Conducting the structural analysis of a model is an imperative step in the design process. The Structural Mechanics Module, an add-on to COMSOL Multiphysics, offers a virtually limitless amount of capabilities for you to do just that.

This video introduces you to the Structural Mechanics Module and walks you through the entire model-building process for setting up and solving a mechanical problem. This includes demonstrations on how to create parameters, named selections for different parts of your geometry, local variables to implement complicated expressions defined in the model, custom meshes, and tabulated results. To demonstrate the workflow for building and solving a structural mechanics problem, the COMSOL Multiphysics version 4.4 tutorial model of a static bracket assembly is used.

The smallest components, while often overlooked in design, can be the most instrumental ones. Brackets serve as a core component of support for many mechanical devices in numerous industries. In this model, a bracket assembly is fixed in place through eight mounting bolts. A load is applied on the two arms of the bracket, which is representative of a pin being placed between the holes in the bracket arms. As a result, the two bracket holes will experience a loading from this pin. After an initial analysis is complete, the direction of the pin load is varied through a parametric sweep to see the variations in force exertion, stress distribution, and deformation.

- Shown in the video: Bracket — Static Analysis (version 4.4)

*Today, we will be learning how to model a structural mechanics problem, in COMSOL Multiphysics. We will conduct a static analysis of a bracket assembly, and in the end, perform a parametric sweep to analyze a bearing load at different angles. So let’s get started.*

We start our modeling by opening COMSOL Multiphysics, bringing us to the New window. Here we have two options for setting up our model. Use the Model Wizard as a guide for specifying the dimension, physics, and studies you want, instead of starting with an empty model. Here we select our space dimension. When modeling in Structural Mechanics, we can work in 3D, 2D, or 2D axisymmetry, but not 1D or 0D. With our model being three-dimensional, choose 3D for the space dimension.

We are brought to the Select Physics window in the Model Wizard. Here we can add the physics that our model will exhibit. When it comes to choosing physics, this is completely dependent on the model and what meaningful information you are trying to extrapolate from the results. It is advantageous to familiarize yourself with all the physics interfaces available to you, because you may want to add more physics that are relevant to your model later on. Since we are doing a structural analysis of a 3D solid, we go under the Structural Mechanics branch, select Solid Mechanics, and add it to our model. Click Study to enter the Select Study window.

There are several different types of studies to choose from, as shown, depending on the physics interfaces used within the model. Selection of the study type is completely dependent on your analysis objectives. For example, in the case of our bracket model we want to compute deformations and stresses at static equilibrium; so the properties are time-independent. Therefore, under Preset Studies we select Stationary. After clicking Done, you are brought to the COMSOL Multiphysics desktop.

To create parameters and constants in COMSOL Multiphysics in the ribbon, add Parameters. Here we define the parameters and constants that we’ll be using later in this model, which are stored in this table.

Before we add the parameters let’s take a look at the problem we are modeling to have an understanding of where our parameters are derived from. We have a bracket, where the mounting bolts of the assembly are assumed to be fixed, and securely bonded to the bracket itself.

*A pin is placed between the two holes in the bracket arms, and the inner surfaces of the two bracket holes will experience a loading from this pin. We will want to vary the direction of the pin load to see the variations in stress distribution and deformation. After an initial analysis is done with the pin load applied to the bracket holes along the negative *y*-axis at zero degrees, we will perform a parametric sweep of the pin load direction, starting at 0 degrees, and rotating 45 degrees, up to 180 degrees. “theta0” will be used to specify the main direction of the load, and will be the parameter used later in our parametric sweep. “P0” is the peak load intensity applied to the bracket holes. Lastly, “y0” and “z0” are the coordinates of the centers of the bracket holes. It’s good practice to use parameters instead of just the numerical values. When you change these global parameters, they will update throughout the entire model.*

Creating a geometry in COMSOL Multiphysics can be done three different ways. The geometry can be manually created within COMSOL, it can be imported from a file, or you can synchronize the geometry you have open in a CAD program to COMSOL, through any of the LiveLink interfaces. In this example, the assembly of both the bracket and mounting bolts, are available to be imported. In the Geometry section of the ribbon, click Import. Change the Geometry import type to COMSOL Multiphysics file and Browse to where the “bracket.mphbin” file is stored on your computer. This should be located in your COMSOL folder under models, Structural Mechanics Module, and then Tutorial Models. Select the file, click Import, and the geometry will appear in your graphics window.

In the Model Builder Window, under the Geometry node, you can see the Form Union node which is the default setting for finalizing your geometry into COMSOL Multiphysics. Since we are dealing with a set of domains that are assumed to be perfectly bonded to each other, and will not move relative to each other, we use the default Form Union to finalize the geometry. Click Build All.

