What Is Sputtering?

Sputtering is a physical process where ionized particles of a plasma or gas are bombarded on a solid surface, or target, to displace the target’s atoms, which can then land on another surface, or substrate. Sputtering occurs naturally in outer space and can be an unwelcome source of wear in precision components. However, it is also useful for thin-film deposition and for physical surface removal in semiconductor manufacturing, optics, and MEMS applications. A schematic of a sputtering setup can be seen in the figure below.

A typical sputter deposition setup. Image in the public domain, via Wikimedia Commons.

Various types of sputtering mechanisms, namely magnetron, reactive, and ion-beam sputtering, are used in the semiconductor industry to deposit thin films, which are crucial for creating the complex multilayer structure required in today’s integrated circuits. These films may form barrier layers, adhesion layers, or conductive layers within chips. Sputtering offers several advantages in this regard. For instance, it enables excellent control over the thickness of the deposited film while being applicable for different materials, such as metals, alloys, and compounds. Moreover, sputtering is the ideal choice for substrates that cannot withstand higher temperatures.

In addition to thin-film deposition, sputtering is also used for the selective removal of material from a wafer in dry etching by employing ionized gas particles guided by a protective mask. Inert gases such as xenon and argon are typically preferred as sputtering gases because of their minimal reactivity and ability to cause higher displacement of the target material due to their significant molecular weight.

A sputtering device, with plasma in a vacuum glass tube.

Understanding Complexities with Simulation

While sputtering is commonly used in chip manufacturing and thin-film processing, complexities arise mainly due to various phenomena simultaneously at play, such as plasma generation and ion bombardment. Hence, maintaining meticulous balance and control over various electrical, chemical, and physical parameters, such as gas pressure, ionization potential, and substrate bias voltage, is critical to achieve the desired outcomes.

Numerical simulation can be used to analyze factors such as pressure, geometry, and material properties, making it beneficial in the optimization of thin-film growth, substrate erosion, and device fabrication. Furthermore, virtual analysis and optimization can help teams save on costly physical experiments.

Let’s look at an example of modeling argon ion sputtering on a silicon surface in the COMSOL Multiphysics^® software.

How Is Argon Sputtering Modeled in COMSOL Multiphysics^®?

The Particle Tracing Module, an add-on to COMSOL Multiphysics^®, provides physics interfaces that can be used in sputtering-related models. In this example, the Charged Particle Tracing interface is used to simulate argon ion trajectories near a silicon surface. It can be used to compute ion and electron trajectories in electric and magnetic fields. The Deformed Geometry interface is used to visualize the surface evolution caused by sputtering and capture the resulting morphological progression.

Representation of the model setup.

As seen in the image above, the sputtering process involves two ion beams incident at 45^° and −45^°, respectively. When argon ions hit the silicon surface, silicon atoms are sputtered. Each particle is assumed to represent 10¹² actual argon ions. The mask height is taken as 0.2 μm, and the etching window width is 0.5 μm. Mask sputtering is ignored in this simplified example.

What the Results Depict

The figure below shows the sputtering yield in relation to the incident angle of the beam. Sputtering yield refers to the number of target atoms sputtered per incident ion. As can be seen, the number of secondary particles varies with respect to the angle of incidence. In this simplified model, the direction of secondary particle emission is assumed to follow a cosine distribution, and elastic-collision-based assumptions are used to estimate the emitted particle velocity. As observed in the graph, sputtering yield increases almost linearly with respect to incident angle between 30^° and 70^°. Accordingly, the sputtering yield can be calibrated by controlling the incident angle of the beam. For example, a lower angle of incidence can be employed for lesser material removal, whereas increasing the angle of the incident beam will result in more removal of target material within this range.

Number of secondary sputter particles with respect to the incident angle.

Particle trajectories and sputtering progression with beams incident at 45^° and −45^°.

The above figure depicts particle trajectories with beams incident at 45^° and −45^° and the resulting sputtering progression. More material is observed to be sputtered at the corners, resulting in faster height reduction.

How Can These Findings Be Improved?

This model can be further extended to incorporate other factors that affect sputtering. For example, in addition to the incident angle, sputtering yield also depends on the energy of the incident ion as well as the absolute and relative mass of the incident ion with respect to the target atoms. Furthermore, the bombardment process is significantly more complex in the real world, with some ions penetrating the substrate to generate cascading collisions and atoms being ejected only after multiple scattering events. These effects are not captured in this simplified model and generally require more specialized sputtering or collision-cascade descriptions.

Try It Yourself

Want to try modeling the sputtering process yourself? Download the related MPH-file in the Application Gallery:

Inducing Streaming with Surface Acoustic Waves

When alternating voltage is applied on the surface of a piezoelectric material, it generates a strain determined by the electrical field, and waves start to propagate on the surface. These waves are called surface acoustic waves (SAWs), and they are distinguishable by how the material deforms relative to the propagation and normal directions. Two types of SAWs include Rayleigh waves and Love waves. This blog post will focus on Rayleigh waves, which make the surface deform in the normal direction. For the generation of the SAWs on the substrate, normally a set of comb-like terminals, or interdigital transducers (IDTs), is used to impose alternating electric potential. IDTs can both generate and receive the SAWs. When used as a filter in electric components, one more set of terminals is placed on the path of generated SAWs. The two terminals of the IDT on the receiver side will have different electric potentials according to the strain the substrate is experiencing, and information about the oscillation can be determined.

Instead of placing the second IDT on the surface as a receiver, we will put a droplet on the propagation path. The droplet will start interacting with the SAWs and absorb their energy. The SAWs attenuate as they travel under the droplet and are called “leaky SAWs” for this behavior. Energy is radiated to the droplet in the form of bulk waves incident at an angle called the Rayleigh angle. In the droplet, the incident waves reflect on the free surface of the droplet while also losing energy due to viscous dissipation, eventually resulting in a steady circulating flow component called acoustic streaming. We can induce steady flow just by oscillations. This plays an important role in the microfluidics area — we can enhance mixing inside the droplet without needing to physically put something in the fluid to stir it; the method is noninvasive. The resulting streaming may have different circulation patterns inside the droplet depending on the energy of the waves, the dimensions of the system, the material properties of the droplet, and so on.

Schematic of SAW-induced streaming. The SAWs (wavy blue line) depart from the IDT. Once they reach the droplet, the energy is transferred into the droplet (solid yellow arrow), and finally, the streaming occurs (dashed yellow arrow).

As we have seen so far, the droplet streaming has multiple physics areas involved. Due to their complexity, these contributing factors are frequently modeled separately by dividing the process into several steps. Note that the analysis of the acoustic field in a droplet takes up considerable RAM (under the current conditions), while the analysis of the other fields usually does not. We will use COMSOL Multiphysics^® to handle the complexity and create a model covering the whole process of transferring energy in the system. In this model, the drop has a wetting diameter of about 2 mm and a contact angle of 78° with the solid surface. The SAW device is excited with a frequency of 20.37 MHz. The droplet is assumed to be a glycerol–water mixture, and the properties of the piezo crystal are taken from the lithium niobate material in the Piezoelectric material library. As a reference, the values and setup are similar to the ones used in the paper: “On the Influence of Viscosity and Caustics on Acoustic Streaming in Sessile Droplets: An Experimental and a Numerical Study with a Cost-Effective Method” (Ref. 1).

Streaming Model Setup in the 2D Configuration

First, we will check what the model looks like in 2D. We know that the streaming has a 3D structure because of the hemispherical shape of a droplet, but it is always a good starting point to create a 2D model to see if the necessary settings and physics are complete and that the phenomenon we hope to model occurs under the 2D assumption. We will use the same cut plane as the schematic shown above for the 2D simulation. We will use the Electrostatics, Solid Mechanics, Pressure Acoustics, and Creeping Flow interfaces to excite the SAWs, and to capture the streaming, the Acoustic Streaming Domain Coupling multiphysics is used. Given that the time scales are quite different between the oscillation and the streaming, the simulation is done in two steps: a Frequency Domain study and a Stationary study.

In piezoelectric analysis, we need to pay attention to the crystal cut of the piezoelectric material. In this model, 128^° YX-cut lithium niobate (LiNbO₃) is used for the substrate; thus, the rotation should be reflected on the material properties. Functionality in COMSOL Multiphysics^® enables us to consider the angle of the crystal cut by defining a coordinate system using, for example, the Rotated System feature and specifying the coordinate system in the Piezoelectric Material node of the Solid Mechanics interface. We also have an entry in the Application Gallery that explains the coordinate settings: Euler Angles Rotation in SAW Modeling. Note how the angles are set differently in the Application Gallery model for a 2D component (XY sagittal plane) vs. a 3D component (XZ sagittal plane). Further below, we will create a 3D model with the XZ plane as the sagittal plane.

Settings of the Rotated System feature in the 2D model. In the 3D model, the value for β is set to -38 [deg].

We also need to make sure that an appropriate loss mechanism is taken into account in the Pressure Acoustics node. In the current setup, it is mostly the Eckart streaming that drives the flow inside the droplet. Therefore, the bulk attenuation of the sound wave should be modeled to capture it. The Fluid model in the Pressure Acoustics node specifies what type of attenuation the acoustic waves will experience. Here, we simply choose Viscous. If we left Linear elastic selected (the default), we would see no streaming in the results.

