Why Activate or Deactivate Material?

Imagine that you want to simulate a structural material that is in a molten state and then solidifies. Alternatively, the material is initially solid and then melts. This can be the case when you want to simulate manufacturing processes such as arc welding, selective laser melting, and selective laser sintering — the last two being common additive manufacturing methods.

Material activation is useful when simulating additive manufacturing processes. Image of a 3D printer by Jonathan Juursema — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

You can use the Activation node to activate or deactivate material in a simple manner during a simulation. As a note, the Activation node is available in the add-on Structural Mechanics Module and MEMS Module as of version 5.4 of COMSOL Multiphysics®.

Activation of Material: The Naive Approach

One approach to emulate that material is structurally absent is to simply reduce its elastic stiffness to a point where it becomes negligible. This way, the rest of the structure is free to deform without “feeling” the structurally weaker material. This is a viable approach as long as we have no desire to actually activate material.

A problem arises if we try to activate the weak material by simply restoring its stiffness to the nominal level at some point during the simulation. When the stiffness is restored, any strains present in the activated material will suddenly produce stresses. In most situations, this is not a desired effect when activating material. Instead, material should be activated in a state of zero stress. This is more physical, as we usually want to emulate that material is deposited or solidified.

Activate Material in a Stress-Free State

The Activation node avoids the problem of the artificially produced stresses described above. This node reduces the stiffness of the inactive material as described before, but, importantly, it also removes any elastic strains that are present at the instant of activation. Simply put, material is activated in a state of zero stress.

The Activation node is located under the Linear Elastic Material node, as shown in the figure below, and it is available for the Solid Mechanics and Membrane interfaces.

The Activation feature and its Settings window.

The Activation panel of the Settings window contains two settings, namely:

• Activation expression
• Activation scale factor

The Activation expression setting is a logical expression that you define. It is used to determine whether the material is active or not, and it is defined per integration point of the mesh elements. For example, an activation expression that reads T<T_s says that the material is active if the expression is logically true (when the temperature, , is less than the solidification temperature, ) and inactive otherwise.

The Activation scale factor setting defines the factor that multiplies the elastic stiffness to emulate that material is not present. It has a default value of 10^-5, but you can change it if you wish. However, a value that is too low could make the stiffness matrix ill-conditioned.

Two built-in variables are provided to describe the active/inactive state, namely:

• isactive
• wasactive

The variable isactive indicates the current active/inactive state of the material, while the variable wasactive indicates whether the material has been active at any previous point during the simulation. In the case of a Solid Mechanics interface with the tag solid, the variable describing the current state of the material is called solid.isactive. The wasactive variable can be used to simplify the formulation of the activation expression in some situations, as we will see below.

Note: If a material undergoes multiple activation/deactivation events, the elastic strains are removed at every instant of activation. This means that the material is always activated in a state of zero stress, regardless of its history, including past activations or deactivations. Inelastic strains, such as plastic strains, are not removed.

Let us look at some examples of how to use the Activation node.

Example 1: Pointwise Activation

As a simple 2D example, suppose that you want to gradually activate a material in the y direction as time, t, progresses. The imagined “activation front” travels at a velocity, vel, and the region of active material is therefore given by . It is entered as the activation expression, as shown in the following figure.

Activation expression for pointwise material activation.

To illustrate this, consider a solid quadrilateral element with four integration points (Gauss points), as in the figure below. With the activation expression above, each integration point is activated individually by evaluating the activation expression. This means, in practice, that a single mesh element can be partially active if it has more than one integration point.

Activation of individual integration points in a mesh element.

Example 2: Elementwise Activation

Now, consider a case where you want to activate entire mesh elements, and not on an individual integration point basis. To do so, you need to phrase the activation expression so that it evaluates equally for every integration point in each mesh element. This can be done using the centroid operator. The activation expression that was used in the previous example is modified, as shown in the following figure. The coordinate Y is now evaluated at the mesh element centroid, which means that the activation expression will evaluate to the same value for all of the integration points in the mesh element.

Activation expression for elementwise material activation.

Inside the mesh element in the figure below, the activation expression is fulfilled for the element centroid, and therefore all four integration points are active.

Activation of all integration points in a mesh element by using the centroid operator.

Example 3: Using the Previous State of Activation

Suppose that you want to simulate a laser cladding process where a filler material is melted and deposited over time. In this situation, the current position of the laser beam defines the location where material is currently deposited. The region of previously activated material is defined by the entire trajectory of the laser beam from the start of the process. (For details on how to model the movement of a laser beam, you can read this blog post on modeling moving loads and constraints.) The variable wasactive can be used to avoid having to describe this trajectory mathematically. You can express an activation expression for this situation schematically as:

(logical expression describing the current position of the laser beam) || solid.wasactive

which states that the material is active if the “logical expression describing the current position of the laser beam” is true or if the material has been active at any previous time (or parameter step) during the simulation. If the activation expression is used without the wasactive variable, the material would become inactive once the laser beam has passed, which is likely contrary to what is intended.

Visualizing the Results

Suppose that you have simulated a time-dependent process where material is deposited over time. It may be interesting to display results only for the active parts of a domain. You can do this by using the variable isactive as the Logical expression for inclusion in the Filter node, as shown in the figure below. Note that depending on the chosen fulfillment type, the isactive filtering may show slight differences compared to the underlying isactive variable defined at the integration points of the mesh elements.

Using the Filter node to display only active parts of a domain.

Concluding Thoughts on the Activation Node

In this blog post, we have described different ways of using the Activation node to activate material during a simulation. The Activation node makes it easy to simulate the deposition of material in simulations of different types of manufacturing processes, such as welding and additive manufacturing. If you want to examine a model that uses the Activation node, click the button below to see the Thermal Initial Stresses in a Layered Plate example in the Application Gallery. Note that you need to have a COMSOL Access account and valid software license to download the related MPH-file.

Achieving the Ideal Frequency of a Tuning Fork

Tuning forks are made out many different materials, but most of them are calibrated to a standard pitch of 440 Hz for an A note. An article in JOM discusses how the frequency of a tuning fork changes with different materials and a fixed geometry (Ref. 1). This made me think: What if we change the geometry and material of a tuning fork to reach the desired frequency?

A tuning fork.

Performing an Optimization Study for Multiple Materials

The Application Library contains several models featuring a tuning fork geometry, as well as a tuning fork simulation app. You can access the Application Library within the COMSOL® software GUI in the File menu and search the keyword “tuning fork”. For this blog post, we’ll start with the simple Tuning Fork model (and accompanying example app).

This model features a parametric geometry, material properties of steel, the Solid Mechanics interface, and two studies. Both studies perform an eigenfrequency analysis to search for the eigenfrequency around 440 Hz. The first study uses a parametric sweep of the tuning fork’s arm length, set as a parameter L, to find the optimal design for 440 Hz. In contrast, the second study applies a mathematical optimization algorithm that uses L as the control variable and the deviation from the target frequency as the optimization objective for fast, precise, and efficient optimization.

Tuning Fork model showing the original settings to search for a tune of 440 Hz via two different strategies.

Let’s return to the initial question: How does the tuning fork’s arm length depend on the material for a tune of 440 Hz?

First, we extend the model with a material switch. This option allows us to set and test various materials for the model. Further, this switch is needed to work with the material sweep in the Study node, as described later in this blog post. We can add available materials from the built-in Material Library, including:

• Aluminum
• Titanium beta-21S
• Copper
• Steel AISI 4340

The switch is assigned to the solid domains of the tuning fork.

Setting the material switch for a multimaterial analysis.

Now that we transformed the original model into a multimaterial model, we can adjust the studies. Due to the combination with a material sweep, the studies can now solve the physical model for all chosen materials and it is possible to analyze all of the results together. For example, we can look at the eigenfrequency and confirm that the eigenfrequency changes with different materials and arm lengths.

Various sweeps can be combined easily within a study, such as the extension of Study 1 with a material sweep. In contrast, when we try to add a material sweep to the optimization study for Study 2, we get an error message. The good news is that there is another way of achieving this by using Study References, as explained below.

Setting the material sweep directly in the study is not supported, since it is only possible to use one Sensitivity, Optimization, Parameter Estimation, or Parametric/Material/Function Sweep study step in each study. These study nodes tend to control the same solver settings and are therefore incompatible with each other. To perform a parametric or nested optimization, we can call a study containing an Optimization node from inside another study via a Study Reference node.

