What if there was a bright side to hitting a pothole? Innovations in vehicle suspension technology could make this possible. Potential developments include a method for converting kinetic energy into electrical energy to power vehicles, softwaredriven shocks that can mitigate potholes, and mechanical suspension settings that adjust with voice commands.
Enhanced suspension systems are not possible without first developing a strong foundation. The suspension system in any vehicle, after all, needs to adapt to load variations, absorb dips and bumps in the road, and more. If not, common suspension problems arise, such as poor wheel alignment, wearing springs, and damaged dampers.
An example of a chassis with a suspension system. Image by Christopher Ziemnowicz — Own work. Licensed under CC BYSA 2.5, via Wikimedia Commons.
By setting up a simplified lumped model in the COMSOL Multiphysics® software, you can analyze and optimize vehicle suspension system designs.
Available as of version 5.3a of COMSOL Multiphysics®, the Lumped Mechanical System interface can be used for modeling discrete mechanical systems in a nongraphical format. This can be in terms of masses, dampers, and springs. You have the option of connecting these systems to a 2D or 3D Multibody Dynamics interface. When modeling a lumped mechanical system, you can use both the Lumped Mechanical System and Multibody Dynamics interfaces within the Multibody Dynamics Module.
In this tutorial, the lumped model of the vehicle suspension system has three main components:
The lumped model of a vehicle suspension system with three main components.
Each wheel has one degree of freedom (DOF) and is represented by a green circle in the image above. Each seat is represented by a blue circle and also has one DOF. At the center of gravity, the body has three DOFs that account for the system’s rotation:
You can use a Rigid Domain node and Prescribed Displacement/Rotation subnode in the Multibody Dynamics interface to restrict the number of DOFs for the body.
To model the wheel and seat, you use the Mass, Spring, and Damper nodes within the Lumped Mechanical System interface. The full vehicle model includes all four wheels and four seats, and both components are defined as a subsystem.
In the schematic below, the mass (m), spring (k), and damper (c) are shown. The lumped model of the wheel accounts for its mass and stiffness, as well as the stiffness and damping of the vehicle suspension. The lumped model of the seat accounts for its stiffness and damping, as well as the mass of the passenger.
The lumped model of a wheel and seat.
The Lumped Mechanical System interface enables you to model the vehicle body as an External Source in the lumped mechanical system. This helps to connect the suspension system with the vehicle body at the wheelbody and bodyseat points.
Through a transient analysis, you can compute both the vehicle motion and seat vibration levels for a given road profile. In this scenario, the bump height for the road is 4 cm and the width is 7.5 cm. The vehicle is assumed to be moving with a constant speed of 40 km/h. The road profile is modeled by assuming a series of bumps on the road, but only the left wheels of the vehicle are assumed to be moving over the bumps.
Let’s take a look at the time history of the vehicle’s roll, pitch, and heave. These results could be useful for designing shocks that intuitively reduce the amount of roll, pitch, and heave after hitting a pothole.
As shown below, the roll rotation is larger than the pitch rotation for the given road excitation as the left side of the vehicle is moving over the bumps given in the road profile. You can also see the corresponding velocities for the roll, pitch, and heave motions in the velocity plot below on the right. Two different frequencies — low and high — correspond to the natural frequencies for the components of the system.
Vehicle roll, pitch, and heave motions at the center of gravity (left) and corresponding vehicle velocities (right).
If you want to harness the kinetic energy induced by hitting a pothole, for example, you need to determine how the vehicle moves and the rate at which it moves. In this case, you can analyze the time history of displacement and acceleration at all four seat locations. The seat displacement results show that the left side of the vehicle has a much larger displacement because this side goes over the bumps in the road, whereas the right side does not.
Time history of seat displacements (left) and seat accelerations (right).
Finally, to determine how soft or hard the suspension is and modify it accordingly, we want to find out what the forces are in the springs. The results show that the force magnitude in the spring and damper of a wheel is much larger than that of a seat. This is because the force is absorbed by the inertia of the wheels and the vehicle body, so only a fraction of the force is transmitted from the wheel to the seat. Additionally, the frequency of vibration is much lower for the forces in the seat compared to the forces in the wheel — making for a smoother ride.
Forces in the springs and damper of the frontleft wheel (left) and frontleft seat (right).
This simplified model provides a solid foundation for analyzing vehicle suspension, which you can then compare to data from experiments. With verified results, you can enhance suspension system designs for realworld performance.
Try the Lumped Model of a Vehicle Suspension System tutorial yourself via the button above. From there, you can download the MPHfile if you have a COMSOL Access account and a valid software license.
]]>
In an interference fit, also known as a press fit, two parts are joined together with minimal space in between. Initially, the inner object has a slightly larger diameter than the outer one. The outer part is heated until it is large enough to fit the inner part. After the inner part is inserted, the outer object shrinks over it during cooling, which strengthens the fastening. A suboptimal interference fit can cause the parts to come loose, bulge excessively because of permanent plastic deformations, or make it impossible for the parts to fit together. However, with the right interference fit, the parts essentially fuse into one piece that can be held together, even when met with a large amount of torque.
A new take on the popular Goldilocks fairy tale: Instead of a bowl of porridge heated to her preferred temperature, Goldilocks finds an interference fit for two pipes that is neither too tight nor too loose.
An optimized interference fit reduces unwanted effects in a structure. For instance, in ball bearing assemblies, the interference fit protects against the bearing sliding on the shaft. If the interference fit is too loose, sliding will still occur, but if it is too tight, the ball bearing will experience increased operation temperature and wear particles.
For the best performance of a structure, the interference fit needs to be optimized with the application in mind. By building a simulation app, it is possible to efficiently evaluate parameters that affect the interference fit between two parts.
The Interference Fit Calculator computes and visualizes the interference fit between two pipes. The app includes an Input section where app users can enter different geometry parameters to quickly and easily test designs.
In this example, the inputs include:
The userfriendly interface of the app makes it easy to compute and visualize the results of the interference fit analysis. You can include buttons such as Solve, Reset to Default, Create Report, and Open Documentation for app users to run and view different analyses.
The different Results tabs of the app enable you to visualize how slight parameter changes affect the interference fit, which are computed by the underlying model. The results show the maximum transferable torque and axial force, as well as the effective stress, contact pressure, and pipe deformation for different inputs.
The Interference Fit Calculator is an example of what you can create with the Application Builder, a builtin tool included with the COMSOL Multiphysics® software. As you are in control of an app’s design, you can include different inputs and outputs to suit your needs.
The Interference Fit Calculator in action.
With a simulation app, you can test different parameters to optimize the interference fit for your specific structural application.
Click the button below to try the Interference Fit Calculator example.
Rolling element bearings typically consist of four parts:
The inner race is connected to the shaft and the outer race is connected to the bearing pedestal. Multiple rolling elements are inserted between the inner and outer races, which enables them to slide relative to each other with the help of the rolling motion of the elements. The cage keeps the rolling elements separated from each other.
Front view of a typical rolling element bearing with an offcentered shaft.
A rolling element bearing helps support loads while permitting the constrained motion of the inner race relative to the outer race. Bearing models are available in COMSOL Multiphysics for the following bearing types:
There is a point contact between the rollers and the races in deep groove ball bearings, angular contact ball bearings, selfaligning ball bearings, and spherical roller bearings. However, in cylindrical and tapered roller bearings, there is a line contact. Generally, line contact bearings have a larger loadcarrying capacity than point contact bearings. To increase the loadcarrying capacity of a bearing, multiple rows of rolling elements are sometimes used instead of a single row.
The bearing geometry plays an important role in deciding its application area. For example, deep groove ball and cylindrical roller bearings cannot support axial loads, whereas angular contact ball bearings and tapered roller bearings can withstand significant axial loads. Selfaligning ball bearings are another special example, where the shafts can tilt within the bearings, thus making them suitable for cases of misaligned mounting. Common application areas for the different bearing types are shown below:
Application areas of the different types of bearings.
Roller bearings in COMSOL Multiphysics are the abstract models for the contact between the rollers and races based on the Hertz contact theory. Therefore, you need to input the geometric parameters to account for the specific characteristics of the bearings. The images below show the geometric parameters of the different types of bearings with two rows of rollers.
Sketches of the deep groove ball bearing (left), angular contact ball bearing (center), and selfaligning ball bearing (right).