One last point to check is the Values of variables not solved for settings in the Stationary step. In the streaming analysis, the Acoustic Streaming Domain Coupling feature couples the Frequency Domain and Stationary studies. When the coupling is activated in a Stationary study, it refers to the variables solved in a Frequency Domain study to calculate the terms contributing to the streaming. Since we use multiple study nodes, the coupling does not know which solution has the frequency domain data of interest, so we need to specify it in the settings of the study step.

Now, let’s run the Frequency Domain and Stationary studies in order. The result should show distributions like the images below. The IDT is placed to the left of the droplet, outside of the images. There are SAWs generated on the piezoelectric material, moving to the right. The wave propagation direction can be checked more clearly using the Animation feature with the Dynamic data extension. The surface waves turn almost invisible after they have traveled over half of the contacting area. In exchange, the bulk waves in the droplet propagate in the upper-right direction, resulting in a complex pressure pattern. As might be expected, the circulating flow field is also confirmed from the stationary result. It has a large vortex across the whole domain area, but note that this might be 2D-specific. In a 2D configuration, we cannot simulate a vortex whose axis is not normal to the screen. However, it is a good start that we have set up a SAW model that induces streaming similar to what we have expected. Let’s move on to the 3D model.

Results computed in the 2D model. Stress in the substrate and acoustic pressure in the droplet (left); displacement of the substrate and velocity distribution of the streaming in the droplet (right).

Tackling the RAM Issue in the 3D Configuration

Despite the difference in the dimension, from 2D to 3D, the principles of the simulation do not change. We use the same interfaces with the same multiphysics couplings. What does differ from the 2D model is the requirement for the computational resources. The 3D model’s geometry dimension is much larger than the wavelength, and it is highly likely to face memory issues. Not only would we experience longer computation, but we would also need to reduce the memory consumption so that the model can fit into the RAM of the computer.

First, we can simplify the geometry to an extent that will not significantly affect the result. In the 3D model, the IDT is modeled as several sets of simple rectangular 2D terminals that are placed in parallel. Moreover, the droplet and the substrate are cut in half by the middle plane to halve the degrees of freedom (DOFs). A periodic condition is used for the transverse direction of the piezo while a symmetry boundary condition is applied to the drop. This is a good approximation for the physics, especially when we are interested in the flow field in the droplet. We would need to widen the substrate domain if the model exhibited a strong dependency on the transversal length.

Second, considering that the resolution levels required to capture the waves are different in the droplet than in the substrate, we will use different mesh and mesh sizes for them. This is achieved by using the Form Assembly method instead of the Form Union method in the geometry sequence. This feature will allow the model to have multiple geometry objects in a component, mesh each object separately, and connect them using the so-called Pair feature that couples nonconforming meshes. Remember to use the Union operation where applicable so that the software can recognize that some of the geometry instances belong to a single object. The domains that belong to the same object will have a conformed mesh, and no pair is used between them.

Magnified view of the nonconforming mesh near the leading edge of the droplet.

Lastly, we need to use an iterative solver to reduce the RAM usage in the Frequency Domain study. A general guideline on the settings of iterative solvers for acoustics problems is available in the Solving Large Acoustics Problems Using Iterative Solvers section of the Acoustic Module’s User Guide documentation. In this model, the number of DOFs for the Electrostatics and Solid Mechanics interfaces is much smaller than for the Pressure Acoustics interface, and direct solvers are expected to work well for solving for the electric potential and the solid displacement fields. Therefore, we activate hybrid preconditioners and use a Direct Preconditioner for the dependent variables of the Electrostatics and Solid Mechanics interfaces. With this feature, we can use a direct solver for small fields while applying efficient solvers to the other large fields. If the model did not have a piezoelectric domain, we could use a Segregated solver to reduce the RAM usage even further. However, as explained in Solving Large Acoustic–Structure Interaction Models section of the Acoustic Module’s User Guide documentation, acoustic–structure interaction models with piezoelectricity need to use the Fully Coupled solver, so the linear solver is the only part we can tweak. The acoustics part of the equations uses the shifted Laplace approach as an efficient form of the multigrid preconditioner.

The Settings window for Direct Preconditioner in the 3D model. Note that hybridization is activated by choosing Multi preconditioner in the Hybridization section.

Now, the Frequency Domain study solves with 130 GB RAM in about 1 hour 10 minutes in our environment. However, even after running the study, there is something we need to keep in mind: rendering. In a large model, rendering a result can take a long time. To simplify working with large models, it is recommended that you enable the Only plot when requested checkbox in the Settings window for the Results node.

The results below clearly show the influence of the 3D geometry. Now the pressure pattern has a more defined peak in the middle slice than the 2D result. In addition, although it is a bit difficult to discern from a single picture, a small vortex is formed near the leading edge, and the rest of the droplet is occupied by a large vortex. These show good agreement with the reference, where a 3D simulation was conducted by decomposing the computation into 2D subproblems. In our model, we have been taking a rather straightforward approach; thus, we can simply set up multiphysics modeling with some modifications in the settings. The strategy to use the nonconforming mesh and the iterative solver would also work for other large problems, such as complicated MEMS devices.

Results computed in the 3D model. Stress on the substrate surface and acoustic pressure in the droplet (left); streamlines of the streaming colored by the velocity magnitude (right).

Now It’s Your Turn

In this blog post, we have dealt with the complexity of modeling streaming in a droplet and the multiphysics involved using 2D and 3D models. We observed the interactions between physics interfaces, even with the 2D model, that solved quickly. Extending a 2D model to a 3D one can sometimes be challenging; it might require trial and error to see what configuration works for each model. We hope the current strategy will be useful to your multiphysics problem. The models are available from the following links:

• SAW-Induced Streaming in a Droplet – 2D Setup
• SAW-Induced Streaming in a Droplet – 3D Setup

As mentioned above, piezoelectric materials are frequently used to excite SAWs, but the definition of the properties requires special attention in terms of the coordinate system. Moreover, the design of the IDTs is closely related to the wave speed to excite; therefore, it is also important to see if SAWs are generated as desired. The following models work as useful references to test your SAW setup before building a complex model:

• Euler Angle Rotation in Surface Acoustic Wave Modeling
• Surface Acoustic Wave Velocity Calculations from a Unit Cell

This blog post does not mention particle movement in the streaming flow. To model acoustic manipulation of such things as particle movement, it might also be necessary to consider acoustic radiation force exerted on the particles. The Particle Tracing for Fluid Flow interface includes this functionality in the Acoustophoretic Radiation Force node, which can be used along with the Drag Force node. This combined use will consider the force due to the steady flow component at the same time. The following models would be useful as the starting point of such applications:

• Acoustic Streaming in a Microchannel Cross Section
• 3D Acoustic Trap and Thermoacoustic Streaming in a Glass Capillary

Reference

1. A. Riaud et al., “On the Influence of Viscosity and Caustics on Acoustic Streaming in Sessile Droplets: An Experimental and a Numerical Study with a Cost-Effective Method,” Journal of Fluid Mechanics, vol. 821, pp. 384–420, 2017. DOI: https://doi.org/10.1017/jfm.2017.178

Cupping Therapy Explored

Suction pressure elevates skin, fat, and muscle to varying extents depending on the pressure level and contact area. Along with cupping therapy (Figure 1), suction is used in medical applications such as cryolipolysis for fat freezing and reduction, lymphatic therapy to promote fluid drainage, and laser therapy, such as in port-wine stain (PWS) laser treatment, where lasers are used to lower blood vessel visibility.

Etim and her team are working to increase understanding of how suction pressure and applicator size influence the stress distribution across tissue layers. Computational modeling can be used to study the stress patterns and specific responses in each layer of skin during suction. By understanding how skin deforms under suction pressure, researchers can better predict the effects of devices and assess extreme loading conditions. Moreover, these insights can help medical device designers develop more reliable models and optimize device design.

Before diving into how Etim and her team used simulation, let’s consider the layers of skin and their relationship with medical applicators.

Figure 1. Cupping therapy, which results in deformation of the skin and underlying tissue due to suction.

Layers of Skin & Aperture Sizes

Skin is made up of three primary layers:

1. Epidermis
2. Dermis
3. Hypodermis

The epidermis is the protective outer layer of skin, consisting of four layers over most of the body and five layers on the palms and soles. The dermis contains blood vessels, nerves, hair follicles, and sweat glands. This layer deforms anisotropically due to the preferential orientation of collagen fibers. The foam-like hypodermis insulates the body, cushions organs, and stores subcutaneous fat, behaving as an energy-absorbing layer because of its collagen matrix, which encloses lipid-filled adipocytes.

Suction devices have typically been made with apertures under 10 mm; however, clinical applications such as cupping therapy, laser-assisted PWS removal, and some cryolipolysis applicators can use larger apertures ranging from 30–65 mm, highlighting the need to evaluate a broader range of loading geometries. Smaller apertures primarily engage the epidermis and dermis layers, where the fiber alignment may play a stronger role, whereas larger apertures engage the hypodermis and muscle underneath.

Now, we’ll dive into how simulation was used to analyze tissue.