Creating definitions in COMSOL Multiphysics will help simplify your model, especially when working with large and or complicated geometries. Let’s take a look at a few of the options.

Go to the ribbon, and under the Definitions tab, in the Selections section, add a Box. Box selections allow you to create groups of geometric entities partially or completely inside the box, that would have the same features applied to them. This makes the process of changing materials, model equations, boundary conditions or constraints to different parts of your model much easier to do. In this example we want to make two box selections: the first is for the bolt domains, and the second is for the load-bearing boundaries of the bracket holes.

In the Box settings window find the Box Limits section. Here we can change the limit values which will serve as the dimensions of the box. We want to change these limit values so that the bolts are contained within the box. Under the Output Entities section, in the Include entity if list, choose Entity inside box.

Click the Wireframe Rendering button on the Graphics toolbar and we see in fact that the bolt domains are selected.

We can add a second box, or a cylinder selection to select the bracket hole boundaries, but we will instead add an explicit selection. From the level list, choose Boundary, in the graphics window, select any one of the interior boundaries of one of the bracket holes. Now we can check the box for Group by continuous tangent, and the rest of the interior boundaries will automatically be selected. In the graphics window, select any one of the boundaries of the other hole, and all four of the boundaries will be added.

*Now that we’ve added selections to our model, we can define expressions for adding the boundary load. Local Variables can be used to introduce short and descriptive names for the complicated expressions defined in the model. Go to the ribbon and under the Definitions tab click Local Variables. In the table to define the load, we need an expression for the angle and load intensity, so we enter the following. The angle variable is used to help define the load intensity. This expression evaluates the radial angle, based on its position along the global *z*-coordinate. Since our loading direction will change in only in the *y* and negative *z* directions; or equivalently the 3 ^{rd} and 4^{th} Cartesian quadrants, we can have COMSOL Multiphysics solve for the angle, by computing the four-quadrant inverse tangent. This enables calculating the arctangent in all four quadrants. The load that the bracket holes experience will be sinusoidal in nature, so the sine function is used. This last part of this expression is added to make sure that the load is only applied to the bottom half of each bracket hole.*

*COMSOL uses a global Cartesian coordinate system by default to specify material properties, loads, and constraints in all physics interfaces and on all geometric entity levels. For this model we want to define the orientation of the load applied to the bracket holes. Since the load direction will be rotating about the negative *z*-axis, we need to create a rotated coordinate system. In the Coordinate Systems section of the ribbon, choose Rotated System. This creates a rotated coordinate system, relative to the global system, that defines the orientation of the load applied to the bracket holes. Under the Euler angles subsection, in the beta field type “-theta0”.*

COMSOL Multiphysics comes with a Material Browser, complete with built-in material properties for common materials, as well as materials for specific applications, and any materials created by you, the user. The addition of the Material Library grants users access to the entire COMSOL Multiphysics database of materials. Under the Built-In node, scroll down to select Structural Steel, click Add to Component, and we are done. The material has been automatically assigned to all domains. Here we can see the properties of the newly assigned material. You are free to create your own materials using the New Material function, and you can also use the Add Material button to stay within the main user interface.

Defining the physics and boundary conditions in COMSOL is made as easy as possible, to let you focus on what matters, the physics. To start go to the ribbon and click the Physics tab. Each selection level comes with the various physical properties that can be applied. You can learn about each physical property by adding it, and clicking the Help button in the top right corner of the window.

We first want to set the constraints acting on the structure. Since the mounting bolts are fixed in place, click on the Domains button and add a Fixed Constraint. Under the Domain Selection section from the Selection list, choose Box 1. This assumes that the bolts are rigid and the displacements are perfectly constrained. Next, we want to define the loads acting on the structure. Since the inner surfaces of the bracket holes experience the pin load, in the Physics tab, click the Boundaries button and choose Boundary Load. Choose Explicit 1 for the Selection. Under the Coordinate System Selection section, from the Coordinate System list, choose Rotated System 2, setting the load orientation with a value of “theta0”. Under the Force section, specify the Load vector with the following.

Whenever building a finite element model, we may want to customize the mesh if we anticipate that higher accuracy is needed in some parts of the model. Although we can solve this model with the default mesh, I will demonstrate how to use the mesh settings to get a finer mesh in some regions. In the ribbon, go to the Mesh tab and select Mesh 1. This shows the bracket geometry with the default Normal mesh applied. Although the elements appear as having straight sides, the default mesh used for solid mechanics problems is a second order, or quadratic, mesh. This means that the elements are conformal to the curved geometry.

