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How can we solve the equation [curl(curl E)-k^2*E=f ] in COMSOL?

Posted Dec 12, 2017, 10:23 p.m. EST RF & Microwave Engineering, Studies & Solvers Version 5.2a 4 Replies

Qlogic Chen
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Hi everyone,

Does anyone know how to solve the vector field (E) from the equation [curl(curl E)-k^2*E=f ] in COMSOL ? The parameters k and f are constants and the curl means curl operator. We have tried to find an appropriate form of equation in the mathematical module of COMSOL but cannot get a satisfactory one to solve the equation. Do you have any suggestion? Thank you for your generous help.

 Best Wishes
 Qlogic
4 Replies Last Post Dec 14, 2017, 4:09 p.m. EST
Vasily Kozhevnikov
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Posted: 6 years ago Dec 13, 2017, 1:39 a.m. EST

Dear friend,

You should first use a rule rot(rotE) = grad(divE) - laplace(E) and then use e.g. Coefficient Form PDE to define the equation you solve

Dear friend, You should first use a rule rot(rotE) = grad(divE) - laplace(E) and then use e.g. Coefficient Form PDE to define the equation you solve
Qlogic Chen
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Posted: 6 years ago Dec 13, 2017, 6:01 a.m. EST

Thanks for your reply. Indeed I tried to use the rule rot(rotE) = grad(divE) - laplace(E) at first. However, I cannot find a suitable Coefficient Form PDE to express the grad(divE) term. That's why I ask the question.

Thanks for your reply. Indeed I tried to use the rule rot(rotE) = grad(divE) - laplace(E) at first. However, I cannot find a suitable Coefficient Form PDE to express the grad(divE) term. That's why I ask the question.
Robert Koslover Certified Consultant
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Posted: 6 years ago Dec 13, 2017, 11:11 p.m. EST
Updated: 6 years ago Dec 13, 2017, 11:19 p.m. EST

FYI, your PDE has a name. It is called the "Inhomogeneous Helmholtz Equation." (See https://en.wikipedia.org/wiki/Helmholtzequation ). If you haven't already done so, I suggest you take a look at the Comsol RF Module, 3D, frequency domain (emw). This module lets you solve the homogeneous version of this equatiion, subject to various boundary and volume conditions. I suspect there is a fair chance that you can re-express your quantity f (which makes the equation inhomogeneous) in terms of an equivalent boundary or volume condition. Good luck.

-------------------
Scientific Applications & Research Associates (SARA) Inc.
www.comsol.com/partners-consultants/certified-consultants/sara
FYI, your PDE has a name. It is called the "Inhomogeneous Helmholtz Equation." (See https://en.wikipedia.org/wiki/Helmholtzequation ). If you haven't already done so, I suggest you take a look at the Comsol RF Module, 3D, frequency domain (emw). This module lets you solve the homogeneous version of this equatiion, subject to various boundary and volume conditions. I suspect there is a fair chance that you can re-express your quantity f (which makes the equation inhomogeneous) in terms of an equivalent boundary or volume condition. Good luck.
John
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Posted: 6 years ago Dec 14, 2017, 4:09 p.m. EST

Hi, are you sure there are no discontinuities in material properties? If so, then there is probably a term that was once inside the curl(curl()) and if so then your field is not best discretized by Lagrange elements but instead by curl elements. They come for free if you have purchased the RF Module :-)

On a different note, you dont mention boundary conditions. For RF fields, the normal vector field is undefined on the boundary, but the tangential field is known. In return there is a constraint on divergence throughout the domain. Curl-elements are appropriate for tangential constraints. If you have an arbitrary divergence but constraints on the full three components on the boundary it starts to look more like Stokes equation.

Which is it?

With regard the coefficient formula for grad(div(E)) maybe it is a bit ugly in 3D. Nicer might be to use a weak formulation of the PDE and do integration by parts in which case you have get an integral of the form div(testE) * div(E)

Regards, J.

Hi, are you sure there are no discontinuities in material properties? If so, then there is probably a term that was once inside the curl(curl()) and if so then your field is not best discretized by Lagrange elements but instead by curl elements. They come for free if you have purchased the RF Module :-) On a different note, you dont mention boundary conditions. For RF fields, the normal vector field is undefined on the boundary, but the tangential field is known. In return there is a constraint on divergence throughout the domain. Curl-elements are appropriate for tangential constraints. If you have an arbitrary divergence but constraints on the full three components on the boundary it starts to look more like Stokes equation. Which is it? With regard the coefficient formula for grad(div(E)) maybe it is a bit ugly in 3D. Nicer might be to use a weak formulation of the PDE and do integration by parts in which case you have get an integral of the form div(testE) * div(E) Regards, J.

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