Sketches of the cylindrical roller bearing (left), spherical roller bearing (center), and tapered roller bearing (right).
In addition to the geometric parameters, material parameters of the rollers and the races play an important role in the bearing characteristics. A nonlinear Hertzian contact law is used to determine the deformation of the rollers and the transmitted force vector from the inner race to the outer race.
The clearance between the rollers and races is a very important parameter that strongly affects the rotor vibration. A smaller clearance reduces the highfrequency vibration but increases the torque required to operate the bearings. A very large clearance, however, can excite the highfrequency vibrations in the rotor with the large forces and moments acting on the bearing foundation, which is best avoided.
Let’s take a look at the effect of the roller bearing clearance on the rotor vibration by considering the rotor assembly of a blower in a continuous casting machine. Continuous casting is a process through which molten metal is solidified into billets. The blower in the casting machine accelerates the cooling of the molten metal stream, which enters into the mold by blowing cold air on the mold.
Schematic of the rotor assembly.
The blower assembly consists of a driving motor connected to a shaft, which is connected to the blower fan. The shaft is supported by two roller bearings placed between the motor and blower. Thus, the blower fan is overhung on the bearing support.
Axial rotation of the rotor in combination with the bending of the shaft due to overhung weight gives rise to the whirling motion in the rotor. Also, due to the contact between the rollers and the races, highfrequency vibrations can be induced in the rotor. A timedependent analysis is performed to capture this vibration in the shaft for different bearing clearances.
The shaft is modeled using the Beam Rotor interface in COMSOL Multiphysics, which uses beam elements based on Timoshenko theory. The shaft at the motor end is considered to be fixed and is modeled using the Journal Bearing feature, and the fan is modeled using the Disk feature using its mass and moment of inertia.
A Radial Roller Bearing feature is used to model the bearing, which requires the geometric and material properties of the bearing components. The rotor rotates at 2000 rpm and the whole system is subjected to gravity load. The effect of the roller bearing clearance on the shaft vibration is analyzed by considering three different clearances values: C = 1e5 m, 1e4 m, and 1e3 m.
Physics features for modeling the rotor system.
A simulation is performed for 1 second with a time step of 1e3 second. The orbit of the shaft at the fan end for different clearances is shown in the figure below:
Orbits of the shaft at the fan end for different bearing clearances (centershifted for C = 1e4 m and C = 0.001 m).
From the orbits, it is clear that for a smaller clearance, the vertical motion is smaller than it is for the larger clearances. However, the horizontal motion of the shaft for the small bearing clearance is larger than it is for larger bearing clearances. Therefore, at smaller clearances, the contact between the races and rollers is maintained at all times. When the clearance increases, contact can become intermittent, causing the impact between the races and rollers. Bearing forces at the different clearances shown below confirm this behavior.
The vertical reaction of Bearing 2 (closer to the fan) is upward, supporting the overhung weight of the fan. However, the vertical reaction of Bearing 1 is downward the entire time because of the bending of the shaft due to the overhung weight. It is also clear that the horizontal reaction of the bearings is very intermittent at the large clearances, indicating seldom contact between the rollers and the races in the horizontal direction.
Horizontal reaction of Bearing 1 (left) and vertical reaction of Bearing 1 (right).
Horizontal reaction of Bearing 2 (left) and vertical reaction of Bearing 2 (right).
The intermittent force may induce highfrequency vibration in the rotor. A frequency spectrum of the shaft’s horizontal motion in the bearing closer to the motor shows the presence of highfrequency vibration at large clearances. As the clearance decreases, the highfrequency vibrations are less significant.
Frequency spectrum of the horizontal motion of the shaft at Bearing 1.
During the rotor’s operation, due to wear, the bearing clearance may increase over a period of time. Because of this, the vibration response of the rotor will have highfrequency content too. If the measured rotor response has some highfrequency content, it could be an indicator that the bearings are worn out and should be replaced.
Learn more about using the COMSOL® software for rotordynamics analysis by clicking the button below.
One common treatment for atherosclerosis is a procedure called percutaneous transluminal angioplasty, which removes or compresses unwanted plaque that has built up in a patient’s coronary artery. This procedure sometimes relies on stents, placed within a blocked artery by an angioplasty balloon.
After reaching the intended location, the balloon inflates the stent, which locks into an expanded position. The balloon is then deflated and removed, while the stent remains in the artery. The expanded stent functions like a scaffold, keeping the blood vessel open and enabling blood to flow normally.
A stent example. Image by Lenore Edman — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
Of course, for the angioplasty procedure to be a success, the tools used must perform as intended. If the ends of the stent expand more than its middle — a common defect known as dogboning — the artery can face serious damage. Another potential issue is foreshortening, which makes it challenging to position the stent and can also damage the artery.
To avoid these issues and make the angioplasty a success, it’s necessary to evaluate stent designs. One step in this process is analyzing the deformation experienced by a stent.
For this example, let’s examine a PalmazSchatz stent model, the geometry of which is seen below. This model looks at the stress and deformation in a stainless steel stent that is expanded via a radial outward pressure on the tube’s inner surface. (The pressure represents the balloon expansion.) The original diameter of the stent is 0.74 mm, but after the expansion period, the middle section has a diameter of 2 mm.
Thanks to the inherent symmetry of the stent’s geometry, we can minimize the computational costs of this simulation by reducing the size of the model to 1/24 of its original geometry.
The full stent geometry. The reduced geometry used in this example is represented by the darker meshed area.
First, let’s look at the various stresses and strains experienced by the stent during operation. Below, we see the stress distribution in the stent at maximum balloon inflation (left) and the residual stress in the stent after balloon deflation (right). As expected, stress in the stent is reduced after the balloon deflates.
Stress in the stent during balloon expansion (left) and after balloon deflation (right).
Moving on, we analyze how the effects of dogboning (blue) and foreshortening (green) change in relation to pressure during balloon inflation. Using this plot, we can check for potentially harmful effects in the stent design and optimize its performance.
Dogboning and foreshortening in the stent vs. the pressure in the angioplasty balloon.
We also examine the effective plastic strains in the tube at maximum dogboning, as seen in the following image.
Effective plastic strains and deformation at the time of maximum dogboning. The peak value is about 25%.
In regard to the recoil parameters, note that the longitudinal recoil is around −0.9%, the distal radial recoil is about 0.4%, and the central radial recoil is approximately 0.7%. These parameters provide more details on how the stent behaves when the inflated balloon is removed.
With the information provided by simulations like this one, engineers can improve the design of stents and optimize their use in biomedical applications. To try this example for yourself, click on the button below.
The process of natural convection, also called buoyancy flow or free convection, involves temperature and density gradients that cause a fluid (like air) to move, leading to the transport of heat. Unlike forced convection, no fans or external sources are needed to generate fluid flow — just differences in temperature and density.
Natural convection in air has a wide range of applications in various industries. In the electronics field, this phenomenon dissipates heat in devices, which helps prevent them from overheating. Additionally, structures like solar chimneys and Trombe walls take advantage of this heat transport method to heat and cool buildings. The agricultural industry also depends on natural convection, which helps in the drying and storage of various products.
Natural convection of air through vertical circuit boards.
With the COMSOL Multiphysics® software, it is possible to study natural convection in air for both 2D and 3D models. Let’s take a look at one example…
The Buoyancy Flow in Air tutorial shows how to model natural convection in air for two geometries:
In both cases, all of the edges are insulated except for the left and right sides, which are set to a low and high temperature, respectively. The temperature difference (around 10 K) leads to density gradients in the air, generating buoyancy flow. Note that the cube has more sides than the square, which influences how the air flows.
To simplify the model setup, there are a couple of builtin features in COMSOL Multiphysics that we can use. First up is the predefined Nonisothermal Flow interface, which couples fluid dynamics and heat transfer in the model. We can also use the Material Library to easily determine the thermophysical properties of air.
Next, we can estimate the flow regime by computing the Grashof, Rayleigh, and Prandtl numbers. The Grashof and Rayleigh numbers suggest that the flow is laminar, with a velocity of around 0.2 m/s. As for the Prandtl number, it indicates that viscosity doesn’t influence the buoyancy of the air and that the shear layer thickness is about 3 mm.
For more details on estimating the flow regime, download the model documentation from the Application Gallery.
