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Average displacement boundary condition

Posted Feb 14, 2024, 5:05 a.m. EST Structural & Acoustics, Equation-Based Modeling, Physics Interfaces 11 Replies

John Smith
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COMSOL v4.2a

I would like to specify the average displacement of a boundary.

In the solid mechanics module, I have used the prescribed displacement boundary condition but as I understand it this requires every point of the boundary to have the specified displacement value. How can I define the average displacement of a boundary.

Can this be achieved with a weak contribution/constraint?

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11 Replies Last Post Feb 22, 2024, 11:45 a.m. EST
Henrik Sönnerlind COMSOL Employee

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Posted: 2 months ago Feb 14, 2024, 10:14 a.m. EST

Even though it is not clear to me what you want to achieve, here is a way of doing it:

  1. Add an average coupling operator to that boundary, say aveop1
  2. Create a variable for the average displacement, like u_a set to aveop1(u)
  3. Use a global constraint to set the value, using something like u_a – 0.2[mm]

The solution that you get will have a displacement distribution on the boundary that fulfills the average displacement criterion, while minimizing the elastic energy in the structure.

Out of curiosity, can you describe the application where you need this?

-------------------
Henrik Sönnerlind
COMSOL
Even though it is not clear to me what you want to achieve, here is a way of doing it: 1. Add an average coupling operator to that boundary, say *aveop1* 2. Create a variable for the average displacement, like *u\_a* set to *aveop1(u)* 3. Use a global constraint to set the value, using something like *u\_a – 0.2[mm]* The solution that you get will have a displacement distribution on the boundary that fulfills the average displacement criterion, while minimizing the elastic energy in the structure. Out of curiosity, can you describe the application where you need this?
John Smith
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Posted: 2 months ago Feb 15, 2024, 7:34 a.m. EST

Even though it is not clear to me what you want to achieve, here is a way of doing it:

  1. Add an average coupling operator to that boundary, say aveop1
  2. Create a variable for the average displacement, like u_a set to aveop1(u)
  3. Use a global constraint to set the value, using something like u_a – 0.2[mm]

The solution that you get will have a displacement distribution on the boundary that fulfills the average displacement criterion, while minimizing the elastic energy in the structure.

Out of curiosity, can you describe the application where you need this?

Hi Henrik,

Thank you very much for your reply. I realise now that it was a bit of a silly question. My motivation was that I was trying to mimic a piezoelectric transducer actuating a resonator. This was described in a research paper where the authors set an average displacement boundary condition. I now see that for this the authors defined an "envelope function" for the boundary condition where the average of the function must have been the desired displacement value. How I can achieve this myself, I am not sure.

Thanks, John

>Even though it is not clear to me what you want to achieve, here is a way of doing it: > >1. Add an average coupling operator to that boundary, say *aveop1* >2. Create a variable for the average displacement, like *u\_a* set to *aveop1(u)* >3. Use a global constraint to set the value, using something like *u\_a – 0.2[mm]* > >The solution that you get will have a displacement distribution on the boundary that fulfills the average displacement criterion, while minimizing the elastic energy in the structure. > >Out of curiosity, can you describe the application where you need this? Hi Henrik, Thank you very much for your reply. I realise now that it was a bit of a silly question. My motivation was that I was trying to mimic a piezoelectric transducer actuating a resonator. This was described in a research paper where the authors set an average displacement boundary condition. I now see that for this the authors defined an "envelope function" for the boundary condition where the average of the function must have been the desired displacement value. How I can achieve this myself, I am not sure. Thanks, John
Acculution ApS Certified Consultant
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Posted: 2 months ago Feb 15, 2024, 8:14 a.m. EST

Probably something like a Gaussian distribution spatially across the boundary, scaled to achieve a particular average value. You can do that in COMSOL.

-------------------
René Christensen, PhD
Acculution ApS
www.acculution.com
info@acculution.com
Probably something like a Gaussian distribution spatially across the boundary, scaled to achieve a particular average value. You can do that in COMSOL.
John Smith
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Posted: 2 months ago Feb 15, 2024, 8:36 a.m. EST
Updated: 2 months ago Feb 15, 2024, 8:38 a.m. EST

Probably something like a Gaussian distribution spatially across the boundary, scaled to achieve a particular average value. You can do that in COMSOL.

Hi René,

Sorry, I am relatively new to COMSOL and am not sure how I would implement the Gaussian distribution as a boundary condition.

The paper that I was trying to reproduce was: https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.8.024020, but in 2 dimensions for simplicity. The function they used as a boundary condtion was

where was the desired average displacement and are the lengths of the surfaces in the x and y directions.

Probably too complicated for me.