We will create a second mesh and customize the mesh via the Element size parameters. Click the Add Mesh button. In the Mesh settings window, change the sequence type to User-controlled mesh. This will generate a Size sub node under our second Mesh. Click the Size node. Under the Element Size section, click on Custom. This automatically expands the Element Size Parameters window where the element parameters can be changed. Reducing the Curvature factor to “0.3″ and rebuilding the mesh, results in a finer mesh around the bracket holes. Rotating the geometry, and zooming in on a bolt, we can see the mesh is quite coarse around these small curved faces. The Minimum element size parameter is preventing the changes from the curvature factor. Reduce it to “0.005″ and rebuild the mesh. The elements around the curved edges are now smaller, but the bulk of the geometry remains relatively unchanged.

You can continue to refine the mesh manually if you want. It is also possible to use Adaptive Mesh Refinement to let the software automatically refine and coarsen the mesh, if desired. For the purposes of this example, we will continue with the default Normal mesh size setting.

We continue our simulation with creating a mesh using the default options. Go to the ribbon and in the Home tab, click Build Mesh. Then, in the Study section, click on the eye glasses icon labeled Study 1. Notice the Generate default plots check box. This will create a plot automatically, based on the structural mechanics physics, so in this case, a Stress plot will be created. To solve a stationary study in COMSOL, it is as simple as clicking Compute. COMSOL Multiphysics also defines a solver sequence for the simulation based on the physics and the stationary study type.

After a model has solved in COMSOL, it is time to postprocess the results. We will show you how to add to an existing plot, create a new plot, and extract information from the results.

Here we see the von Mises stress in the bracket and an exaggerated picture of the deformation, which is occurring mostly in the bracket arms. We also want to visualize the plot with vectors, so we can better see the pressure distribution on the inner surfaces of the bracket holes. Under the contextual Stress tab, in the Add Plot section click Arrow Surface. In the Arrow Surface settings window you’ll see an Expression section. From the menu choose Solid Mechanics, Load, and then Spatial load. In the Coloring and Style section, under Number of arrows, the default setting is 200. Increasing the number of arrows will give you a larger volume of arrows that are smaller in size, but heavier in concentration, which allows you to better visualize the load on the bracket holes. Go ahead and experiment with the number to see this yourself. Three thousand seems to give a quality visual. You can now see the load that was applied is displayed.

In this model we’ll also be interested in any displacement that occurs within the bracket geometry. To make a plot showing this, go to the ribbon and in the Results tab under Plot Group, click 3D Plot Group. This will open the newly generated 3D Plot Group 2 contextual tab in your ribbon. In the Add Plot section, click on Surface. The plot for the total displacement experienced by the bracket is automatically generated. Go to the ribbon, and under the Results tab you’ll see different dimensional types for plot groups. In this example we stick to two plot groups, but you are virtually limitless as you can make as many 3D, 2D or 1D plot groups you want for any type of visualization desired.

*Because the mounting bolts are fully constrained, use a volume integration over those domains to accurately calculate the reaction forces. On the Results tab, click More Derived Values and choose Integration, Volume Integration. In the Volume Integration settings window, locate the Selection section and from the Selection list, choose Box 1 to add the bolts. Click Replace Expression here in the upper-right corner of the Expression section, and from the menu choose Solid Mechanics, Reactions, Reaction Force, and the *x* component of the reaction force. Click the Evaluate button. Let’s do this again for the *Y* and *Z* components as well. To save time you can edit the expression, in this case, by changing the component letter.*

*Click Evaluate and the results are shown in Table 1 under the Graphics window. They match what we would expect them to be; the entire load is in the *y* direction while negligible in the *x* and *z* directions.*

It’s often necessary to solve several iterations of a model to find the optimal properties for its design. Instead of manually changing parameter values, and resolving each time, a parametric sweep can be used. A parametric sweep allows you to change the values of a parameter by sweeping the parameter values through a range defined by the user.

Adding a parametric sweep to this model will enable us to solve for different load angles. Go to the ribbon, and in the Study tab, click Parametric Sweep. In the Parametric Sweep window, under the Study Settings section, click the plus sign button to add the load direction as a parameter. To the right of that, click the Range button to define the range for this sweep. We’ll start at zero degrees, and rotate the load forty-five degrees, up to 180 degrees. Click Add and then the Compute button to re-solve the model.

We are automatically brought back to our stress plot. In the 3D Plot Group window, under Data you’ll notice the Parameter value list. Now we have the five different solutions dependent on the angle of the load and can alternate between them by selecting the different values and then clicking Plot.

*After performing a parametric sweep, you can create a table that lists the solutions for each parameter value. This way you can view the different solutions all at once. In the Volume Integration 1 node, click Evaluate and then New Table. The reaction forces at the different parameter values are computed. The reaction force in the *x*-direction is always zero, while the *y* and *z* directions share the load, depending on the angle.*

Traditionally, the way to calculate the effective mass of a particle is to push on it and measure how it reacts to the applied force. One University of Alberta research team (Brad Hauer, Callum Doolin, Kevin Beach, and John Davis) uses simulation as an efficient and noninvasive tool to achieve thermomechanical calibration.