Note: The Buoyancy Flow in Water tutorial model demonstrates a similar model setup with water instead of air.
Let’s take a look at the results, starting with the velocity magnitude of air in the 2D square. In the left image below, we see that the velocity increases as the air nears the left and right edges, with a maximum velocity of 0.05 m/s. While this is a bit lower than the estimated velocity calculated using the Grashof and Rayleigh numbers, it is still in the same order of magnitude. Further, the shear layer thickness (3 mm) corresponds with the estimate from the Prandtl number.
The velocity magnitude (left) and velocity profile (right) of air in the 2D square.
As shown below, the results for the velocity magnitude in the 3D cube are similar to those for the 2D square.
Velocity magnitude in the cube.
Next up, let’s look at the temperature results for the 2D geometry. A single convective cell fills the square, with the air flowing around the edges. We see that the flow of air is faster at the left and right sides, where the temperature differences are the greatest.
The temperature field in the square.
The 3D results show a slightly different scenario. There are small convective cells in the cube at the corners of a vertical plane perpendicular to the heated sides. As mentioned, this difference is likely due to how the front and back sides in the cube affect the airflow.
The temperature and velocity fields in the 3D cube.
The model geometries in the Buoyancy Flow in Air tutorial are rather simple, but the example provides you with a solid foundation for modeling natural convection in more detailed models that represent realworld applications.
For more details about this example, go to the Application Gallery via the button above. From there, you can download the MPHfile and stepbystep instructions on how to build the model.
]]>
During the TPV cell energy production process, fuel burns within an emitting device that intensely radiates heat. Photovoltaic (PV) cells capture this radiation and convert it into electricity, with an efficiency of 1–20%. The required efficiency depends on the intended application of the cell. For example, efficiency is not a major factor when TPVs are used to cogenerate electricity within heat generators. On the other hand, efficiency is critical when TPVs are used as electric power sources for vehicles.
Left: Simplified schematic depicting the electricity generation process of a TPV. Right: An image from a prototype TPV system. Right image courtesy Dr. D. Wilhelm, Paul Sherrer Institute, Switzerland.
To improve the efficiency of TPV systems, engineers need to maximize radiative heat transfer, but this comes with a catch. The more radiation in the system, the less radiation converted to electric power. These losses — as well as conductive heat transfer — raise the temperature of the PV cell. If the temperature increases too much, it can exceed the operating temperature range of the PV cell, causing it to stop functioning.
One option for increasing the operation temperature of a TPV system is to use highefficiency semiconductor materials, which can withstand temperatures up to 1000°C. Since these materials tend to be expensive, engineers can reduce costs by combining smallerarea PV cells with mirrors that focus radiation onto the cells. Of course, there is a limit to how much the beams can be focused, since the cells overheat if the radiation intensity gets too high.
Engineers designing TPV devices need to find optimal system geometries and operating conditions that maximize performance, minimize material costs, and ensure that the device temperature stays within the operating range. Heat transfer simulation can help achieve these design goals.
This example uses the Heat Transfer Module and the SurfacetoSurface Radiation interface to determine how operating conditions (e.g., the flame temperature) affect the efficiency of a normal TPV system as well as the temperature of the system’s components. The goal is to maximize surfacetosurface radiative heat fluxes while minimizing conductive heat fluxes. In this model, the effects of geometry changes are also evaluated.
The model geometry includes an emitter, mirrors, insulation, and a PV cell that is cooled by water on its back side. For details on setting up this model — including how to add conduction, surfacetosurface radiation, and convective cooling — take a look at the TPV cell model documentation.
The TPV system model geometry.
To minimize the computational costs of the simulation, we use sector symmetry and reflection to reduce the computational domain to one sixteenth of the original geometry. When modeling the surfacetosurface radiation, we expand this view to account for the presence of all of the surfaces in the full geometry.
First, let’s check the voltaic efficiency of the PV cell for a range of cell temperatures. In doing so, we see that the efficiency decreases as the temperature increases. When the temperature of the cell exceeds 1600 K, the efficiency is 0. As such, the maximum operational temperature for the PV cell design is 1600 K.
Plotting PV cell voltaic efficiency versus temperature.
In the next plots, we see how the temperature of the emitter affects the temperature of the PV cell and the electric output power. The cell temperature plot (left image below) indicates that the emitter temperature must be under ~1800 K to keep the PV cell below its maximum operating temperature of 1600 K.
Keeping this in mind, let’s take a look at the electric power output results (right image below). From the results, we conclude that the maximum electric power is achieved when the emitter temperature is ~1600 K.
Plotting PV cell temperature (left) and electric output power (right) against operating temperature.
Moving on, let’s examine the temperature distribution in the PV cell for the optimal operating condition (left image below) and compare it to a temperature that exceeds this operating temperature (right image below). The two plots highlight how the device’s temperature distribution varies due to operating conditions.
The stationary temperature distribution in the full TPV system when the emitter temperature is 1600 K (left) and 2000 K (right).
Looking closer at the plot of the optimal emitter temperature of 1600 K, we see that the PV cells are heated to a sustainable temperature of slightly above 1200 K. It is important to note that the outside part of the insulation reaches a temperature of 800 K, indicating that a large amount of heat is transferred to the surrounding air. In addition, the irradiative flux significantly varies around the PV cell circumference and insulation jacket.
To determine the cause of this variation, we generate a plot of the irradiative flux for a single sector of symmetry at a temperature of 1600 K. The graph indicates that the variation is caused by shadowing and is related to the mirror positions. Using this plot, we could optimize the cell size and placement of the mirrors for a PV design.
The irradiation flux at the TPV cell, insulation inner surface, mirrors, and emitter.
Using models like the one discussed here, engineers can efficiently find optimal operating conditions for TPV devices, minimizing prototype development and testing.
To try this TPV cell example yourself, download the model files above.
]]>
Picture a micromirror as a single string on a guitar. The string is so light and thin that when you pluck it, the surrounding air dampens the string’s motion, bringing it to a standstill.
Because this damping effect is important to many MEMS devices, micromirrors have a wide variety of potential applications. For instance, these mirrors can be used to control optic elements, an ability that makes them useful in the microscopy and fiber optics fields. Micromirrors are found in scanners, headsup displays, medical imagers, and more. Additionally, MEMS systems sometimes use integrated scanning micromirror systems for consumer and telecommunications applications.
Closeup view of an HDTV micromirror chip. Image by yellowcloud — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
When developing a micromirror actuator system, engineers need to account for its dynamic vibrating behavior and damping, both of which greatly affect the operation of the device. Simulation provides a way to analyze these factors and accurately predict system performance in a timely and costefficient manner.
To perform an advanced MEMS analysis, you can combine features in the Structural Mechanics Module and Acoustics Module, two addon products to COMSOL Multiphysics. Let’s take a look at frequencydomain (timeharmonic) and transient analyses of a vibrating micromirror.
We model an idealized system that consists of a vibrating silicon micromirror — which is 0.5 by 0.5 mm with a thickness of 1 μm — surrounded by air. A key parameter in this model is the penetration depth; i.e., the thickness of the viscous and thermal boundary layers. In these layers, energy dissipates via viscous drag and thermal conduction. The thickness of the viscous and thermal layers is characterized by the following penetration depth scales:
where is the frequency, is the fluid density, is the dynamic viscosity, is the coefficient of thermal conduction, is the heat capacity at constant pressure, and is the nondimensional Prandtl number.
For air, when the system is excited at a frequency of 10 kHz (which is typical for this model), the viscous and thermal scales are 22 µm and 18 µm, respectively. These are comparable to the geometric scales, like the mirror thickness, meaning that thermal and viscous losses must be included. Moreover, in real systems, the mirrors may be located near surfaces or in close proximity to each other, creating narrow regions where the damping effects are accentuated.
The frequencydomain analysis provides insight into the frequency response of the system, including the location of the resonance frequencies, Qfactor of the resonance, and damping of the system.
The micromirror model geometry, showing the symmetry plane, fixed constraint, and torquing force components.
In this example, we use three separate interfaces:
By modeling the detailed thermoviscous acoustics and using the Thermoviscous Acoustics, Frequency Domain interface, we can explicitly include thermal and viscous damping while solving the full linearized NavierStokes, continuity, and energy equations. In doing so, we accomplish one of the main goals for this model: accurately calculating the damping experienced by the mirror.