Therefore, we add an additional empty study and fill it with a Material Sweep node and a study reference pointing to the optimization study. In the Optimization node, we can define the settings needed for the optimization. This is possible as long as all entries are globally available. For that, we leave Study 2 containing the optimization as is.

Use of an additional study to create nested studies.

Optimizing a Tuning Fork for Various Materials in COMSOL Multiphysics

With all the discussed adjustments, we can now run the multimaterial optimization study by computing Study 3. This study controls the material assignment and runs individual optimization procedures for each by starting Study 2 automatically. Hence, we can extract and postprocess the individual design changes for the different materials. This can be done, for example, by a global evaluation of the parametric data set. Evaluating the tuning fork arm length, set as the control variable L, gives us the needed design changes to tune each tuning fork design to 440 Hz.

Top: Settings of the global evaluation. Bottom: Results table for the different materials identified via the switch index.

Testing Tuning Fork Design Parameters with an App

You can build an app from a tuning fork model that includes a customized user interface and restricted inputs and outputs. Take the Tuning Fork app in the Application Library as an example. This app can be used to quickly compute the frequency of a tuning fork with the prong length as an input or the optimal prong length with the frequency as an input.

The user interface of an example tuning fork app.

The example featured above can be used as inspiration for building apps of your own via the Application Builder tool in COMSOL Multiphysics.

Concluding Thoughts on Multimaterial Optimization

In this blog post, we used material sweeps in combination with an optimization study to find the best geometry for tuning forks made of different materials. Note that the approach discussed here is generic. You might want to combine other study steps that have the same hierarchy.

With this approach, you can combine all of the Sensitivity, Optimization, Parameter Estimation, or Parametric/Material/Function Sweep features into nested studies, thus improving your simulations, results, and products.

Next Steps

Try it yourself: Access the Tuning Fork model and app by clicking the button below, which will take you to the Application Gallery. Then, you can download the MPH-files if you have a COMSOL Access account and valid software license.

Read More on the COMSOL Blog

• Multiparameter Optimization with a Least-Squares Objective
• Tuning Fork Application

Reference

1. T.D. Burleigh and P.A. Fuierer, “Tuning Forks for Vibrant Teaching“, JOM, pp. 26–27, Nov. 2005.

Synthesizing Randomized Inhomogeneous Material Data

We have already discussed how to generate random-looking geometric surfaces by using sum and if operators in combination with uniform and Gaussian random distribution functions. The idea is that by summing up a set of spatially varying waves with careful choices of amplitudes and phase angles, we can mimic the type of randomness frequently found in natural materials and many natural phenomena in general.

To generate synthetic material data in 2D, we can use the same double-sum expression that we used to create randomized CAD surface data in the previous blog post:

Note that this sum could also be used to generate random data for use in boundary conditions on surfaces in a 3D model.

For the 3D volume case, we will need triple sums:

The frequency-dependent amplitudes will take their values from a random distribution according to

and

for the 2D and 3D cases, respectively.

The functions g(k,l) and g(k,l,m) have a random Gaussian, or normal, distribution and h(k,l) and h(k,l,m) are frequency-dependent amplitude functions with values that taper off for higher frequencies in accordance with the spectral exponent β. The higher the value of the spectral exponent, the smoother the generated data will be. For a variety of reasons, many natural processes have this property, which is characterized by slower variations that are more dominant than fast.

The phase angles φ are sampled from a function u. The function has a uniform random distribution that is between and :
φ(k,l) = u(k,l) and φ(k,l,m) = u(k,l,m)

The sums run over a set of discrete frequencies. More nonzero terms in a sum imply a larger number of higher-frequency contributions, resulting in data that contains finer details. The maximum frequencies in each direction are determined by the integers K, L, and M, respectively. These types of sums are closely linked to discrete inverse cosine transforms. They essentially correspond to an inverse cosine transform of the amplitude coefficients g(k,l) and g(k,l,m), with some additional manipulations of the phase angles. For details, see the previous blog post on how to generate random surfaces in COMSOL Multiphysics.

Expressions for Double and Triple Summation

In COMSOL Multiphysics, the following double-sum expression can be entered in various edit fields, such as 2D material properties or 3D boundary conditions:

0.01*sum(sum(if((k!=0)||(l!=0),((k^2+l^2)^(-b/2))*g1(k,l)*cos(2*pi*(k*s1+l*s2)+u1(k,l)),0),k,-N,N),l,-N,N)

We can use a similar expression for the triple sum, which can be used for 3D material data, loads, sources, and sinks:

0.01*sum(sum(sum(if((k!=0)||(l!=0)||(m!=0),((k^2+l^2+m^2)^(-b/2))*g1(k,l,m)*cos(2*pi*(k*x+l*y+m*z)+u1(k,l,m)),0),k,-N,N),l,-N,N),m,-N,N)

where we have set K = L = M = N.

For more details about the underlying theory and syntax used here, see the blog post mentioned above.

Working with models that contain triple sums is computationally quite expensive. It is more efficient to first generate the data and export it to file and then import it again as an interpolation function, perhaps in a separate model. This interpolation function can then be used in a variety of ways, as we will explain later. Alternatively, you can use external software to generate the data by means of inverse FFT.

Let’s now take a look at how to generate 3D material data.

Generating a Randomized Interpolation Table in COMSOL Multiphysics®

Creating a 3D volume matrix of random data is surprisingly easy. It amounts to creating a couple of random functions, some parameters, a Grid data set, and an Export node.

Start by creating a random function for the amplitudes with 3 input arguments based on a normal, or Gaussian, distribution. This corresponds to the function g(k,l,m) in the mathematical description above. In this case, we arbitrarily use the default settings for a random function with the mean value set to 0 and the standard deviation set to 1.

Next, we create a random function for the phase angles with 3 input arguments based on a uniform distribution between and corresponding to the function u(k,l,m) above.

Now, create a data set of the type Grid 3D that references the random functions as a source. We need this data set to give an evaluation context to the triple-sum expression that we will define later in the Export node.

We will use two results parameters, N and b, for the spatial frequency resolution and spectral exponent, respectively.

To make it easier to work with the large data sets that are generated, you can turn off the Automatic update of plots option. This setting is available in the Settings window of the Results node. Turning it off avoids recomputing the expressions each time you click on a plot group under Results.

To visualize the data before exporting to file, add a Slice plot and type (or paste) the expression:

0.01*sum(sum(sum(if((k!=0)||(l!=0)||(m!=0),((k^2+l^2+m^2)^(-b/2))*g1(k,l,m)*cos(2*pi*(k*x+l*y+m*z)+u1(k,l,m)),0),k,-N,N),l,-N,N),m,-N,N)

To export the data, add a Data node under Export and type in the same expression as for the Slice plot above. In the Settings window of the Data node, make sure to set the data set to Grid 3D and to specify a file name that the data will be written to. Here, we can let the points be evaluated in a way that is independent of the Grid 3D data set.

For the setting Points to evaluate in, select Regular grid. For Data format, select Grid. You can freely choose the number of x, y, and z points to evaluate in. In the figure below, these points have each been set to 50. Note that data generation corresponding to numbers higher than 50 may take a very long time to generate. For a 50x50x50 grid, we already get 125,000 data points.

The text file that is generated and exported can now be imported to a new file for the purpose of setting up a physics analysis where we use the generated data in material properties. This can be done for any type of physics, including electromagnetics, structural mechanics, acoustics, CFD, heat transfer, and chemical analysis. By using the COMSOL® API in model methods or applications, for example, such export and import operations can be automated and set in the context of a for-loop in order to generate statistics over a larger sample set. In this example, we only generate one set of data.

Using the Generated Data in a Heat Transfer Analysis

To illustrate how this type of data can be used, let’s create a test model of the simplest possible kind, based on a heat transfer analysis.

Start by creating a new 3D model using a Heat Transfer in Solids interface.

Now, import the data from file as an Interpolation function. This function will be available under the Global Definitions node.

The Interpolation function is given the name cloud and can later be accessed using expressions like cloud(x,y,z).

To make unit handling easy, when using this interpolation function, we will set the input argument units to m and the function unit to 1. The Function unit corresponds to the unit of f(x,y,z)=cloud(x,y,z) and setting it to 1 makes it dimensionless.