According to Hauer, “The proper calibration of resonators is extremely important, especially in industries where precision is nonnegotiable.”

Because of its accuracy, thermomechanical calibration enables equipment to function both correctly and optimally. The thermal motion of a resonator is proportional to its energy, which is in turn proportional to its effective mass and time-dependent displacement squared. The computation of the effective mass takes into account both the mass and mode shape, and consequently, the displacement of a resonator. Simply put, an accurate prediction of the effective mass of a resonator design allows for proper calibration.

Atomic force microscopy is one field in which very fine measurements are needed. Atomic force microscopy is a way for instruments to inspect surfaces. It works by creating high-resolution images of objects by running a physical probe along them. One downside to this process is that measurements can be completely thrown off by manufacturing errors in the equipment. A device as sensitive as an atomic force microscopy tip requires precise calibration.

The University of Alberta researchers analyzed the fundamental mode shape with the Eigenfrequency Study available in the Structural Mechanics Module. They then derived the effective mass by performing a volume integration of the resonator’s density multiplied by the normalized displacement squared over its entire geometry.

*Simulation of atomic force microscopy tip mode shapes, where light reflected off a cantilever is measured by a photodiode.*

With so many kinds of sensors in use that need to be calibrated, it is a huge benefit to be able to model all geometries in the same software. In the future, the researchers at University of Alberta will work on some cutting-edge designs involving optomechanics. Naturally, they will continue to use COMSOL Multiphysics® to model their designs.

There is a broad range of uses for companies working with nanostructures, nanostrings, and everything in between. The best part is that anyone with the Structural Mechanics Module can get the effective mass of nanoelectronic and nanomechanical devices in a more efficient and scalable way.

*A force transducer able to measure force in increments as small as Attonewtons (10 ^{-18}N).*

- Check out the University of Alberta paper: “Effective Mass Calculations Using COMSOL Multiphysics for Thermomechanical Calibration“
- Familiarize yourself with the Structural Mechanics Module

Heat transfer will take place when materials at different temperatures come into contact with one another. It may initially appear that the surface of each material is entirely in direct contact. However, upon closer inspection, you’ll find that many materials have a surface roughness measurable at the micron or nanometer scale.

When materials are in direct contact, thermal conductivity is determined by the properties of the two materials. However, surface roughness introduces gaps between contacting materials, which are usually filled with air. The thermal conductivity of gasses, such as air, is typically much lower than the conductivity of common solid materials. Therefore, the heat flux due to conduction is smaller in noncontacting regions, leading to increased thermal resistance at the interface.

Yet, if you increase the structural stress over the gap, you’re going to decrease the size and extent of the gaps and, therefore, influence the thermal resistance. Most of the time, there is also surface-to-surface radiation in the gap, however, it can be neglected in many common applications as the temperature difference between the materials is usually sufficiently small.

In the Model Gallery, you can find the pre-solved model “Thermal Contact Resistance Between an Electronic Package and a Heat Sink“, which can be used to investigate the effect of thermal contact resistance on heat transfer in an electronic package.

The model is based off of a study by M. Grujicic, C.L. Zhao, and E.C. Dusel of Clemson University titled “The effect of thermal contact resistance on heat management in the electronic packaging“. In their paper, the authors use finite element analysis (FEA) to investigate the effect that thermal contact resistance has on heat management in a simple central processing unit (CPU) and heat sink design. They explore the effect of surface roughness, the mechanical and thermal properties of the contacting materials, the contact pressure, and the effect of the materials on the maximum temperature experienced by the CPU in detail in their paper.

In the COMSOL Multiphysics model, part of the Grujicic et al. study is reproduced, where we take a look at the influence of four main parameters on thermal contact resistance, and, thereby, the heat transfer:

- Contact pressure
- Microhardness
- Surface roughness
- Surface roughness slope

The model geometry is composed of a cylindrical electronic package that is located inside a heat sink constructed of eight cooling fins. Device efficiency is dependent on the eight cooling fins of the heat sink, as well as on the efficiency of heat transfer between the electronic package and heat sink. The device geometry is shown below, where radial symmetry has been used to reduce the geometry to one sixteenth of its original size.

*Left: Heat sink and electronic package geometry, showing the eight cooling fins around the cylindrical package. Middle and right: Radial symmetry and simplification of the geometry.*

The electronic package is modeled as a cylinder with a radius of 1 centimeter and a height of 5 centimeters and is made of silicon. The heat sink is made of aluminum with fins reaching a distance of 2 centimeters from the cylinder axis. The electronic package produces a total heat source of 5 W. In order to dissipate this heat, a cooling fan blows room-temperature air at 8.5 m/s across the heat sink.

