Nice video ! As I know that Simp is the most common method in TO, I 've read some documentation about it but still don't understand how it deals with gray elements (with density between 0 and 1). Please explain it in more detail, and try to tie this to the schematic diagram you showed in video. I am new to this field, so I really want to get your explainations, or some related documents. Thank you !
Good question, Xuân If the density of an element is zero, apparently its stiffness shall be zero. If its density is one, let's say its stiffness is 1E. For intermediate densities we need to an interpolation function to get their stiffness. The first idea one may come up with is to use a linear interpolation. Then a density of 0.5 leads to a stiffness of 0.5E. However, it is known that if the ratio of solid material in a microstructure volume is 50%, theoretically its stiffness is lower than 0.5E. This is known as Hashin-Shtrikman bounds. For a solid material with a Poisson's ratio 0.3, this theoretical bound can be approximated by a cubic function. Thus comes the popular power law interpolation with a power of 3 (also referred to as the penalization parameter). Under this physically realistic interpolation, since with 0.5 material one gets less than 0.5E stiffness, it is not economical to use 0.5 material, and thus the optimization automatically converges to a solution with densities either close to zero or close to one, i.e., a binary design. I suggest to download the popular 99 or 88 line topopt code, and play with the 'penal' value to see the effects.
So Simp method solves the gray elements by calculating its density value. Through iterations, these intermediate density values will gradually progress to the value 0 or 1. Am I correct in this ? If Yes, then I have 2 assumptions about determining the density value of the element: 1 : Initial density value before iteration of the element is determined based on calculating its displacement (corresponding to steps 1 and 2 in your diagram), then the density value will change through iterations in the direction towards 0 or 1. However, I have not found any relationship between the displacement of an element and its density value. If this is true can you show it ? 2 : The initial density value of an element is randomly determined, and the change of this value through the iterations is also random. The structural displacement (or stiffness) formula is just to compare how the element's contribution to the overall stiffness changes as its density value changes. If the contribution is small then the element will be removed and vice versa. This is consistent with your interpretation of sensitivity. But if this is true, is the statement "is the density value of the element really approaching 0 or 1" guaranteed ? Or the change of density value is no rule at all ? I find this to be a very interesting field, but there is still a lot to understand about it. Looking forward to hearing more from you. Thanks very much
great work , please can You recommend great book to learn TO from scratch im new in this field .. my first year in phd (Topology optimization for additive manufacturing) what i should learn first please help me
What do you mean when you relax the design variables to be continuous? Does it mean the design variables can be anywhere within 1 to 0, which means there will be a density gradient? Also I am currently undertaking a project that seeks to apply topology optimisation for lattice structures, do you have any research papers you recommend for a person with zero prior no knowledge like me. Thanks
yes, relaxation refers to allowing the design variables to be anywhere within [0,1]. In compliance minimization which is the case in my example, you get a pretty black and white design, meaning that most density values are very close to 1 or 0. In some special cases, e.g, if you have a distributed load, it is likely that there is a visible density gradient. For getting multi-scale structures, I recommend our review article on this topic homepage.tudelft.nl/z0s1z/projects/2021-multiscale-review.html
Great suggestion! Stress constraints have been a topic of interest in topology optimization since late 1990s. There are a number of recent developments. I hope to share some results soon.
Okay great, also when i read the literature about stress constraints in topology optimization, they use aggregating techniques like p norm or ks function. I have already perform the stress constraints topology optimization under minimise volume objective function in Ansys using simp technique, but now i want to validate my Ansys results with the stress constraints imposition techniques used in literature. It is really a great help if you make a small demonstration that how to impose global stress constraints using p norm or ks function.
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Great explanation of the basic theory behind it. Would love for much more in depth videos into the mathematics behind it. Thanks.
thanks for your great explains!
good explanation sir, keep it up!
Thanks a lot for great demonstration.
