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Eigenvalue problem with Dirichlet Boundary condition

Posted Nov 19, 2013, 6:32 a.m. EST Modeling Tools & Definitions, Parameters, Variables, & Functions, Studies & Solvers Version 4.3b 6 Replies

Jason Pan
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Hi all,

I am a newbie of comsol and try to solve eigenvalue problem in a finite region using the PDE module. When I try to impose Dirichlet boundary condition, it seems not working. The eigen-wavefunction's value at boundary does not equal to the value I set. Even when I set u(at boundary)=1, the eigen-wavefunction appears to be 0 at boundaries. What kind of stupid errors have I made in the attachment? I am really looking forward to your response.

Thank you for your time!

jason
6 Replies Last Post Nov 20, 2013, 8:38 p.m. EST
Josh Thomas Certified Consultant
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Posted: 1 decade ago Nov 19, 2013, 12:04 p.m. EST
Jason-

For Eigenvalue problems in COMSOL, in my experience, any prescribed values of the dependent variable (no matter the magnitude) are taken to be zero.

Since you don't get magnitude information (only mode shape results) from an eigenvalue problem anyway, are you sure you really are looking for a difference in results between setting r=1 and r=0 along the outside boundary?

--
Best regards,
Josh Thomas
AltaSim Technologies
Jason- For Eigenvalue problems in COMSOL, in my experience, any prescribed values of the dependent variable (no matter the magnitude) are taken to be zero. Since you don't get magnitude information (only mode shape results) from an eigenvalue problem anyway, are you sure you really are looking for a difference in results between setting r=1 and r=0 along the outside boundary? -- Best regards, Josh Thomas AltaSim Technologies
Jason Pan
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Posted: 1 decade ago Nov 19, 2013, 8:06 p.m. EST
Hi Josh,

Thank you so much for your reply. The Dirichlet Boundary condition (DBC) is very important for my eigenvalue problem. The BC I am interested in is actually more complex, i.e. u(at boudary)=f(ux,uy,lambda), it turns out to be u(at boundary)=0. Then I apply a simpler case, like mentioned above, u(at boundary)=1, I get the same results.

Do you have some suggestions about this kind of Boundary condition? I really appreciate your help.

Regards,
Jason
Hi Josh, Thank you so much for your reply. The Dirichlet Boundary condition (DBC) is very important for my eigenvalue problem. The BC I am interested in is actually more complex, i.e. u(at boudary)=f(ux,uy,lambda), it turns out to be u(at boundary)=0. Then I apply a simpler case, like mentioned above, u(at boundary)=1, I get the same results. Do you have some suggestions about this kind of Boundary condition? I really appreciate your help. Regards, Jason
Josh Thomas Certified Consultant
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Posted: 1 decade ago Nov 20, 2013, 10:08 a.m. EST
Jason-

I'm not sure about applying Dirichlet conditions for eigenvalue problems. I know that any solution independent source/load conditions are ignored by the Eigenvalue solver. Not sure how that relates to Dirichlet conditions where you are directly specifying the dep. variable. Perhaps your problem is analogous to the pre-stressed eigenfrequency problems solved in the Solid Mechanics physics area? First solve a static form of the equations and then solve the eigenvalue problem on the solved static problem? Those are the best thoughts I have for this problem. Also, for understanding what COMSOL is doing under the hood, I recommend reading the documentation on the Eigenvalue/Eigenfrequency steps since your problem is special.

--
Best regards,
Josh Thomas
AltaSim Technologies
Jason- I'm not sure about applying Dirichlet conditions for eigenvalue problems. I know that any solution independent source/load conditions are ignored by the Eigenvalue solver. Not sure how that relates to Dirichlet conditions where you are directly specifying the dep. variable. Perhaps your problem is analogous to the pre-stressed eigenfrequency problems solved in the Solid Mechanics physics area? First solve a static form of the equations and then solve the eigenvalue problem on the solved static problem? Those are the best thoughts I have for this problem. Also, for understanding what COMSOL is doing under the hood, I recommend reading the documentation on the Eigenvalue/Eigenfrequency steps since your problem is special. -- Best regards, Josh Thomas AltaSim Technologies
Nagi Elabbasi Facebook Reality Labs
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Posted: 1 decade ago Nov 20, 2013, 11:57 a.m. EST
Hi Jason,

Interesting discussion. A constant Dirichlet boundary condition in an Eigenfrequency analysis is treated as a 0 value regardless of what constant you use. If it’s a pre-stress type problem look into the good example that Josh pointed out.

However, if you set the value in the Dirichlet BC to be a function of u instead of a constant then COMSOL recognized that. Try for example setting the value to “u” instead of “1”. Using a function of lambda (the eigenvalue) is tricky though. I don’t think (but I’m not 100% sure) that the solver can handle it because it is a form on nonlinearity not built-in eigenfrequency solvers.

Nagi Elabbasi
Veryst Engineering
Hi Jason, Interesting discussion. A constant Dirichlet boundary condition in an Eigenfrequency analysis is treated as a 0 value regardless of what constant you use. If it’s a pre-stress type problem look into the good example that Josh pointed out. However, if you set the value in the Dirichlet BC to be a function of u instead of a constant then COMSOL recognized that. Try for example setting the value to “u” instead of “1”. Using a function of lambda (the eigenvalue) is tricky though. I don’t think (but I’m not 100% sure) that the solver can handle it because it is a form on nonlinearity not built-in eigenfrequency solvers. Nagi Elabbasi Veryst Engineering
Jason Pan
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Posted: 1 decade ago Nov 20, 2013, 8:32 p.m. EST
Hi Josh,

Thank you for your remind. I will look into the documentation you mentioned.

Here I encounter another problem. Under the PDE, I try to solve the coupled PDEs of the first order. You can refer to the attachment for the equation. I found that the solution is very nonphysical as the eigenvalue should be real. Moreover, the eigen wavefunction looks very strange (there should be a standing wave). Would you please take a look?

Regards,
Jason
Hi Josh, Thank you for your remind. I will look into the documentation you mentioned. Here I encounter another problem. Under the PDE, I try to solve the coupled PDEs of the first order. You can refer to the attachment for the equation. I found that the solution is very nonphysical as the eigenvalue should be real. Moreover, the eigen wavefunction looks very strange (there should be a standing wave). Would you please take a look? Regards, Jason

Attachments:

Jason Pan
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Posted: 1 decade ago Nov 20, 2013, 8:38 p.m. EST
Hi Nagi,

Thank you for your information about 'lambda'. I recognized that 'lambda' can be eliminated by solving a coupled PDE equivalently (refer to the attachment in the reply to Josh). However, this time another problem appears, which I don't know how to deal with. The eigenvalue should be real (there are always imaginary parts in the results) and the eigen wavefunction looks strange.

Would you please take a look at the attachment above? Thank you so much for your help.

Regards,
Jason
Hi Nagi, Thank you for your information about 'lambda'. I recognized that 'lambda' can be eliminated by solving a coupled PDE equivalently (refer to the attachment in the reply to Josh). However, this time another problem appears, which I don't know how to deal with. The eigenvalue should be real (there are always imaginary parts in the results) and the eigen wavefunction looks strange. Would you please take a look at the attachment above? Thank you so much for your help. Regards, Jason

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