To keep things simple, let’s use a Block geometry object that matches the imported data exactly, with the corner at the origin and sides at 1. This corresponds to the size and position of the Grid 3D data set used earlier for generating the data.

For a “real” case, you can instead import or create a CAD geometry, which can be used to truncate the interpolation function in a suitable way. This truncation of data is automatic in COMSOL Multiphysics. The figure below shows such an interpolation of randomized data over a CAD model of a wrench. When evaluating over an arbitrary geometry, it can be useful to scale the coordinate values in the triple-sum. In the wrench example, instead of k*x+l*y+m*z, as in the expressions above, the scaled expression k*(x/0.05)+l*(y/0.05)+m*(z/0.05) is used.

This type of irregular material data may have uses in statistical modeling of materials such as those found in additive manufacturing, where perfect material homogeneity of a 3D-printed component may not always be possible to achieve. The data can be used for any type of material property, such as conductivity, permeability, and elasticity properties, to name a few.

Getting back to our unit cube example, we now add a Blank Material node. We will, somewhat arbitrarily, set the Density to 2000 kg/m³ and the Heat capacity to 1 J/kg/K. Since we are performing a stationary analysis, the Heat capacity is irrelevant. The Thermal conductivity is set to the expression 1+2[W/m/K]*cloud(x,y,z). We can see from the earlier Slice plot visualization that the values for the interpolation table are roughly between -0.2 and 0.2. This means that this expression will generate an interesting spatial distribution of thermal conductivity values between about 0.6 and 1.4.

The coefficient 2[W/m/K] is used to assign a consistent unit to the expression. The constant 1 will be automatically converted to the correct unit: [W/m/K].

Let’s define some simple boundary conditions. Set the temperature at the top surface to 393.15[K] and the bottom surface to 293.15[K], corresponding to a 100-K temperature difference.

Now, let’s generate a default mesh.

COMSOL Multiphysics will automatically interpolate values such as those from the material properties to this unstructured mesh from the imported interpolation function. Alternatively, we could generate a swept mesh with hexahedral elements of the same size as the original data, 50x50x50. Such a representation would be more “true” to the original data.

You can experiment with different element orders, such as linear and quadratic types. Unless you use a very fine mesh that “oversamples” the data, the results will depend somewhat on the element order.

Running the Study will produce a couple of temperature plots, the second of which is an Isosurface plot.

Notice how the Isosurface plot looks a little bit jagged, which is due to the underlying irregularity of the material data. We can create another Slice plot to yet again visualize the data. This time, we do so under the guise of thermal conductivity by using the variable ht.kmean, which equals the expression 1+2[W/m/K]*cloud(x,y,z) defined earlier.

Here, the data is sampled at a lower sampling density than the original interpolation function, since we used the default mesh with the Element size set to the default Normal setting. Successively refining this unstructured mesh will ultimately sample the data at more or less the same level of detail as the original synthesized data.

As mentioned earlier, the approach used here for heat transfer is applicable to virtually any other type of simulation. For example, in a porous media flow simulation, the randomized quantity would be permeability rather than thermal conductivity. In the case of porous media flow, a more advanced random distribution may be needed, but let’s save that discussion for a future blog post.

Binary Data and Percolation Effects

We can also use the synthesized data in a different way: by using Boolean expressions to convert it to binary data. This method can be used for simulating two or more materials where the material interface is randomized and the material properties change abruptly from one point to another. COMSOL Multiphysics will automatically handle the sharp interpolations needed for this case.

The following picture shows a visualization of the Boolean expression cloud(x,y,z)>-0.03, which evaluates to 1 at points where the inequality is true and 0 at the other points.

To get a nicer plot, you can set the resolution of the Slice plot to Extra fine. This setting is available in the Quality section of the Settings window for the Slice plot.

We would now like to use this type of binary information in a simulation. It can be interesting, for example, to use it in a heat transfer simulation to see the so-called percolation effects. For certain threshold values, you get a large connected component in the material so that the entire slab of material starts conducting much more efficiently.

To try this, change the expression for the thermal conductivity to 1-0.9[W/m/K]*(cloud(x,y,z)>thold), where thold is a global parameter. Start by defining thold in Parameters under Global Definitions.

Then, change the material data accordingly.

For each point in space, the Thermal conductivity will, in a binary fashion, evaluate to 1 or 0.1, depending on the value of the inequality.

Now, let’s see how different values of this Boolean threshold will affect the simulation. For this purpose, run a parametric sweep over the parameter thold from -0.2 to 0.2.

Add a Surface Integration node under Derived Values to integrate the total heat flux that goes through one of the surfaces. This will be given by the surface integral of -ht.ntflux or +ht.ntflux, depending on if you are integrating over the top or bottom surface. In the figure below, we used the top surface.

The resulting Table plot shows the amount of heat power transferred (in watts). We can see that for threshold values around 0, the conductivity rises quickly from a low value to a high value. This is due to the sudden appearance of one or more large connected components where the expression 1-0.9[W/m/K]*(cloud(x,y,z)>thold) evaluates to 1.

The figures below show a Volume plot with a Filter attribute for three threshold values around 0. The filter shows the parts of the domain for which cloud(x,y,z)<thold corresponds to the locations of higher conductivity.

We can see from these figures how the highly conductive parts start connecting for the higher threshold values.

The corresponding Filter settings are shown in the figure below.

A similar type of percolation effect, seen here for binary data, is also happening for the continuous data case shown earlier. However, when using binary data, the effects are easier to see.

Point Cloud Visualization

Finally, let’s look at an alternative way of visualizing this type of random data. We will visualize the data set using a large number of randomized points (or rather, small spheres) and let the radius and color of the points vary according to the interpolation function cloud(x,y,z). In addition, we will only allow the points to be visualized for positive values of cloud(x,y,z). This technique will allow us to “see inside the data” in a way that is difficult to achieve using other methods. Note that this visualization technique works for any type of data, including real measured data.

Start by generating three random variables with uniform distribution, with the Range set to 1 and the Mean value set to 0.5.

To generate this type of plot, we use a Scatter Volume plot type. This is available by right-clicking a 3D Plot Group and selecting More Plots > Scatter Volume.

In the Settings window of the Scatter Volume plot, set the expression for the X-, Y-, and Z-components: rn1(x), rn2(x), and rn3(x), respectively. Here, we are using the x-coordinate in an unusual way, in that we are using it merely as a long vector of arbitrary values.

Next, in the Evaluation Points section, set the Number of points for the X grid points to 100,000; 1,000,000; or more, depending on how many points your computer can handle. Set each of the Y grid point and Z grid point values to 1. This is a trick for getting a long vector of values that we can feed into the random functions in order to generate a lot of random points within the unit cube.

To make the plot appear as in the above figure, go to the Radius section and set Expression to cloud(rn1(x),rn2(x),rn3(x)) and the Radius scale factor to 0.3. In addition, in the Color section, set the Expression to cloud(rn1(x),rn2(x),rn3(x)) and the Color table to GrayScale.

One additional noteworthy fact about this plot: negative values will be ignored. This helps our visualization, since roughly half of the generated data is negative and we can more easily see through the data and get an intuitive feel for the variations. This method will only work for a rectangular block. To instead generate this type of plot over an arbitrary CAD geometry, you can use the Particle Tracing Module, which allows you to generate random points inside any type of CAD model.

By the way, a similar-looking plot can be achieved in a 2D model by simply creating a 2D Surface plot using a double-sum expression, as shown in the figure above.

Conduct Material Sweeps to Automate Comparing Materials

When assigning materials to your model geometry, you may want to experiment with a few options and see how different materials affect your simulation results. In COMSOL Multiphysics, you can automate this process via the Material Sweep parametric study and Material Switch feature. As such, you do not need to add several materials one at a time and compute for the corresponding solution. In addition to saving you time during model set up, this facilitates the comparison of results during postprocessing.

Screenshot from the material sweeps video, showcasing the ability to switch between different results based on the material.

The Material Switch node houses the materials that you want to sweep over and provides functionality to automatically switch materials while your model is solving.

In the five-minute tutorial video below, we outline the procedure for performing a material sweep in your model and then walk you through the steps for doing so. This includes adding a Material Switch node; specifying parts of the geometry that the material sweep will be applied to; selecting the materials to switch between; adding the Material Sweep parametric study; and finally postprocessing the sweep’s results. We also briefly discuss how you can customize the materials being swept over as well as how to easily toggle between different sets of results obtained from your material sweep.