To define the cooling due to the air flow, we use the built-in heat transfer coefficient in COMSOL Multiphysics. The four parameters — contact pressure, microhardness, surface roughness, and surface slope — can all be modeled using parametric sweeps set up in the *Thermal Contact* interface. Both a free triangular mesh and a swept mesh are used in the model.

Tip: You can find more information about the mesh in the model documentation.

The figure below shows the temperature profile obtained using reference values:

*Temperature profile with reference values for the parameters.*

Closer to the fan (on the left side of the model), the temperature of the fins reaches about 483 K. The temperature increases with greater distance from the fan, reaching 490 K at the other extremity.

Next, we further analyze the model to determine the effect of contact pressure, microhardness, surface roughness, and surface slope on constriction and gap resistance within the model. The amount that each of these four parameters affect both the constriction resistance and gap resistance directly influences the material characteristics at the surface of the heat sink and electronic packaging. Thus, the heat dissipation from the electronic device is altered.

Below are the results from this analysis:

*Left: Constriction resistance depending on contact pressure (*x*-axis) and microhardness (*y*-axis). Right: Constriction resistance depending on roughness (*x*-axis) and roughness slope (*y*-axis).*

*Left: Gap resistance depending on contact pressure (*x*-axis) and microhardness (*y*-axis). Right: Gap resistance depending on roughness (*x*-axis) and roughness slope (*y*-axis).*

Contact pressure, roughness, roughness slope, and microhardness all affect the constriction resistance within the model. However, roughness slope has little to no effect on the gap resistance. We can see this in the bottom-right image, where the plot shows constant values in the vertical direction.

In their paper, Grujicic et al. make the conclusion that surface roughness and mechanical and thermal properties can have a significant effect on thermal contact resistance, and, therefore, on thermal management. According to Grujicic et al., thermal contact resistance, and the parameters that influence it, can play a major role in the heat management of electronic devices. Therefore, it may significantly affect device performance, reliability, and life cycle.

- Read the paper: “The Effect of Thermal Resistance on Heat Management in the Electronic Packaging” by M. Grujicic, C.L. Zhao, and E.C. Dusel
- Download the model: Thermal Contact Resistance Between an Electronic Package and a Heat Sink

The coil heat exchanger we’ll consider is shown in the figure below.

*A copper coil carries hot water through a duct carrying cold air.*

Copper tubing is helically wound so that it can be inserted along the axis of a circular air duct. Cold air is moving through the duct, and hot air is pumped through the tubing. The air flow pattern and the temperature of the air and copper pipes will be computed using the *Conjugate Heat Transfer* interface. Since the geometry is almost axisymmetric, we can simplify our modeling by assuming that the geometry and the air flow are entirely axisymmetric. Thus, we can use the *2D axisymmetric Conjugate Heat Transfer* interface. Since the airspeed is high, a turbulent flow model is used; in this case, it is the k-epsilon model.

We can assume that the water flowing inside of the pipe is a fully developed flow. We can also assume that the temperature variation of the water is small enough that the density does not change, hence the average velocity will be constant. Therefore, we do not need to model the flow of the water at all and can instead model the heat transfer between the fluid and the pipe walls via a forced convective heat transfer correlation.

The Convective Heat Flux boundary condition uses a Nusselt number correlation for forced internal convection to compute the heat transfer between the water and copper tubing. This boundary condition is applied at all inside boundaries of the copper piping. As inputs, it takes the pipe dimensions, fluid type, fluid velocity, and fluid temperature. With the exception of the fluid temperature, all of these quantities remain constant between the turns of the tubing.

As the hot water is being pumped through the copper coils, it cools down. However, since the model is axisymmetric, each turn of the coil is independent of the others, unless we explicitly pass information between them. That is, we must apply a separate Convective Heat Flux boundary condition at the inside boundaries of each coil turn.

This raises the question: How do we compute the temperature drop between each turn and incorporate this information into our model?

Consider the water passing through one turn of the copper coil. The heat lost by the water equals the heat transfer into the copper pipes. Under the assumption of constant material properties, and neglecting viscous losses, the temperature drop of the water passing through one turn of the pipe is:

\Delta T = \frac{Q}{\dot m C_p} = \frac{\int q'' dA}{\dot m C_p}

where \dot m is the mass flow rate, C_p is the specific heat of water, and Q is the total heat lost by the water, which is equal to the integral of the heat flux into the copper, integrated over the inside boundaries of the coil. This integral can be evaluated via the Integration Component Coupling, defined over the inside coil boundaries.

*The Integration Component Coupling defined over a boundary. Note: The integral is computed in the revolved geometry.*

Using these coupling operators, we can define a set of user-defined variables for the temperature drop:

`DT1 = intop1(-nitf.nteflux/mdot0/Cp0)`

This evaluates the temperature drop along the first turn of the pipe. We can define a different temperature drop variable for each turn of the pipe and use them sequentially for each turn.