Setting Up the Model

Etim and her team used a phenomenological approach to estimate the properties of skin, fat, and muscle for different participants, also considering different aperture sizes. They began with a single-phase solid mechanics model in COMSOL Multiphysics^®, with the governing momentum balance equation

where ρ is the density, u is the displacement field, f_v represents the volume forces, F is the deformation gradient, and S is the second Piola–Kirchhoff stress tensor. The Green–Lagrange strain tensor is defined as

Stress and strain are connected by the strain energy density function W as

Skin () was modeled using the polynomial model:

and fat () was modeled using the Mooney–Rivlin formulation:

The muscle was modeled with an Ogden formulation:

The team implemented a multilayered, axisymmetric finite element model of skin, fat, and muscle in COMSOL Multiphysics^®. Skin thickness was set to 2 mm, muscle thickness was set to 10 mm, and fat thickness was set to values specific to the results of participant ultrasound measurements gathered in a study. The simulation represented suction loading for applicators with aperture sizes of 50, 30, and 16 mm. The layers are assumed to be bonded without friction. Etim explained that a uniform suction pressure was applied at the skin surface over each aperture radius, and a boundary condition was imposed to enable the skin to slide along the rim of the cup.

Etim and her team used the Optimization Module, an add-on to COMSOL Multiphysics^®, to estimate the properties of skin and fat while keeping the properties of muscle constant. was assumed to be zero, while , , and were optimized by minimizing the deviation between the deformations seen in experiments and those in the model. A time-dependent model involving a wide pressure range was used, but the material properties in this optimization were initialized with the results of an optimization for a single high pressure. This methodology enabled the team to capture nonlinear pressure-displacement behavior.

Simulation Results

The results of the simulation showed that tissue deformation increased with suction pressure, with the magnitude and distribution influenced by aperture size and fat thickness (Figure 2). As expected, the simulation showed larger deformation for both larger suction pressure and larger aperture size, with the smaller apertures increasing the engagement of the upper skin layers.

Since the results were based on real data from individual participants with varying levels of fat thickness, the results also showed that differences in fat thickness of approximately 2 mm between participants led to distinct shifts in deformation profiles.

Figure 2. Displacement of skin and fat at 27091.109 pascals (Pa) for a participant with a 4.0-mm fat thickness when aperture sizes of 50 mm (a), 30 mm (b), and 16 mm (c) were used.

The effect of different material property combinations was investigated, and it was found that variations in the fat stiffness of the participant had a stronger influence on deformation with the 50-mm aperture than the 30-mm and 16-mm apertures. These results highlight the importance of taking into account the applicator geometry as well as the composition of the tissue in the individual participant.

Let’s take a closer look at a stress analysis in one individual participant who has a 6.3-mm fat thickness. The skin reacted with the highest stresses near the center of the suction cup, particularly in the circumferential and axial directions when a pressure of 27091.109 Pa (or 203.2 mmHg) was applied. In contrast, the fat layer showed low circumferential and axial stresses, despite large deformation. Shear stress remained minimal across conditions. The muscle experienced less stress, although it was more widely spread out. Other participants who had thinner fat thickness resulted in higher and more widely distributed values. In the muscle layer, the largest stress component is the axial stress (Figure 3). This stress analysis suggested to Etim and the team that fat thickness modulates load transfer and acts as a protective layer, providing a buffer to deeper tissue during suction loading.

Figure 3. Three components of the stress tensor are plotted at the mid-depth of the skin, fat, and muscle layers. The results are for a pressure of 27091.109 Pa in a cup with a 50-mm aperture.

Insights from Simulating Tissue

The results of this study indicate that both aperture size and fat thickness significantly influence the deformation caused by suction. The larger aperture, 50 mm, engaged more of the fat layer, deeper into the tissue, and produced more consistent results. The smaller apertures, 30 mm and 16 mm, engaged primarily the upper skin layers and showed higher variability. As Etim explained, these results highlight the importance of characterizing tissue under multiple loading geometries and pressures. Accurate representation of both skin and fat properties is crucial in the design and performance of medical devices that use suction.

Further Learning

For more information on this work, read the UML team’s full paper, which won a Best Paper award at the COMSOL Conference 2025 Boston! The paper outlines the team’s modeling approach and results.

Creating Surrogate Models: The Workflow

How do you create a surrogate model in COMSOL^®? The workflow can be described in a few steps as follows, starting from a completed parametric single physics or multiphysics model:

1. Add and run a Surrogate Model Training study, which is based on design of experiments (DOE) methods to sample the model parameter space.
2. Add a suitable surrogate model and train it on the simulation data stored in a Design Data table. Optionally train the surrogate model on, for example, experimental data.
3. Use the surrogate model in an app or digital twin, or for other purposes.

This workflow is also illustrated in the figure below.

The workflow for using surrogate models.

Using Surrogate Models in Simulation Apps

One practical use of a surrogate model is to speed up a simulation app created with the Application Builder. In the example of the microstrip patch antenna app shown below, a function call to a surrogate model replaces the need to solve the full finite element model, resulting in near-instantaneous response times when varying the antenna’s dimensions or material properties. In the image below, we can see how the app displays the antenna gain pattern from the trained surrogate model, shown alongside the result from the original model for comparison.

A simulation app comparing results from a surrogate model with those from a full simulation model.

The image below shows another example of a simulation app that has been accelerated using surrogate models. In this case, a set of surrogate models is used to reconstruct the electric potential, temperature, and stress in a MEMS actuator. The user can interactively adjust four geometric dimensions of the CAD model, along with the applied voltage, using sliders. Thanks to the surrogate models, the app responds quickly, enabling a much more interactive experience than would be possible with a full simulation model.

A simulation app of a MEMS actuator that uses multiple surrogate models to quickly visualize and evaluate physical field quantities such as the temperature, stress, and electric field.

In the MEMS app shown above, the visualizations and result evaluations are generated from a set of surrogate model functions that are called upon behind the scenes. A function corresponding to the temperature field is shown in the image below.

A function call to a deep neural network (DNN) surrogate model function, used behind the scenes in the MEMS app.

The syntax dnn1_T(x, y, z, dw, gap, wv, L, DV) calls a DNN function named dnn1_T, with the eight input arguments listed in parentheses:

• Three spatial coordinates: x, y, and z
• Four CAD dimensions: dw, gap, wv, and L
• The applied voltage: DV

This type of function call replaces calls to the field quantities defined by the full simulation model, which in this case is a finite element model.

Each surrogate model can define multiple functions, each with any number of input arguments. These functions typically represent physical quantities such as an electric field, temperature, or stress. Since surrogate model functions can be differentiated, they are well suited for use in gradient-based optimization workflows, such as inverse modeling, where sensitivities with respect to input parameters are required.

Types of Surrogate Models

There are three types of surrogate models available in COMSOL^®: DNN, Gaussian process (GP), and polynomial chaos expansion (PCE). The DNN surrogate model is included in the platform product and does not require any add-on products. The GP and PCE surrogate models are part of the Uncertainty Quantification Module, where they are automatically created and trained using dedicated solvers or studies for uncertainty quantification. However, any of the three surrogate model types can be trained on any kind of simulation or experimental data.

A response surface for a GP surrogate model showing the uncertainty estimate (standard deviation).

Once trained, surrogate models are available as functions under the Global Definitions node, ready to be used throughout the model. Each type of surrogate model comes with its own advantages. The choice of surrogate model depends on the problem at hand: DNNs are powerful for complex, high-dimensional problems with large training sets, while GP and PCE models are better suited when you need access to the confidence or uncertainty in a prediction.

A DNN surrogate model definition that includes six functions in eight input arguments. The user interface makes it possible to customize the number of layers as well as the number of nodes per layer.

Note that for small datasets, GP models may be easier to create, and they perform better than DNN models.

Now let’s take a closer look at how the data used to train these models is generated.

Generating the Training Data

The Surrogate Model Training study is used to generate the data needed to train surrogate models. It performs a parametric sweep using methods based on DOE, and it can be configured to sweep over virtually any combination of input and output parameters. The result is a table of simulation data that serves as the basis for training. Surrogate models are not limited by physics; they can be used in applications across electromagnetics, structural mechanics, acoustics, fluid flow, heat transfer, chemical engineering, or any combination of multiphysics.

The first few rows of a data file generated by the Surrogate Model Training study.

Training Surrogate Models

Training is the step where the collected data is fitted to a surrogate model. Once trained, the surrogate model can be used in place of the original simulation, achieving significant speedup while maintaining sufficient accuracy in many cases. Surrogate models can either be trained automatically after the data generation step or added and trained manually in a separate step.

The fidelity of a surrogate model is controlled by the amount and quality of training data. A higher-fidelity model generally requires more data, which can be obtained from simulations, physical experiments, or a combination of both.

When working with surrogate models, the data-generation step is typically more time-consuming than the training step. However, both steps can be accelerated. Data generation can be sped up by running simulations on a cluster, allowing multiple design points to be computed in parallel. Training DNN surrogate models can also be accelerated using GPUs, which can significantly reduce training time for large datasets.

An app for analyzing the test cycle of a battery, highlighting how a DNN surrogate model can be used to reconstruct time-varying physical quantities.