Video Tutorial: How to Sweep and Compare Materials in COMSOL Multiphysics

Easily Define Material Properties Using Material Functions

As mentioned earlier, COMSOL Multiphysics features a large collection of built-in materials that are available regardless of which modules you hold a license for. Upon adding any of these materials to your model, you will notice that the material properties are provided with certain default values.

In some cases, material properties are constant. In other cases, they may vary in space or be dependent on a physics variable such as temperature. If you want to make a constant material property variable, or if the built-in variation is not what you want to use, you can define your own function. In COMSOL Multiphysics, there are three types of functions that you can use to define a material property: Interpolation, Analytic, and Piecewise functions.

Data table and plot for an Interpolation function.

Interpolation functions are used to define a material property through reading in data from a table or file that contains values of the function at discrete points. You can enter this data manually or import it from an external file. This is useful when you have material properties that are obtained from experiments. COMSOL Multiphysics will automatically evaluate and then generate a function that fits the data you provide. Then, you can also choose how the function interpolates between the measured values or extrapolates outside of your specified range of data.

Input fields and plot for an Analytic function.

Analytic functions are used to define a function using built-in mathematical functions or other user-defined functions. You can enter an expression, specify the input arguments, and define the value range for each of the arguments in your equation.

Settings for a Piecewise function.

Piecewise functions are used to define a material property using different expressions over different intervals. The start and end point for each set of values, as well as the function applicable to that interval, can be entered manually or imported from an external file. The intervals that you define cannot overlap and there cannot be any holes between the intervals. That way, you have a continuous function uniquely defined in terms of the independent variable.

In the following seven-minute tutorial video, we discuss how to create and define Interpolation, Analytic, and Piecewise functions for any material property in your model, the advantages of using each type, and best practices to keep in mind when creating them. We also go over the settings for each function type, demonstrate how the selection of options such as Extrapolation will change your data plot, and show how you can call out your function in the Material Contents table.

Video Tutorial: Use Functions to Define a Material Property

Utilize Global Materials and Material Links for Multiple Components

While creating a model in COMSOL Multiphysics, you will at some point need to identify the materials that your objects are made of. Normally, this requires completing a series of steps in which you open the Add Material or Material Browser windows; choose the material; select and add it to your component; and then go into the material node’s settings to select the parts of the geometry to which the material applies. You would then need to repeat this procedure for each unique material that you want to include in your simulation. In COMSOL Multiphysics, you can expedite the above process using global materials and material links.

Screenshot displaying use of the global materials and material links functionality.

When a material is added under the Global Materials node, it is available to use anywhere throughout the model. Further, global materials can be used for any geometric entity level, whether you assign them to domains, boundaries, edges, or points.

Material links are used locally under a component’s material node to refer to a global material. This is advantageous when you have a COMSOL Multiphysics file that contains multiple components that are made up of similar materials, as you only need to specify the material once under the Global Materials node and can then link to it under each individual component. It is also beneficial for models in which the same material is assigned to different geometric entity levels such as domains and boundaries. In this case, you would again only need to add the material once and could also add a separate Material Link node for each geometric entity type.

In the six-minute tutorial video below, we show you how to use the global materials and material links functionality. We begin by demonstrating how to add global materials to your model and discuss the differences between adding materials globally and locally. Then, we walk through the steps of how to add material links to your model components and assign them to the geometry. After watching this video, we encourage you to try out this functionality yourself and see firsthand the ease with which you can assign materials in a model that contains multiple components or when you want to use the same materials on multiple parts.

Video Tutorial: Use the Same Material for Multiple Components in COMSOL Multiphysics

Efficiently Define the Materials Used in Your Simulation Studies

You can significantly expedite the process of assigning materials to your model geometry using the features and functionality discussed here. To complement these tools, we’ve created instructional videos to help you learn how to utilize them in your own simulations. Whether you have a model file that involves multiple components, need to define a complicated material property, or have to test different materials in your simulation, COMSOL Multiphysics features built-in tools that make this process simpler and more efficient for you.

Browse Additional Tutorial Videos Relating to Materials

• To learn more about specifying and defining materials in your COMSOL Multiphysics models, watch this introductory video series: How to Use Materials in Your COMSOL Multiphysics Models
• Head over to our Video Gallery to check out other videos on the topic of material-based features and functionality

What Is Thermal Hysteresis and How Is It Modeled?

A material with thermal hysteresis will exhibit a solidification temperature that is different from the melting temperature. Such materials have applications in heat sinks and thermal storage systems and are even used by living organisms, such as fish and insects living in cold climates. We won’t concern ourselves here with the exact physical mechanisms by which thermal hysteresis happens, but rather focus on how to model it.

We will begin by considering a representative material with hysteresis that is incompressible and plot out the enthalpy of the material as a function of temperature, as shown below. When the material is in the solid state and is being heated, the enthalpy is given by the bottom curve or path. As the material passes the melting temperature, it becomes completely liquid. When this material is subsequently cooled in the liquid state, it will follow the upper path, thus the material remains liquid at temperatures below the melting temperature. Once the freezing temperature is reached, the material becomes completely solid. If the material is then heated back up, it will follow the bottom path, and so on. In the completely molten or completely solid state, the two enthalpy curves overlap. The latent heat of melting and solidification is the jump in these curves.

Enthalpy versus temperature for an idealized incompressible material.

Now, the above curve represents a bit of an idealized case that would only occur in the real world if we had a perfectly pure material. It is also a bit impractical for computational modeling purposes since this immediate transition between states represents a discontinuity that is quite difficult to solve numerically.

However, if we introduce a small transition zone over which the enthalpy varies smoothly, then we have a model that is much more amenable to numerical analysis. The physical interpretation of this is that the material changes phase over some finite temperature and in the intermediate range, the material is a mixture of both solid and liquid. Only once the material is fully outside of the transition zone will it switch over to following the other curve.

Note that we have centered the smoothing around the nominal melting and freezing temperatures, so the fully molten state is at a temperature slightly higher than the nominal melting temperature and the fully solid state is slightly below the freezing temperature. The plot below shows a gradual smoothing, but this transitional zone can be made very narrow to better approximate the behavior if we really did have a perfectly pure material.

A smoothed enthalpy curve is more amenable to numerical analysis.

Since the material is assumed to be incompressible, the enthalpy depends only on the temperature. The above plot will also give us the specific heat, which is the derivative of enthalpy with respect to temperature. The specific heat is constant except for a small region around the melting and freezing temperatures.

The specific heat is the derivative of the enthalpy with respect to temperature and is different if the material is heated or cooled.

This temperature-dependent specific heat data can be put directly into the governing equation for heat transfer and, along with an appropriate set of boundary conditions, can be solved in COMSOL Multiphysics. In fact, the existing Application Gallery example, Cooling and Solidification of Metal, makes use of such a temperature-dependent specific heat, albeit without hysteresis. The only additional requirement for modeling thermal hysteresis is to introduce a switch to determine which path to follow. Let’s now look at how to implement this in COMSOL Multiphysics.

Implementing Thermal Hysteresis in COMSOL Multiphysics

Here, we will look at a simple example model of a phase-change material within a thin-walled container. One side wall is perfectly insulated and the wall on the other side is held at a known temperature that varies periodically over time. A schematic of this is shown below. We are interested in computing the temperature as a function of time and position through the thickness and can reduce this to a one-dimensional model to get started.

Schematic of the thermal model. A time-varying temperature is applied at one side.

Our modeling begins by setting up some physical constants via the Global Parameters that define the melting and freezing temperatures and the smoothing to apply to the enthalpy functions. The two smoothed enthalpy functions plotted earlier are implemented as shown in the screenshot below. We can here take advantage of the built-in Step function, which additionally features the option to apply a user-defined smoothing.

Implementation of the enthalpy functions shown above, using the smoothed step function. Note that the units have been defined.

The geometry of our model is simply a 1D interval representing the phase-change material region. The Heat Transfer in Solids interface is used, since we are assuming that there is no fluid flow. The material properties are as shown below.

Screenshot showing the material properties definitions in the phase-change material.

The thermal conductivity and the density are constants. The specific heat (the heat capacity at constant pressure) is defined as:

SorL*d(H_StoL(T),T)+(1-SorL)*d(H_LtoS(T),T)

where the differentiation operator takes the derivatives of the two different enthalpy functions with respect to temperature and the SorL variable defines the local material behavior as either Solid or Liquid. The SorL variable can be either zero or one and can be different in each element.