To set up and combine the three interfaces, we use the AcousticsThermoviscous Acoustics Boundary and ThermoviscousAcousticsStructure Boundary multiphysics couplings. We then solve the model using a frequencydomain sweep and an eigenfrequency study. These analyses enable us to study the resonance frequency of the mirror under a torquing load in the frequency domain.
Let’s take a look at the displacement of the micromirror for a frequency of 10 kHz and when exposed to the torquing force. In this scenario, the displacement mainly occurs at the edges of the device. To view displacement in a different way, we also plot the response at the tip of the micromirror over a range of frequencies.
Micromirror displacement at 10 kHz for phase 0 (left) and the absolute value of the zcomponent of the displacement field at the micromirror tip (right).
Next, let’s view the acoustic temperature variations (left image below) and acoustic pressure distribution (right image below) in the micromirror for a frequency of 11 kHz. As we can see, the maximum and minimum temperature fluctuations occur opposite to one another and there is an antisymmetric pressure distribution. The temperature fluctuations are closely related to the pressure fluctuations through the equation of state. Note that the temperature fluctuations fall to zero at the surface of the mirror, where an isothermal condition is applied. The temperature gradient near the surface gives rise to the thermal losses.
Temperature fluctuation field within the thermoviscous acoustics domain (left) and the pressure isosurfaces (right).
The two animations below show a dynamic extension of the frequencydomain data using the timeharmonic nature of the solution. Both animations depict the mirror movement in a highly exaggerated manner, with the first one showing an instantaneous velocity magnitude in a cross section and the second showing the acoustic temperature fluctuations. These results indicate that there are highvelocity regions close to the edge of the micromirror. We determine the extent of this region into the air via the scale of the viscous boundary layer (viscous penetration depth). We can also identify the thermal boundary layer or penetration depth using the same method.
Animation of the timeharmonic variation in the local velocity.
Animation of the timeharmonic variation in the acoustic temperature fluctuations.
When the problem is formulated in the frequency domain, eigenmodes or eigenfrequencies can also be identified. From the eigenfrequency study (also performed in the model), we can determine the vibrating modes, shown in the animation below (only half the mirror is shown as symmetry applies). Our results show that the fundamental mode is around 10.5 kHz, with higher modes at 13.1 kHz and 39.5 kHz. The complex value of the eigenfrequency is related to the Qfactor of the resonance and thus the damping. (This relationship is discussed in detail in the Vibrating Micromirror model documentation.)
Animation of the first three vibrating modes of the micromirror.
As of version 5.3a of the COMSOL® software, a different take on this example solves for the transient behavior of the micromirror. Using the same geometry, we extend the frequencydomain analysis into a transient analysis. To achieve this, we swap the frequencydomain interfaces with their corresponding transient interfaces and adjust the settings of the transient solver. In the simulation, the micromirror is actuated for a short time and exhibits damped vibrations.
The resulting model includes some of the most advanced air and gas damping mechanisms that COMSOL Multiphysics has to offer. For instance, the Thermoviscous Acoustics, Transient interface generates the full details for the viscous and thermal damping of the micromirror from the surrounding air.
In addition, by coupling the transient perfectly matched layer capabilities of pressure acoustics to the thermoviscous acoustics domain, we can create efficient nonreflecting boundary conditions (NRBCs) for this model in the time domain.
Let’s start with the displacement results. The 3D results (left image below) visualize the displacement of the micromirror and the pressure distribution at a given time. We also generate a plot (right image below) to illustrate the damped vibrations caused by thermal and viscous losses. The green curve represents the undamped response of the micromirror when the surrounding air is not coupled to the mirror movement. The timedomain simulations make it possible to study transients of the system, like the decay time, and the response of the system to an anharmonic forcing.
Micromirror displacement and pressure distribution (left) and the transient evolution of the mirror displacement (right).
We can also examine the acoustic temperature variations surrounding the micromirror. The isothermal condition at the micromirror surface produces an acoustic thermal boundary layer. As with the frequencydomain example, the highest and lowest temperatures are located opposite to one another.
In addition, by calculating the acoustic velocity variations of the micromirror, we see that a noslip condition at the micromirror surface results in a viscous boundary layer.
Acoustic temperature variations (left) as well as acoustic velocity variations for the xcomponent (center) and zcomponent (right).
These examples demonstrate that we can analyze micromirrors using advanced modeling features available in the Acoustics Module in combination with the Structural Mechanics Module. For more details on modeling micromirrors, check out the tutorials below.
]]>
Modeling the transport of heat and moisture through porous materials, or from the surface of a fluid, often involves including the surrounding media in the model in order to get accurate estimates of the conditions at the material surfaces. In the investigations of hygrothermal behavior of building envelopes, food packaging, and other common engineering problems, the surrounding medium is probably moist air (air with water vapor).
Moist air is the environing medium for applications such as building envelopes (illustration, left) and solar food drying (right). Right image by ArianeCCM — Own work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
When considering porous media, the moisture transport process, which includes capillary flow, bulk flow, and binary diffusion of water vapor in air, depends on the nature of the material. In moist air, moisture is transported by diffusion and advection, where the advecting flow field in most cases is turbulent.
Computing heat and moisture transport in moist air requires the resolution of three sets of equations:
These equations are coupled together through the pressure, temperature, and relative humidity, which are used to evaluate the properties of air (density ; viscosity ; thermal conductivity ; and heat capacity ); molecular diffusivity and through the velocity field used for convective transport.
With the addition of the Moisture Flow multiphysics interface in version 5.3a, COMSOL Multiphysics defines all three of these equations in a few steps, as shown in the figure below.
Singlephysics interfaces and multiphysics couplings for the coupled resolution of singlephase flow, heat transfer, and moisture transport in building materials and moist air.
Whenever studying the flow of moist air, two questions should be asked:
If the answer is “yes” for at least one of these questions, then you should consider using the Moisture Flow multiphysics interfaces, found under the Chemical Species Transport branch.
The Moisture Flow group under the Chemical Species Transport branch of the Physics Wizard, with the singlephysics interfaces and coupling node added with each version of the Moisture Flow predefined multiphysics interface.
The Laminar Flow version of the multiphysics interface combines the Moisture Transport in Air interface with the Laminar Flow interface and adds the Moisture Flow coupling. Similarly, each version under Turbulent Flow combines the Moisture Transport in Air interface and the corresponding Turbulent Flow interface and adds the Moisture Flow coupling.
Besides providing a userfriendly way to define the coupled set of equations of the moisture flow problem, the multiphysics interfaces for turbulent flow handle the moisturerelated turbulence variables required for the fluid flow computation.
One advantage of using the Moisture Flow multiphysics interface is its usability. When adding the Moisture Flow node through the predefined interface, an automatic coupling of the NavierStokes equations is defined for the fluid flow and the moisture transport equations by the software (center screenshot in the image below) by using the following variables:
User interfaces of the Moisture Flow coupling, Fluid Properties feature (SinglePhase Flow interface), and Moist Air feature (Moisture Transport in Air interface).
The performance of the Moisture Flow multiphysics interface is especially attractive when dealing with a turbulent moisture flow.
For turbulent flows, the turbulent mixing caused by the eddy diffusivity in the moisture convection is automatically accounted for by the COMSOL® software by enhancing the moisture diffusivity with a correction term based on the turbulent Schmidt number . The KaysCrawford model is the default choice for the evaluation of the turbulent Schmidt number, but a userdefined value or expression can also be entered directly in the graphical user interface.
Selection of the model for the computation of the turbulent Schmidt number in the user interface of the Moisture Flow coupling.
In addition, for coarse meshes that may not be suitable for resolving the thin boundary layer close to walls, Wall functions can be selected or automatically applied by the software. The wall functions are such that the computational domain is assumed to be located at a distance from the wall, the socalled liftoff position, corresponding to the distance from the wall where the logarithmic layer meets the viscous sublayer (or would meet it if there was no buffer layer in between). The moisture flux at the liftoff position, , which accounts for the flux to and from the wall, is automatically defined by the Moisture Flow interface, based on the relative humidity.
Approximation of the flow field and the moisture flux close to walls when using wall functions in the turbulence model for fluid flow.
Note that the LowReynolds and Automatic options for Wall Treatment are also available for some of the RANS models.
For more information, read this blog post on choosing a turbulence model.
