Topological Optimization

jacob

I don't see any downward force in the picture.
All 3 cylinders are bonded?

Amin LAK

jacob
the downward distributed load is not shown in the picture. Imagine a distributed load pulling down the lower face of the magenta cylinder.
All 3 cylinders are bound together. The volume of the middle cylinder has been introduced to allow a smooth transform of the geometry from the tube above to the large cylinder underneath.

jacob

So the pulling force is like gravity?
The structure stands on the ground and is loaded by bending moment?

Amin LAK

jacob
The pulling force is actually coming from a surcharge deposited above the magenta cylinder but we can arrange the structure of the magenta cylinder to let the surcharge to rest on the lower skin of the magenta cylinder and therefore directly transferred to the elastic bedding. I think in the model, a distributed load of 1,5e5 [force/area] can be applied on the lower face of the magenta cylinder to represent this surcharge.

The structure is only loaded by this distributed load (1,510⁵ [force/area] on the lower skin of the magenta cylinder) and a bending moment of 1,310⁹ [force.length] applied on top of the tube.

jacob

Can you draw some 2d figure with all the forces where exaxtly the are applied and units?

Amin LAK

jacob

Have you tried to solve it like this?
If the bottom cylinder will be whole in contact anyway then there us no need for contact.

Amin LAK

jacob
If the magenta cylinder is very rigid in comparison with the stiffness of the elastic bedding, then no gapping due to uplift occurs and thus there is no need for modelling "only-compression" contact. But if the optimum topology of the magenta cylinder be not very stiff in comparison with the elastic bedding, then there is a risk of local gapping between the magenta cylinder and the bedding.

Maybe as a first try, we can neglect the contact and find the optimum shape of the magenta cylinder, then we can see if there is any significant tension in the elastic bedding or not. If there is no tension applied to the elastic bedding or the gapping zone is limited in size, then we can consider that the topology found for the structure is sufficiently correct.

jacob

or the distributed load q is high enough

Amin LAK

jacob
the distributed load is probably high enough to resist the applied bending moment and to avoid overturning.
But if there is any difficulties with the convergence, I think we can reduce the value of the overturning moment of 1,3e9 to even one-tenth of it because the amplitude of the overturning moment perhaps doesn't affect the optimum shape of the magenta cylinder (as far as there is no uplift thus the problem remains linear).

jacob

Here is you case solved for E = 6.25 and 6000.25

The stiffer the ground is, the lower the prying.

Amin LAK

jacob
Thank you Jacob for your effort!
If I understood correctly, you have modeled the elastic bedding as a layer of 5 m thick with an elastic modulus of E=6,25 MPa in order to reach a stiffness of 1e9 N/m. I think it is not a bad idea. Then I would assume that the lower face of this auxiliary 5 m thick should be fixed and doesn't move. Right?

The main question is how the optimum structure of the 8 m high magenta cylinder is? I cannot see in these snapshot the optimum shape of the structure. Does your code allow to optimize the magenta cylinder?

Thanks,
Amin

jacob

yes, because with solids and contact it's the simples way. Probably it could be done other way with compression-only springs etc., but the fixed support adds very small number of degrees of freedom anyway.

I only tried to solve it like it is yet.
The behavior of 2 m deflection and 1.5 m prying is all right?

Amin LAK

jacob
Please increase the distributed load to 2e5.

jacob

I forgot to devide the moment by 2, since I use symmetry, so the difference without contact is not that big

without contact for 20 % of the volume

Amin LAK

jacob
Thanks a lot Jacob!
The optimized shape looks reasonable to me. I also used 2D optimization tools of DTU TOPOPT and found a similar truss-form shape for the magenta cylinder. See a snapshot below.
$w40-h8-fraction-18-percent-distributed-support-2.png$

Amin LAK

jacob
This problem is very iterative because the contact conditions are changing in each iteration by the update of the structure. So the interaction of the magenta cylinder and the supporting media is also changing.
I am wondering how the optimum shape would look like if we consider the contact? In this case I assume on the negative X side of the foundation, we must get a full surface in order to have downward distributed load which helps to balance overturning of the structure.

jacob

there is not much difference with a contact

Amin LAK

jacob
Thanks a lot Jakub!
Can you please share with me your contact details?
My email address is leuven8607@gmail.com

Amin LAK

jacob
is it possible to share the model with me so I can further play with it?

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