. An AI agent can plan a sequence of tasks, execute them through software tools, inspect the results, and decide what to do next. In simple terms, an agent can be thought of as a wrapper or “harness” around a large language model (LLM). The harness applies the actions derived by the LLM. The LLM writes a piece of code and wraps it in a specific format to perform a tool call, which triggers the harness to, for example, open a command line and a command to run the piece of code.

Understanding Interactions Between Agents and COMSOL^®

COMSOL Multiphysics^® is a great fit for exploring the shift toward agentic workflows thanks to the COMSOL API, the software’s rich programming interface. Here, we will look at how agents can leverage the COMSOL API to assist engineers in their modeling workflows by acting as collaborators.

There are different ways to set up an agentic workflow in COMSOL Multiphysics^®. For interactive workflows, it is recommended to use a client–server setup to inspect an agent’s progress and the state of the COMSOL model, which lives on a COMSOL Multiphysics^® server (not to be mistaken with COMSOL Server™, an add-on product for deploying simulation apps). Both the COMSOL Multiphysics^® user and the agent can connect to the COMSOL Multiphysics^® server, see the model state, and make changes.

In a typical interaction between an agent and a live COMSOL model, the agent would:

1. Generate the source code for a Java program using the COMSOL API for use with Java
2. Compile and run the code using the Java commands bundled with the COMSOL Multiphysics^® installation. Through the Java program, the agent would then:
   1. Connect to the COMSOL Multiphysics^® server
   2. Query model properties, run studies, and/or make controlled changes
   3. Disconnect from the server
3. Reason over the output, plan, and perform next steps

How to Connect COMSOL^® and Codex

Now that we have covered the basics of how an agent interacts with a COMSOL^® model, we’ll go over the steps for setting up an agentic workflow in the software.

Since we have an agent available, we won’t have to complete the setup manually. Instead, we can ask the agent for help based on the architecture description. To do this, we start with a COMSOL Multiphysics^® server instance, connect the COMSOL^® user interface (UI), and load a model. (Also see: Running COMSOL Multiphysics^® in Client–Server Mode.) Then, we install and set up an agent. For this example, we will use Codex from OpenAI. Codex can create a skill that enables us to manipulate the model object located on the server. A skill bundles instructions and scripts in a package that you can reuse and share.

Here is the prompt we used to create a simple version of the skill:

Create a lightweight, local Codex skill named comsol-api. Install it as a repo-scoped skill. The skill lets Codex inspect and modify a COMSOL Multiphysics model located on a COMSOL Multiphysics server through the COMSOL API *for* use with Java.
 
The skill should work in a client-server configuration. The model is located on an already running COMSOL Multiphysics Server. The main part of the skill is a small wrapper script that compiles Java code using the COMSOL’s bundled Java compiler and API plugin classpath, and runs this program. The Java code should connect to the server, manipulate the model object and disconnect again. Before performing any changes, connect to the server and read the model tag.
 
Save the .java file of each agent interaction with the server in a workspace folder alongside a log of the output of the Java programs and the COMSOL progress log (showing computation time, memory requirement, degrees of freedom) for later reference.
 
Research here to learn more about the COMSOL API.
 
https://www.comsol.com/support/learning-center/article/overview-of-the-comsol-api-107912
 
Based on that, create an implementation plan and implement it. The skill should be intentionally light and only focus on running Java code against the already running server. No need for tests. Don’t install additional packages. Keep it package free.
 
Test the skill by adding a parameter and reading back the same parameter.

Luna — the most affordable current-gen OpenAI model, with a reasoning setting of medium — was able to implement a skill in under ten minutes.

The Codex agent implementing and validating a skill.

To test the skill, the agent added a global parameter and then read the same parameter back from the live model on the COMSOL Multiphysics^® server. During the test, the parameter appeared in the COMSOL Multiphysics^® UI.

Possibilities and Challenges of Agents

Now that we have established a connection between COMSOL^® and Codex, what can we do with it? Every click in the UI has a corresponding sequence of COMSOL API commands. Due to the comprehensive scope of the COMSOL API (which makes it possible to perform any part of the modeling workflow for any physics), Codex can perform any action.

The API can expose a broad variety of information for an agent to inspect, such as:

• Loaded model tags and component structure
• Parameters, expressions, descriptions, and units
• Feature types, labels, and tags
• Geometry and mesh sequences
• Physics interface and boundary condition settings
• Study and solver configuration
• Numerical results and evaluated expressions
• Warnings, errors, and solver logs

This information can be extracted, reasoned over, and manipulated, enabling agents to plan, run, and adjust experiments as necessary, as well as summarize the results in a report. This way, the agent can, for example, define parameters and settings, add physics interfaces, and create plots or numerical evaluations. Additionally, this agentic workflow enables you to perform sweeps that are usually not possible with the Parametric Sweep study. With the right setup, you can keep such a system running for an extended period of time.

Demonstration: Heat Sink Model

To demonstrate this capability, we will use the agent to experiment with nonlinear solver parameters in order to reduce the number of nonlinear iterations and speed up model computation, using the heat sink tutorial model as an example. The task is to analyze the available settings, select the three most promising ones, and run them with three values each. While the agent can traverse the model object and identify the corresponding node or search the COMSOL^® documentation, we extracted the relevant code using the Copy as Code to Clipboard submenu to help the agent. This functionality is especially useful here, as the connection we implemented is quite bare bones. The prompt we used is as follows:

I am looking to speed up the computation, specifically reduce the number of nonlinear iterations necessary to achieve convergence. Analyze the parameters of the segregated solver, identify which might be good candidates to tune. Pick the most promising three and run three values each. Analyze the computation time, number of iterations in a report. Make only changes that are meaningful. I extracted the values that can be set in the segregated solver node.
 
model.sol("sol1").feature("s1").feature("se1").set("segterm", "tol");
model.sol("sol1").feature("s1").feature("se1").set("maxsegiter", 200);
model.sol("sol1").feature("s1").feature("se1").set("ntolfact", 1);
model.sol("sol1").feature("s1").feature("se1").set("segtermonres", "auto");
model.sol("sol1").feature("s1").feature("se1").set("segreserrfact", 1000);
model.sol("sol1").feature("s1").feature("se1").set("segstabacc", "segcflcmp");
model.sol("sol1").feature("s1").feature("se1").set("segcfltech", "interp");
model.sol("sol1").feature("s1").feature("se1").set("subinitcfl", 5);
model.sol("sol1").feature("s1").feature("se1").set("submincfl", 10000);
model.sol("sol1").feature("s1").feature("se1").set("subforcecfl", true);
model.sol("sol1").feature("s1").feature("se1").set("subkppid", 0.65);
model.sol("sol1").feature("s1").feature("se1").set("subkipid", 0.15);
model.sol("sol1").feature("s1").feature("se1").set("subkdpid", 0.15);
model.sol("sol1").feature("s1").feature("se1").set("subcfltol", 0.2);
model.sol("sol1").feature("s1").feature("se1").set("subadaptcfltol", true);
model.sol("sol1").feature("s1").feature("se1").set("segcflaa", "on");
model.sol("sol1").feature("s1").feature("se1").set("segcflaacfl", 9000);
model.sol("sol1").feature("s1").feature("se1").set("segcflaaset", "pid");
model.sol("sol1").feature("s1").feature("se1").set("segcflaadim", 10);
model.sol("sol1").feature("s1").feature("se1").set("segcflaamix", 1);
model.sol("sol1").feature("s1").feature("se1").set("segcflaadelay", 0);
model.sol("sol1").feature("s1").feature("se1").set("segcflaafact", 1);
model.sol("sol1").feature("s1").feature("se1").set("segcfljtech", "off");
model.sol("sol1").feature("s1").feature("se1").set("consistencycheck", "strictinitially");
model.sol("sol1").feature("s1").feature("se1").set("plot", "off");
model.sol("sol1").feature("s1").feature("se1").set("probesel", "none");

Your prompt can, of course, include additional resources, including Learning Center articles, blog posts, or Knowledge Base articles from the COMSOL^® website.