Dive Deeper into Surrogate Models

In this blog post, we have provided a brief introduction of the functionality in COMSOL^® for creating and building surrogate models. To get a more comprehensive overview of this functionality, check out our Learning Center course on surrogate modeling, which is an 8-part self-guided course that covers an introduction to creating surrogate models, fitting data with a DNN, evaluating model uncertainties, geometry sampling, and more.

Tip: To learn about the theoretical background of surrogate modeling, check out this Learning Center course: “Surrogate Modeling Theory”.

MM MaschinenMarkt “Design Engineering” Award: Best of Industry 2025

The Best of Industry Award, presented annually by MM MaschinenMarkt and determined by a public vote from their readers, celebrates outstanding achievements that move the development of industrial technologies forward. As a leading trade publication for mechanical engineering and manufacturing, the MM MaschinenMarkt community is comprised of engineering experts, designers, and business leaders who are up to date on the latest innovations, rendering their recognition of the Electric Discharge Module an even greater honor.

At left: Phillip Oberdorfer, technology communication manager at COMSOL (second from right), receiving the Best of Industry Award 2025 on behalf of the company, among various other award recipients. Photo courtesy of the Vogel Communications Group. At right: A close-up of the Best of Industry Award 2025 for the “Design Engineering” category.

MM MaschinenMarkt presents this award in association with trade media outlets such as Automobil Industrie, PROCESS, Labor Praxis, DeviceMed, and konstruktionspraxis, covering a comprehensive list of achievements in manufacturing and construction, process manufacturing, laboratory and medical technology, and automotive development.

Speakers presenting on stage at the MM MaschinenMarkt Best of Industry Awards 2025. Photo courtesy of the Vogel Communications Group.

We are grateful to be recognized alongside so many leading voices in technical advancement and are incredibly excited about the role that the Electric Discharge Module will continue to play in the development of next-generation electrical systems.

Understand, Analyze, and Predict Electric Discharges and Breakdown

The Electric Discharge Module extends the capabilities of COMSOL Multiphysics^® for assessing electrical insulation performance, adding specialized functionality for modeling electric discharges in gases, liquids, and solid dielectrics. Engineers and researchers can simulate phenomena such as streamer, corona, dielectric barrier, and arc discharges; lightning-induced electromagnetic pulses; surface charge accumulation; and more — all within a unified multiphysics environment. The Electric Discharge Module seamlessly integrates with other products in the COMSOL product suite, including those for electromagnetics, structural mechanics, and fluid dynamics, enabling users to explore the full spectrum of multiphysics effects often associated with electric discharges.

“The combination of specialized functionality and multiphysics integration reflects COMSOL’s long-standing focus on bridging the gap between physics understanding and practical simulation,” said Lipeng Liu, technical product manager for the Electric Discharge Module at COMSOL. “Built on the COMSOL Multiphysics^® platform, the Electric Discharge Module gives engineers and researchers robust, validated capabilities for modeling electrical insulation and discharge phenomena while helping reduce reliance on costly experiments.”

That focus is especially important in industries like electronics, aerospace & defense, automotive, healthcare, and manufacturing, with applications ranging from consumer electronics to high-voltage power system components. For many teams, the Electric Discharge Module serves as an important tool for product development, helping to reduce costs associated with experimental testing and prototyping while garnering a better understanding of electrical insulation design.

A streamer discharge propagating in transformer oil simulated with the Electric Discharge Module.

We would like to thank the MM MaschinenMarkt community and the other award recipients for continuously pushing technical boundaries in the spirit of developing better products and more efficient manufacturing solutions.

Next Steps

Learn more about the award-winning Electric Discharge Module add-on to COMSOL Multiphysics^® via the button below.

History of Estimating Pi

Historically, pi has been calculated through a variety of geometric methods by several ancient cultures. Later, during the 17^th and 18^th centuries, infinite series like the Gregory–Leibniz and Machin-like formulas advanced pi calculations, allowing mathematicians to compute up to hundreds of digits by hand.

Using modern technology, the calculation of pi has reached extraordinary precision using powerful computers and advanced algorithms. In a previous blog post, we discussed how the Monte Carlo method can also be used to estimate the value of pi. Today, state-of-the-art calculation methods involving rapidly converging infinite series, such as those derived from Srinivasa Ramanujan and the Chudnovsky brothers (Ref. 1), have allowed us to compute trillions of digits of pi.

The Simple Pendulum

In this blog post, however, we are moving away from sophisticated technology and will instead rely on an experiment you can easily replicate in real life. A simple pendulum consists of a mass (ideally point-like) called a bob, attached to one end of a string (ideally massless). The time period T of such a pendulum’s swing, which is the time it takes to complete one oscillation back and forth, can be formulated as

where L is the length of the pendulum and g is the acceleration due to gravity. If the time period of a pendulum’s swing is known, the value of pi can be estimated as

By measuring the time period of a pendulum’s oscillations, you can estimate the value of pi from direct observation. The equation assumes that the motion is simple harmonic — which is only true when the angle is small enough that sin⁡θ ≈ θ — so it works best for small release angles. You may notice that the time period is evidently independent of the mass of the bob. In realistic systems, however, the string will have a nonzero mass and the bob will have a nonzero radius. This means that the accuracy of the estimated value of pi improves when the pendulum mimics an ideal case, i.e., when the string is massless, when all of the mass is concentrated at a point-like bob, and when the initial angle is small (<15 degrees). A pendulum comprising a string of length L and a bob of radius R_b being released at an initial angle θ.

Building the Simulation App

A simulation app based on a COMSOL model can serve as a predictive tool, allowing you to explore how different design parameters, such as pendulum length or mass, affect the results before performing the actual experiment. To build a model of a simple pendulum, we’ll use the Multibody Dynamics interface, available in the Multibody Dynamics Module add-on product, as shown in this example model of a double pendulum. In the model the Events interface is used to track the number of oscillations in the simulation using a Discrete State variable called count. The process occurs in two stages, represented by two implicit events. When the bob’s velocity in the x direction becomes negative, count is incremented by a positive fractional value less than one. When the x-component of the velocity becomes positive again, count is rounded up to the next integer using the ceiling function. A stop condition is implemented as another implicit event that is triggered when count reaches the specified number of oscillations. Once the simulation terminates, the total simulation time (T_sim) and the number of oscillations (N) are used to calculate pi according to the equation

Now let’s use the Application Builder to create an app that estimates pi for different configurations of the simple pendulum. You can easily build custom simulation-based apps without needing extensive programming by using the drag-and-drop form design and the Method Editor for short code snippets. The app serves as an interactive tool to estimate pi for different pendulum parameters by computing the time period. This approach makes it especially useful in educational settings. For instance, a teacher could use the simulation to design a pendulum of appropriate size and timing for classroom demonstrations, helping students connect theoretical predictions with hands-on measurement. 

The app’s UI.

The app contains sliders to control the length of the pendulum, the radius of the bob, the initial angle of release, and the number of oscillations to solve. It also contains a field where you can provide a numeric value of the ratio of the bob’s density to that of the string. 

Once you have chosen the desired parameters for your pendulum, clicking the Plot Pendulum button displays what your pendulum looks like based on the chosen values. The Compute button can then be used to simulate the pendulum, during which the Graphics window and the energy plot are updated in real time. After the solve, the estimated value of pi and the error from the true value are displayed to reflect the results obtained from the current solve. Feel free to share the set of input values that gave you the best estimate of pi in the comment section below!

A screen recording of the app in use.

Building an app like this in the COMSOL Multiphysics^® software using the Application Builder can be done by leveraging templates and without extensive programming experience. This way, engineers and scientists can link theory and simulation practically to make their models accessible to users who are not familiar with modeling.

Next Steps

You are welcome to download the MPH file with the app design and related files from the Application Gallery via the button below!

Further Reading

Want to see how simulation apps are being used in the real world? Check out a few examples below:

• Read about a smartphone app powered by multiphysics simulation: Forecasting Fruit Freshness with Simulation Apps
• Take a look at how a university professor uses apps within the classroom: Bringing Lab Courses to Remote Learning Students with Simulation Applications

Reference

1. Borwein, J.M. and Borwein, P.B., 2004. Ramanujan and Pi. In Pi: A Source Book (pp. 588-595). New York, NY: Springer New York.

The Future Is Here

CES offers inventors a stage to showcase the tech of the future, and this year, real-world applications of artificial intelligence (AI) emerged as the prevailing theme.

Throughout the 4-day event, it was clear to see that AI is expanding beyond chatbots to “physical AI”, often in the form of endearing, humanoid designs. According to a CES press release, these breakthroughs are turning AI into adaptable machines capable of delivering complex, real-world outcomes, evolving from simple task completion to more collaborative, analytical assistants. Autonomous robots could be spotted around the conference hall performing advanced tasks such as playing tennis, cooking customized meals, and deep-cleaning footwear. Additional exhibitions highlighted the use of robotics for home, industrial, medical, mobility, and supply chain applications.

At left: CES 2026’s venue, the Las Vegas Convention Center. At right: Dr. Lisa Su of AMD and Daniele Pucci of Generative Bionics unveiled GENE.01 at CES, a robot that can not only execute instructions but also learn from physical experience and respond fluidly to changing conditions in industrial settings. Photo courtesy of the Consumer Technology Association^®.