A Domain ODEs and DAEs interface is used to define this variable, with interface settings as illustrated below. Note that the dependent variable and the source term are both dimensionless and the shape function is of the type Discontinuous Lagrange — Constant, meaning that the SorL variable will take on a different constant value within each element.

Settings for the Domain ODEs and DAEs interface that tracks the material state.

The Source Term settings.

The above screenshot depicts the equation that is being solved for in the Domain ODEs and DAEs interface. Let’s examine in detail the Source Term equation used:

SorL-nojac(if(T> T_top,0,if(T< T_bot,1,SorL)))

This equation is evaluated at the centroid of each element and implements in a single line the following:

If the current temperature is greater than the temperature of complete melting (the nominal melting temperature plus half the smoothing temperature), then the material has passed its solid-to-liquid phase-change temperature, so set SorL = 0. This means that the liquid-to-solid enthalpy curve is followed. If the current temperature is less than the temperature of complete solidification (the nominal freezing temperature minus half the smoothing temperature), then the material has passed its liquid-to-solid phase-change temperature, so set SorL = 1. This means that the solid-to-liquid enthalpy curve is followed. Otherwise, when neither of the previous two conditions are satisfied, leave the SorL variable at its previous value, meaning that there is no change of path while in the intermediate zone.

The nojac() operator tells the software to exclude the enclosed expression from the Jacobian computation, thus it does not try to differentiate the enclosed expression but merely evaluates the expression itself at each time step.

The SorL variable is used in one place in the model: in the definition of the specific heat in the phase-change material domain, as shown earlier. It is also important to set the initial value of the variable appropriately. If the initial temperature of the phase-change material is above or below the temperature of complete melting or solidification, then this choice is unambiguous; otherwise, you must choose the initial state of the material. In the example here, we will consider the initial temperature of the system to be below the freezing temperature, so the material will initially follow the solid-to-liquid path. Therefore, the initial value of the SorL variable is set to one.

The solver settings showing the usage of the Previous Solution operator.

In terms of solving the model, we only need to keep in mind that the SorL variable needs to be evaluated at the previous time step using the Previous Solution operator in the solver sequence, as shown in the screenshot above. We can also use a segregated solver and, of course, we should investigate tightening the time-dependent solver tolerances, the scaling of the dependent variables, and study the convergence of the solution with mesh refinement.

Let’s now look at some results. In the plots below, we see the temperature through the thickness of our modeling domain for the heating and cooling of the phase-change material. Observe that the slope of the temperature as a function of position changes as the material passes through the melting and freezing points. This is due to extra heat that must be added or removed as the material changes phase.

Temperature over time in the phase-change material during heating. The blue line is the melting temperature.

Temperature over time in the material during cooling. The red line is the freezing temperature.

An animation showing the combined temperature profile over time during heating and cooling. The blue and red lines are the melting and freezing temperatures.

Closing Remarks on Modeling Materials with Thermal Hysteresis

Today, we have introduced an approach appropriate for modeling materials exhibiting thermal hysteresis under the assumption of constant density and that the material must go completely above and below the melting and freezing temperatures to change phase. This modeling approach makes use of the Previous Solution operator and equation-based modeling. For more details on the usage of the Previous Solution operator and further examples, please see:

• Using the Previous Solution Operator in Transient Modeling
• Tracking Material Damage with the Previous Solution Operator

The approach shown here is a bit simplified for the sake of explanation. If you are interested in the modeling of heat transfer with phase change, either with or without hysteresis, we recommend that you look to the Heat Transfer Module, which has a built-in interface for modeling heat transfer with phase change, as introduced here.

If you are instead interested in the modeling of irreversible changes in phase, then you may also want to take a look at our previous blog post on thermal curing as well as tracking material damage with the Previous Solution operator.

Looking to model thermal hysteresis in COMSOL Multiphysics or have other questions about this process? Please contact us.

Heating a Wafer to Remove a Thin Film of Material

Let’s consider a wafer of silicon with a very thin layer of material coated on the surface. This thin film may have been introduced in a previous processing step, and we now want to quickly “bake off” this material by heating the wafer with a laser. The wafer is mounted on a spinning stage while the laser heat source traverses back and forth over the surface.

We will consider a layer of material that is very thin compared to the wafer thickness. We can thus assume that the film does not contribute to the thermal mass of the system, nor will it provide any additional conductive heat path. However, this coating will affect the surface emissivity. If the coating is undamaged, the emissivity is 0.8. Once the coating has baked off, the emissivity of that region of the wafer will change to 0.6. This will alter both the amount of heat absorbed from the laser heat source and the heat radiated from the wafer to the surroundings.

A laser beam traversing over a rotating wafer ablates a thin surface coating when the temperature is high enough.

We will not concern ourselves too greatly with the process by which the coating is removed from the wafer. Although the actual process may include phase change, melting, boiling, ablation, and chemical reactions, we are, in this case, dealing with a very thin layer of material. Thus, we can simply say that once the temperature of the wafer surface exceeds 60°C, the coating immediately disappears. Under the assumption of very fast dynamics of the material removal process relative to the heating of the wafer, this is a valid approach.

Implementation of the Material Removal Model

We will begin with the previously developed model of a rotating wafer exposed to a moving heat source. An additional boundary equation will be added to our existing model. Hence, we need to model this in a coordinate system that moves with the rotation of the wafer.

The equation we add will track the surface emissivity on the top boundary of the wafer. The Previous Solution operator is used since we simply want to change the surface emissivity once the temperature gets above the specified value and leave it otherwise unchanged. We have already introduced the use of the Previous Solution operator in a previous blog entry. We will now focus more specifically on modeling the removal of the film from the wafer surface.

The settings for the Boundary ODEs and DAEs interface. Note the shape function settings.

The domain settings and initial values settings for the Boundary ODEs and DAEs interface, which models the emissivity of the surface of the wafer.

The settings for the Boundary ODEs and DAEs interface are shown above. Note that a Constant Discontinuous Lagrange discretization is used to solve for the field “emissivity”, the surface emissivity. This discretization is equivalent to saying that the emissivity will have a constant value over each element and that the field can be discontinuous across different elements. We are assuming that the film is either present or not present, so the surface emissivity will have two discrete states. The initial value of the field variable is the undamaged value of the surface emissivity.

The settings for the Heat Flux boundary condition use the computed surface emissivity.

The settings for the Diffuse Surface boundary condition use the computed surface emissivity.

The computed surface emissivity is used in two places within the Heat Transfer in Solids interface, as shown above. The applied Heat Flux boundary condition and the radiation of ambient temperature via the Diffuse Surface boundary condition both reference the emissivity field.

Since the surface emissivity is constant across each element, a finite element mesh size of 0.3 mm is used to obtain a smoother representation of the damage field. Also, the relative solver tolerance is set to 1e-6.

The results of the simulation are depicted in the animation below. As the temperature rises, certain portions of the wafer surface rise above the ablation temperature and the surface emissivity changes. The process is complete once the entirety of the wafer surface has been heated above the desired temperature.

An animation of the temperature field of the laser heating the rotating wafer (left). The dark gray color indicates the damage zone (right).

Summary

We have demonstrated how to model an irreversible change in the state of a material. In this case, we have analyzed the removal of a thermodynamically negligible thin layer of material from the surface of a wafer and modified the resultant surface emissivity as a consequence. The technique outlined here for using the Previous Solution operator can also be used in many other cases. What comes to your mind?

If you are interested in downloading the model related to this article,
it is available in our Application Gallery.

An Introduction to Modeling Laser-Material Interactions

While many different types of laser light sources exist, they are all quite similar in terms of their outputs. Laser light is very nearly single frequency (single wavelength) and coherent. Typically, the output of a laser is also focused into a narrow collimated beam. This collimated, coherent, and single frequency light source can be used as a very precise heat source in a wide range of applications, including cancer treatment, welding, annealing, material research, and semiconductor processing.

When laser light hits a solid material, part of the energy is absorbed, leading to localized heating. Liquids and gases (and plasmas), of course, can also be heated by lasers, but the heating of fluids almost always leads to significant convective effects. Within this blog post, we will neglect convection and concern ourselves only with the heating of solid materials.