The results of the agent performing solver tuning studies.

Running nine experiments, the agent identified a relevant setting and reduced the number of nonlinear iterations by 25%, leading to a 30% faster computation time!

Another test we can run is to try out the difference between the three different comprehensibility options in the Laminar Flow interface. Although we cannot use the Parametric Sweep study in the UI to switch between the options in the dropdown menu directly, we can ask Codex to test each option, print the temperature in the chip, and list the number of nonlinear iterations. To instruct Codex to perform these tasks, here is the prompt we used:

There are three different compressibility options in the Laminar Flow interface. Run a test and compare all of them. I added an Evaluation Group that evaluated the temperature in the chip that you can run with the command:
 
model.result().evaluationGroup("eg2").run();
 
Create a report with the average temperature, computation time and number of iterations for the three different compressibility options.

There is no substantial change between the options, and the computation time is almost identical. So, we will stick with the default option. Please note that this was only an example on how to perform sweeps, and any settings in the interfaces should be taken based on physical considerations, not purely on computational speed.

Considerations When Using Agents

Although our experiments were successful, there are still challenges associated with agentic workflows. One of the challenges is spatial perception and spatial reasoning. For a human user, it is easy to select a boundary for a boundary condition by looking at the geometry. Under the hood, the number of the boundary is assigned to the selection of the respective boundary condition. Identifying this number purely by using the API can be difficult. Possible ways to circumvent this challenge is the rigorous use of named selections. Alternatively, you can activate the entity number in the view and instruct the agent to automatically export images from different view angles. This method is helpful as long as the geometry is not too complex.

It is also important to check the output to see if the changes are really meaningful. Considering the example of nonlinear solver tuning, the agent could simply change the termination technique to Iterations instead of Tolerance and set the maximum number of iterations to one. This method would reduce the number of iterations but would not give a numerically sound result. Monitoring key quantities, like the temperature of the chip, and ensuring that they do not change substantially is one possible way to filter out such bad candidates.

Since agents like Codex can do more than simply output text, there are risks associated with running them. They can make incorrect assumptions, delete files, expose sensitive data, or read malicious prompts on the websites they ingest. Because agents are inherently stochastic systems, they can make mistakes. To use these tools more safely, the agents should be operated in a sandbox, only have access to the files they really need, and require approval for high-risk actions.

Final Thoughts on Connecting Codex to COMSOL^®

What we showed here is a minimal working example of a connection of Codex to COMSOL Multiphysics^® for agentic workflows. This skill and wrapper script can be further extended to enable a more robust interface handling, which would be especially useful for identifying overly long simulation or mesh creation times for robust recovery.

These advancements can change how engineers interact with simulation software. The general scope of the COMSOL API makes COMSOL Multiphysics^® well suited for updating simulation workflows to meet the era of AI.

OpenAI is a trademark of OpenAI, Inc. Oracle and Java are registered trademarks of Oracle and/or its affiliates.

Read the previous blog post here.

Why Accessibility Matters

Color vision deficiency (CVD) can make it difficult to distinguish between certain colors. The most common form is red–green color blindness, which affects the ability to differentiate between shades of red and green. People with CVD may find it difficult to analyze or may misinterpret results presented in a standard rainbow color table or grayscale. Because simulation results are often shared with a wide audience, choosing an accessible color table can help make the results easier for more people to interpret as intended.

The electric potential of an atmospheric busbar shown using the Cividis color table option available in COMSOL Multiphysics^®.

Built-In Accessible Color Tables

COMSOL Multiphysics^® includes several built-in color tables that are both perceptually uniform and well suited for accessible scientific visualization including Cividis, Viridis, Inferno, Magma, and Plasma.

The tables being “perceptually uniform” means equal changes in your data appear as equal changes in the colors. Accessible color tables provide smooth, consistent changes in both color and brightness, making it easier to identify gradients. These color tables provide a range of options for presenting simulation results clearly while remaining accessible to a broad audience. Whether you’re visualizing temperature distributions, stress concentrations, fluid flow, or electric fields, your choice of color table plays an important role in communicating the model’s underlying physics.

Example mesh plots, propagating waves, wake regions, singularities, and smooth gradients for each color table.

While all five color tables can be used across a wide range of applications, some may be better suited for different types of simulation results. Viridis and Cividis are good choices for stress, strain, pressure, velocity, and concentration, where smooth gradients help reveal variations. Inferno and Magma can help emphasize temperature distributions and thermal gradients, providing strong contrast. Plasma can be useful for quantities such as electric and magnetic field magnitude and current density, where its distinctive color progress can help highlight variations in the field.

In addition to these perceptually uniform color tables, COMSOL Multiphysics^® includes the GrayScale linear color table. This option can be particularly useful when hue should not influence the interpretation of a result, or when you want to emphasize the intensity of a quantity without introducing additional color information. However, note that humans can only distinguish between about 30 different shades of gray, so it may be more challenging to notice small changes in a grayscale format.

A plot showing the interpolation function (left) of a model resembling the Alpine mountain the Matterhorn. The grayscale image (right) shows the height of the mountain. Note that the color bar is normalized to go from 0 to 1.

Transforming Color Tables

Color table transformations available in the settings of results plots, including Reverse, Nonlinear, and Nonlinear symmetric options, provide additional flexibility in how colors can be used to represent simulation results. They can be used to adjust the appearance of a color table and change how variations in the data are displayed. A transformation can be used to adjust the appearance of a color table and change how variations in the data are displayed. It can be used to emphasize or de-emphasize different parts of a result and provides another way to tailor a color table to a particular application or type of data.

The available color table transformations.

The von Mises stress in a bracket without a color table transformation applied (left). The von Mises stress in a bracket with the Nonlinear color table transformation applied. (right)

Custom Color Tables

In addition to the built-in color tables, version 6.1 of COMSOL Multiphysics^® introduced support for custom color tables. You can import color table files, edit the RGB components of a color table directly, and refer to color tables stored on disk or in a database.

This flexibility makes it possible to develop custom visualization styles for specific applications while maintaining control over how simulation results are represented.

To add a Color Tables node to the model tree, open the Show More Options dialog and select the Color Tables checkbox.

The Color Tables node used to define a custom table. When values are defined for a custom color table, a preview will appear in the Settings window.

Accessible Visualizations

Choosing an appropriate color table is an important part of communicating simulation results. By using perceptually uniform and accessible color tables, you can create visualizations that accurately represent your data while making your results easier for a wider audience to interpret. With the built-in Cividis, Viridis, Inferno, Magma, and Plasma color tables, as well as GrayScale and support for custom color tables, COMSOL Multiphysics^® provides a range of options for creating clear and accessible simulation visualizations.

Read Part 1 of this two-part blog series here.

The Importance of Acoustics in a Meeting Room

Have you ever been in a meeting where you can barely hear what the other person is saying? This is often due to excessive reverberation in the room. It is well known that a bad sound environment in the workplace can lead to poor communication, focus, and productivity. The best solution to this problem is to install acoustic absorbers, such as a suspended porous ceiling or an absorbing curtain, in a suitable place in the room. These absorbers have an air layer behind them to absorb low-frequency sound, making them highly extended reactive. It is interesting to see the difference that using different sound-absorbing boundary models makes when simulating a room with such extendedly reacting sound-absorbing functions.