Accessibility technology proved to be another major theme at CES, with a focus on improving quality of life and removing everyday barriers. Wearable tech, like artificial reality glasses and smart watches and rings, provides consumers with advanced health metric tracking and personalized alerts. Smartphones can be integrated with features such as at-home hearing tests and object identification, while smart appliances and security systems in the home can help older adults age safely in place.

Attendees take a photo with a product exhibited at CES. Photo courtesy of the Consumer Technology Association^®.

In addition to AI and accessibility tech, exhibitors represented a range of industries, from streaming services to energy to construction. No matter the industry, they can all utilize the capabilities of COMSOL Multiphysics^® — simulation software to optimize R&D and accelerate innovation.

In the Arena

The COMSOL booth provided access to cutting-edge simulation work from various industries, demo stations for trying out COMSOL Multiphysics^®, and even fun giveaways (like playing cards and tote bags). When asked what draws attendees into the booth, Bjorn Sjodin, sr. vice president of product management at COMSOL, replied, “It could be as simple as they are looking for an easy-to-use simulation software like COMSOL Multiphysics^®. Or it could be that they are currently using simulation, but they’re only doing single-physics simulation…and they are now looking into COMSOL Multiphysics^® to incorporate additional physical phenomena to get a higher fidelity representation of their product or device.”

CES 2026 attendees explore the many use cases of modeling and simulation at the COMSOL booth.

There were also many conversations at the booth about the use of simulation in battery design, reflecting on the rapid expansion of electrification beyond electric vehicles and into electric aviation, boats, and other varieties of novel applications. “Batteries are multiphysics by nature,” said Niloofar Kamyab, product management lead of electrochemical engineering at COMSOL. Revolutionizing these applications with battery power requires rethinking design metrics such as performance, battery life, and safety analysis. “Having an optimal design requires a long experimentation and iteration process,” she explained. “That’s where simulation comes into play. Engineers can use multiphysics simulation not only to understand these multiphysics aspects but also to achieve an optimized and safe design.”

Building multiphysics models of batteries across scales provides insight into how batteries perform under various conditions as well as their thermal characteristics, which helps engineers develop the optimal EV battery designs. Image courtesy: COMSOL.

With more than 55,000 international attendees, CES also offered a unique opportunity to connect with COMSOL’s global user base. “It was exciting to connect with professionals from a wide range of industries and backgrounds, all looking to understand how simulation can support their innovation efforts,” said Katherine Leiva, sales development representative at COMSOL.

We spoke to customers who were looking to expand the use of simulation to other departments in their organization. Simulation could optimize processes and accelerate development in areas such as manufacturing, production facilities, or on the factory floor, as well as even sales and marketing. “Two COMSOL Multiphysics^® tools, the Application Builder and COMSOL Compiler™, allow simulation experts to create easy-to-use interfaces for non-simulation experts to use in other departments,” said Sjodin.

Learn more about the possibilities of simulation apps and how they can be brought into the field, factory, and lab.

Oscar Littmarck, vice president of marketing at COMSOL, also had several conversations with attendees about COMSOL Multiphysics^® version 6.4. “At CES, we showcased some of the updates from our latest software release, such as GPU acceleration for all physics.” Version 6.4 introduced new capabilities for accelerating simulations using NVIDIA^® GPUs, including NVIDIA cuDSS, NVIDIA’s CUDA^® accelerated direct sparse solver. This solver performs matrix factorizations with one or more GPUs on a single computer, taking advantage of the high memory bandwidth and massive parallelism provided by recent GPU hardware. GPU support for NVIDIA CUDA^® direct sparse solver (cuDSS) is fully integrated into the standard solver framework in 6.4, enabling users to take advantage of GPU acceleration for existing models without needing to make changes to the underlying physics settings.

GPU acceleration with NVIDIA cuDSS also benefits conventional structural finite element analyses on standard workstation hardware. In this wheel rim example, the effective stress is visualized, and the GPU-based solve on an NVIDIA RTX^™ 5000 Ada Generation workstation GPU achieved a 2× speedup compared with a CPU-based solve on an Intel^® W5-2465X processor.

Learning from Leaders

CES featured a range of keynote talks from leaders and executives across a wide range of industries. Some highlights included a talk by the founder and CEO of NVIDIA, Jensen Huang. Huang shared the stage with Yuanqing Yang of Lenovo to discuss how AI is reshaping how we live, play, and work — sensing the 3D world, learning from data uniquely complex to the individual, and interacting with reality in ways we’ve never seen before as a “self-reinventing entity”. Other keynote speakers included Joe Creed of Caterpillar, who shared how the company is transitioning from a traditional equipment manufacturer to a high-tech innovator, and Dr. Lisa Su of AMD, who explored how much further AI will evolve and expand into our everyday lives. Other opportunities to hear from prominent speakers included a live podcast taping where Bob Sternfels of McKinsey and Hemant Taneja of General Catalyst explored how AI is reshaping strategy, investment, and innovation.

A live taping of the All-In podcast at CES. Photo courtesy of the Consumer Technology Association^®.

We left the conference feeling inspired by the cutting-edge technology we witnessed, insights shared by industry leaders, and interesting conversations with attendees at our booth. Thank you to CES for hosting this dynamic event!

Looking Ahead

We have more exciting events lined up for the rest of 2026! Check out a few of our stops and be sure to swing by our booth:

• International Battery Seminar & Exhibit
• OFC Conference
• CLEO

Consumer Technology Association is a registered trademark of the Consumer Technology Association. Intel is a trademark of Intel Corporation in the U.S. and/or other countries. NVIDIA, CUDA, and RTX are trademarks and/or registered trademarks of NVIDIA Corporation in the U.S. and/or other countries.

Open Floor Plans, High Noise Levels

One common complaint with open layouts is noise level. The architecture, engineering, and construction industries have a growing interest in mitigating noise issues while designing open layouts. Acoustics modeling enables engineers in these industries to simulate noise levels efficiently and optimize designs ahead of physical construction of an office space.

Pressure distribution depicted on the walls of a small open-plan office with a pulse emitted from a point 1.5 m above the floor.

Building a Digital Office

In the Acoustics of an Open-Plan Office Space tutorial model, the acoustics of an open-plan office space are analyzed using a wave-based approach in the time domain. An initial pulse is emitted and the room response is analyzed. Realistic frequency-dependent impedance conditions for several aspects of the room — including the ceiling, carpet, and gypsum bolts on the walls — are included using the General Local Reacting (Rational Approximation) impedance option. The Partial Fraction Fit function is then used to fit the input data. An open window is also modeled using the Absorbing Layer feature. This model is solved with the Pressure Acoustics, Time Explicit interface and uses the accelerated solver formulation, which can be run on one or more GPUs.

COMSOL Multiphysics^® version 6.4 extends the existing single-GPU support to multi-GPU configurations, for accelerated simulations of pressure acoustics in the time domain, allowing speedups of 25x and more compared to using CPUs. To run the model, 12 gigabytes (GB) of GPU memory are required and a NVIDIA^® GPU are required. The version used here is 6.4 build 343.

Solving 20 periods of the 500-Hz carrier signal (mesh resolution up to 750 Hz) on a CPU with 12 cores takes about 10 hours (hardware dependent), whereas solving the same 20 periods on two GPUs takes 9 minutes. This initial analysis gives the propagation for the first 25 ms, and is useful for visualizing the initial propagation of the pulse.

Pressure distribution is depicted in red and blue on the walls of an open-plan office space. A pulse is emitted from a point 1.5 meters above the floor, near the bookshelf along the back wall. Frequency-dependent impedance conditions are used to describe the boundaries.

The model includes several impedance conditions, which use frequency-dependent data. The complex-valued impedance data is generated using the Acoustic Treatment Boundary Calculator, a simulation app for analyzing dispersion. The app can compute several properties or boundary properties that can then be used in room acoustics simulations. The data for various aspects of the model, such as the carpet, ceiling, and gypsum bolts on the walls, is computed using the app.

The open window along the left wall is included in the model to highlight the use of absolving layers together with the GPU. You can add half a sphere with absolving layers on the outside of the window to model the open window in this space.

A half sphere with the absolving layers, added onto the outside of the open window.

Studying Microphone Points

This open-plan office space acoustics model solves for 46 million degrees of freedom ( DOFs) in two studies. In study one, the model solves for 20 periods of the 500-Hz carrier signal, storing the solution twice per period on all boundaries. This study takes 9 minutes to solve on two GPUs (15 minutes on one GPU) and looks at the sound field in the beginning of the propagation of the pulse. To analyze the impulse response of the system, a second study is run, where the solution is only stored in the selected microphone points.

In study two, the model solves for a duration of 0.4 s (300 periods), storing the solution with a higher resolution 30 times per period only, on selected receiver positions. This study takes 2 hours and 56 minutes to solve on two GPUs. The signal is again received at the microphones (see image below). By using the impulse response plots, the level decay curves for 1/3 octave bands can be computed and analyzed. This enables you to compute room acoustics metrics that would typically be done using ray acoustics, with a more accurate full-wave method (more accurate in the low to medium frequency limit).

Impulse response for the microphone position closest to the manikin.