Solid materials can be either partially transparent or completely opaque to light at the laser wavelength. Depending upon the degree of transparency, different approaches for modeling the laser heat source are appropriate. Additionally, we must concern ourselves with the relative scale as compared to the wavelength of light. If the laser is very tightly focused, then a different approach is needed compared to a relatively wide beam. If the material interacting with the beam has geometric features that are comparable to the wavelength, we must additionally consider exactly how the beam will interact with these small structures.

Before starting to model any laser-material interactions, you should first determine the optical properties of the material that you are modeling, both at the laser wavelength and in the infrared regime. You should also know the relative sizes of the objects you want to heat, as well as the laser wavelength and beam characteristics. This information will be useful in guiding you toward the appropriate approach for your modeling needs.

Surface Heat Sources

In cases where the material is opaque, or very nearly so, at the laser wavelength, it is appropriate to treat the laser as a surface heat source. This is most easily done with the Deposited Beam Power feature (shown below), which is available with the Heat Transfer Module as of COMSOL Multiphysics version 5.1. It is, however, also quite easy to manually set up such a surface heat load using only the COMSOL Multiphysics core package, as shown in the example here.

A surface heat source assumes that the energy in the beam is absorbed over a negligibly small distance into the material relative to the size of the object that is heated. The finite element mesh only needs to be fine enough to resolve the temperature fields as well as the laser spot size. The laser itself is not explicitly modeled, and it is assumed that the fraction of laser light that is reflected off the material is never reflected back. When using a surface heat load, you must manually account for the absorptivity of the material at the laser wavelength and scale the deposited beam power appropriately.

The Deposited Beam Power feature in the Heat Transfer Module is used to model two crossed laser beams. The resultant surface heat source is shown.

Volumetric Heat Sources

In cases where the material is partially transparent, the laser power will be deposited within the domain, rather than at the surface, and any of the different approaches may be appropriate based on the relative geometric sizes and the wavelength.

Ray Optics

If the heated objects are much larger than the wavelength, but the laser light itself is converging and diverging through a series of optical elements and is possibly reflected by mirrors, then the functionality in the Ray Optics Module is the best option. In this approach, light is treated as a ray that is traced through homogeneous, inhomogeneous, and lossy materials.

As the light passes through lossy materials (e.g., optical glasses) and strikes surfaces, some power deposition will heat up the material. The absorption within domains is modeled via a complex-valued refractive index. At surfaces, you can use a reflection or an absorption coefficient. Any of these properties can be temperature dependent. For those interested in using this approach, this tutorial model from our Application Gallery provides a great starting point.

A laser beam focused through two lenses. The lenses heat up due to the high-intensity laser light, shifting the focal point.

Beer-Lambert Law

If the heated objects and the spot size of the laser are much larger than the wavelength, then it is appropriate to use the Beer-Lambert law to model the absorption of the light within the material. This approach assumes that the laser light beam is perfectly parallel and unidirectional.

When using the Beer-Lambert law approach, the absorption coefficient of the material and reflection at the material surface must be known. Both of these material properties can be functions of temperature. The appropriate way to set up such a model is described in our earlier blog entry “Modeling Laser-Material Interactions with the Beer-Lambert Law“.

You can use the Beer-Lambert law approach if you know the incident laser intensity and if there are no reflections of the light within the material or at the boundaries.

Laser heating of a semitransparent solid modeled with the Beer-Lambert law.

Beam Envelope Method

If the heated domain is large, but the laser beam is tightly focused within it, neither the ray optics nor the Beer-Lambert law modeling approach can accurately solve for the fields and losses near the focus. These techniques do not directly solve Maxwell’s equations, but instead treat light as rays. The beam envelope method, available within the Wave Optics Module, is the most appropriate choice in this case.

The beam envelope method solves the full Maxwell’s equations when the field envelope is slowly varying. The approach is appropriate if the wave vector is approximately known throughout the modeling domain and whenever you know approximately the direction in which light is traveling. This is the case when modeling a focused laser light as well as waveguide structures like a Mach-Zehnder modulator or a ring resonator. Since the beam direction is known, the finite element mesh can be very coarse in the propagation direction, thereby reducing computational costs.

A laser beam focused in a cylindrical material domain. The intensity at the incident side and within the material are plotted, along with the mesh.

The beam envelope method can be combined with the Heat Transfer in Solids interface via the Electromagnetic Heat Source multiphysics couplings. These couplings are automatically set up when you add the Laser Heating interface under Add Physics.

The Laser Heating interface adds the Beam Envelopes and the Heat Transfer in Solids interfaces and the multiphysics couplings between them.

Full Wave

Finally, if the heated structure has dimensions comparable to the wavelength, it is necessary to solve the full Maxwell’s equations without assuming any propagation direction of the laser light within the modeling space. Here, we need to use the Electromagnetic Waves, Frequency Domain interface, which is available in both the Wave Optics Module and the RF Module. Additionally, the RF Module offers a Microwave Heating interface (similar to the Laser Heating interface described above) and couples the Electromagnetic Waves, Frequency Domain interface to the Heat Transfer in Solids interface. Despite the nomenclature, the RF Module and the Microwave Heating interface are appropriate over a wide frequency band.

The full-wave approach requires a finite element mesh that is fine enough to resolve the wavelength of the laser light. Since the beam may scatter in all directions, the mesh must be reasonably uniform in size. A good example of using the Electromagnetic Waves, Frequency Domain interface: Modeling the losses in a gold nanosphere illuminated by a plane wave, as illustrated below.

Laser light heating a gold nanosphere. The losses in the sphere and the surrounding electric field magnitude are plotted, along with the mesh.

Modeling Heat Transfer, Convection, and Reradiation Within and Around a Material

You can use any of the previous five approaches to model the power deposition from a laser source in a solid material. Modeling the temperature rise and heat flux within and around the material additionally requires the Heat Transfer in Solids interface. Available in the core COMSOL Multiphysics package, this interface is suitable for modeling heat transfer in solids and features fixed temperature, insulating, and heat flux boundary conditions. The interface also includes various boundary conditions for modeling convective heat transfer to the surrounding atmosphere or fluid, as well as modeling radiative cooling to ambient at a known temperature.

In some cases, you may expect that there is also a fluid that provides significant heating or cooling to the problem and cannot be approximated with a boundary condition. For this, you will want to explicitly model the fluid flow using the Heat Transfer Module or the CFD Module, which can solve for both the temperature and flow fields. Both modules can solve for laminar and turbulent fluid flow. The CFD Module, however, has certain additional turbulent flow modeling capabilities, which are described in detail in this previous blog post.

For instances where you are expecting significant radiation between the heated object and any surrounding objects at varying temperatures, the Heat Transfer Module has the additional ability to compute gray body radiative view factors and radiative heat transfer. This is demonstrated in our Rapid Thermal Annealing tutorial model. When you expect the temperature variations to be significant, you may also need to consider the wavelength-dependent surface emissivity.

If the materials under consideration are transparent to laser light, it is likely that they are also partially transparent to thermal (infrared-band) radiation. This infrared light will be neither coherent nor collimated, so we cannot use any of the above approaches to describe the reradiation within semitransparent media. Instead, we can use the radiation in participating media approach. This technique is suitable for modeling heat transfer within a material, where there is significant heat flux inside the material due to radiation. An example of this approach from our Application Gallery can be found here.

Summary

In this blog post, we have looked at the various modeling techniques available in the COMSOL Multiphysics environment for modeling the laser heating of a solid material. Surface heating and volumetric heating approaches are presented, along with a brief overview of the heat transfer modeling capabilities. Thus far, we have only considered the heating of a solid material that does not change phase. The heating of liquids and gases — and the modeling of phase change — will be covered in a future blog post. Stay tuned!

The Beer-Lambert Law and Material Heating

When light is incident upon a semitransparent material, some of the energy will be absorbed by the material itself. If we can assume that the light is single wavelength, collimated (such as from a laser), and experiences very minimal refraction, reflection, or scattering within the material itself, then it is appropriate to model the light intensity via the Beer-Lambert law. This law can be written in differential form for the light intensity as:

where z is the coordinate along the beam direction and is the temperature-dependent absorption coefficient of the material. Because this temperature can vary in space and time, we must also solve the governing partial differential equation for temperature distribution within the material:

where the heat source term, , equals the absorbed light. These two equations present a bidirectionally coupled multiphysics problem that is well suited for modeling within the core architecture of COMSOL Multiphysics. Let’s find out how…

Implementation in COMSOL Multiphysics

We will consider the problem shown above, which depicts a solid cylinder of material (20 mm in diameter and 25 mm in length) with a laser incident on the top. To reduce the model size, we will exploit symmetry and consider only one quarter of the entire cylinder. We will also partition the domain up into two volumes. These volumes will represent the same material, but we will only solve the Beer-Lambert law on the inside domain — the only region that the beam is heating up.

