Including Operators and Expressions in a Multiphysics Simulation Is Easier Than You Think

Linus Andersson December 11, 2009
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As most skilled COMSOL users, I am sure you know that you are not limited to just selecting what is in our drop-down lists. Say that you have invented your own measure of structural stress. You want it to be equal to the quadratic mean of the Tresca and von Mises stresses. Go to Plot Parameters to find out what these predefined stresses are called (tresca_smsld and mises_smsld if you are modeling in 3D with the Structural Mechanics Module). Now all you need to do is enter sqrt(0.5*(tresca_smld^2+mises_smld^2)) in any of the Expression fields and click OK to see your new stress distribution.

Operators in COMSOL Multiphysics

You probably didn’t think of it, but in the expression I just mentioned, sqrt, ^, and even + are all examples of operators. COMSOL offers a whole range of useful ones, not all equally obvious. Did you for instance know that the letter d will differentiate any variable or expression with respect to time or space? d(c,z) gives the derivative of a concentration c with respect to the z-coordinate. d(sqrt(0.5*(tresca_smld^2+mises_smld^2)),t) is the time-derivative of your stress. If you have created your own subdomain expression my_stress containing your stress definition, d(my_stress,t) gives the same results.

The at operator lets you access the solution at any time in postprocessing. This is handy if you want to see changes over a time interval. Plotting the expression at(20,p)-at(10,p) overrides the Solution at time setting and shows you the pressure increase between 10 and 20 seconds. The with operator lets you postprocess more than one parametric or eigensolution in a similar fashion.

Another handy pair of operators is up and down. They live on boundaries and help you evaluate anything with discontinuities. Consider for example a temperature gradient on a boundary between two subdomains with different conductivities. gradT_ht will silently evaluate this gradient on both sides of the boundary and give you the average. With up(gradT_ht) and down(gradT_ht) however, you can decide which side you are interested in.

If you work with electromagnetics, you might have plotted the magnetic field in an eigenmode analysis only to find that it appears to be identically zero. Chances are it is non-zero but perfectly imaginary due to its 90-degree phase difference with a real-valued electric field. Use the imag operator to show its imaginary part, abs to plot the norm, or arg to see the phase angle. Note that the default plot for complex fields shows the real part.

Next Steps

This is just the tip of the iceberg. You can find the complete list of mathematical and other operators in the COMSOL Multiphysics Quick Start and Quick Reference.



  1. Jing Zhou September 6, 2012   4:48 pm

    Many thanks to the author for this useful information!
    By the way, do you how to express an integral? Thank you!

  2. Linus Andersson, COMSOL Employee September 7, 2012   2:11 am

    Hi Jing, glad you found it useful. To express an integral, just create and use an integration operator. This is done from Definitions > Model Couplings.

  3. Jesus Lucio December 4, 2012   8:30 am

    Hello, Linus,
    Please, could you tell me how to access (for instance with the WITH() operator, or by means of the JOIN node for data) to any value of two different parameterized solutions (say in one solution the parameter goes from 0 to 1 and in the other the parameters goes form 1 to 2)? Each solution comes from one step function.
    Thanks in advance.

  4. Linus Andersson, COMSOL Employee December 5, 2012   8:17 am

    Hello Jesus,
    I believe a Join data set should do the trick. Once you have created it, make sure that it has Data 1 set to Solution 1 and Data 2 set to Solution 2. Set “Solutions” to “One” in both the Data sections, and you will be able to pick any combination of parameter values. As usual, the Combination Method gives you various options for how to combine the solutions.
    I hope this helps. If not, please contact with your model, and we should be able to give you some more specific advice.

  5. Randhir Singh July 3, 2013   5:29 pm

    Following up Jing Zhou: how to perform definite integral where the result is a field variable f(x,y) and where bounds are defined by the arbitrary space coordinates? say, integrating an field-expression within bounds of (x1,,y1) and (x,y) in a 2d model (x1<x and y1<y).

  6. Linus Andersson, COMSOL Employee July 8, 2013   2:11 am

    Hi Randhir,
    Suppose you have an integration operator intop1 and an expression u that you want to integrate over the rectangle x1<x<x2,y1<y<y2. The expression to evaluate then becomes intop1(u*(x>x1)*(xy1)*(y<y2)). The way this works is that comparisons evaluate to 1 if they are true and 0 if they are false.

  7. Hossein Mousavi August 5, 2014   12:06 am

    I really like the new COMSOL capability to join two separate solutions into one dataset using the Join functionality. However, it seems data2 special operator returns only the real part of its argument but data1 can handle both components. Is that a bug or it this meant to be?

  8. Mahvash Afzal September 29, 2016   12:58 am


    You’ve mentioned how the ‘d’ operator differentiates a variable wrt to time and space but how do we differentiate a variable wrt another dependent variable. I am using the d operator but it isn’t working for me.

  9. Amir Hossein Kazemi July 27, 2018   2:39 am

    Hello my freinds,
    (^2) can change negetive sign to positive but why we should use ”abs” before for example; abs(ewfd.s11)^2 to calculate norm of S11?

  10. Athena Serra February 22, 2019   3:40 am

    Please: How to calculate this in Comsol 5.4:-8*Al*Al*integralof(dT/dz(r,0))*r*dr) from(0 to 1/2/Al). My Email:

  11. Brianne Christopher February 25, 2019   8:59 am

    Hello Athena,

    For questions related to your modeling, please contact our Support team.

    Online Support Center:


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