editied:
Hi,
I deleted the original question because it was ill-posed.
The question was: How does the Volume Constraint work and how the understanding of it can help to better interpret optimization results. Here is what I learned after a conversation with Mr Grasmannsdorf from Altair:
Volume Constraint means you want OptiStruct to calculate for example the stiffest (minimum compliance) structure with a certain volume fraction (VF), for arguments sake let's say 25% of the original volume of your design space.
The VF references a virtual volume for the calculations, not your actual structure's volume! This is a bit confusing at first, because the help defines the VF as total volume at current iteration divided by the initial design volume. However the 'volume at current iteration' is a virtual volume!!! Similar to the Young's Modulus, the virtual volume is a function of the element density.
V_virtual_i = V_element_i * density
Therefore the virtual volume is the integral over the design domain of the density values. The volume constraint then is
V_virtual <= VF * Initial Volume
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What's the meaning of this?
The integral can be satisfied by either many elements with small density values or a lesser amount of elements with higher density values. The density value of the elements is updated via the strain energy during the iterations. Due to the optimization's surge for a maximum stiffness, the optimization solution strives to higher density values hence trying to eliminate elements with lower density values.
For the interpretation best-practice has show that all elements above a density value of circa 0,3 corresponds to the desired VF of the structure. The exact density threshold for the VF depends on the amount of intermediate densities in the solution a therefore varies a little.
I hope the helped you to understand how the volume constraint worked and helps you to a better understanding of topology optimizations
Regards
Hauke