Room Model and Absorption Setting

The simple room (3.5 m 6.8 2.5 m) below was analyzed with octave bands ranging from 63 to 500 Hz (approximately 40 to 710 Hz) using the Pressure Acoustics, Time Explicit interface.

A diagram showing the geometry of an analyzed meeting room.

The Gaussian pulse was applied to the source point (0.7 m, 4.15 m, 1.2 m), and at the receiver point (2.8 m, 2.35 m, 1.2 m), the pressure response was evaluated up to 0.8 s with a sampling frequency of 10000 Hz.

To solve the acoustic wave equation in the time domain, the Pressure Acoustics, Time Explicit interface uses the discontinuous Galerkin finite element method (dG-FEM), which is a memory-efficient and scalable method. The room model has typical sound absorption functions, i.e., the suspended porous ceiling and the curtain. In this blog post, these absorbers are modeled using both locally and extendedly reacting models, and the resulting differences are examined. The image above is for the extended reaction model, which includes the thickness of the porous material and the air space behind the absorbing materials. In the case of the local reaction model, the thickness and air space are not modeled explicitly, while the surface impedance at normal incident condition is assigned to the surface boundaries of the materials. The surface impedances of those sound absorbers were evaluated with submodels in the frequency domain, assuming a two-dimensional periodic structure.

The suspended ceiling is made of a 50 mm porous material. The air space behind the porous material is 180 mm and was made to absorb low-frequency sound. Since the thickness of the porous layer is large, an equivalent fluid model is used to model the ceiling absorber, including the extended reaction. The Johnson–Champoux–Allard (JCA) model is used to model the fluid property in the porous material. With this model, the complex effective density and bulk modulus of the porous material are expressed as follows:

Therein, , , and are respectively the absolute pressure, the density, the ratio of specific heats, and the dynamic viscosity of the fluid in the porous material. , , , and represent the porosity, the flow resistivity, the viscous characteristic length, the thermal characteristic length, and the tortuosity factor of the porous material. is the angular frequency. The table below summarizes the parameters of the porous material used in the example model.

Parameter (Unit)	(1)	(Pa·s/m2)	(mm)	(mm)	(1)
Value	0.99	17000	0.14	0.15	1.01

Porous matrix properties used in the example model. These values are from Ref. 1.

The Poroacoustics feature, available in the Pressure Acoustics, Time Explicit interface, can be used to model the porous layer based on the equivalent fluid model. The Porous Absorber with Local and Extended Reacting Approximations for Time-Domain Modeling document is a good reference for our implementation of the time-domain equivalent fluid model. To use this feature, we should prepare the frequency-dependent data of the complex effective density and compressibility (a reciprocal of the bulk modulus) of the porous material. In the example model, these properties were calculated using the same submodel used to evaluate the surface impedances for local reaction modeling. The following screenshot shows the Poroacoustics settings for the submodel.

Poroacoustics feature settings for submodeling of the porous material.

The following screenshots show more calculations of the material properties from the result of the submodeling.

Calculation of the complex effective density (left) and compressibility (right) of the porous material from the result of the 2D frequency-domain analysis.

You could also use the measured data of these material properties. Then, the frequency-dependent properties of the porous materials are imported and approximated to the rational function forms with the Partial Fraction Fit function for the analytic inverse Fourier transformation.

Use of the Partial Fraction Fit function to approximate the complex effective density (left) and compressibility (right) to the rational function forms.

The fitted results are further imported in the Poroacoustics feature in the Pressure Acoustics, Time Explicit interface.

Poroacoustics feature in the Pressure Acoustics, Time Explicit interface.

On the other hand, the extended reacting absorbing curtain was modeled using a transfer impedance, on the assumption that it was very thin. We assumed that the flow resistance (transfer impedance) of the curtain was 416 Pa·s/m. Note that the mass effect was neglected here for simplicity. The curtain was placed on a window frame with an air space of 200 mm. The Interior Impedance feature was used for modeling.

Note that to compare the different boundary models, the boundary surfaces of the spaces behind the extended reactive materials were assumed to be rigid. Other boundaries were assumed to be plaster board and flooring. They were modeled using the frequency-dependent impedance boundary. The random incidence absorption coefficients for all materials are shown below:

Random incidence absorption coefficients for all materials used in the room model.

The ceiling and curtain show the different absorption coefficients for the different boundary models due to the high extended reactivity of these absorbers. The local reaction models show larger absorption coefficient values than the extended models over a wide frequency range, except above 630 Hz. You can see the angle-dependent absorption coefficients for the absorbing materials from the model file.

Normalizing and Filtering of the Impulse Response

The Gaussian pulse is a typical source signal in wave-based, time-domain modeling, but its frequency spectrum is not flat, as will be shown later. Thus, to use dG-FEM results for room acoustics evaluation, the source spectrum needs to be normalized. Here, we introduce how to perform the normalization in the frequency domain.

First, we introduce the frequency characteristic of the Gaussian pulse. The Gaussian pulse was excited to sound fields with the following initial sound pressure distribution:

Here, represents the sound pressure. is the distance from the source, and is the parameter characterizing the source spectrum. Substituting this initial sound distribution function to the wave equation for a spherical wave yields the following relation in a free field (Ref. 2).

Therein, is the time variable, is the coordinate vector, and is the sound speed. With the Green’s function for the three-dimensional free field, the volume acceleration of the point source is expressed as follows (Ref. 3):

Here, is the density of the medium. By Fourier transformation, the spectrum of the source signal using the Gaussian pulse is expressed as follows:

Here, is the frequency variable.

The spectrum of the source signal used in the example model.

The source was designed so that the gain of -3 dB from the peak was obtained at the upper-limit frequency in the 500 Hz octave band.

Thus, we can obtain the normalized impulse response with a source strength of 1 m³/s² volume acceleration (around 0.14 mW source power in the air) by dividing the results by the analytic spectrum above. However, this normalization also amplifies the unwanted frequency components, such as a higher frequency component that is not resolved in the dG calculation. To remove such components, we also perform filtering. The following equation expresses the normalization procedure:

Here, and represent the normalized and dG-FEM-calculated sound pressure. is the Fourier transformation operator. and are the high-pass and the low-pass filters in the frequency domain, respectively.

Since the filtering process may result in a noncausal signal, zero padding is recommended before converting the dG results to the frequency domain via discrete Fourier transformation. The example model exported the pressure waveform calculated by the dG-FEM to a WAV file and imported it as an interpolation function. Then, zero padding was performed over the Grid 1D dataset as follows:

Definition of an interpolation function using a WAV file exported from the calculation result. To perform the zero padding, the extrapolated value is set to 0.

Zero-padded Grid 1D for discrete Fourier transformation. The negative part of the interval is the zero-padded region. The resolution is set so that the sampling frequency corresponds to 10,000 Hz.

In the example model, the normalization and filtering were done with the following Global ODEs and DAEs interface.

Here, pE_normalized represents the normalized sound pressure, and rpE(freq) and ipE(freq) express, respectively, the real and imaginary part of the Fourier coefficient of the dG-FEM-calculated sound pressure for the extended reaction model. pL_normalized, rpL(freq), and ipL(freq) represent those same values, but for the local reaction model. source(freq) denotes the spectrum of the source signal defined with an analytic function. HP(freq) and LP(freq) are the high-pass and low-pass filters defined using step functions. You can see more detail of the setting from the example model. The frequency domain data of pE_normalized and pL_normalized are calculated using the Frequency Domain study step. Then, the normalized impulse responses are reconstructed via the Frequency to Time FFT study step.