By using the Moisture Flow interface, an appropriate mass conservation is granted in the fluid flow problem by the Screen and Interior Fan boundary conditions. A continuity condition is also applied on vapor concentration at the boundaries where the Screen feature is applied. For the Interior Fan condition, the mass flow rate is conserved in an averaged way and the vapor concentration is homogenized at the fan outlet, as shown in the figure below.
Average mass flow rate conservation across a boundary with the Interior Fan condition.
Let’s consider evaporative cooling at the water surface of a glass of water placed in a turbulent airflow. The Turbulent Flow, Low Reynolds kε interface, the Moisture Transport in Air interface, and the Heat Transfer in Moist Air interface are coupled through the Nonisothermal Flow, Moisture Flow, and Heat and Moisture coupling nodes. These couplings compute the nonisothermal airflow passing over the glass, the evaporation from the water surface with the associated latent heat effect, and the transport of both heat and moisture away from this surface.
By using the Automatic option for Wall treatment in the Turbulent Flow, Low Reynolds kε interface, wall functions are used if the mesh resolution is not fine enough to fully resolve the velocity boundary layer close to the walls. Convective heat and moisture fluxes at liftoff position are added by the Nonisothermal Flow and Moisture Flow couplings. The temperature and relative humidity solutions after 20 minutes are shown below, along with the streamlines of the airflow velocity field.
Temperature (left) and relative humidity (right) solutions with the streamlines of the velocity field after 20 minutes.
The temperature and relative humidity fields have a strong resemblance here, which is quite natural since the fields are strongly coupled and since both transport processes have similar boundary conditions, in this case. In addition, heat transfer is given by conduction and advection while mass transfer is described by diffusion and advection. The two transport processes originate from the same physical phenomena: conduction and diffusion from molecular interactions in the gas phase while advection is given by the total motion of the bulk of the fluid. Also, the contribution of the eddy diffusivity to the turbulent thermal conductivity and the turbulent diffusivity originate from the same physical phenomenon, which adds further to the similarity of the temperature and moisture field.
Learn more about the key features and functionality included with the Heat Transfer Module, and addon to COMSOL Multiphysics:
Read the following blog posts to learn more about heat and moisture transport modeling:
Get a demonstration of the Nonisothermal Flow and Heat and Moisture couplings in these tutorial models:
]]>
The French scientist Barré de SaintVenant formulated his famous principle in 1855, but it was more of an observation than a strict mathematical statement:
“If the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally, but has a negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed.”
B. SaintVenant, Mém. savants étrangers, vol. 14, 1855.
Portrait of SaintVenant. Image in the public domain, via Wikimedia Commons.
Many great minds within the field of applied mechanics — Boussinesq, Love, von Mises, Toupin, and others — were involved in stating SaintVenant’s principle in a more exact form and providing mathematical proofs for it. As it turns out, this is quite difficult for more general cases, and research on the topic is still ongoing. (The argumentation has at times been quite vivid.)
Let’s start with something quite simple: a thin rectangular plate with a circular hole at some distance from the loaded edge, which is being pulled axially. If we are interested in the stress concentration at the hole, then how important is the actual load distribution?
Three different load types are applied at the rightmost boundary:
As seen in the plots below, the stress distribution at the hole is not affected by how the load is applied. The key here is, of course, that the hole is far enough from the load.
Von Mises stress contours for the three load cases.
Another way of visualizing this scenario is by using principal stress arrows. Such a plot emphasizes the stress field as a flux and gives a good feeling for the redistribution.
Principal stress plot for the three load cases. Note that there is a singularity when a point load is used.
By graphing the stress along a line, we can see that all three cases converge to each other at a distance from the edge, which is approximately equal to the width of the plate.
Stress along the upper edge as a function of the distance from the loaded boundary. The distance is normalized by the width of the plate.
If the hole is moved closer to the loaded boundary, we get another situation. The stress state around the hole now depends on the load distribution. But even more interesting is that the distance to where the three stress fields agree now is twice as far from the loaded boundary. The application of SaintVenant’s principle requires that the stresses are free to redistribute. In this case, that redistribution is partially blocked by the hole.
Stress along the upper edge with the hole closer to the loaded boundary.
Note that SaintVenant’s principle tells us that there is no difference in the stress state at a distance that is of the order of the linear dimension of the loaded area. The loaded area to be taken into consideration, however, may not be the area that is actually loaded! This statement may sound strange, but think of it this way: When the hole is far away, we may compute the stress concentration factor using a handbook (mine says 4.32) rather than by an FE solution. The handbook approach contains an implicit assumption that the load is evenly distributed as in the first load case. So even if the actual load was applied to only a small part of the boundary, the critical distance in that case is related to the size of the whole boundary.
When solving the problem using the finite element method (FEM), then the hole can be arbitrarily close to the load. What sets the limit is that from the physical point of view, the load distribution is well defined. As soon as we make assumptions about redistribution, however, there is an implicit assumption about the load distribution, which may differ from the actual one.
So far, we have said that the stresses are the same independent of the load details at some suitable distance. Since we are dealing with linear elasticity here, it is always possible to superimpose load cases. When working with proofs of SaintVenant’s principle, it is easier to formulate a principle along these lines: The stresses caused by a load system with no resulting force or moment will be small at a distance that is of the same order of magnitude as the size of the loaded boundary.
Thus, we study the stress caused by the difference between the two load systems with equal resultants. Most modern proofs are based on estimates of the decay of the strain energy density for such a zeroresultant system.
Returning to the problem above, we can compute the difference between the load cases. Doing so allows us to study the actual decay of stress or strain energy density for the difference of the stress fields.
Logarithm of strain energy density for the zeroresultant load cases.
The strain energy density along the plate for the zero resultant load cases. The energy is integrated along the vertical direction in order to produce a quantity that is only a function of the distance from the load.
The decay in the logarithm of the strain energy density is more or less linear with the distance from the loaded boundary. This is actually in line with what modern proofs predict: an exponential decay of the strain energy density. We can also clearly see how the hole temporarily reduces the decay rate.
For thinner structures like shells, beams, and trusses, it is well known that SaintVenant’s principle cannot be applied the same way as for a more “solid” object. Disturbances travel longer distances than what we expect, because the load paths in a thin structure are much more limited. This is the same phenomenon we see with the hole in the example above, but more prominently.
Here, we study a beam with a standard IPE100 cross section. The end of the beam is subjected to an axial stress, with an amplitude that has a linear distribution in both crosssectional directions.
Load distribution, displayed as contours and arrows.
Due to the symmetries, this load has a zeroresultant force, as well as zero moment around all axes. The height of the cross section is 100 mm, so if the standard form of SaintVenant’s principle is applicable, then the stresses should be small at a distance of approximately 100 mm from the end section.
Effective stress in the beam. The red contour indicates where the stress is less than 5% of the peak applied stress.
It turns out that in order for the stress to be below 5% of the peak applied stress, we have to travel almost a meter along the beam. Thus, the load redistribution is much less efficient here, since the equilibration between the top and bottom flanges requires moment transfer through the thin web.
If you are familiar with the theory for nonuniform torsion of beams (i.e., warping theory or Vlasov theory), you will recognize that the applied load has a significant bimoment. The bimoment is a crosssectional quantity with the physical dimension force X length^{2}.
Maybe (this is just my personal speculation), an efficient SaintVenant’s principle for this case should require not only force and moment but also a bimoment of zero. This can be accomplished by adding four point loads that provide a counteracting bimoment. The result of such an analysis is shown below.
Effective stress with four point loads that also provide a zero bimoment. The 5% stress contour is now much closer to the loaded boundary.
The applied point loads, which are not optimally placed on purpose, give extremely high (actually singular) local stresses. However, the stress does drop off much faster and is below 5% after about 100 mm. The 5% limit is still in terms of the applied distributed load, so it is not adjusted for the new local stresses. The logarithmic decay rate of the strain energy density is three times faster after the point loads are added.
In some cases, you can intuitively consider SaintVenant’s principle to be applicable to the FE discretized problem. Here, we look at distributed loads and nonconforming meshes.
In the FE model, loads are always applied at the mesh nodes, even though you specify them as a continuous boundary load. The load is internally distributed to the nodes of the element using the principle of virtual work, as shown in the example below.
A linearly distributed load and how it is applied at the nodes of a secondorder Lagrange element with side length L.