Extending the simulation time from 20 periods to 300 periods would suggest that it takes 2 hours and 15 minutes to solve study two, instead of the 2 hours and 55 minutes that it takes. This is due to a small overhead as a result of storing the solution more often. Storing the solution 30 times per period, compared to 2 times per period, creates a relatively small overhead and time difference, as well as some additional communication between the GPU and CPU.

The locations of the microphone points throughout the open-plan office space.

The level decay curves (for three 1/3 octave bands) computed for the microphone location closest to the manikin (see figure above) is depicted in the figure below. For the three bands the T20 reverberation time is about 0.65 s.

Level decay curves for the three 1/3 octave bands centered at 500 Hz.

Fitting the Impedance Data

In some sets, the Partial Fraction Fit function is used for rational approximations instead of nominal approximations. In the Frequency Domain Data graph shown below, there is a high number of points in the graph and both real and complex values of the admittance. The data needs to be fit using an approximation, and the approximation data is done with a partial fraction fitting. This equation approximates the data with a sum of fractions.

When the frequency-domain data is approximated with this formula, there is an analytical inverse Fourier transform of the data. With the Fourier transform, the data can then be used in the time domain in a more practical way. This translates into a lumped system or a system of memory ODEs that can be solved together with domain equations. This setup is handled automatically in COMSOL^®.

After the data has been fitted, the fitting parameters can be used in the time domain to model frequency-dependent properties, for example, an impedance condition or porous materials.

The frequency-domain data is approximated with this formula, creating an analytical inverse Fourier transform of the data.

What’s Next?

As the trend of open-plan office spaces continues to grow, mitigating noise levels will be a top concern in layout design. Try out the Acoustics of an Open-Plan Office Space tutorial model or Acoustic Treatment Boundary Calculator demo app yourself.

Further Reading

If you’d like to learn about a real-world use of modeling and simulation for improving the acoustical conditions of a workplace, you can check out our article, featuring Swiss consultancy Zeugin Bauberatungen, “Harmonizing Sound and Style in Open-Plan Offices”.

Tackling the Wind-Noise Spectrum

MEMS technology boasts a high signal-to-noise ratio that makes it possible for some environmental noise to be filtered, such as the subtle hum of an HVAC system or the background chatter of an office space, but wind noise will often introduce a level of distortion that can make otherwise clear recordings unintelligible. This distortion becomes an even larger issue for electronic devices that rely on voice assistants, such as smartphones and speaker systems, as wind disruptions can cause operational problems.

In an effort to reduce this obstacle, the audio engineering team at Sonos, Inc. developed a simulation workflow for MEMS microphones to predict how cross-flow velocity shapes the wind-noise spectrum and to evaluate the efficacy of different port geometries.

A microphone port model used to analyze the effects of wind noise.

Sonos’ Simulation Approach

A microphone converts acoustic energy into electrical signals, but in order to properly do so, the microphone diaphragm must be open enough for acoustic sound waves to reach it. The geometry of the port dictates how the microphone interacts with potential moving air around it. To observe how airflow behaves spectrally when flowing across a microphone system, the Sonos team built a microphone model in the COMSOL Multiphysics^® software.

The model is composed of elements that directly affect the microphone’s acoustic performance — with the microphone itself, adhesives, and circuit board thickness remaining consistent across testing. The software’s Aeroacoustic Flow Source multiphysics coupling was used to combine fluid dynamics (using large eddy simulation, or LES) and acoustics (using pressure acoustics) analysis to compute the flow-induced noise. The model predicts the total sound pressure level and pressure spectra at the microphone diaphragm, depending on the air velocity across the microphone inlet and changes in the inlet port geometry.

The simulation consisted of wind flowing from an inlet, traveling across the microphone, and exiting through an outlet. In each test, the team changed the diameter of different cylindrical and conical microphone ports. The results suggested that the shape of the microphone port does, in fact, impact the wind noise present in the microphone cavity, although port shape does not have as large of a role in affecting wind noise performance as other factors, such as acoustic mesh. The results also proved that in an optimized design, there would be minimal pressure at the microphone and thus less wind noise captured.

Additionally, the team evaluated how the noise spectra changed with the following wind speeds:

• 2.4 m/s
• 3.0 m/s
• 4.0 m/s
• 5.0 m/s

The 2.4-m/s simulation resulted in the lowest levels, while 5.0 m/s produced the highest, as expected. The result were in good agreement with the team’s analytical scaling-law-based predictions, as the simulation successfully captured the velocity-driven shift in the corner frequencies.

The sound pressure level of a microphone cavity when exposed to 5.0-m/s wind.

Lab Experiment Verification

Validating predicted results against real-world prototypes is a crucial step in the development process. For Sonos’ purposes, a series of physical lab experiments was conducted to confirm the modeling results.

The team produced laminar flow across the prototype ports in an effort to establish a consistent and repeatable wind flow evaluation. The microphone stackup was mounted to allow the exterior surface to be flush with a surrounding hard plane, while a singular angle of wind flow traveled perpendicular to the microphone. Multiple port geometries were tested in this setup.

Comparing the Results

The combined results from the simulations and lab experiments provided valuable insights — with both approaches confirming that wind velocity largely influences the wind-noise spectra scale. More specifically, it was confirmed that high velocities shift the shedding frequency. The agreement between the results also validates the COMSOL^® model’s ability to capture the dominant physical mechanisms at play within wind-induced noise.

The normalized sound pressure level of four wind speeds for a cylindrical port.

However, when analyzing the effects of different port geometries, the team realized that in contrast to the simulation results, the laboratory measurements produced negligible differences across geometries under similar but not identical conditions (for example, the simulation included a slip-wall condition to simplify the modeling, whereas the lab experiment involved a no-slip condition). This disagreement shows that although the model offered a reliable representation of velocity-driven spectral trends, how the geometry influenced the aeroacoustic couplings was not fully captured. A factor that may elucidate the divergence between the model and lab results is the exclusion of an acoustic mesh over the microphone port in the COMSOL^® model. Incorporating an acoustic mesh was found to substantially reduce wind-induced fluctuations in real-world systems, suggesting that modeling acoustic mesh could be an essential addition in future studies to improve predictive accuracy between model and experiment.

Ultimately, the audio engineers at Sonos concluded that the modeling framework is beneficial for early guidance in the design of MEMS microphones, can help with identifying worst-case scenarios, and can reduce the need for costly prototyping. Along with the study providing insight into the relationship between flow velocity and wind-noise spectra, it also offered a salient example of the importance of validating modeling results and modeling assumptions.

Further Learning

For more information on this work, read the Sonos team’s full paper, which won a Best Paper award at the COMSOL Conference 2025 Boston! The paper outlines the team’s modeling approach and results.

To get hands-on experience with modeling cavity flow noise in COMSOL Multiphysics^®, check out the Cavity Flow Noise tutorial model in the Application Gallery. Want to share your work at the next COMSOL Conference? Learn more on our COMSOL Conference 2026 landing page.

Lithium-Ion Batteries

Let’s consider a 1D model of a lithium jelly roll consisting of graphite for the negative electrode, NMC 111 for the positive electrode, and 1.0 M LiPF₆ for the electrolyte. The degradation is calculated based on a generalized volumetric Butler–Volmer expression in the negative graphite electrode current:

where is the electrolyte salt concentration, while and are the electric and electrolyte potentials, respectively. The charge loss can then be computed by integrating over the negative electrode 

and this is a good measure of the battery degradation during a charge cycle. We then compute the number of cycles resulting in a 10% degradation and constrain this variable based on the maximum number of allowed cycles.

Time-Optimal Control

The purpose of the optimization is to minimize the charging time while constraining the degradation charge. This is achieved by changing the charging profile. A Control Function feature is used to allow the charging current to vary with time. The problem can be regularized in different ways, and here we use a function based on five second-order Bernstein polynomials where the shared points are taken as the average of the midpoints so that the slope of the function becomes continuous. The time-dependent solver must stop when the battery is full, so the stop condition is set based on the integral of the control function.

The animation below illustrates how the progress of an optimization might look like for a battery designed to last 2000 charging cycles.

The MMA (method of moving asymptotes) optimization solver reduces the charging time without violating the constraint on the degradation charge. Note that the number of optimization iterations has been artificially increased using a move limit for visualization purposes.

The Pareto optimal front of the charging time versus the degradation can be traced by varying the degradation charge.

The maximum number of cycles is plotted versus the charging time for charging to 90% and 100% state of charge (SOC). The result of a constant charging current (dashed line) is also included to better illustrate the benefit of a time-varying current.

The degradation charge has an exponential dependence on the charging time; a ~4-minute-longer charging time increases the maximum number of cycles by a factor of 10. The result of using a constant current is included for reference, and this shows that it takes 38 minutes to charge the battery 90% with a constant current, while it only takes 22 minutes with an optimized charging profile. Finally, one can get down to 18 minutes if the battery only has to last 200 cycles.

Takeaways

The degradation of a battery is highly sensitive to the charging rate, so significant improvements in battery longevity can be gained with modest sacrifices in the charging time. The functionality for gradient-based optimization with a condition-based final time can be used for a wide range of optimal control problems, including time-optimal control.

