Discussion Forum

In 3D, add a 'Weak Form Boundary PDE' interface (under the Mathematics branch) with 3 dependent variables, say u, v and w. In the Weak Expressions section enter:

test(u)*(u-nx)
test(v)*(v-ny)
test(w)*(w-nz)

Now solve the model. The Gaussian curvature is then given by uTx+vTy+wTz. I think the mean curvature would be half of this.

You can also refer to the attached (V4.2a) model.

Hello,

Thanks for your answer. Can you explain to me why you solve this to obtain the Gaussian curvature? According to wikidedia the Gaussian curvature is the product of the principal curvatures K=k1*k2.
But I need the mean curvature, that's k1+k2. In 2-D maybe both are the same, but now you talked about 3-D.

For me what you do is finding the vector u so that div(u-n)=0. But I still miss something because when I apply divergence theorem now in 3-D for general answer:

Vintegral(phi*div(u-n))= - Vintegral(Grad(phi)*(u-n))+Sintegral(phi*(u-n))

You're only calculating the second integral on the right side, why? Am I right with what I say here?
If yes in 3-D I'm able to solve this by adding a weak expression in the volume and another at the boundary. Do I need to do that?

Hi Karl,

Why are you applying the divergence theorem? The equations that Daniel provided apply only on the surface to elegantly calculate the curvature. That curvature is actually the mean curvature though not the Gaussian curvature.

Nagi Elabbasi
Veryst Engineering

Because I want to understand why this calculation gives the surface curvature. Can you explain this for me?

Ok, I've understand.
First step: calculate n in weak form
second step: calculate div(n).
I don't need divergence theorem.
Mr. Smith can you post the curvature file for Comsol 4.2 only please? I don't have 4.2a, only 4.2. I was wrong.
Thank you!

Hi Daniel, I find very interesting the way you calculate the surface curvature. I have a similar problem:

- I am simulating an sphere shell with pressure from the inside, which makes the sphere expand.
- I am using the Structural Mechanics Module
- The expanded surface presents some irregularities in the topography of the surface, which I want to evaluate by measuring the curvature.
- I want to calculate the curvature in a different manner. I want to make cuts that pass through the center, and evaluate the curvature of the resulting x,y plane. This means that i want only to consider the curvature on the radial direction of the sphere.

Could you help me to build the Weak Form PDE? I appreciate your help!!

José

Hello Mr. Smith,

I know that this converstation came long ago but hopefuly I can bring it back to life again.

I was trying your expression for the calculation of the curvature. As expected, it works really good. Then, I tried to use the value to model a surface diffusion process and then problems started.

In my model (deformed geometry) I define the normal displacement of a surface as a function of its curvature. The problem is, that the values of the curvature have small oscillations (likely due to the discretization of the geometry in finite elements). You can see those oscillations also in your moder (for example by ploting the curvature of the smallest cicumference of 0,75m which should be aroung 1.33 1/m). Those oscillations lead to small differences on the displacements of the points of the surface, which leads to huge differences on the curvature of the new surface. You can guess that after a few time steps the model collapses.

My question is: Can I by any means reduce the oscillations anyhow by somekind of smoothing?

I would also appreciate some enlightment about the relation between the uTx vTy and the geometry and gradient of the variables. I cannot find them defined in the equation view!

Thank you.

--
M. Sc. Oscar Banos