Another benefit of the normalization is that the results can be combined with the ray acoustics model results to obtain the broadband impulse response using a hybrid approach. This is done by setting the ray release condition to be omnidirectional with the following total source power :

Here, is the unit volume acceleration of 1 m³/s².

Results

The normalized band-limited room impulse responses and their spectra (room transfer function) for two sound-absorbing boundary models are shown below.

Normalized room impulse responses for the extended and the local reaction models.

Normalized room transfer function for the extended and the local reaction models.

The differences between the two models using different sound absorbing models are clear. The local reaction model overestimates the absorption by the sound-absorbing materials over a wide frequency range, which results in a faster decay of the impulse response. According to the comparison of the spectra, the local reaction model has more absorption below 160 Hz and around 500 Hz, while it underestimates the absorption above 600 Hz. These tendencies correspond to the differences in the random incidence absorption coefficients between two sound-absorbing boundary models.

The following comparisons of the room acoustics metrics also indicate the differences between the two models.

A comparison of the reverberation parameters (EDT and T20) between the extended and local reaction models.

A comparison of the clarity (C50) between the extended and local reaction models.

As for the reverberation parameters (EDT and T20), the differences between the two models are larger than the two times of just noticeable difference (JND) of 5% in a broadband. The clarity parameter (C50) also showed a larger difference than the JND (1.1 dB) at most frequencies, with the maximum difference reaching 11 dB. These results indicate that the difference between the local and the extended models is audible. You can hear the normalized and unnormalized room impulse responses for the different absorbing boundary models in the following:

Playback audio waveforms of the normalized impulse responses for the extended (left) and the local (right) reaction models.

Playback audio waveforms of the unnormalized impulse responses for the extended (left) and the local (right) reaction models.

You can feel the longer reverberation for the extended model. Also, the unnormalized results will be heard as emphasizing high-frequency sounds due to the frequency characteristic of the Gaussian pulse.

These results are at one receiver location, and it is important to set up more receiver locations for the practical evaluation of room acoustics. However, the clear differences at the possible receiver location are sufficient to demonstrate the effect of the type of the sound-absorbing boundaries on the meeting room.

The wave-based methods are considered to be inherently very accurate because they include all wave phenomena. However, the results of the example model showed that depending on the characteristics of the sound absorber, a simple use of the local reaction model may lead to erroneous results. In order to perform a more reliable simulation, we should carefully confirm the configuration and the absorption characteristics of the absorbers.

Conclusion

As a continuation of the previous blog post, this blog post examines the effects using different sound-absorbing boundary types on a small meeting room. Typical sound absorbing elements, like the suspended ceiling and the curtain, are highly extended reactive. As a result, applying the conventional local reaction model (the impedance boundary) to these absorbers produced a less accurate result. The demonstration revealed that the capability of the extended sound-absorbing boundary model is very crucial to perform a proper simulation. Of course, the impedance boundary is still convenient because of its lower computational cost than the extended model. For efficient modeling, the use of different absorbing models, depending on the design stage or the characteristics of absorbers, is very important. This blog post also introduced the procedure for obtaining a band-limited impulse response from the dG-FEM analysis with the Gaussian pulse excitation. Note that when exciting a sound field with the initial pressure distribution condition, the source location must be apart from any boundary surfaces reflecting sounds.

The Pressure Acoustics, Time Explicit interface in the Acoustics Module, an add-on to the COMSOL Multiphysics^® software, is well suited for wave-based room acoustics modeling because of its high-memory efficiency and accuracy, thanks to the ability to model both locally and extendedly reacting boundary models, including the frequency dependency. In addition, you can significantly reduce the computation time by using a GPU-accelerated formulation (currently available for only local reaction model). Even with an entry-level GPU (NVIDIA^® T400), the local reaction model (1,307,650 DOFs and 40,000 timesteps) can be solved in about an hour. If you have a higher-grade graphics card, the reduction of computation time is expected to be much greater. You can find more details about the performance gain by GPU in the following document: Acoustics of an Open-Plan Office Space.

Let’s model room acoustics with the exceptionally accurate and efficient simulation capabilities available in COMSOL Multiphysics^®!

Next Steps

Interested in trying out the model for yourself? Download the related MPH file below:

Reference

1. H. Wang and M. Hornikx, “Extended reacting boundary modeling of porous materials with thin coverings for time-domain room acoustic simulations,” J. Sound Vib., vol. 548, 117550, 2023; https://doi.org/10.1016/j.jsv.2022.117550.
2. S. Sakamoto, “Phase-error analysis of high-order finite difference time domain scheme and its influence on calculation results of impulse response in closed sound field,” Acoust. Sci. Technol., vol. 28, 295-309, 2007; https://doi.org/10.1250/ast.28.295.
3. T. Okuzono, T. Otsuru, R. Tomiku, and N. Okamoto, “Application of modified integration rule to time-domain finite-element acoustic simulation of rooms,” J. Acoust. Soc. Am., 1vol. 32, 804–813, 2012; https://doi.org/10.1121/1.4730920.

What Is Agentic AI?

Before diving into the discussion, let’s clarify the distinction between a couple of commonly used AI technologies:

• Large language model (LLM) copilots, or chatbots, are conversational AI tools that can answer questions, summarize information, generate code or text, and help users interpret or apply information based on prompts. As they require human input, they are not agentic.
• AI agents, on the other hand, are goal-oriented systems that can use tools and carry out multistep workflows, evaluating intermediate results and adjusting their approach as needed, with or without human input.

How Can Users Integrate AI into Modeling Workflows in COMSOL Multiphysics^®?

Sjodin began the webinar by discussing how users of COMSOL Multiphysics^® can utilize AI in their workflows thanks to the software’s powerful Java-based API, which enables communication with third-party LLMs.

The Chatbot Window

To provide LLM copilot functionality, the COMSOL Desktop^® UI includes the Chatbot window. Through the Chatbot window, users can connect directly to an LLM — such as GPT-5™, DeepSeek™, Google Gemini™, or an on-premise LLM — to ask modeling questions and/or enter prompts to have the LLM generate and debug code. Code generated in the Chatbot window can then be run in the Java Shell window to apply changes to a model.

To determine which type of response you would like the chatbot to return, the Chatbot window provides three distinct subject options: General, Programming, and Modeling, each of which allows for more customized answers. For example, if you select the Modeling subject, the LLM will answer with guidance in text form, accessed from model information, guides, and relevant provided documentation, whereas if you select the Programming subject, the LLM will output Java API code.

In the webinar, Sjodin showed an example of how to use the Chatbot window to ask a specific question about modeling a pretensioned bolt with a gasket, using a deliberately broad prompt: “How can I reduce the risk for leakage in this design?”

The Chatbot window providing guidance on gasket leakage after an entire model was attached as context.

Sjodin attached a model by selecting the Send to Chatbot option in the Model Builder and added some supporting screenshots, and the LLM was able to provide a detailed response that referenced specific geometric entities and variables in the model. Its response suggested ways to evaluate and improve the design, including checking gasket compression, contact pressure, bolt preload, material stiffness, and pressure boundary conditions. Sjodin noted that the response was reviewed by COMSOL’s structural mechanics engineers, who found it to be a very strong answer.

For more information on the Chatbot window and what it can do, check out the following Learning Center article and blog post.