There is, however, an infinite number of load distributions that give the same nodal loads as long as they share the same resultant force and moment. Obviously, the solution to the finite element problem is the same for all of these cases. From SaintVenant’s principle, however, we can conclude that all such loads should give essentially the same stress field as soon as we are some distance away.
Since the size of the area over which we redistribute loads is an element face, the linear dimension after which there is no difference is essentially one element layer inside the structure. Thus, the solution in the outermost layer of elements may not correspond to the actual load, but further in, it does.
As an example, we can load a rectangular plate with a boundary load that has an exponential stress distribution. The stress computed with a fine mesh is shown below.
Contour plot of the axial stress distribution.
Because of SaintVenant’s principle, the stress field is redistributed to a pure bending state at some distance from the loaded edge, just as we expect. This, however, is not the target of the current discussion. Rather, we investigate the difference between the stress distribution above, and what we get with a number of coarse meshes.
Error in axial stress for three different meshes. Note the different scales. As expected, the error is smaller when the mesh is finer.
As can be seen in the figure, the error quickly decreases after the first element layer. What we see here is actually a combination of mesh convergence and the redistribution of stresses implied by SaintVenant’s principle.
A nonconforming mesh occurs when the shape functions in two connected elements do not match. The most common case is when an assembly is connected using identity pairs and continuity conditions. To exemplify this, we can study a straight bar with an intentionally nonmatching mesh. With a simple load case, such as uniaxial tension, it is possible to study the stress disturbances caused by the transition.
Axial stress at a nonconforming mesh transition. Secondorder elements are used.
The forces transmitted by the nodes at the two sides do not match the assumption of constant stress. Again, this can be seen as a local load redistribution over an area that is the element size. Using the reasoning of SaintVenant, the disturbance should fade away at an “elementsized” distance from the transition. Let’s investigate what happens if the mesh is refined in the axial direction.
Region with more than 0.1% error in stress. Three different discretizations are used in the axial direction.
It turns out that the region of disturbance is not affected much by the discretization in the direction perpendicular to the transition boundary. This is exactly what SaintVenant’s principle tells us.
Without making use of SaintVenant’s principle, many structural analyses are difficult to perform, simply because the detailed load distribution is not known.
The principle is formally only valid for linear elastic materials. In practice, we also intuitively use it on a daily basis for other situations. If, for example, the material in the “plate with a hole” example were elastoplastic, we would expect the two distributed loads to give equivalent results, as long as the yield stress is above the stress applied at the boundary so that there is only plastic deformation around the hole. The point load, however, always gives a different solution, since the material yields around the loaded point. For a longer discussion, read this blog post on singularities at point loads.
Learn more about using the COMSOL Multiphysics® software for FEA.
Born in 1707 in Basel, Switzerland, Leonhard Euler (pronounced “oiler”) was a prolific mathematician who published more than 800 articles during his lifetime. He studied under the famous Johann Bernoulli and received his master’s degree in philosophy from the University of Basel. Before moving to St. Petersburg, Russia, to work at the university, Euler submitted his first paper to the Paris Academy of Sciences, coming in second place at only 19 years old.
A portrait of Leonhard Euler. Image in the public domain, via Wikimedia Commons.
Euler quickly rose through the academic ranks and in 1733 succeeded Bernoulli as the chair of mathematics in St. Petersburg. Euler moved to Berlin in 1741 at the invitation of King Frederick II. In his 25 years there, he wrote around 380 articles and the first volume of his seminal book Introductio in Analysin Infinitorum, which formally defined functions for the first time; introduced the notation; popularized the and notation; and established the critical formula .
JosephLouis Lagrange (pronounced “luhgronj”) was born Giuseppe Lodovico Lagrangia in Turin. Today, this city is the capital of the region of Piedmont in Italy, but when Lagrange was born in 1736, it was ruled by the Duke of Savoy as part of the Kingdom of Sardinia. Lagrange developed an interest in mathematics and, after working independently on novel topics, began corresponding with Euler, whom he succeeded when Euler left Berlin.
A portrait of JosephLouis Lagrange. Image in the public domain, via Wikimedia Commons.
In Berlin, Lagrange developed most of the mathematics for which he is famous today. He played an important role in the development of variational calculus and came up with the Lagrangian approach to mechanics. Although Lagrangian mechanics makes the same predictions as Newton’s laws of motion, the Lagrangian functional introduced by Lagrange allows the classical mechanics of many problems to be described in a mathematically more straightforward and insightful manner than in Newtonian mechanics. Lagrange also developed the method of Lagrange multipliers, which allows constraints on systems of equations to be introduced easily in a variational approach.
The mathematical formulations of Euler and Lagrange are fundamental to the finite element method, which is used to solve equations in COMSOL Multiphysics.
In the Eulerian method, the dynamics of a system are considered from the viewpoint of an observer measuring the system’s evolution with respect to a fixed system of coordinates. This coordinate system is called the spatial frame in COMSOL Multiphysics. It could be understood to correspond to the laboratory frame in physical analysis, since the system of coordinates is oriented according to a fixed set of axes without any reference to the orientation of the components of the physical system itself.
The figure below illustrates a thin plate of material whose structural mechanics are modeled in a 2D plane. The plate is fixed to a rigid wall at the lefthand side and is deformed under its own weight, as gravity acts downward. With the results plotted in the spatial frame, we see the deformation of the object, as we would expect to observe in the laboratory.
A thin plate fixed to the gray block at the left deforms under its own weight, as viewed in the spatial (lab) frame. The deflection at the tip is about 5 mm for the given mechanical properties.
Formulating physical equations seems very natural in the Eulerian method. Indeed, this is the common formulation for problems such as electromagnetics and fluid physics, in which the field variables are expressed as functions of the fixed coordinates in the spatial frame.
For mechanical problems, though, the Lagrangian method offers a helpful alternative. In the Lagrangian method, the mechanical equations are written with reference to small individual volumes of the material, which will move within an object as it displaces or deforms dynamically. To put it another way, the object itself always appears undeformed from the point of view of the Lagrangian coordinate system, since the latter stays attached to the deforming object and moves with it, but external forces in the surroundings appear to change their orientation from the deforming object’s perspective. The corresponding coordinate system, which moves along with the deforming object, is called the material frame in COMSOL Multiphysics.
A point within the object, as measured in the spatial frame, is displaced from the position of the same point as expressed in the material frame by the mechanical displacement of that point. In the image below, we focus our view on the tip of the deforming plate in the example above and animate its deformation as the density of the object increases so that the weight increases too. As you can see, the material frame coordinate system (red grid and arrows) deforms together with the object, as the object’s dimensions in the spatial frame change. This means that anisotropic material properties — such as mechanical properties of composite materials — can be expressed conveniently in the material frame.
Zoomedin view of the tip of a thin plate deforming under its own weight, as its density is increased. The red grid denotes the material frame coordinates, tied to the object, as viewed in the spatial (lab) frame. The red and green arrows show the x and ycoordinate orientations of the material frame, as viewed in the spatial frame.
In the limit of very small strains for this type of mechanical problem, the spatial and material frames are nearly coincident, because the mechanical displacement is small compared to the object’s size. In this case, it is common to use the “engineering strain” to define the elastic stressstrain relation for the object, and the resulting stressstrain equations are linear. As the mechanical displacement increases, though, the linear approximation used to evaluate the engineering strain is increasingly inaccurate — the exact GreenLagrange strain is required. In COMSOL Multiphysics, the term “geometric nonlinearity” means that the GreenLagrange strain is used.
For further details on the mathematics, see my colleague Henrik Sönnerlind’s blog post on geometric nonlinearity.
Geometric nonlinearity is handled in COMSOL Multiphysics by allowing the spatial frame to be separated from the material frame, according to a frame transformation due to the computed mechanical displacement. It remains convenient to access the material frame to express properties such as anisotropic mechanical material properties, since these properties will usually remain aligned with the material frame coordinates, even as the object deforms.
By contrast, external forces such as gravity have a fixed orientation in the spatial frame. From the perspective of the material frame, external forces like gravity change direction as the object deforms. The image below shows the tip of the thin plate as above, but here, the displacement magnitude is plotted with colors. Arrows are used to illustrate the force due to gravity, as expressed in the material frame coordinates. Since the material frame coordinates remain fixed with respect to the object, the dimensions of the object appear not to change. However, the displacement magnitude increases with the object’s weight and the gravity force increasingly changes direction with respect to the deformed material in conditions of greater deformation.