Thanks, John

>Probably something like a Gaussian distribution spatially across the boundary, scaled to achieve a particular average value. You can do that in COMSOL. Hi René, Sorry, I am relatively new to COMSOL and am not sure how I would implement the Gaussian distribution as a boundary condition. The paper that I was trying to reproduce was: https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.8.024020, but in 2 dimensions for simplicity. The function they used as a boundary condtion was \int_0^{L_p,x/2} \frac{2}{L_{p,x}}dx \int_0^{L_{p,y}/2} \frac{2}{L_{p,y}}dy (u_z-d_0)(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) where d_0 was the desired average displacement and L_{p,x}, L_{p,y} are the lengths of the surfaces in the x and y directions. Probably too complicated for me. Thanks, John
Acculution ApS Certified Consultant
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Posted: 2 months ago Feb 15, 2024, 8:53 a.m. EST
Updated: 2 months ago Feb 16, 2024, 2:19 a.m. EST

Okay, that is not what I expected. Henrik can probably help better here, as I would say this is not uniquely defined as a boundary conditions, but his first answer also described a way of inputting an average that I would not have thought to be proper one.

-------------------
René Christensen, PhD
Acculution ApS
www.acculution.com
info@acculution.com
Okay, that is not what I expected. Henrik can probably help better here, as I would say this is not uniquely defined as a boundary conditions, but his first answer also described a way of inputting an average that I would not have thought to be proper one.
Henrik Sönnerlind COMSOL Employee

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Posted: 2 months ago Feb 19, 2024, 5:21 a.m. EST
Updated: 2 months ago Feb 19, 2024, 5:22 a.m. EST

I assume that the intended expression is

If so, it differs from my original answer only by the tanh() weighting function. So instead of an average coupling operator, you need to use an integration coupling operator. The right hand side is obtained by using the form

-------------------
Henrik Sönnerlind
COMSOL
I assume that the intended expression is \frac{2}{L_{p,x}} \frac{2}{L_{p,y}} \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} (u_z-d_0)(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx = 0 If so, it differs from my original answer only by the tanh() weighting function. So instead of an average coupling operator, you need to use an integration coupling operator. The right hand side is obtained by using the form \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} u_z(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx = d_0 \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} (-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx
John Smith
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Posted: 2 months ago Feb 19, 2024, 7:02 a.m. EST

I assume that the intended expression is

\frac{2}{L_{p,x}} \frac{2}{L_{p,y}} \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} (u_z-d_0)(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx = 0

If so, it differs from my original answer only by the tanh() weighting function. So instead of an average coupling operator, you need to use an integration coupling operator. The right hand side is obtained by using the form

\int_0^{L_p,x/2} \int_0^{L_{p,y}/2} u_z(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx = d_0 \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} (-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx

Thank you for this reply, Henrik. I followed the steps you outlined in your previous comment, but wondered if a global constrain (u_a-0.2[mm]) in step 3 is correct as this doesn't seem to produce any displacement in a frequency study. Should this be a weak constraint on the boundary?

Thanks, John

>I assume that the intended expression is > > \frac{2}{L_{p,x}} \frac{2}{L_{p,y}} \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} (u_z-d_0)(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx = 0 > >If so, it differs from my original answer only by the tanh() weighting function. So instead of an average coupling operator, you need to use an integration coupling operator. The right hand side is obtained by using the form > >\int_0^{L_p,x/2} \int_0^{L_{p,y}/2} u_z(-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx = d_0 \int_0^{L_p,x/2} \int_0^{L_{p,y}/2} (-\tanh \bigg(\frac{x-\frac{1}{2}L_p}{\Delta L_p} \bigg)) dydx Thank you for this reply, Henrik. I followed the steps you outlined in your previous comment, but wondered if a global constrain (u_a-0.2[mm]) in step 3 is correct as this doesn't seem to produce any displacement in a frequency study. Should this be a weak constraint on the boundary? Thanks, John
Henrik Sönnerlind COMSOL Employee

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Posted: 2 months ago Feb 20, 2024, 11:19 a.m. EST
Updated: 2 months ago Feb 20, 2024, 11:21 a.m. EST

There should not be a problem with prescribing the displacement in a frequency domain study in the suggested way (as long as it is direct frequncy domain, not mode superposition).

If the distribution of the prescribed displacement is allowed to vary as freely as suggested by the integral expression, then there is only one number to fix, thus the global constraint. In some sense, this is a (very) weak way of prescribing the displacement. Any displacement distribution that fulfills the value of the integral is allowable. As I indicated above, something more is needed to determine the exact distribution. That, I think, will lead to some kind of smooth distribution based on energy minimization.

In a classical FE context, the integral will essentially be interpreted as a 'multipoint constraint'.

where the summation index i ranges over all nodes on the boundary. This resembles what is sometimes termed an RBE3 coupling, in which you prescribe the value of the 'master' DOF.

A true 'weak constraint' prescribes the displacement distribution. It is just another method of prescribing the value everywhere when compared to a pointwise constraint. Rather than directly setting the value of a dof, an extra equaton (containing the Lagrange multiplier) is added.