AI Agents

Sjodin also explained that you can download an AI agent (typically to the same computer where you are running COMSOL Multiphysics^®), install it, and then connect it to COMSOL Multiphysics^® via the COMSOL API. He demonstrated how a standard OpenAI™ Codex agent could be used with COMSOL Multiphysics^® to set up and solve an equation-based model from a single prompt. In his example, the agent was given the task of computing the surface-distance field on a shell geometry in 3D with the following prompt:

I would like a COMSOL example applied to the attached shell diffusion geometry for the wall distance equation, but implemented so that we can obtain the distance along the surface. I have also attached a model example as a Java file, along with reports on computing the distance in COMSOL throughout a volume.

In response, the agent created the model, defined the equations, ran the study, and generated the resulting visualizations, showcasing how agentic AI can automate workflows — not just by answering questions but by planning and carrying out multistep simulation tasks.

Setting up and solving an equation-based model from a single prompt.

How Can Cosmon’s AI Agent Help Simulation Engineers?

Rui Aguiar introduced Cosmon’s AI agent, “Nexus,” which engineers can use in tandem with the COMSOL Multiphysics^® software to automate repetitive simulation tasks, including CAD preparation and geometry cleanup, simulation setup, solver troubleshooting, parametric sweeps, evaluation and visualization of results, and reporting directly in COMSOL Multiphysics^®.

Although agents can operate with or without human input, Aguiar explained that Nexus is not intended to replace engineers’ expertise; instead, it uses a human-in-the-loop design and knows when to ask for the engineer’s input or approval.

Nexus can support engineers with capabilities that include interpreting natural language instructions, interacting with COMSOL Multiphysics^® via the COMSOL API, intent-driven setup of materials and physics, running automated parametric studies, resolving common solver issues, and organizing results for review. Another common use case is recreating simulations from academic papers in COMSOL Multiphysics^®; to see a detailed example, please refer to this Cosmon case study here.

The Nexus window (right) assisting the engineer with setting up a thermal simulation according to attached specifications.

Aguiar emphasized that this new way of conducting simulation with agentic AI helps engineers spend less time on manual setup, debugging, and documentation. He also highlighted Nexus’s support for processes upstream and downstream of simulation, such as its capabilities for automated CAD translation and defeaturing and generation of charts and reports. With agentic support available at every point in the simulation workflow and beyond, engineers have more time to focus on higher-level decision-making, including physics and design decisions that require domain knowledge.

Getting into the Details

During a Q&A at the end of the webinar, Sjodin and Aguiar gave the following responses to attendee questions.

Does the Chatbot window in COMSOL Multiphysics^® require a separate module or license?

“No,” Sjodin said. “The chatbot functionality is provided as part of the COMSOL Multiphysics^® core package.” However, he went on to state that one requirement for enabling the Chatbot window is having your own API subscription with an AI provider or using an on-premise model. This can be established in the Preferences settings, where you input an API key.

Is the Cosmon AI agent a separate program?

The Cosmon AI agent is a separate software program that lives on your desktop. It can connect to an existing COMSOL license when both the Cosmon AI agent and Chatbot windows are open.

Why should I use an AI agent and not just use a chatbot?

According to Sjodin and Aguiar, there are a lot of reasons why one might opt to use a particular type of AI tool over another, and part of learning to utilize AI is knowing when certain AI technologies are better than others. They agreed that a chatbot is definitely useful, but an agent is often capable of more. A chatbot, they explained, is powerful enough and easy enough to use for a quick answer to an engineering or physics question; similarly, someone debugging code to input into the Java Shell window may also find a chatbot useful. However, if an engineer wants an AI tool to take on actions such as troubleshooting, fixing geometry, meshing, or plotting results, Sjodin and Aguiar agreed that using an AI agent would be a better choice.

They said if you’re looking to use an AI tool to perform a task such as creating an MPH file, you’ll need to use an AI agent because a chatbot typically doesn’t know the entire COMSOL API perfectly. “If there’s some iteration needed, you need to go the agent way,” Sjodin explained. Aguiar went on to mention the benefits of Nexus for iterative tasks. “A great thing about agents is that you can [train them to develop customized] skills that perform a workflow over and over,” said Aguiar. With the Cosmon agent, best practices and guidelines can be baked into a skill.

Can the Cosmon agent analyze assemblies?

According to Aguiar, yes. The agent can import assemblies from other software programs and perform actions on them. “Luckily, the COMSOL API is very comprehensive,” Aguiar stated. “Our agent has the ability to execute operations [such as geometry healing] over assemblies. We can do geometry healing or cleanup in COMSOL^® natively or within a CAD model, and we can do it on a part or on the assembly level as well.”

How does an AI agent learn?

“Like many things in life, it depends,” said Aguiar. He talked about how you can set the agent to learn alongside the user based on what tasks it typically performs and gets human feedback on. However, most of this happens within a single user interface. When it comes to cross-team memory, the agent predominantly operates on a skill-file level. “For example, if you have a team of engineers and one engineer who knows how to run a specific type of simulation, and you want to share that knowledge with another team member, what you can do is create a skill file, say ‘This is how it’s run, and this is how it’s set up,’ and export that as a skill file for another engineer to load into their agent.” This functionality within Nexus is especially valuable to organizations and teams who wish to democratize simulation so that more engineering staff can run more simulations with the COMSOL Multiphysics^® software.

The Future of Agentic AI

According to Aguiar, agentic AI is a rapidly advancing field. His studies have shown that Cosmon’s AI copilot for engineering, Nexus, has helped make simulation workflows — from setup and troubleshooting to evaluation and visualization of results — between two to three times faster, enabling users to spend less time on manual tasks such as meshing, adjusting parameters, and repairing or healing geometry. This extra time allows engineers to create more impactful simulations faster.

Aguiar explained that, as it works today, copilot tools require engineers to act as operators who configure most steps of the engineering and simulation process. In the future, Aguiar expects AI agents to design, simulate, analyze, and iterate autonomously, with engineers using human intuition and expertise to interpret and interrogate AI-generated conclusions.

Next Step

If you’re interested in watching the complete webinar, you can access it via the button below!

Further Resources

Interested in learning more about how to use AI tools in COMSOL Multiphysics^® or the advancements of AI in the engineering space? Check out these additional resources on the COMSOL website:

• Using the Chatbot Window in COMSOL Multiphysics^®
• Simulation Summit Panel: The Multiphysics Roadmap

 
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The Exhibition Floor

At an interactive demo station at the COMSOL booth, engineers spoke with visitors about how they could use the latest features in the COMSOL Multiphysics^® software for their RF modeling scenarios. New functionality with practical applications for RF and microwave engineers includes simplified transmission line modeling, a dedicated feature for streamlined metamaterial design, and enhanced far-field functionality for optimization and polarization analysis.

Zhangxing Li was one of the applications engineers who worked with attendees at the demo station. “When I chatted with people, they were happily surprised by how streamlined the process is to set up multiphysics simulations in the software. Everything is done in one user interface, and if the user knows the workflow for one physics, they practically know the general workflow for the others,” Li said. “This streamlined workflow is particularly relevant for anyone in the RF and microwave community who needs to consider multiphysics interactions between electromagnetics, heat transfer, and solid mechanics for a robust representation of their device.”

The COMSOL booth at IMS2026 featured a demo station (left), a video featuring multiphysics models (center), and informational materials focused on how the COMSOL Multiphysics^® software can be used to develop RF and microwave applications (right).

The theme of IMS2026 was “Revolutionizing RF”, and this year’s conference took a similarly unprecedented approach, splitting itself into three symposia focused on RF integrated circuits, systems and applications, and technology and techniques, respectively.