Zoomedin view of the tip of a thin plate deforming under its own weight as its density increases. The plot is in the material frame as used for the Lagrangian formulation, so the deformation is not apparent, although displacement increases. The red arrows indicate the apparent direction of gravity (which is constant in the spatial frame) as perceived from the material frame of reference within the deforming object.
Neither the Lagrangian nor Eulerian formulation is more “physical” or “correct” than the other. They are simply different mathematical approaches to describing the same phenomena and equations. Through coordinate transformation, we can always transform the physical equations for any phenomenon from the material frame to the spatial frame or vice versa. From the perspective of interpretation and implementation, though, each approach has certain advantages and common applications. Some of these are summarized in the table below:
Strengths  Common Applications  

Eulerian Method 


Lagrangian Method 


What about multiphysics problems, such as fluidstructure interaction (FSI) or geometrically nonlinear electromechanics? In these cases, one physical equation might be formulated most naturally with the Eulerian method, while another might be better expressed with the Lagrangian method. This is where the ALE method comes in. This method solves the equations on a third coordinate system, which is not required to match either the spatial frame or the material frame coordinate systems.
The third coordinate system is called the mesh frame in COMSOL Multiphysics. There is one mathematical mapping between the spatial frame and the underlying mesh frame, and one between the material frame and the underlying mesh frame, so at all points in time, the equations formulated in the spatial and material frames can be transformed into the mesh frame to be solved.
In domains representing solids in a model, mechanical displacement is predicted using structural mechanics equations in the Lagrangian formulation. Here, the relation of the spatial and material frames is given by the mechanical displacement, as above. The ALE method adds more equations to allow the apparent positions and shapes of mesh elements in neighboring domains to displace in the spatial frame. That is in order to account for how mechanical deformation can change the shape of the boundaries of any domain where the physics are described in the Eulerian formulation. These additional equations are called a Moving Mesh or Deformed Geometry in COMSOL Multiphysics.
At boundaries between Lagrangian and Eulerian domains, a boundary condition for these additional equations requires that the displacement of the spatial frame (as defined through the moving mesh) for the Eulerian domain must match the mechanical displacement of the spatial frame away from the material frame in the Lagrangian domain. Even where no mechanical equations are solved, such that no Lagrangian method is used, the ALE method can still be used to express moving boundaries due to deposition or loss of material.
If you find the ALE method quite mathematical, that’s OK! It’s a difficult concept to follow in the abstract. To better understand the way the ALE method works, let’s take a look at an example within COMSOL Multiphysics.
The ALE method plays an important role in modeling FSI. In COMSOL Multiphysics, this method enables the automated bidirectional coupling of fluid flow and structural deformation, a capability demonstrated in our Micropump Mechanism tutorial model.
At the heart of this micropump mechanism are two cantilevers, which perform the same function as valves in conventional pumping devices. These cantilevers are flexible enough that the fluid flow causes them to deform. As fluid is alternately pumped into or out of the channel at the top, the force of the fluid flow causes the two cantilevers to deform so that fluid flows out to the right or in from the left.
The micropump mechanism. Pumping fluid into or out of the top tube produces opposite reactions in the two cantilevers, pushing fluid in or out of the chamber. Even though there is no timeaveraged net flow into the upper tube, there is a timeaveraged net movement of fluid from left to right.
The cantilevers deform enough that there is an appreciable change in the position of the boundary where the fluid and solid meet: a geometrically nonlinear case. The selfconsistent handling of the fluid’s pressure on the solid and the solid’s force on the fluid, together with the deformation of the mesh, are handled automatically by the FluidStructure Interaction interface. The interface employs the ALE method to account for the change in shape in the solid and fluid regions.
For solids, the mechanical equations with geometric nonlinearity define the displacement of the spatial frame with respect to the material frame. In the fluid equations, it’s necessary to deform the mesh on which the equations are solved in order to express the displacement of the solid boundaries in the spatial frame where the fluid equations are formulated. The deformation at the boundaries is controlled by the mechanical displacement from the solution to the structural problem. Within the fluid, though, the exact position or orientation of mesh nodes isn’t important, as the equations are formulated in the fixed spatial frame. Instead, the deformation of the mesh is smoothed in order to ensure that the numerical problem remains stable with highquality mesh elements.
To explain the ALE method for the FSI problem, we could paraphrase a common explanation for general relativity: forces due to fluid flow (Eulerian) tell the structure how to deform in the material frame (Lagrangian), while the structural deformation (Lagrangian) tells the mesh how to move in the spatial frame (Eulerian).
Top: The micropump’s operation, including pressure, flow, and cantilever deformation, as plotted in the spatial frame. Bottom: Mesh deformations calculated by the ALE method.
As of COMSOL Multiphysics version 5.3a, the Moving Mesh feature to define mesh deformation in this type of problem is located under Component > Definitions. This allows consistency in the definition of material and spatial frames between all physics included in a model, even if several physics interfaces are included. The screen capture below shows where these settings are located in the COMSOL Multiphysics Model Builder tree.
Screen capture showing Moving Mesh features under Component > Definitions, and physical coupling between two physics interfaces through Multiphysics > FluidStructure Interaction.
Turning to an electrochemical problem, the Copper Deposition in a Trench tutorial model shows that the ALE method can be vital for simulating electrodeposition problems. In this model, copper is deposited onto a circuit board that has a small “trench”. The deposited copper layer becomes thick compared to the overall size of the trench, so the size and orientation of the copper surface change appreciably as deposition proceeds. Since the rate of copper deposition at different points on this surface is nonuniform, the shape and movement of the boundary cannot be neglected.
A schematic of the physical problem being solved in the electrodeposition model.
To calculate the rate of deposition at a given point on the copper electrodeelectrolyte interface, we need the concentration of the species and the electrolyte potential of the solution adjacent to that point. As the deposition progresses and the boundary moves, the shape of the electrolyte volume has to change continuously. Similarly, the concentration and potential distributions on the altered shape must be recalculated.
The coupling of the deposition rate to the boundary motion rate and the calculation of the changing shape are accomplished with the ALE method and fully automated multiphysics couplings with the Tertiary Current Distribution and Deformed Geometry interfaces. Here, the Deformed Geometry displaces the copper surface in the spatial frame at a rate proportional to the local current density for electrodeposition, as computed from the electrochemical interface.
With this model, we can accurately account for the deposition process in order to optimize its parameters. We can also experiment with different applied potentials and deposition surface geometries to improve the uniformity of the deposition, which produces a more efficient process and a higherquality end product.
Animations showing the evolution of the deposition process in time. It is clear that the deposition happens unevenly, resulting in a pinching of the trench opening at its top.
Thermal ablation, discussed in this previous blog post, involves a very high temperature applied to an object, causing the surface to melt and vaporize. Examples of thermal ablation include the removal of material by lasers — such as in the etching process, laser drilling, or laser eye surgery — and a spacecraft’s heat shield as it reenters the atmosphere.
Animation showing the effect of thermal ablation on a material.
Since we expect that an object’s shape will change when some of its material is removed, deforming meshes are clearly a key part of thermal ablation simulation. What we need to know is how the shape of the object will change. This depends on how we balance the applied heat with heat lost to ablation and heat dissipation throughout the structure by mechanisms such as conduction.
To obtain this information, we can predict the temperature profile as a function of space and time by solving the heat transfer equations using the Heat Transfer interface. Because the mass and shape of the object are changing, the Heat Transfer interface is coupled to a Deformed Geometry interface, using the ALE method to displace the boundary according to the rate of ablation. The Heat Transfer equations predict the temperature distribution in the object as its shape evolves.
By performing these steps, we can attain accurate calculations for the thermal ablation process. Moreover, we can determine the final shape of the object after ablation is complete. This might enable us to check whether a laser weld will fall within acceptable tolerances or whether a spacecraft will survive an emergency landing.
The contributions of Leonhard Euler and JosephLouis Lagrange in the field of mathematics have paved the way for simulating a variety of systems involving multiphysics applications. The combination of their individual methods has led to the development of the ALE method, which can be used to predict physical behavior when objects deform or displace. By properly accounting for these movements, you can set up highly accurate models. Remember to thank Euler and Lagrange as you investigate these and other models that exploit the ALE method!