Nice video ! As I know that Simp is the most common method in TO, I 've read some documentation about it but still don't understand how it deals with gray elements (with density between 0 and 1). Please explain it in more detail, and try to tie this to the schematic diagram you showed in video. I am new to this field, so I really want to get your explainations, or some related documents. Thank you !
Good question,
Xuân
If the density of an element is zero, apparently its stiffness shall be zero. If its density is one, let's say its stiffness is 1E. For intermediate densities we need to an interpolation function to get their stiffness. The first idea one may come up with is to use a linear interpolation. Then a density of 0.5 leads to a stiffness of 0.5E. However, it is known that if the ratio of solid material in a microstructure volume is 50%, theoretically its stiffness is lower than 0.5E. This is known as Hashin-Shtrikman bounds. For a solid material with a Poisson's ratio 0.3, this theoretical bound can be approximated by a cubic function. Thus comes the popular power law interpolation with a power of 3 (also referred to as the penalization parameter).
Under this physically realistic interpolation, since with 0.5 material one gets less than 0.5E stiffness, it is not economical to use 0.5 material, and thus the optimization automatically converges to a solution with densities either close to zero or close to one, i.e., a binary design.
I suggest to download the popular 99 or 88 line topopt code, and play with the 'penal' value to see the effects.
So Simp method solves the gray elements by calculating its density value. Through iterations, these intermediate density values will gradually progress to the value 0 or 1. Am I correct in this ? If Yes, then I have 2 assumptions about determining the density value of the element:
1 : Initial density value before iteration of the element is determined based on calculating its displacement (corresponding to steps 1 and 2 in your diagram), then the density value will change through iterations in the direction towards 0 or 1. However, I have not found any relationship between the displacement of an element and its density value. If this is true can you show it ?
2 : The initial density value of an element is randomly determined, and the change of this value through the iterations is also random. The structural displacement (or stiffness) formula is just to compare how the element's contribution to the overall stiffness changes as its density value changes. If the contribution is small then the element will be removed and vice versa. This is consistent with your interpretation of sensitivity. But if this is true, is the statement "is the density value of the element really approaching 0 or 1" guaranteed ? Or the change of density value is no rule at all ?
I find this to be a very interesting field, but there is still a lot to understand about it. Looking forward to hearing more from you. Thanks very much
Plz can you provide mi some notes of this lecture and tell me the software which you are using in this video.
Thank you 😊
great work , please can You recommend great book to learn TO from scratch im new in this field .. my first year in phd (Topology optimization for additive manufacturing) what i should learn first please help me
What do you mean when you relax the design variables to be continuous? Does it mean the design variables can be anywhere within 1 to 0, which means there will be a density gradient? Also I am currently undertaking a project that seeks to apply topology optimisation for lattice structures, do you have any research papers you recommend for a person with zero prior no knowledge like me. Thanks
yes, relaxation refers to allowing the design variables to be anywhere within [0,1]. In compliance minimization which is the case in my example, you get a pretty black and white design, meaning that most density values are very close to 1 or 0. In some special cases, e.g, if you have a distributed load, it is likely that there is a visible density gradient. For getting multi-scale structures, I recommend our review article on this topic homepage.tudelft.nl/z0s1z/projects/2021-multiscale-review.html
Clear explanation, but what does mean U to the power of T? What T does mean ?
The superscript "T" refers to the transpose of a vector. U is a column vector, and its transpose U^T is a row vector.
Nice one...hello, i am new in the topology optimization field. Can you please make a video describing stress constraints in topology optimization?
Great suggestion! Stress constraints have been a topic of interest in topology optimization since late 1990s. There are a number of recent developments. I hope to share some results soon.
Okay great, also when i read the literature about stress constraints in topology optimization, they use aggregating techniques like p norm or ks function. I have already perform the stress constraints topology optimization under minimise volume objective function in Ansys using simp technique, but now i want to validate my Ansys results with the stress constraints imposition techniques used in literature. It is really a great help if you make a small demonstration that how to impose global stress constraints using p norm or ks function.