Papers, presentations, and workshops were categorized into one or more of these symposia, helping attendees quickly assess the application areas. After meeting with visitors to our booth and attending a wide variety of events at IMS2026, we picked up on the following RF and microwave core industry trends.

This Year’s RF and Microwave Trends

Fellow attendee Jiyoun Munn, senior technical product manager at COMSOL, noted that this year’s topics marked a move toward system-level thinking, with more emphasis on real-world applications, the integration of AI and machine learning (ML), and tighter links between academia and industry. A main point of discussion at IMS2026 was the continued development of 6G wireless networks and the technical requirements needed to make the transition from 5G to 6G possible.

“The exhibition area reflected where the market is headed: integration, higher frequencies, better packaging and thermal design, and stronger links between research and commercialization,” Munn said.

AI Integration

Discussions about AI and ML centered on taking these fields of study from hypothetical to practical applications, for example, by integrating them into established modeling workflows, thereby cutting down on the time required to develop RF designs using simulation. In the dedicated AI/ML Bootcamp session, attendees were presented with the basics of AI and ML and how they could be used in microwave engineering. Meanwhile, hardware and connectivity solutions for deploying and integrating AI infrastructure were discussed and displayed across the different symposia and the exhibition hall.

Next-Generation Wireless

The transition from 5G to 6G wireless networks is far from imminent. Implementing 6G will require ultrahigh frequencies, new hardware, and significant infrastructure investment from private and public funds alike. Nonetheless, 6G was on many people’s minds at IMS2026. Andrew Strikwerda, senior applications manager at COMSOL, heard a shift in the conversations about 6G this year.

“Discussions about 6G networks were much more practical than in previous years; participants’ interests in the specific technologies that would be required were more concrete,” Strikwerda said. “Inherently, these conversations touched on simulation of such devices to ensure high-fidelity representation of performance at higher frequency bands.”

The next generation of networks will require the use of the subterahertz spectrum. To make such technology feasible, numerous physics challenges such as attenuation and molecular loss need to be addressed. At IMS2026, there were sessions and papers that discussed finding a balance between bandwidth and system capacity. Notable examples included the workshop “Broadband and Spectrally Agile RF Front-Ends for Advanced Software-Defined Radios” and the article E-Band Wireless Radio Links Deliver High Capacity Backhaul Solutions for 5G Networks from Analog Devices.

An example model in the COMSOL Multiphysics^® software showing a millimeter-wave analysis of a 5G cellphone.

As it stands today, organizations and companies are working hard to overcome the hurdles involved in the transition to 6G, and the developments shown at IMS2026 were promising.

Integrated Sensing and Communication

Integrated sensing and communication (ISAC) combines radar and wireless communications into a cohesive platform and unlocks new capabilities for wireless networks, giving those using it environmental sensing data that wasn’t available before. While deployment of ISAC remains years away, its potential was still a topic of conversation at IMS2026.

“At IMS2026, ISAC was discussed in a speculative way, similarly to how 6G was talked about at previous conferences,” Strikwerda said. “There was a general agreement that ISAC would be coming at some point and in some capacity, but it was still unclear precisely what that would entail.”

The panel discussion “Integrated Sensing and Communication (ISAC)—Enabling the Future of Radar and Wireless Systems” gave an end-to-end view of the topic, featuring invited speakers from organizations such as the Mitre Corporation, Samsung Electronics, and Nokia. Also featured at this year’s conference was a panel session titled “The Rise of ISAC — Opportunity or Hype?”, with industry experts from academia and the commercial sector. Both these sessions and several papers on the topic presented at IMS2026 gave attendees a comprehensive overview of ISAC and its promise.

IMS Heads to San Antonio, Texas

IMS2026 proved once again to be at the forefront of the rapidly evolving RF and microwave sector, with thousands of industry experts sharing their work and discussing the latest developments.

COMSOL was represented at IMS2026 by (from left to right) Keon Campbell, Zhangxing Li, Andrew Strikwerda, Christine Le, Walter Frei, Madison Grover, Ian Woods, and Jiyoun Munn.

COMSOL was privileged to take part in this year’s conference, and we look forward to IMS bringing the RF revolution to San Antonio in 2027.

Matthew Hancock, PhD, principal at Veryst Engineering, said, “As AI and machine learning become part of the regulatory submission process, the real test isn’t just how accurate a surrogate model is compared to the reference simulation; it’s whether we can show, not just assume, that any AI-introduced error is small enough to preserve the integrity of the decision it’s informing. We call this decision reliability.”

Using Simulation to Mitigate Risk

When medical devices interact with blood, there is a risk of them causing unintentional blood damage. This risk poses a design challenge for devices such as instrumentation loops or device–blood vessel connections, including pacemakers, heart valves, and catheters.

Hemolysis occurs when red blood cells rupture and release hemoglobin into the surrounding blood plasma. When modeling medical devices, this type of blood damage can be measured by the increase in unbound hemoglobin in the blood circulating through the device. Quantifying the hemoglobin, while considering sublethal damage and thrombosis, can lead to more informed design decisions.

Veryst’s goal was to create a simulation that can aid medical device designers in understanding the relationship between device geometry and blood damage. To meet this goal, the team turned to multiphysics simulation and surrogate models, which also offer the benefit of significantly reducing the resources needed to conduct real-world experiments.

In a poster presentation from a COMSOL Conference, engineers at Veryst shared how they trained and deployed surrogate models to provide visualizations of blood damage in real time. They were able to change aspects of the design and different operating parameters and observe the effects that occurred. These analyses helped them determine the most influential parameters and their tolerance levels.

Building and Training Surrogate Models

For their simulation work, the Veryst engineers used deep neural network (DNN) surrogate models to predict blood velocity and hemolysis. This work references and builds upon Veryst’s previous work modeling blood damage in the FDA benchmark nozzle (Ref. 2), which was presented in an invited talk at the COMSOL Conference 2020. The team was able to use that data from the previous full-model simulations to help train the DNN models.

Additionally, the FDA has benchmark datasets for validating numerical simulations of blood flow through medical devices (Ref. 3), and requirements must be met for a medical device to receive FDA approval. One benchmark dataset is for a nozzle that has a tube with a contraction, neck, and expansion, or a conical change in diameter at one end of the throat and sudden change at the other end (Figure 1).

In order to prepare their nozzle model for submission to the FDA, the Veryst engineers chose fluid properties that match the FDA benchmark protocol. They set up four adjustable input parameters:

1. Flow rate
2. Converging length
3. Neck diameter
4. Diverging length

The surrogate model predicts the blood velocity and estimates hemolysis using a power-law relationship as a function of stress (Ref. 2). The hemolysis model output represents the prediction of how much hemoglobin is released to the blood stream due to cell rupture or leakage.

Figure 1. A diagram showing nozzle setup and inputs for the DNN model.

To build the hemolysis model, the team computed mean hemolysis across the flow’s exit with a design of experiments sample of 9000 simulations. A surrogate model was built to train the velocity components u and w on raw data. The DNN surrogate model included 3 hidden layers, each with 20 nodes using tanh activation functions. To train the model, 5000 epochs were used. The velocity network included a DNN trained on 500 simulations at 3500 epochs, with 32 nodes in each of the 3 hidden layers. The velocity training data utilized all mesh points after expansion, with the diverging length set to 0 in order to focus on recirculation zones.

Predictions Align with Previous Data

After creating a surrogate model to train u and w, the team developed a modified DNN with a physically informed no-slip hard constraint. This network was trained on the deviation of axial velocity, w’, from the analytic laminar flow solution w’ = w – (2*Q/A)(1–(r/R)²). In this DNN, Q represents the flow rate, A represents the cross-sectional area, r represents the radial coordinate, and R represents the end radius (Figure 2).

