[greybeard]

Cameron - What happens if the figure for the density of any one size is wrong by say 5% ?

Looking at the densities given for quartz, I've seen 2.5, 2.6, and 2.66. Given that a source of one person's sand may be quite different to another's, I wondered how much their final mix might be off of the "ideal" that all this effort is aimed at.
I think that most readers are aware that throwing almost any sort of mix together will give a result, but if the "tweaking" is to have any useful purpose, how accurate do we need to be in these measurements ?

I've just checked the kitchen balance which has a division of 1% of full scale.
How reproducible that is, is anyone's guess.
With a lab balance and glassware in my workshop, I have no problem in working to 1 part in a 10000( I never guessed my early training as a chemical analyst would come in so useful ), but I can see this may not be that common.

John

[Gizmot]

I think that most readers are aware that throwing almost any sort of mix together will give a result, but if the "tweaking" is to have any useful purpose, how accurate do we need to be in these measurements ?

go with the end in mind. Don't forget the goal is to be able to find a mix to create a couple hundred pounds machine. Even with a laboratory perfect mix, it is useless if you are unable to practically mix a very large batch and cast it. In my opinion, maybe it would be best to optimixe a mix that you don't need to vibrate. That is for sure if somebody know how to easily vibrate in the basement and move it afterward a 800lbs structure.

[greybeard]

Hi Gizmot.
I agree with you 100% re the end point being for most people who need large castings.
I think we need to know the % accuracy for weighing or measuring volumes.
My mentioning the accuracy I am able to work to was only to underline that it's not common for everyone else. If your weighing scales can measure to 1%, and your figure for the density of your sand is 5% out, the recipe your working with may be within the best you can achieve, or it may be way out. It depends on the maths from which the recipe is derived.
A 1% error in weighing may only represent a 1% loss of packing density, but a 5% error in the figure for the density of the sand may produce only another 1%, or it may produce a 10% loss. We need to know.

Regards
John

[lerman]

Ken, if you don't mind me chipping here as a non-mathematician, where did that come from ?
I am under the impression that the de Larrard type recipes will have equal volumes of each size.

Any over-abundance of the smallest particle is "allowable" if it's actually much smaller than the recipe predicts for that position.
With my own recipe needing to start at 1mm maximum, my fifth component should be about 12microns, but opting to use cabosil at ~3microns, I'm pushing the volume of that up. It might also have beneficial effects when mixing, but that is yet to be proven.
Regards
John

I guess it came from the "common sense" (which is neither common, nor necessarily sensible) idea that if you start by filling a container with the largest size sphere, the space volume will be less than half the total volume. Therefore, there will be more of the largest size than of the next to largest size. By induction, we (wrongly) conclude that relationship will continue to hold for successive sizes. This is wrong because:
1 -- after the first size, the spaces become irregular and you might be able to have more small gaps that can be of large volume
2 -- we are *not* following a procedure where we put in as much as each size as we can and then go on to the next size.

Thanks for your correction.

Ken

[lerman]

Cameron,
I'm trying to write a C program to look at some of this. What values did you use for the calculation in post 2732?

I need the betas and diameters for:
#6 AlO2
#4 Quartz
#2 Quartz
#2/0 Quartz
G800 Zeospheres

Post 2697 has ranges of diameters for some of these (not all), but I'd like to match the values *you* used. That way, if I get different answers, I know to look further.

My initial version will plug in values of Phi for the proportions shown in post 2732 and print the value of K. If all is right, a value of Phi of .88 will give a K of 9.0.

I guess I then need to write a solver to solve for Phi given K (9.0) and the other parameters.

Ken

[ad_bfl]

I do not want to complicate this needlessly, but need to ask this anyhow.

May we want to "consider" creating solvers for optimizing against a $cost of Materials?

I have not checked materials cost, but when one starts talking about several hundred pound castings (which is my usecase), then it's may be something to keep in the back of our minds.