Next Steps

• Interested in the details of this model’s setup in COMSOL^®?
  • Download and explore it for yourself: Minimizing the Charging Time of a Lithium-Ion Battery
• Want to learn more about optimization?
  • Check out these previous blog posts.
  ]]> https://www.comsol.com/blogs/minimizing-the-charging-time-of-a-battery/feed/ 0 https://www.comsol.com/blogs/optimizing-battery-pack-lifetime-using-simulation-aided-design https://www.comsol.com/blogs/optimizing-battery-pack-lifetime-using-simulation-aided-design#respond Tue, 03 Feb 2026 14:40:44 +0000 https://com.staging.comsol.com/blogs?p=492161 Guest bloggers André Gugele Steckel and Thomas Bisgaard discuss using traditional model-order-reduction techniques and the surrogate modeling techniques in the COMSOL Multiphysics^® software for efficiently designing battery systems.

  Lifetime predictions are paramount when designing battery systems, large and small. In this blog post, we present a method for performing these investigations efficiently and quickly with simulation by using new reduced-order models. This method is a paradigm shift from the traditional build-and-test method or using generalized design principles, which can be expensive, slow, and imprecise when trying to design systems that are to last decades. In the scope of the new method, the COMSOL Multiphysics^® software shines bright with its multiphysics modeling capabilities and, very importantly for this showcase, its implementation of surrogate modeling for model order reduction.

  Introduction

  Battery Market Growth

  Sales of electric cars surpassed 10 million in 2022, largely contributing to the 65% increase in demand for automotive lithium-ion (Li-ion) batteries that same year (Ref. 1). The use of batteries for large-scale energy storage is also gaining more interest because of more consistent scale-up compared to traditional gravity-reliant methods like hydropower. Battery pack efficiency, longevity, and recyclability are critical design targets in the goal to meet sustainability targets, which also entails responsible use of essential raw materials such as lithium and nickel.

  Why Simulation?

  Simulation is an enabling tool that helps engineers reach design targets at low resource and material cost, as it reduces experimental iterations and prevents designs from having unneeded overcapacity. However, engineers are often forced to rely on potentially impairing assumptions and nonphysical parameters to bridge small-scale (single-cell electrochemistry) simulations and large-scale (battery pack) simulations. In the revolutionizing integrated approach discussed here, deep neural networks (DNNs) are used to bridge the scales in such a way that a model’s high fidelity is maintained at a low computational cost.

  This approach makes it possible to calculate degradation at every point in a 3D battery pack model at full time resolution during discharge and charge cycles. For example, with these simulation tools, design engineers can incorporate realistic use patterns to optimize battery management systems (BMS) and temperature control systems as well as to improve profit by balancing capacity, lifetime, and application specifications. An overview of the model scales and features can be seen in Figure 1.

  Methodology

  The methodology presented here is for multiscale and multiphysics modeling of batteries. The approach allows for modeling down to the finer details and using those results to expand the model to the whole battery pack of a car. This multiscale modeling spans from detailed electrochemical transport of lithium between the anode and cathode — including diffusion into storage particles — to system-level modeling of a full battery and multiple interconnected batteries in a pack.

  We have used a combination of traditional model-order-reduction techniques and the surrogate modeling techniques in COMSOL^®. By using the DNNs, we have been able to model multiscale systems in a way that we have not been able to do before.

  
  Figure 1. A multiscale approach was used for simulating battery packs, and each model scale offered different features.

  Cell and Particle Scale: Battery Electrochemistry

  A lithium-ion battery (LIB) consists of a repeating sequence of layers: anode, separator, cathode and current collectors on both sides (Figure 2). Each layer typically has a thickness in the range of 5 µm to 60 µm. The anode, separator, and cathode are porous structures. Within the matrices there is a liquid electrolyte. Graphite is the most common anode material. A graphite anode can be combined with a NMC (nickel–manganese–cobalt) cathode or a LFP (lithium iron phosphate) cathode. The nonconducting separator allows for electrolyte transfer. The electrolyte consists of dissociated salt (commonly LiPF6) in a solvent (commonly alkyl carbonates). Note that next-generation batteries that have different chemistries are ongoing work, including sodium-ion batteries (lithium-ion replacement) and solid-state batteries.

  Figure 2. The structure of a lithium-ion battery.

  The Doyle–Fuller–Newman model (Ref. 2) is the most commonly used model in LIB simulations. The electrode domains are treated as homogeneous when it comes to Li-ion transport but contain an additional dimension representing the radius of the electrode particles within the electrode domain. Hence, when modeling a battery in 1D, incorporating a radial dimension increases the dimension by one, thereby arriving at the common term “pseudo-2D” (Figure 3). Li-ions in the electrolyte solution act as charge carriers and transfer freely in the electrolyte solution.

  
  Figure 3. The pseudo two-dimensional (P2D) model of the lithium-ion battery, the DFN model. This multiscale model (cell and particle scales) is available in P2D, P3D, and P4D in the Battery Design Module. It is also available as a lumped P1D version.

  When a battery discharges rapidly, lithium near the surface of the electrode particles is depleted faster than it can be replenished by diffusion from the interior. This imbalance causes a significant voltage drop and ultimately reduces the amount of energy available, especially at low temperatures since the diffusion rate will be rate limiting. Assuming spherical electrode particles, intercalated or deintercalated Li-ion reacts with the electrode particle surface at rate :

  \frac{\partial c_{P,j}}{\partial t}=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2D_{eff,S,j}\ \frac{\partial c_P}{\partial r}\right)

   

  BC1: \left (-4 \pi r^{2} D_{eff, S} \frac{\partial c_P}{\partial r}\right)|_{r=S_{P,j}} = -4\pi \delta {\tiny \overset{2}{P},J} \frac {1}{F} i_{tot,j}

   

  BC2:\left(\frac{\partial c_P}{\partial r}\right)|_{r=0}=0

   

  Intercalated lithium of concentration  diffuses inside the electrode particles of radius  (radial coordinate ) and diffusion coefficient . is the universal Faraday’s constant,  is time, and  is an index representing the anode () or cathode () in order to differentiate between the physical properties of the domains.

  Charge capacity will decrease over time for an LIB. Depending on its operating history, the internal resistance increases due to several degradation mechanisms. For the capacity and power loss observed over time, some of the important factors are the temperature, state of charge, and load profile. Because of this, as the battery is used, both capacity and power loss occurs. The primary degradation mechanism considered is the solid electrolyte interphase (SEI) that is forming during the charge–discharge cycles of a battery. By using the approach of Safari et al. from Ref. 3 the SEI formation can be modeled. Ethyl carbonate (EC) diffuses through the SEI layer and reacts at the electrode surface. SEI formation consumes electrolyte solvent and cyclable lithium (Figure 4). Other degradation mechanisms, such as lithium plating, cathode breakage, and electrode fracture, are not included in this work. To extend the 2D domain defined for the electrode particle, the SEI layer growth can be included using the following model:

  \frac{\partial c_{EC,P}}{\partial t}=\frac{\partial}{\partial r}\left(D_{EC,P}\frac{\partial c_{EC,P}}{\partial r}\right)-\frac{d\delta_{SEI}}{dt}\frac{\partial c_{EC,P}}{\partial r}

   

  BC1:\ c_{EC,P}|_{r=\delta_{P,j}+\delta_{SEI}}=\varepsilon_{SEI}c_{solv}

   

  BC2:\ \left(-D_{EC,P}\frac{\partial c_{EC,P}}{\partial r}+\frac{d\delta_{SEI}}{dt}c_{EC,P}\right)|_{r=\delta_{P,j}}=\frac{i_s}{F}

   

  This set of equations makes it possible to track the SEI layer thickness (degradation status), , in any position of a battery using the growth rate . The SEI layer thickness is assumed to be small compared to the particle radius, and the loss of lithium in the electrolyte is assumed to be small compared to the initial content. The spatially dependent solvent concentration in the SEI layer is , the diffusion coefficient , the electrolyte lithium concentration , and the SEI layer porosity . The consumption of Li-ion in the SEI layer formation contributes to .

  

  Figure 4. Battery degradation due to solid electrolyte interphase (SEI) formation.

  Fitting Parameters to Real Data

  Battery models involve a large number of physical parameters. Some of these parameters can be determined in controlled experiments, for example, by measuring heat transfer properties calorimetrically and electrode characteristics microscopically. Another subset of the parameters is considered as system-specific, and these parameters are fitted for experimental conditions of cycling performance of single-cell batteries. Databases like Battery Archive offer standardized and cleaned experimental data of various battery chemistries for various conditions, such as temperatures, charge current, discharge current, and state of health. System knowledge was used to devise a parameter-fitting strategy that allows sequential parameter fitting based on the data type. For example, the time series data of the first cycle was used to fit kinetic data to potential time data, and cycling time series data was used to estimate degradation parameters. Arrhenius-type expressions (two parameters) were used for the electrode diffusion coefficient and the exchange current density in the Butler–Volmer kinetics to account for temperature dependence.

  Since the parameter fitting involves highly nonlinear dynamics and time integration, a brute-force method was adopted, where many samples based on a design of experiments defined by random samples were simulated, and the objective function (cost function) was evaluated for each simulation. The mean squared error between the simulated and experimental cell potentials was used as an objective function. Latin hypercube sampling (LHS) was used to sample from uniformly distributed parameter ranges.

