Nested Optimization or Variable x Sizing?

Hi there,

I'm trying to optimize a composite laminate plate and would like to automate the selection process using optimtool functions. I have already created a program that can optimize the strength of the plate with a fixed number of layers, using either fmincon or fminunc. However, I would like the program to be able to minimize the number of layers used while still meeting a minimum strength requirement, but I'm not sure how to do this.

For clarification, I have:

n=number of layers

theta=[ theta_1 theta_2 ... theta_n ] (these are the laminate ply orientations)

I want the program to minimize the number of layers n but still have a safety factor above 2 (this would be a non-linear constraint). What the program does is it uses the genetic algorithm to choose the number of layers (since ga can handle integer optimization) and then it uses fmincon with the non-linear constraint function to maximize the strength of the particular number of layers, where the SF must be over 2 at least. It's kind of strange having the x value for the genetic algorithm also used as the output to minimize (if the SF is met), and that may be causing problems. It has been running for several hours now, but I'm not sure if it's working.

Is this nested optimization strategy the only way to do this or is there another way, for example by varying the number of inputs to the function or having dummy variables that aren't counted towards the theta angles when there are less layers?

Thanks! ~K

Answer 1

[Matt J]

I don't immediately see how you can avoid the nested optimization, but simple bisection should be more efficient than ga(). You know that the maximum SF is a monotonically increasing function of n (I assume that more layers always means more strength and higher SF), so you should be able to use bisection over n to find the threshold n where SF_max(n) crosses 2.

[Matt J]

Incidentally, once you've solved for the strength-maximizing theta_1...theta_n for a fixed n, do you really have to start again from scratch for n+1 when you add another layer? Can't you assume that the first n thetas are still valid strength-maximizers in an n+1 layer system? If so, that should simplify things too. It means for each new layer, you only have one new theta to find.

[Kiwi]

Hi,

Thanks for your answers! I guess nested optimization is what I'll have to do, then. I'm running another trial as I type and hopefully it will work out.

I'm not sure that bisection would work. Yes, more layers will be stronger in the end, but it also depends on the fiber angles and loading (for example, an 8 layer 90 deg layup will be weaker in 1 direction loading than a 7 layer 0 deg layup). I'm not sure if bisection can take this into account. Any thoughts? Maybe I'm missing something.

You're probably right that taking the optimal layup for less layers and using it as a starting point for the next test would yield good results. I'll try to take that into account.

Thanks again! ~Kieran

[Matt J]

Yes, more layers will be stronger in the end, but it also depends on the fiber angles and loading

But isn't this taken that into account already when you maximize over theta? My point was that an 8-layer system with optimized thetas might always be stronger than a 7-layer system with optimized thetas. This isn't my field, but it just doesn't make sense to me that the maximum achievable strength decreases with additional layers.

[Kiwi]

Yep, you're absolutely right! So in this case, I would just need to test progressively smaller numbers of layers until the performance criterion isn't met. When you put it that way, I guess I do only need bisection, or a for loop.

[Matt J]

I would just need to test progressively smaller numbers of layers until the performance criterion isn't met.

Or progressively larger numbers of layers (starting from n=1). If you do it that way, your problem appears in fact to be a Dynamic Program where the value function V_n is the optimized strength of the first n layers.