-------------------
Henrik Sönnerlind
COMSOL
There should not be a problem with prescribing the displacement in a frequency domain study in the suggested way (as long as it is direct frequncy domain, not mode superposition). If the distribution of the prescribed displacement is allowed to vary as freely as suggested by the integral expression, then there is only one number to fix, thus the global constraint. In some sense, this is a (very) weak way of prescribing the displacement. Any displacement distribution that fulfills the value of the integral is allowable. As I indicated above, something more is needed to determine the exact distribution. That, I think, will lead to some kind of smooth distribution based on energy minimization. In a classical FE context, the integral will essentially be interpreted as a 'multipoint constraint'. \sum_i \alpha_i u_i = u_{prescribed} where the summation index *i* ranges over all nodes on the boundary. This resembles what is sometimes termed an RBE3 coupling, in which you prescribe the value of the 'master' DOF. A true 'weak constraint' prescribes the displacement distribution. It is just another method of prescribing the value everywhere when compared to a pointwise constraint. Rather than directly setting the value of a dof, an extra equaton (containing the Lagrange multiplier) is added.
John Smith
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Posted: 2 months ago Feb 21, 2024, 5:49 a.m. EST
Updated: 2 months ago Feb 21, 2024, 8:40 a.m. EST

There should not be a problem with prescribing the displacement in a frequency domain study in the suggested way (as long as it is direct frequncy domain, not mode superposition).

If the distribution of the prescribed displacement is allowed to vary as freely as suggested by the integral expression, then there is only one number to fix, thus the global constraint. In some sense, this is a (very) weak way of prescribing the displacement. Any displacement distribution that fulfills the value of the integral is allowable. As I indicated above, something more is needed to determine the exact distribution. That, I think, will lead to some kind of smooth distribution based on energy minimization.

In a classical FE context, the integral will essentially be interpreted as a 'multipoint constraint'.

\sum_i \alpha_i u_i = u_{prescribed}

where the summation index i ranges over all nodes on the boundary. This resembles what is sometimes termed an RBE3 coupling, in which you prescribe the value of the 'master' DOF.

A true 'weak constraint' prescribes the displacement distribution. It is just another method of prescribing the value everywhere when compared to a pointwise constraint. Rather than directly setting the value of a dof, an extra equaton (containing the Lagrange multiplier) is added.

Thanks for the reply, Henrik.

I must be going wrong somewhere.
I have:
1. Defined a global parameter d0 (prescribed displacement)
2. Defined an average coupling operator aveop1 at the boundary to be displaced
3. Defined a variable u_a (aveop1(v))
4. In the physics interface defined a global constraint u_a-d0
5. Run a frequency study

but the resultant displacement field is 0 over the entire domain.
If there is anything you think I have missed or done incorrectly please let me know. Otherewise, thank you for all of your advice.

Thanks, John

>There should not be a problem with prescribing the displacement in a frequency domain study in the suggested way (as long as it is direct frequncy domain, not mode superposition). > >If the distribution of the prescribed displacement is allowed to vary as freely as suggested by the integral expression, then there is only one number to fix, thus the global constraint. In some sense, this is a (very) weak way of prescribing the displacement. Any displacement distribution that fulfills the value of the integral is allowable. As I indicated above, something more is needed to determine the exact distribution. That, I think, will lead to some kind of smooth distribution based on energy minimization. > >In a classical FE context, the integral will essentially be interpreted as a 'multipoint constraint'. > > \sum_i \alpha_i u_i = u_{prescribed} > >where the summation index *i* ranges over all nodes on the boundary. This resembles what is sometimes termed an RBE3 coupling, in which you prescribe the value of the 'master' DOF. > >A true 'weak constraint' prescribes the displacement distribution. It is just another method of prescribing the value everywhere when compared to a pointwise constraint. Rather than directly setting the value of a dof, an extra equaton (containing the Lagrange multiplier) is added. Thanks for the reply, Henrik. I must be going wrong somewhere. I have: 1. Defined a global parameter d0 (prescribed displacement) 2. Defined an average coupling operator aveop1 at the boundary to be displaced 3. Defined a variable u_a (aveop1(v)) 4. In the physics interface defined a global constraint u_a-d0 5. Run a frequency study but the resultant displacement field is 0 over the entire domain. If there is anything you think I have missed or done incorrectly please let me know. Otherewise, thank you for all of your advice. Thanks, John
Henrik Sönnerlind COMSOL Employee

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Posted: 2 months ago Feb 21, 2024, 11:10 a.m. EST
Updated: 2 months ago Feb 21, 2024, 11:10 a.m. EST

That sounds correct. Can you upload the model file?

-------------------
Henrik Sönnerlind
COMSOL
That sounds correct. Can you upload the model file?
John Smith
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Posted: 2 months ago Feb 22, 2024, 11:45 a.m. EST

That sounds correct. Can you upload the model file?

Hello Henrik,

I think I have achieved what I wanted. Instead of the global constraint I set a pointwise constraint (u_a-d0) on the boundary to have the average displacement. Upon running the frequency study I have a non-zero displacement field and an average probe on the boundary-in-question shows that it has the desired average displacement value.

Thanks for all of your help and patience.

John

>That sounds correct. Can you upload the model file? Hello Henrik, I think I have achieved what I wanted. Instead of the global constraint I set a pointwise constraint (u_a-d0) on the boundary to have the average displacement. Upon running the frequency study I have a non-zero displacement field and an average probe on the boundary-in-question shows that it has the desired average displacement value. Thanks for all of your help and patience. John

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