  A custom program was developed with the Application Builder in COMSOL Multiphysics^® that allowed for automatically looping through parameter sets and datasets and reporting the objective function. Table A highlights the six experimental datasets, each consisting of hundreds of thousands of points in the time series. An excellent match was achieved for all datasets using the same parameter set. Figure 5 illustrates a comparison between the simulation and experimental degradation performance of a battery cell in terms of charge and discharge energy (W*h) and capacity (A*h).

  fID	Chemistry	Temperature	Charge Current	Discharge Current	SOC min.	SOC max.
  1	NMC	Medium	High	High	Low	High
  2	NMC	Medium	Low	Low	Low	High
  3	NMC	Low	Medium	Medium	Low	High
  4	NMC	Low	Low	Low	Low	High
  5	NMC	High	Medium	Medium	Low	High
  6	NMC	High	Low	Low	Low	High

  Table A. The datasets included for parameter fitting with qualitative conditions.

  Figure 5. Cycling data for a single NMC battery cell, comparing simulated and experimental charge and discharge energy and charge and discharge capacity. Data source: Ref. 4.

  Battery Simulation in 3D: Building Upon the Electrochemical Model

  Many phenomena are only present in 3D, and to fully and accurately model these phenomena, the modeling therefore also needs to be expanded to 3D. This includes the modeling of, for example, the cooling on the side of the battery packs, the heating of the busbar, and the current distribution across many of the different individual batteries. Another aspect to consider is that temperature gradients across the battery pack can lead to spatially nonuniform degradation within the battery. An important thing to also note is that there are significant computational costs from going up in dimensions. Also, for the models at the scale of a full battery, it would not be possible to have the same level of detail as in the 1D chemistry, as the computational cost would simply be too great. Some of the information that was calculated in the 1D chemistry needs to be taken up in the full 3D model in order for it to accurately represent the functions of the battery, but much of the information is not needed at all. Therefore, the chemistry can likely be described with a reduced-order model, such that it is computationally feasible to include in a 3D model.

  The DNN architecture that was implemented in COMSOL Multiphysics^® version 6.1 was adopted in this work. For the most important inputs and outputs from the P2D model, in the span that we expect the model to run in, we are able to generate a dataset that was needed in the training of the DNN.

  
  Figure 6. Conceptual workflow for data generation, DNN training, and approximation of the P2D model using a DNN-based surrogate model.

  In order to obtain a representative span of the parameters (Figure 6), LHS was performed for the most important control parameters, the temperature, current (to be used with the recorded voltage to train the DNN), state of charge, and SEI layer thickness. While it is possible to make LHS in the Model Builder in COMSOL Multiphysics^®, we opted to use the Method Editor in the Application Builder in order to control each individual simulation. That way, it was possible to distribute the simulation across many parallel COMSOL^® sessions and collect the results automatically in one file. Moreover, this method allowed for reducing the time it would take to generate the necessary data each time some changes were made inside the P2D model. The scripting in the Application Builder also makes it easier to automate the entire sequence of operations, which would be beneficial if this was going to be packaged into an easy-to-use COMSOL model or potentially a COMSOL app to be used by clients.

  After each iteration of solving for the P2D model, certain outputs were stored. Specifically, we chose values that would be integrated into the full 3D model, e.g., predicting the SEI layer growth speed in order to time integrate it into the SEI layer thickness.

  Having this dataset, it was then a task of finding a configuration of the neural network in terms of activation functions, depth and width of the neuron layers that would give the most optimal performance, and learning rate. The DNN was evaluated against how closely it would predict the out-of-sample dataset. Another key performance metric was the time required to evaluate neural networks of a given width and depth. For example, having a very wide and deep DNN not only took a long time to train but also led to a significantly longer wait time for results.

  

  Figure 7. Animation of a battery cell under fast discharge. From left to right: temperature, state of charge, and SEI layer thickness. The system exhibits spatial inhomogeneities due to a constant room-temperature boundary condition at the far-right end of the battery.

  Figure 7 illustrates a 3D model and simulation that couple temperature, state of charge, and SEI layer thickness. The figure gives an indication of how much stress the battery is undergoing during its use. Among other things, the results also illustrate how the heat is developed and distributed inside the battery and how different areas inside the battery degrade in terms of SEI layer growth. This is a coupled dynamic effect: degradation alters future charge–discharge behavior, which modifies the temperature field and, in turn, drives further degradation through SEI growth.

  For the system in Figure 7, temperature gradients lead to spatial variations in both the discharge rate and SEI layer growth.

  Larger Battery Packs and Multiphysics Modeling

  In practical applications, batteries are typically operated as part of packs rather than individually (Figure 8). It therefore becomes much more interesting to model entire packs of batteries. Specifically, the modeling could be performed for battery packs for an electric vehicle (EV). Modeling the full battery module or pack increases computational cost but provides a more accurate representation of thermal behavior and degradation during operation. By including flow and current distributions in the model, it gives a more realistic picture of how heat is generated and cooled in the system. By computing the temperature and state of charge, the model can predict where SEI deposition and associated degradation occur within the battery. Nonuniform temperatures across the battery pack lead to nonuniform degradation.

  Figure 8. The figure illustrates the scaling of a battery system from a single cell to a module and a full battery pack for energy storage. It also highlights the phenomena that can be more accurately captured at the module and pack levels, including flow, current distribution, and ohmic losses and heating.

  The current, temperature, state of charge, and SEI layer thickness are all modeled using the Weak Form PDE interface in COMSOL Multiphysics^®.

  Flow and temperature around the pack are modeled using the fluid flow and heat transfer interfaces in the CFD Module and Heat Transfer Module. The flow and temperature fields are coupled in the domain, while fixed-temperature and no-slip boundary conditions are imposed.

  

  Figure 9. Animation of a battery module under fast discharge. From left to right: temperature, state of charge, and SEI layer thickness. A constant room-temperature boundary condition at the outer edges of the module leads to spatial inhomogeneities.

  From these models (Figure 9), we can see that we can calculate temperature, charge, and SEI layer degradation distributions in the batteries. This information can provide battery pack manufacturers with valuable insight into how to run their systems as well as the lifetime expectancy.

  Summary

  By using surrogate models in COMSOL Multiphysics^®, we are able to simulate large systems while retaining key electrochemical behavior that would otherwise be computationally infeasible. This has enabled us to obtain information on full battery packs and how they evolve over multiple charges and discharges, as well as how degradation evolves throughout the system. This information can be used to gain important learnings before the battery is put into production or undergoes a several-thousand-hour test.

  The modeling technique we presented here is not unique to batteries. The methodology can be applied to a wide range of systems that differ in their underlying physics but share similar computational challenges, such as large scales and long transient behavior. In such cases, the surrogate modeling and DNN functionality in COMSOL^® can indeed be an important tool for investigations.

  About the Authors

  André Gugele Steckel is a senior modeling specialist at resolvent, a COMSOL Certified Consultant based in Denmark. He holds an MSc in physics and nanotechnology and a PhD in engineering physics from the Department of Physics at the Technical University of Denmark (DTU). His work spans multiphysics simulation, with a particular emphasis on acoustofluidics, electromechanical and piezoelectric transducers, and thin-film technologies. He has contributed to COMSOL Conference proceedings on battery modeling, covering topics such as methods for predicting lifetime degradation and analyzing the impact of operation and manufacturing processes on battery performance using high-fidelity models and surrogate models.

  Thomas Bisgaard was a senior simulation specialist at resolvent. He holds an MSc and a PhD in chemical engineering from DTU. His expertise spans multiphysics and multiscale mathematical modeling, process dynamics, optimization, and control, with doctoral research focused on heat-integrated distillation and optimal control. He has contributed to technical work on battery performance and lifetime modeling, combining mechanistic electrochemical models with surrogate approaches to assess the impact of operation and manufacturing processes.

  References

  1. “Trends in batteries,” IEA; https://www.iea.org/reports/global-ev-outlook-2023/trends-in-batteries
  2. M. Doyle, T.F. Fuller, and J. Newman, “Modelling the Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell,” Journal of the Electrochemical Society, 140(6), 1993; https://iopscience.iop.org/article/10.1149/1.2221597/pdf
  3. M. Safari et al., “Multimodal Physics-Based Aging Model for Life Prediction of Li-Ion Batteries,” Journal of the Electrochemical Society, 156(3): 2009, A145-153; https://iopscience.iop.org/article/10.1149/1.3043429
  4. Battery Archive; http://www.batteryarchive.org/

  Additional Resources from Resolvent

  To learn more about the topic discussed in this blog post, as well as the work of Resolvent, see:

  • “Extending battery pack lifetime using virtual design and testing”, an article by Resolvent
  • Resolvent’s COMSOL Certified Consultants page

  Acknowledgements

  The authors would like to acknowledge the financial support by the European M-ERA.NET 3 call (project9468 LaserBATMAN), Innovation Fund Denmark (grant number 1139-00001), and the Swedish Governmental Agency for Innovation Systems (Vinnova grant number 2022-01257). The project aims to optimize battery pack manufacturing with a focus on joining processes. The consortium is comprised of the following companies and institutions: University of Skövde (Sweden), Technical University of Denmark (Denmark), Volvo Group Trucks Operations (Sweden), Aurobay Powertrain Engineering Sweden (Sweden), and Resolvent (Denmark).

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