*The water temperature in the sixth turn considers the temperature drop in the first five turns.*

*Flow field and temperature plot (left) and the temperature along a line through the center of the coils (right).*

Since this is a 2D axisymmetric model, it will solve very quickly. We can examine the temperature and the flow fields and plot the temperature drop along a line down the center of the coils. We can observe that the water cools down between each turn of the coil, and the air heats up.

This can be considered a parallel-flow heat exchanger, since the hot and cold fluids flow in the same overall direction. If we wanted to change this model to the counter-flow configuration, we could simply switch the air inlet and outlet conditions so that the fluids travel in opposite directions.

What other kinds of heat exchanger configurations do you think this technique can be applied to?

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When a structure undergoes vibrations, its components experience stresses and strains, which are amplified by the excitation of the natural frequencies of the structure. In addition to potential damage to the structure itself, these oscillations can also be a source of discomfort and disruption for occupants.

Whether a rare or persistent hurdle, seismic and wind-induced vibrations, and their effects, are an important consideration in the design process. *Damping* is one influence that has proved valuable in helping to reduce such vibrations, particularly in tall buildings and bridges, and preserve the longevity of these structures.

Did you know that you can model viscoelastic structural dampers using COMSOL Multiphysics? To get you started, we have created a tutorial model. The Viscoelastic Structural Damper model can be found in the Model Gallery as well as within the Structural Mechanics Module Model Library.

The model first analyzes the frequency response of a viscoelastic structural damper. Comprised of two layers of viscoelastic material, the damper is restrained between steel mounting elements.

*Image depicts the two layers of viscoelastic material in bold, with the steel mounting elements shown in light gray.*

Here, two of the mounting elements are subjected to periodic forces, with frequencies ranging from 0 to 5 Hz. Meanwhile, one of the mounting elements remains fixed. The figure below shows the displacement of the damper at 5 Hz. The second figure highlights the relationship between the applied frequency and the storage modulus and loss moduli, a representation of the viscoelastic properties of the material.

*Displacement at 5 Hz.*

*Plot of storage (blue line) and loss moduli (green line).*

Next, we can run a transient analysis to find out the displacement field as a function of time, as seen in the figure below.

*Surface plot of the z-component of displacement field after 1 second of forced vibrations. *

Plotting the applied force versus the displacement at one of the loaded points shows hysteresis loops, which are characteristic for damped problems. Energy is dissipated since the force and displacement are out of phase with each other.

*2D plot relating the displacement to the applied force.*

*Phase change* is a transformation of material from one state of matter to another due to a change in temperature. Phase change leads to a sudden variation in the material properties and involves the release or absorption of latent heat. We can use the Heat Transfer Module to model this type of phase change. Let’s start with an example.

In the continuous casting process, liquid metal is poured into a cooled mold and starts to solidify. As the metal leaves the mold, the outside is solidified completely, while the inside is still liquid. To further cool down the metal, spray cooling is used. When the metal is completely solidified, it can be cut into billets. This is a stationary, time-invariant, process. The rate at which the metal enters and leaves the modeling domain does not vary with time, and neither does the location of the solidification front.

Here is an illustration of the continuous casting process:

*Sketch of a continuous casting process.*

In order to optimize and improve this process, we can turn to simulation. With COMSOL software, we can predict the exact location of the phase interface.

COMSOL Multiphysics and the Heat Transfer Module together offer a tailored interface for modeling phase change with the *Apparent Heat Capacity method*. The method gets its name from the fact that the latent heat is included as an additional term in the heat capacity. This method is the most suitable for phase transitions from solid to solid, liquid to solid, or solid to liquid. Up to five transitions in phase per material are supported.

When implementing a phase transition function, \alpha(T), a smooth transition between phases takes place, within an interval of \Delta T_{1\rightarrow2} around the phase change temperature, T_{pc, 1\rightarrow 2}. Within this interval, there is a “mushy zone” with mixed material properties. The smaller the interval, the sharper the transition.

The below figure shows the phase change function for the continuous casting model:

*COMSOL Multiphysics settings for phase change. Keep in mind that phase 1 is below T_{pc, 1\rightarrow 2} and phase 2 is above.*

The material properties for the solid and liquid phase are specified separately. These values are combined with the phase transition function so that there is a smooth transition from solid to liquid. The heat capacity of the material is expressed as C_p=C_{p,solid}\cdot(1-\alpha(T))+C_{p,liquid}\cdot\alpha(T), and similarly for the thermal conductivity and density. For a pure solid, \alpha(T)=0, and for a pure liquid, \alpha(T)=1. Within the transition interval, the material properties vary continuously.