To implement the Beer-Lambert law, we will begin by adding the General Form PDE interface with the Dependent Variables and Units settings, as shown in the figure below.

Settings for the implementing the Beer-Lambert law. Note the Units settings.

Next, the equation itself is implemented via the General Form PDE interface, as illustrated in the following screenshot. Aside from the source term, , all terms within the equation are set to zero; thus, the equation being solved is . The source term is set to Iz-(50[1/m]*(1+(T-300[K])/40[K]))*I, where the partial derivative of light intensity with respect to the z-direction is Iz, and the absorption coefficient is (50[1/m]*(1+(T-300[K])/40[K])), which introduces a temperature dependency for illustrative purposes. This one line implements the Beer-Lambert law for a material with a temperature-dependent absorption coefficient, assuming that we will also solve for the temperature field, T, in our model.

Implementation of the Beer-Lambert law with the General Form PDE interface.

Since this equation is linear and stationary, the Initial Values do not affect the solution for the intensity variable. The Zero Flux boundary condition is the natural boundary condition and does not impose a constraint or loading term. It is appropriate on most faces, with the exception of the illuminated face. We will assume that the incident laser light intensity follows a Gaussian distribution with respect to distance from the origin. At the origin, and immediately above the material, the incident intensity is 3 W/mm². Some of the laser light will be reflected at the dielectric interface, so the intensity of light at the surface of the material is reduced to 0.95 of the incident intensity. This condition is implemented with a Dirichlet Boundary Condition. At the face opposite to the incident face, the default Zero Flux boundary condition can be physically interpreted as meaning that any light reaching that boundary will leave the domain.

The Dirichlet Boundary Condition sets the incident light intensity within the material.

With these settings described above, the problem of temperature-dependent light absorption governed by the Beer-Lambert law has been implemented. It is, of course, also necessary to solve for the temperature variation in the material over time. We will consider an arbitrary material with a thermal conductivity of 2 W/m/K, a density of 2000 kg/m³, and a specific heat of 1000 J/kg/K that is initially at 300 K with a volumetric heat source.

The heat source itself is simply the absorption coefficient times the intensity, or equivalently, the derivative of the intensity with respect to the propagation direction, which can be entered as shown below.

The heat source term is the absorbed light.

Most other boundaries can be left at the default Thermal Insulation, which will also be appropriate for implementing the symmetry of the temperature field. However, at the illuminated boundary, the temperature will rise significantly and radiative heat loss can occur. This can be modeled with the Diffuse Surface boundary condition, which takes the ambient temperature of the surroundings and the surface emissivity as inputs.

Thermal radiation from the top face to the surroundings is modeled with the Diffuse Surface boundary condition.

It is worth noting that using the Diffuse Surface boundary condition implies that the object radiates as a gray body. However, the gray body assumption would imply that this material is opaque. So how can we reconcile this with the fact that we are using the Beer-Lambert law, which is appropriate for semitransparent materials?

We can resolve this apparent discrepancy by noting that the material absorptivity is highly wavelength-dependent. At the wavelength of incident laser light that we are considering in this example, the penetration depth is large. However, when the part heats up, it will re-radiate primarily in the long-infrared regime. At long-infrared wavelengths, we can assume that the penetration depth is very small, and thus the assumption that the material bulk is opaque for emitted radiation is valid.

It is possible to solve this model either for the steady-state solution or for the transient response. The figure below shows the temperature and light intensity in the material over time, as well as the finite element mesh that is used. Although it is not necessary to use a swept mesh in the absorption direction, applying this feature provides a smooth solution for the light intensity with relatively fewer elements than a tetrahedral mesh. The plot of light intensity and temperature with respect to depth at the centerline illustrates the effect of the varying absorption coefficient due to the rise in temperature.

Plot of the mesh (on the far left) and the light intensity and temperature at different times.

Light intensity and temperature as a function of depth along the centerline over time.

Summary and Further Refinements

Here, we have highlighted how the General Form PDE interface, available in the core COMSOL Multiphysics package, can be used for implementing the Beer-Lambert law to model the heating of a semitransparent medium with temperature-dependent absorptivity. This approach is appropriate if the incident light is collimated and at a wavelength where the material is semitransparent.

Although this approach has been presented in the context of laser heating, the incident light needs only to be collimated for this approach to be valid. The light does not need to be coherent nor single wavelength. A wide spectrum source can be broken down into a sum of several wavelength bands over which the material absorption coefficient is roughly constant, with each solved using a separate General Form PDE interface.

In the approach presented here, the material itself is assumed to be completely opaque to ambient thermal radiation. It is, however, possible to model thermal re-radiation within the material using the Radiation in Participating Media physics interface available within the Heat Transfer Module.

The Beer-Lambert law does assume that the incident laser light is perfectly collimated and propagates in a single direction. If you are instead modeling a focused laser beam with gradual variations in the intensity along the optical path then the Beam Envelopes interface in the Wave Optics Module is more appropriate.

In future blog posts, we will introduce these as well as alternate approaches for modeling laser-material interactions. Stay tuned!

Different Material Models

All material models are mathematical approximations of a true physical behavior. Material models can, however, not always be derived from physical principles, like mass conservation or equations of equilibrium. They are by nature phenomenological and based on measurements. The laws of physics will, however, enforce limits on the mathematical structure of material models and the possible values of material properties.

It is well known, even from everyday life, that different materials exhibit completely different behavior. A material can be very brittle, like glass, or very elastic, like rubber. Choosing a material model is not only determined by the material as such, but also by the operating conditions. If you immerse a piece of rubber into liquid nitrogen, it will become as brittle as glass — a popular educational experiment. Also, if you heat up glass, it will start to creep and show viscoelastic behavior.

When analyzing structural mechanics behavior in COMSOL Multiphysics, you can choose between about 50 built-in material models, many of them featuring several options for their settings. You can also set up and define your own material models, or combine several of the material models to, for example, describe a material exhibiting both creep and plasticity at the same time.

Some of the available classes of materials are:

• Linear elastic
• Hyperelastic
• Nonlinear elastic
• Plasticity
• Creep
• Concrete

Without going into details about how you should actually come to the correct decision about an appropriate material model, here are some questions you should ask yourself before you start modeling:

• How large are the stress and strain ranges?
• Will the loading speed be important?
• What is the operating temperature and will it be constant?
• Is there a predefined material model targeted specifically at my material, such as concrete or soil plasticity?
• Is the load constant, monotonously increasing, or cyclic?
• Is the stress state predominantly uniaxial or is it fully three-dimensional?

Based on these considerations, you can then make a choice of a suitable material model. Determining the correct parameters to use in this material model will then be more or less difficult.

On one end of the spectrum, there are common materials (such as steel at room temperature) where many engineers know the material data by heart (E = 210 GPa, ν = 0.3, ρ = 7850 kg/m³) and where data is easily found in the literature or through a simple web search.

On the other end of the spectrum, finding the high temperature creep data for a cast iron to be used in an exhaust manifold can be a major project in itself. Many tests at different load levels and at different temperatures are required. A complete test program for this may take half a year and have a price tag of several hundred thousand dollars.

Tensile testing equipment. “Inspekt desk 50kN IMGP8563″ by Smial. Original uploader was Smial at de.wikipedia — Transferred from de.wikipedia; transferred to Commons by User: Smial using CommonsHelper. (Original text: eigenes Foto). Licensed under CC BY-SA 2.0 de via Wikimedia Commons.

Common Types of Tests

Before starting your simulation with COMSOL Multiphysics, it is not enough to import the geometry of the specimen, select the material model, and apply the loads and other boundary conditions; you should also provide the parameters for the chosen material model in the operating stress-strain and temperature range. These parameters are typically obtained from one or more tests.

