I'm working on p-norm topology optimization in plane stress using a MATLAB code adapted from the article An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization" by Hao Deng, Praveen S. Vulimiri and Albert C.To. I've noticed small sensitivity values (e.g., 4.54e-05, -7.30e-09) with a stress norm parameter (p) of 5. Are such values typical in this context, and should negative sensitivity values be expected? The relevant codes are attached.
Your experiences and recommendations would be greatly appreciated.
Thanks!
Hi Mr. Azar,
in contrast to the gradient for compliance in TopOpt that only has negative entries (assuming the standard case of a linear model and positive material stiffness...), the sensitivities for stress can have both signs. For compliance this simply means that adding material anywhere always will reduce the overall compliance of the part so the performance is always increased. For stress however, adding material can either reduce the stresses in some areas (negative gradient sign) but sometimes also increase the stresses (positive gradient sign), so more material is not always better in a stress-based TopOpt especially around sharp corners. Now for the magnitudes: Using p-norm will introduce a weighting of the stress values and also their sensitivities to derive a single global stress measure from a large number of local stress values. The currently highest stress value will get the highest weight and all other values will quickly get very low weights the further away their stress values are compared to the current maximum stress value. This is the “trick” used to replace the maximum stress by a differentiable expression using the p-norm. A very low sensitivity magnitude means that a certain design variable has negligible effect on the change of stresses the currently highest stressed regions. Locally it may still have a significant effect on local stresses in other regions but not on the highest stress values that make up the largest contribution to the global stress measure.
So yes, you have to expect everything (negative and positive values and high and very low magnitudes of sensitivities) in a stress-based TopOpt using p-norm.
Best regards,
Olaf
Hi Mr Azar, you may check the following paper "Reformulation for stress topology optimization of continuum structures by floating projection" recently published in CMAME. I believe this paper can answer your question.
https://www.sciencedirect.com/science/article/pii/S0045782524001269
Best regards,
Tao
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