The ALE method is one of many builtin physics capabilities in the COMSOL Multiphysics® software. See more of them:
Biological cells are essential for life as we know it. They not only store and replicate hereditary information in the form of DNA but also are instrumental in biological processes. In most, if not all, of these processes, the mechanical behavior of cells is a main factor in ensuring normal physiological functions.
It goes without saying that we would not exist or function without cells. Vertebrates utilize the circulation of red blood cells, erythrocytes, to deliver oxygen to body tissue. Fibroblasts use their contractile machinery to migrate to — and start the healing process of — wounds. The endothelial cells lining our blood vessels serve as filtration barriers. These cells not only rely on biochemical/transport mechanisms but also on their mechanical behavior to ensure normal physiological functions.
The structural entity responsible for providing cellular stiffness is an interconnected network known as the cytoskeleton, visualized in the image below. This cytoskeleton primarily consists of three types of polymerized filaments, each with their own distinct structure and mechanical characteristics:
This complex foundation provides cells with the ability to adapt their mechanical properties to the environment, both instantaneously and over time.
A fibroblast cell with the cytoskeleton visualized, including actin (blue), intermediate filaments (green), and microtubules (red). Used with permission from Rathje et al. from the paper “Oncogenes induce a vimentin filament collapse mediated by hdac6 that is linked to cell stiffness”.
Both cells and cytoskeletal networks are highly viscoelastic, as you can see in the plot below of a relaxation curve from a cell indentation experiment by atomic force microscopy (AFM).
Force relaxation curve of a fibroblast cell.
Numerous examples exist in which a diseased cell exhibits abnormal mechanical properties, promoting the progression of pathology. The cytoskeleton found in these cells is often found to behave differently compared to healthy cells. For example, cancer cells are known to exhibit significant stiffness variations compared to control cells. In many cases, this can be linked to the cytoskeleton. The intermediate filament network could be collapsing around the nucleus or there could be increased cell spreading (closely linked to the actin cytoskeleton through focal adhesions).
As mentioned, the cytoskeleton is a dynamic entity with the capability of remodeling itself on a time scale from milliseconds to hours. A consequence of this is a pronounced viscoelastic behavior, due to the nature of the constituent networks. For example, a solution of actin filaments behaves like a solid at short time scales and a liquid at longer time scales. This is due to the link between the thermal fluctuations of semiflexible filaments and their propensity to slide between each other; i.e., they are more or less kinematically constrained at short time scales. The temperature is also an important factor, partly because it affects the thermal behavior, but also because of various linking proteins in the solution.
Taken together, the mechanical behavior of this type of underlying polymer network, together with other cell constituents (e.g., cell nucleus and membrane), it is clear that a detailed analysis accounting for all of these factors is nearly impossible. However, it is possible to circumvent this challenge and obtain results by considering the cell at a macroscopic level.
By creating a finite element model in the COMSOL Multiphysics® software, you can essentially ignore the heterogeneous intracellular structure and instead view it as a continuum; i.e., the displacement field is continuous. This is an acceptable approximation if your goal is to quantify the macroscopic cell response to external stimuli.
The computational model described in this blog post is that of a relaxation test. A rigid indenter is pressed into the soft, viscoelastic cell, and the resulting relaxation of the indentation force is measured and compared to experimental data.
A model of a cell with typical dimensions is seen below. Note that the domain is created around the centerline. The semicircular section is the cell nucleus, which will also influence the mechanical response. We also create an indenter in the geometry and neglect the cell membrane in this analysis. For simplicity, we perform a 2D analysis by assuming the cell is axisymmetric.
The model is meshed with 2D elements and refined under the indenter.
The choice of material model for the cell cytoplasm and nucleus should reflect both the instantaneous and longterm response of the material. A linear elastic model is far too simple, as cells can typically withstand large strains and exhibit significant strain hardening. For the cytoplasmic response, we can choose a simple hyperelastic material model, the neoHookean model, in which stresses and strains are computed from a strain energy density function Ψ on the form
In this form, where the material is assumed (nearly) incompressible, the shear modulus µ, elastic volume ratio J_{el}, bulk modulus κ, and isochoric first invariant are included. To incorporate the viscoelastic behavior, two generalized Maxwell branches are also included. The nucleus has been found to be mainly elastic and is therefore modeled without viscoelastic branches.
The chosen material parameters are shown in this table:
Domain  Shear Modulus  Bulk Modulus  Energy Factor 1  Relaxation Time 1  Energy Factor 2  Relaxation Time 2 

Nucleus  5 kPa  5000 kPa  N/A  N/A  N/A  N/A 
Cytoplasm  0.07 kPa  1000 kPa  10  0.5 s  10  50 s 
The bottom of the cell is constrained vertically. While in reality, the cell adheres to the substrate through focal adhesions, this should be a local effect and not significantly influence the force response.
Contact between the indenter and the cell is enforced by a penalty formulation, using the indenter as the source boundary. The indenter domain is prescribed a velocity of 0.1 µm/s, until the total vertical displacement is 4.6 µm. It is subsequently held fixed for the remainder of the analysis, up to a total time of 30 s.
The local deformation of the cell after indentation is shown in the plot below.
Deformation of the cell under the indenter.
The equivalent von Mises stresses at times 0.5 s and 30 s are shown below. Naturally, the stresses decrease due to stress relaxation because of the inclusion of viscoelastic branches for the cytoplasmic material model.
Stress distribution at 0.5 s (left) and 30 s (right).
The vertical reaction force on the indenter can be extracted from COMSOL Multiphysics and compared with experimental data.
Results for the indentation force of the cell, both experimental (blue) and computed (red).
The relaxation, as measured by experiments, typically exhibits at least two distinct regimes. These values are reasonably well predicted by the simple neoHookean model, along with its two viscoelastic branches. It should be noted that the initial indentation regime exhibits severe strain hardening prior to the constant slope (apparent in the plot above).
As discussed, COMSOL Multiphysics can be easily used to replicate the viscoelastic behavior of cells by (comparatively) simple material models. Naturally, an increasing level of complexity can be obtained by using more complicated material models. In this case, using other hyperelastic models, such as the MooneyRivlin or Ogden models, in combination with a greater number of viscoelastic branches may yield even more accurate results. Keep in mind that as more material parameters are needed, more experimental data points must be available for the material in question.
The cell is in reality a far more complex system than modeled here. There is a constant exchange of mechanical and biochemical signals that constantly alter the intracellular structure, cell shape, and locomotion behavior. Suffice to say, modeling the cell as a continuum is a major simplification, but such an approximation can serve us well in many cases. If we were to analyze metastasizing cells, for example, it may be enough to characterize their macroscopic stiffness in order to assess their capability to squeeze through tissue or arteries. In such a case, the stiffness of the cell as a whole in comparison to the obstacle would be the determining factor, not the detailed interactions of, say, the cytoskeleton and cell nucleus.
It should also be mentioned that the cell is not only a complex system but also far from deterministic and not uniquely characterized by a set of geometrical and material parameters. The response between individual cells varies depending on their health, state of locomotion, and cell cycle state, among other factors. To properly assess the mechanical cell response of a cell type experimentally, a greater number of individual cells would need to be probed. However, we are content with evaluating the capability of modeling the response of an individual cell.
In general, not only cells but also other biological materials can often be modeled by utilizing hyperelastic material models. Depending on the particular material and time scale, viscoelastic behavior can also be included. This opens up some interesting opportunities in the field of biomechanical modeling.
For example, a common type of cardiovascular condition is atherosclerosis in which white blood cells accumulate on the arterial wall, reducing blood flow and increasing the risk of a heart attack due to a blood clot. A common procedure to alleviate this condition is angioplasty, when a balloon is inserted into the artery and inflated. A mechanical stent is then often used to stabilize the artery section. Using COMSOL Multiphysics, we could capture the hyperelasticviscoelastic behavior of the arterial wall, as well as composite characteristics due to collagen fiber directions, and compute the instantaneous and transient development of stresses and strains.
Björn Fallqvist is a consultant at Lightness by Design working with product development based on numerical analysis. He obtained a PhD from the Royal Institute of Technology in 2016, working with developing constitutive models to capture the mechanical behavior of biological cells. His main professional interest and specialization is in the fields of material characterization and using various material models to capture physical phenomena.