The latent heat is included by an additional term in the heat capacity. Let us take a look at the derivative of the phase transition function:

*Derivative of the phase transition function.*

Integrating this function over \Delta T_{1\rightarrow2} gives 1 and multiplying by the latent heat L_{1\rightarrow 2} gives the amount of latent heat that is released over \Delta T_{1\rightarrow2}.

Consider the stationary heat transfer equation with a convective term, of the form:

\rho C_p\cdot \nabla T=\nabla\cdot\left(k\nabla T\right)

The Apparent Heat Capacity method uses the following expression for the heat capacity:

C_p=C_{p,solid}\cdot(1-\alpha(T))+C_{p,liquid}\cdot\alpha(T)+L_{1\rightarrow 2}\frac{d\alpha}{dT}

The advantage of this method is that the location of the phase interface does not need to be known ahead of time.

With the help of the *Heat Transfer with Phase Change* interface, the implementation is straightforward. Axial symmetry is assumed, and the model is reduced to a 2D domain. The casting velocity is constant and uniform over the modeling domain.

To get a sharp transition and thereby the exact location between the solid and liquid phase, we need a small transition interval, \Delta T_{1\rightarrow2}. Resolving such a small interval properly requires a fine mesh. However, we do not know the location of the solidification front in advance, so we first solve the model with a gradual transition interval, and then use adaptive mesh refinement to get better resolution of the solidification interface. The transition interval can then be made even smaller.

The results are compared below for two different transition intervals. As the transition interface is made smaller, the model better resolves the transition between liquid and solid. This information can be used to improve the continuous casting process, and this same approach can be used for similar applications involving phase change.

- Download the model: Cooling and Solidification of Metal
- Read a user story: Optimizing the Continuous Casting Process with Simulation

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

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

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

W=\frac{KF_N L_T}{H}

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

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

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

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

*Modification of contact gap calculation: w is the wear depth, g is the gap, and \lambda is the contact pressure.*

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

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

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

*Pin-on-disc wear test model.*

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

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

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

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

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

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

*Typical brake pad wear depth profile.*

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

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

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Continuous casting, also known as *concasting*, is a process by which a continuous strand of steel is produced during one casting sequence that is subsequently cut into pieces for rolling. Unlike the batch process of ingot casting, which entails the casting of a single ingot at a time, continuous casting allows for the production of waste metal to be reduced, is more energy efficient than ingot casting, and produces products of a superior quality.

During ingot casting, the head of each beam must be cropped after each casting process, producing waste metal. By producing a long continuous strand of steel, on the other hand, cropping is only required at the very beginning and very end of the casting sequence (during which hundreds of ingots or “blooms” are produced), thereby reducing waste.

During continuous casting, molten steel poured from a ladle into a tundish from where it is drawn through a copper mold by a series of rollers, as shown in the diagram below.

*Diagram of the continuous casting process.*

After solidifying, these metal strands are cut into 3- to 15-meter long pieces and left to cool. In continuous casting, the shape of the strand can be cast closer in shape to the final product than what would be produced during ingot casting, greatly reducing the cost of further processing the strands. In the image below, billets are exiting the concast machine onto a discharge table, where the strands are being cut.

*There are four types of semi-finished casting products: ingots, billets, blooms, and slabs. Depicted here are billets exiting a continuous casting machine onto a discharge table.*

Nicolas Grundy, head of Metallurgy & Process Continuous Casting at SMS Concast, has found that simulation can be a valuable tool for better understanding and optimizing the continuous casting process. In a recent article that appeared in *COMSOL News*, Grundy explained how COMSOL Multiphysics was used during his research. “We are constantly pushing the limits and the only way to understand something that we have never done before is to simulate it,” Grundy describes in the article.

In his research, Grundy and his team used simulation to analyze every step of the process. One of the main goals of their study was to learn more about the solidification process and the mechanical deformations that can take place during both quenching and slow cooling of the metal slabs. The team found that by minimizing the segregation of alloying elements in the center of the strand, removing any non-metallic substances, and improving the microstructure of the solidifying steel, they could improve the quality of the final product. They achieved these goals by implementing a stirring process, during which the molten steel is mixed through the use of electromagnetic stirrers and by optimizing the design of the tundish.

In the figures below, you can see the COMSOL Multiphysics simulation the team used to help ensure that the liquid steel flow in the tundish is designed correctly to achieve the best quality steel.

*Model of the tundish created with the CFD Module.*

In addition to analyzing the effect of using electromagnetic stirrers, Grundy also simulated hot charging. This is a recent trend in steelmaking where steel strands are charged to the rolling mill while still hot, instead of leaving them to cool and then reheating them in a reheating furnace. Using their COMSOL Multiphysics model, the team explored the heat exchange process that takes place during the first solidification of the molten steel. Their results were used in the design of a new type of mold that forms billets with large, rounded corners that stay warm after casting, resulting in a more even distribution of temperature on the billet’s surface. This technique was employed at Tung Ho Steel in Taiwan, allowing them to completely forgo the use of a reheating furnace.