Uniaxial Tension

The most fundamental test is the uniaxial tensile test. This is also what most engineers in daily life refer to when they state that they have a “Stress-Strain curve.” If you look at the list of questions above, it is evident that even this seemingly simple test can leave many loose ends:

• A material may exhibit time dependence even at constant loads, giving creep or viscoelastic effects. Many tests, often at different temperature and stress levels, are needed to give reliable data.
• Material parameters obtained from an ordinary tensile test at low speed may not be representative of the material behavior at high strain rates. A crash analysis might show strain rates as high as 10 s^-1, while conventional uniaxial testing machines can use strain rates as low as 10^-3 s^-1.
• Is the material isotropic or would tests in several directions be required?
• If you only have a tension test, what would happen in compression? With a single curve, you cannot really tell.
• A tensile test will supply stress versus strain in the tested direction, but it will not always contain data about the deformations in the transverse direction. Without that data, you have no information at all about the cross-coupling between the directions in the 3D case.
• When curve fitting experimental measurements, perhaps not all data should be given equal weight. It may so be that the response in a certain strain range has a larger impact on your simulation results.

Uniaxial Compression

Some materials, like concrete, have little or no capacity to carry loads in tension. Here, the uniaxial compression test is the most fundamental test. It has many properties in common with the tensile test.

Other materials, like steel and rubber, can also be tested in compression. It is actually a good idea to do so, as we will demonstrate later in this blog post.

When using only uniaxial testing (whether it is in tension, compression, or both), you can however not achieve the full picture of the properties of a given material. You will need to combine it with some other assumptions like isotropy or incompressibility. For many materials, such assumptions are well justified by experience, though.

We have illustrated how the range of a test will affect your conception of the material behavior in the animation below.

• If you just do the onloading part, it is not possible to discriminate between elastic and plastic behavior.
• By unloading, you can distinguish plastic from elastic behavior, but until the specimen is in a state of significant compression, it is not possible to determine whether an isotropic or a kinematic hardening model would give the best representation.

Biaxial Tension

It is significantly more difficult to design testing equipment that can create a homogenous biaxial stress state. Biaxial testing is often used for materials that are available only in thin sheets, like fabrics, for instance. By controlling the ratio between the loads in two perpendicular directions, it is possible to extract much more information than from a uniaxial test.

Triaxial Compression

For soils, which generally need to be confined, triaxial compression is a common test. Triaxial compression tests could in principle be applied to a block of any material, but the testing equipment is difficult to design. The low compressibility of most solid materials also makes triaxial testing less attractive, since the measured displacements will be small when the material is compressed in all directions.

The Triaxial Test model shows a finite element model of a triaxial compression test.

Torsion

The torsion test, where a cylindrical test specimen is twisted, is a rather simple test that generates a non-uniaxial stress state. The stress state is, however, not homogenous through the rod. Therefore, some extra processing is needed to translate the moment versus angle results to stress-strain results.

Testing Hyperelastic Materials

In an upcoming blog post in this series, we will make an in-depth demonstration of how to fit measured data to a number of different hyperelastic material models. In the example here, we will assume that you have been able to fit your data to the tests. The raw data consists of two measurements: one in uniaxial tension and another in equibiaxial tension, as shown below.

The nominal stress (force divided by original area) is plotted against stretch (current length divided by original length).

Measured stress-strain curves by Treloar.

Since the data covers a wide range of stretches, the experimental results are clearly nonlinear. The simplest hyperelastic models with one or two parameters will probably not be sufficient to fit the experimental data. The Ogden model with three terms is a popular model for rubber, and it is the model we used here.

A least squares fit will give the results below when assigning equal weights to both data sets. As we can see in the graph, it is possible to fit both experiments very well with a single set of material parameters.

Fitted material parameters using a three terms Ogden model.

But what if the biaxial test had not been available? Fitting only the uniaxial data will give a different set of material parameters, which will of course fit that set of experimental data even more closely, but it would deviate from the biaxial results. This is shown below.

Analytical results for uniaxial and biaxial tension when only the uniaxial data was used to fit the model parameters.

Clearly, the prediction for a equibiaxial stress state will differ between the two sets of parameters. As we can see, the error in stress in the biaxial curve is more than 20% at some stretch levels.

What about other stress states? Two stress states that can be simulated in a simple finite element model are uniaxial compression and pure torsion. The uniaxial stress-strain curve over a wide range of stretches is shown below. The results on the tensile side are not as sensitive to the data set used for obtaining the material parameters as the compressive side is. This is not surprising as tensile data is used for parameter fitting in both cases, whereas neither of the experiments contain any information about the compressive behavior.

Uniaxial response ranging from compression to tension. The scale on the x-axis is logarithmic.

Note that operating conditions of rubber parts, such as seals, are often under predominantly compressive states. If the data sets used for parameter fitting contain only tension data, this may be a source of inaccuracy when modeling multiaxial stress states.

Finally, let’s have a look at a simulation where a circular bar is twisted. The same type of discrepancies between the results from two sets of material parameters as above can be seen below.

Computed torque as function of the twist angle.

Finally, it should be noted that many hyperelastic models are only conditionally stable. This means that even though the estimated material parameters are perfectly valid for a certain strain range, a unique and continuous stress-strain relation may not even exist for other strain combinations. We often come across such problems in support cases. This is unfortunately rather difficult to detect a priori, since it would require a full search of all possible strain combinations.

Concluding Remarks and Next Up

Measured data must be processed and analyzed before being used as input for simulations. For material models other than the simpler linear elastic model, it is a good idea to make small examples with a unit cube to assess the behavior under different loading states before using the material model in a large-scale simulation.

So the answer to the question: “I would like to just enter my measured stress-strain curve directly into COMSOL Multiphysics” is that such an approach is not recommended. That would make the software a black box where the user really must take a number of active decisions in order to obtain meaningful results.

Up next in our Structural Materials series: We will discuss nonlinear elasticity and plasticity.

Power Law

The nonlinear stress-strain behavior in solids was already described 100 years ago by Paul Ludwik in his Elemente der Technologischen Mechanik. In that treatise, Ludwik described the nonlinear relation between shear stress and shear strain observed in torsion tests with what is nowadays called Ludwik’s Law:

(1)

For , the stress-strain curve is linear; for , the curve is a parabola; and for , the curve represents a perfectly plastic material. Ludwik just described the behavior (Fließkurve) of what we now call a pseudoplastic material.

In version 5.0 of the COMSOL Multiphysics simulation software, beside Ludwik’s power-law, the Nonlinear Structural Materials Module includes different material models within the family of nonlinear elasticity:

• Ramberg-Osgood
• Power Law
• Uniaxial Data
• Bilinear Elastic
• User Defined

In the Geomechanics Module, we have now included material models intended to represent nonlinear deformations in soils:

• Hyperbolic Law
• Hardin-Drnevich
• Duncan-Chang
• Duncan-Selig

An Example with Uniaxial Data

The main difference between a nonlinear elastic material and an elastoplastic material (either in metal or soil plasticity) is the reversibility of the deformations. While a nonlinear elastic solid would return to its original shape after a load-unload cycle, an elastoplastic solid would suffer from permanent deformations, and the stress-strain curve would present hysteretic behavior and ratcheting.

Let’s open the Elastoplastic Analysis of a Plate with a Center Hole model, available in the Nonlinear Structural Materials Model Library as elastoplastic_plate, and modify it to solve for one load-unload cycle. Let’s also add one of the new material models included in version 5.0, the Uniaxial data model, and use the stress_strain_curve already defined in the model.

Here’s a screenshot of what those selections look like:

In our example, the stress_strain_curve represents the bilinear response of the axial stress as a function of axial strain, which can be recovered from Ludwik’s law when .

We can compare the stress distribution after laterally loading the plate to a maximum value. The results are pretty much the same, but the main difference is observed after a full load-unload cycle.

Top: Elastoplastic material. Bottom: Uniaxial data model.

Let’s pick the point where we observed the highest stress and plot the x-direction stress component versus the corresponding strain. The green curve shows a nonlinear, yet elastic, relation between stress and strain (the stress path goes from ). The blue curve portraits a hysteresis loop observed in elastoplastic materials with isotropic hardening (the stress path goes from ).

With the Uniaxial data model, you can also define your own stress-strain curve obtained from experimental data, even if it is not symmetric in both tension and compression.

Further Reading

• P. Ludwik. Elemente der Technologischen Mechanik
• “Hypoelasticity“, Chapter 3.3 of Applied Mechanics of Solids
• Download the Elastoplastic Analysis of a Plate with a Center Hole model