The raw data DNN and the hard-constrained DNN both accurately replicate flow profiles in turbulent circumstances, which represent most of the underlying training data. These DNNs are less quantitative when it comes to matching the laminar simulation or the transition to turbulence regimes.

Figure 2. Velocity at Reynolds number Re = 6500, which replicates the FDA geometry, showing recirculation after expansion.

When the hemolysis predictions are trained to match the log of hemolysis, the results are within 10% accuracy across a wide range of inputs. Veryst found that changing the converging and diverging lengths of the geometry had minimal effect on hemolysis in both the finite element analysis (FEA) and DNN predictions. To validate the surrogate model, the team compared the hemoglobin concentrations predicted by FEA and the DNN for parameters based on the experiments of Herbertson (Ref. 4)(Figure 3).

Figure 3. The fraction of released hemoglobin at the nozzle endpoint for the FDA geometry with variable neck radius and flow rate.

“Getting hemolysis predictions within 10% of the full CFD model, while cutting runtime from hours to milliseconds, is what makes real-time design exploration possible,” said Joseph Barakat, PhD, lead engineer at Veryst Engineering.

Veryst’s work is based around the publicly available FDA benchmark geometry, but it could be expanded for a wide variety of future uses. The methodology could potentially be used to estimate blood damage in a range of medical devices, such as instrumentation loops and device–blood vessel connections.

Hancock added, “Formally combining surrogate approximation error, numerical error, and validation discrepancy into a single total-error estimate to assess this decision reliability — grounded in ASME’s Verification and Validation 20 and 40 standards and the broader verification, validation, and uncertainty quantification literature — is a natural next step, particularly as the FDA’s Center for Devices and Radiological Health continues building out regulatory science tools for AI-accelerated device simulation.”

Next Step

Want to learn more about Veryst’s work on predicting blood damage with surrogate models? Click the button below to see the team’s poster from the COMSOL Conference:

References

1. A. Spann, J. Barakat, and M. Hancock, “Deep Neural Network Surrogate Model for Blood Damage Modeling in FDA Hemolysis Benchmark,” COMSOL Conference 2024 Boston, 2024; https://www.comsol.com/paper/deep-neural-network-surrogate-model-for-blood-damage-modeling-in-fda-hemolysis-benchmark-136422
2. A. Kermani, A. Vanegas, and A. Spann, “Blood Damage Modeling of FDA Benchmark Nozzle,” COMSOL Conference 2020 Boston, 2020; https://www.comsol.com/paper/blood-damage-modeling-of-fda-benchmark-nozzle-93171
3. U.S. Food and Drug Administration, Benchmark dataset for validating computational fluid dynamic (CFD) simulation of blood flow through generalized medical device geometries (RST24CV11.01), 2024; https://cdrh-rst.fda.gov/benchmark-dataset-validating-computational-fluid-dynamic-cfd-simulation-blood-flow-through
4. L.H. Herbertson et al., “Multilaboratory study of flow-induced hemolysis using the FDA benchmark nozzle model,” Artificial Organs, vol. 39, no. 3, pp. 237–248, 2015; https://pubmed.ncbi.nlm.nih.gov/25180887/

What is a Superellipse?

An ordinary ellipse is commonly described by the equation

where a and b are called the semiaxes of the ellipse. When a = b, the equation of a circle with radius a is recovered.

The superellipse is a generalization, where an exponent other than 2 is used, so that

The fountain in the center of Stockholm uses n = 2.5 and the shape factor a/b = 6/5.

The fountain at Sergels Torg (left) and an outline of its shape (right).

Examples of other superellipses are shown below.

You can find more details about the mathematics of superellipses here.

Stress Concentration Factor

The stress concentration factor, K_t, is an important concept in solid mechanics. It is used to describe the increase in stress around a geometric discontinuity in a structure. K_t relates the maximum stress to a suitably defined nominal stress (stress without the geometrical discontinuity) through

Traditionally, stress concentration factors for common cases have been tabulated in handbooks. With today’s easy-to-use finite element (FE) programs, computing a stress concentration factor is often faster and more accurate than looking it up in a graph or table.

The most well-known result is likely that K_t = 3 for a circular hole in a uniaxially loaded large plate.

A model in COMSOL Multiphysics^® of von Mises equivalent stress around a circular hole in a plate subjected to a unit axial stress in the horizontal direction.

Another analytical result is the stress concentration factor for an elliptic hole in a large plate:

Here, the semiaxis with length b is the one perpendicular to the stress field so that the stress concentration increases with the ratio b/a.

A model of von Mises equivalent stress around an elliptical hole with b/a = 3 in a plate subjected to a unit axial stress in the horizontal direction.

Stress Analysis of the Superelliptic Hole

A superelliptic curve can easily be constructed in the COMSOL^® software using the Parametric Curve feature.

A superelliptic curve in the Model Builder made using the Parametric Curve feature settings.

Here, a representation of the superellipse in polar coordinates is used for the parameterization. It would also be possible to use a simpler parameterization based on the original equation expressed in x and y, but it becomes less accurate, since the relation between yand x is highly nonlinear.

The parameterization used (left), and an alternative, simpler but less efficient, parameterization (right).

First, let’s take a look at the results for a hole with a geometry similar to the aforementioned fountain. The peak stress depends on the orientation of the larger semiaxis, but in both cases the stress concentration factor is lower than the value of 3 for a circular hole.

Models of von Mises equivalent stress in a plate with a superelliptic hole having n = 2.5 and a/b = 6/5.

This looks promising. From here on, it is easy to set up a parametric sweep and study the stress pattern for many different shapes.

For any value of the superellipse exponent n less than 2, there will be sharp corners on the hole edge. That would cause stress singularities. Such cases are not of interest in this context. Thus, n is kept in the range 2–8. At the highest values, the hole is almost rectangular with corner fillets. For the axis ratio b/a, values ranging from 0.2 to 5 are tested. The computed stress concentration factors are shown in the diagram below.

Stress concentration factors for superelliptical holes for a range of exponents n and axis ratios q = b/a. The markers on the curves show the minimum values.

For n = 2, it can be seen that the values are as expected for an ellipse:

If the curves are normalized using this factor, we can see how a superellipse differs from an ordinary ellipse having the same axis ratio:

Normalized stress concentration factors for superelliptical holes.

As can be seen, there is always a superellipse with the same axis ratio as a certain ellipse that will give a smaller stress concentration factor. In particular, if we replace a circle with the best possible symmetric superellipse (q = 1), then it is possible to reduce the stress concentration factor by 14% using n = 3. This is quite a significant improvement. Such a decrease in stress could improve the fatigue life by a factor of 2.

The optimal replacement for a circular hole.

Strengthening Structures While Increasing Hole Area

The area enclosed by a superellipse can be expressed as

where Γ is the gamma function.

For comparison, the area of an ordinary ellipse is

This means that the relation between the area of a superellipse and an ordinary ellipse having the same semiaxes is independent of the axis ratio and can be expressed as

In the plot below, the function is shown. 

Relative area increase as a function of n when compared to an ellipse. The asymptotic value 4/π is indicated by the dashed line.

It can be seen that the area of any superellipse with n > 2 is always larger than that of the ellipse with the same semiaxes. Somewhat surprisingly, this means that it is always possible to lower the stress concentration factor by changing a circular or elliptic hole to a superelliptic one that has a larger area, that is, by removing material. In my previous blog post, Making Structures Stronger by Removing Material, some other cases where material removal is beneficial are presented.