The subsequent reduction in yearly emissions by the company is equal to the exhaust of about *20,000 cars* (that’s 40,000 tons of CO_{2})!

Interested in learning more about how simulation can be used to explore and optimize concasting?

The Continuous Casting model, available in the Model Gallery, is an already-solved model that you can download and run using COMSOL Multiphysics and the Heat Transfer Module. The model analyzes the casting of a metal rod from molten metal through both the thermal and fluid dynamic aspects of the process, including heat transfer, the melt flow field, and phase change. Using the model, you can explore how the casting process can be optimized by altering the casting and cooling rates.

*Model of the continuous casting process. Top: Velocity field with streamlines. Bottom: Temperature distribution. Download the model from the Model Gallery.*

With the use of COMSOL Multiphysics, Tingcheng Wu, Guillaume Escamez, Clement Lorin, and Philippe J. Mason from the Department of Mechanical Engineering at the University of Houston were able to perform simulations to analyze how individual components of the structure impacted its overall performance. By applying various parameters to the structural design, they were able to conclude which factors had the greatest effect on the machine, both structurally and thermally. Thus, they could determine how to achieve a balance between the two.

The team of researchers modeled the structure as a rotor shaft separated into five individual parts connected by bolts. As a means to provide thermal insulation, different materials were used on particular areas of the shaft, as you can see below.

*Bolts were used to connect stainless steel with G10, a glass fiber material characterized by low thermal conductivity and a high yield stress. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their poster submission.*

Seeking to analyze the issues of heat transfer and solid mechanics in these machines, the researchers used the Heat Transfer Module and Structural Mechanics Module to create their simulations. The figure below highlights the team’s findings regarding temperature, depicting that the G10 components take on the greatest temperature gradient.

In the next figure, the connection bolts underwent the most stress within the structure, a factor that was found to decrease as the cross-section area was decreased.

*Simulations highlight the thermal and structural pressure that the shaft endured, especially along its connection bolts. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their presentation.*

With the continued funding and efforts of NASA and other research teams, progress continues to be made in the design of aircraft. As torque transfer components within fully superconducting rotating machines continue to be optimized, researchers gain momentum in their quest for developing structures with greater power densities and the potential for electric propulsion. In addition to making air travel a quieter and more energy efficient process, implementing this technology paves the way for its potential use within modes of ground transportation as well.

- Access the paper, presentation, and poster: “FEA Mechanical Modeling of Torque Transfer Components for Fully Superconducting Rotating Machines“

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The researchers at Argonne National Lab (Argonne) turned to multiphysics simulation and trial-and-error prototyping to optimize the effectiveness of their acoustic levitator. When we want to move an object, sound may not be the tool we would typically reach for. So how does it have the power to float or levitate objects in a lab setting? It’s all about combining forces in just the right way to create lift.

When sound vibrations travel through a medium like air, the resulting compression is measurable and real. By combining acoustophoretic force, gravity, and drag, the pressure is just enough to not only lift a material like liquid medicine, but to also allow the medicine to be positioned, rotated, and moved according to the needs of the operator.

*Pressure pockets created by waves between the transducers of the acoustic levitator do the heavy lifting on a particle scale.*

By keeping the droplets in a steady rotation, researchers are able to work on the chemical reactions while the medicine stays liquid and amorphous. This is key for creating a safe, steady environment where medicine will form correctly.

Every material and measurement in the acoustic levitator will change both whether the device works in its final design and how finely it can be adjusted according to the needs of the scientists who use it.

The geometry of the device includes two small piezoelectric transducers that stand like trumpets above and below the working area where medicine is created, like this:

*The acoustic levitator’s wave patterns are controlled by pieces of Gaussian profile foam located on evenly-spaced transducers.*

Possibly the most important part of the design is the Gaussian profile foam, which consists of polystyrene and coats the ends of each transducer. This foam works to remove acoustic waves that fall outside the required range. It acts as a filter to maintain even, well-defined standing waves.

Using COMSOL Multiphysics together with the Acoustics Module, CFD Module, and Particle Tracing Module, the team at Argonne modeled the acoustic levitator. Working cohesively with simulation, they were able to narrow down the shape of the acoustic field and location of floating droplets.

*The simulation above shows that at T = 0.75 seconds droplets formed from the particles. On the left, the simulation shows the expected particle distribution and on the right, a photograph depicts the actual distribution of the droplets.*

As advances in acoustic levitation expand, the ability to work with finer and finer chemical reactions will allow members of the pharmaceutical science community to expand their reach, perhaps discovering many new medicines with life-saving qualities.

- Learn more about how Argonne improved their acoustic levitation technology.