It should, however, be noted that we have only investigated uniaxial stress states that are aligned with one of the semiaxes of the hole. For some other stress states, a stress reduction effect cannot be obtained. The reduced radii of the corners of the superellipse will instead raise the stresses.

Real-World Applications of Superellipses

This is, of course, mainly a fun theoretical discussion. In practice, it is often much easier to drill circular holes. But if, for example, additive manufacturing or casting is used, then it can be beneficial to choose other shapes.

The approach of using a superelliptic shape is not only applicable to holes. The same idea can be used to reduce stress concentrations at fillets.

You can download the model used in the examples above by clicking the button below.

Precursors to Glasses

Reading Stones

One of the earliest examples of people using a lens to assist vision came not in the form of glasses but in the form of a small quartz stone. Reading stones, as they were dubbed, are small, hemispherical stones carved out of quartz, beryl, or glass and shaped in such a way to function as a convex lens. When placed upon paper, this lens would magnify the text written on it for those whose vision was failing. The invention of these stones is often attributed to Ibn Sahl, a 9^th-century Persian mathematician, but they have been found all across the world, including in Viking graves in Sweden.

A glass reading stone in Archeon historical theme park in the Netherlands. Image licensed under the Creative Commons Attribution-Share Alike 4.0 International license via Wikimedia Commons.

Sunglasses

Another early precursor to the corrective glasses we know today? Sunglasses, which were developed by multiple cultures around the world, some of the earliest of which were found in 12^th-century China and featured flat panes of smoky quartz. Another example of early noncorrective eyewear is traditional Inuit snow goggles. These goggles, typically carved from antler, driftwood, or ivory, reduced exposure to sunlight and helped prevent glare from the reflective snow.

Two pairs of Inuit snow goggles from Alaska. Image licensed under the Creative Commons Attribution-Share Alike 4.0 International license via Wikimedia Commons.

Early Corrective Lenses

Most sources attribute the invention of the first corrective lenses to late 13^th-century Italy, although their inventor is unknown. Early evidence lines up with this, as paintings and sermons involving lenses began appearing around this time throughout Europe. There is further evidence of rules and regulations having been set in place for early lens makers by the 14^th century, during the Venetian Renaissance. These “glasses” looked a good deal different than those that exist today; they were often held by hand, made of two magnifying glasses attached to a pair of riveted handles to form an almost scissor-like shape.

A pair of French “scissors glasses”. Image licensed under the public domain via Wikimedia Commons.

How Do Glasses Work?

How Do We See?

Before looking into how glasses work, it helps to first understand vision itself. Seeing is a complex process that takes place in milliseconds. At a high level, the eye can be loosely thought of as two focusing lenses that project incoming light into an image on the retina. The cornea, which is the outer layer of the eye, refracts the incoming light toward the crystalline lens, which in turn focuses the light onto the retina. The retina is where the light is absorbed into our eye’s photoreceptors, the rods and cones. These photoreceptors then send electrical signals to our brain, which can then interpret what we see. To focus on objects at different distances, tiny muscles deform the crystalline lens to change its focus. The last crucial elements are the iris and pupil, which are between the cornea and the crystalline lens. The iris is muscular tissue in the shape of an annular disc, and the pupil is the hole in its center. The iris expands or shrinks the size of pupil to adjust the amount of light that reaches the retina.

Issues with vision can happen during any part of this process but most commonly stem from natural imperfections in eye shape. These include things like the eyeball being misshapen (e.g., excessively oblong), the cornea itself being uneven or not convex enough, or the crystalline lens being damaged. As we age, the crystalline lens also becomes stiffer, causing presbyopia.

How Do Glasses Help?

Glasses use lenses to bend beams of light before they reach the eye, refocusing them based on the eye condition. In the case of myopia, or nearsightedness, the light naturally comes to focus before hitting the retina, causing objects that are far away to appear blurry. Lenses with a concave shape help send the focal point of the light farther away from the retina. In cases of hyperopia, or farsightedness, the opposite happens, so a convex lens is used to move the focal point of the light closer to the retina. When the cornea is uneven, also known as having an astigmatism, corrective lenses are shaped in more complex ways.

Top: By moving the focal point of light beams closer to the retina, a convex lens corrects farsightedness. Bottom: By moving the focal point farther from the retina, a concave lens corrects myopia. Image licensed under the Creative Commons Attribution-Share Alike 4.0 International license via Wikimedia Commons.

How Have Glasses Evolved?

Bifocals

The invention of bifocals is often attributed to Benjamin Franklin, who used them in court to read the lips of French speakers while also taking and reading notes. While Franklin is rumored to have fashioned his bifocals by sawing in half the lenses of two pairs of his glasses and then attaching them together in a new frame, modern bifocals are made by molding a high magnification segment onto a regular corrective lens.

A pair of bifocal glasses. Image licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license via Wikimedia Commons.

Progressive lenses take bifocals a step further, providing three or more magnification levels in a single pair of glasses and allowing wearers to see at multiple distances.

Modern Lens Technology

By the late 19^th century, glass had long since replaced costly quartz and beryl as the primary material for lenses, and in Germany, physicist Ernst Abbe and chemist Otto Schott experimented with adding other elements into the melted glass to increase factors like durability, clarity, and weight. In the 21^st century, many lenses are made of specialized plastics that further reduce cost and weight while increasing durability.

Additional modern developments include blue-light, scratch-resistant, anti-glare, and photochromic technologies that involve coating and/or impregnating lenses with various materials. Anti-glare lenses were originally developed for military optics during World War II but were made available for public use in the late 1940s. These coated lenses help glasses to not appear distorted in photographs and improve visual acuity in situations with inconsistent or bright light, such as driving at night or viewing a computer screen. Photochromic lens material (which may be applied as a coating or within the lens) is UV reactive and darkens in sunlight, removing the need for a person to own a pair of prescription sunglasses and glasses.

An uncoated pair of glasses (top) compared to one with an anti-glare coating (bottom). Notice the reflection of the photographer in the top frame and the blue-tinted lights reflected in the bottom pair. Image licensed under the Creative Commons Attribution-Share Alike 3.0 Unported via Wikimedia Commons.

Contact Lenses

Compared to glasses, contact lenses offer better peripheral vision and are less impacted by weather conditions. The precursor to contact lenses was proposed in the 16^th century by Leonardo da Vinci, who suggested wearing water-filled glass hemispheres over the eyes. The first contact lenses, developed in the 19^th century, were rigid and made of glass and wax. It wasn’t until the early 1960s that modern soft lenses were invented.

Looking Forward

While we’ve mentioned a few recent developments in corrective eyewear here, other innovations are on the horizon. For example, researchers have used simulation to develop liquid lenses that dynamically change optical power and reduce eye fatigue when using AR/VR devices. Others are using virtual modeling to investigate lighter and more durable frames and specialized coatings for glasses. As advancements to eye care continue to be made and wearable tech devices like smart glasses come closer to everyday availability, we only have to wait and see where this technology will take us in the future.

To drive innovation in eye care technology and other optical systems, many engineers turn to ray optics modeling. The Ray Optics Module, an add-on to COMSOL Multiphysics^®, enables the modeling of electromagnetic wave propagation through ray tracing. With this approach, propagating waves are represented as rays that can reflect, refract, or be absorbed. Learn more about the module via the button below.

Reference

1. O. Wilson, “What Percentage of People Wear Glasses? The Global Vision Correction Reality,” GLASSON, 18 Dec. 2025; https://www.glasson.app/blog/what-percentage-of-people-wear-glasses-the-global-vision-correction-reality/.