[lerman]

Cameron,

Am I mistaken, or is the formula for K a little abbreviated.

You have 9.0 = K (that's OK)
then K = summation from j=1 to i-1 of K sub i
I think that should be K = summation from i=1 to i=n of K sub i.
That is, we sum over all of the K sub i.
Then K sub i = summation j=1 to i-1 of the expression you have.
(but that can't be correct either. What does is mean to sum from j=1 to i-1 in the case where i=1?) Also, the last expression doesn't have a j in it, so it doesn't seem to make sense to have j as the index of the summation.

[On the other hand, it's been a while since I did any real math, so for all I know that his expression is using Kolmogorov-Einstein notation and I'm just ignorant.]

I'm working my way through this by writing code and got some negative values for K as phi approaches one. That's what is leading me to taking a very close look at this. BTW: I haven't had this much fun with math in a long time. Thanks.

Regards,

Ken

[ckelloug]

Ken,

The summation for K_i iswrong then. I'm in windows right now so I can't go look at the paper but I cut and pasted another line in the latex source so I'm sure I pasted the wrong one. That line was implying the K_i are all given by the following equation and they sum to K.

I'll update the paper with the reported errors.

B=.72 for zeeospheres since they are round and .62 for everything else. I got all the data from the appropriate manufacturer data sheets including agsco for their quartz. I'm in windows right now so I don't have the aggregate matrix avaiable. I fudged a little in the exact matrix but if you construct one, then you might come up with a better procedure than the hand fudging I used. I'll publish the matrix when I get done with my real work and boot back into linux.

The easiest comparison data is to use Dmax=10000, n=10 and the formula d_i=lambda^i*D with a Beta of 0.72. This is the book approximation formula for the Applonian mixture from ancient greek concrete which is constructed by the common sense process of taking an arrangement of spheres, filling the gaps with smaller spheres and filling those gaps with even smaller spheres. This is a particular solution to the problem but that size progression is not the only one.

I picked simulated annealing because the description I read of it seemed to "fit" the bumpy nature of the problem. It was by random chance. It also helps me use a good optimizer because I am using the most naive and bad possible way of solving the model for Phi. (increasing phi and rerunning until it balances). Binary search should be better by a factor of 1000 or so.

---

Gizmoto et. al.

You're absolutely right that a sensitivity analysis needs to be done. I suspect for the minimum segregation mixture that it will be forgiving but I've got to prove that with the simulator and haven't started working on it.

Thanks for all the help and commentary!

--Cameron

[greybeard]

I do not want to complicate this needlessly, but need to ask this anyhow.

May we want to "consider" creating solvers for optimizing against a of Materials?

I have not checked materials cost, but when one starts talking about several hundred pound castings (which is my usecase), then it's may be something to keep in the back of our minds.

ad NFL - Should this be part of the "fun" of creating our own machines ?
I would guess that most people will be looking for their materials close to home, if only to keep transportation costs down. Quite a lot of info has already been posted on theoretical (but real) sources, and I'm sure that as more practical work is done, the list of options will grow.

Regards
John

[ckelloug]

John,

I think you too ad_bfl's post int he wrong sense. He was suggesting adding a weighting function to the simulator so that it tries to produce a high density mixture without using any more expensive stuff like zeeospheres than necessary. That is an awesome idea to me.

Once you're running numerical optimizations, adding a weighting function is easy.

Regards all,

Cameron

[greybeard]

Cameron - I'm with your explanation of ad_bfl's suggestion, and my apologies to ad bfl.

........ without using any more expensive stuff like zeeospheres than necessary......

I fear I'm still missing something though.

I realize that you have to start somewhere, inputting 'real' numbers into a formula that will eventually spit out proportions, and that choosing any particular material is somewhat arbitrary.
But if start and finish sizes are given, along with the number of components, (or any two of three), earlier work suggested that the size ratio of each step would be set.
I know you are refining the original work to minimize segregation effects, but will this alter the size ratio or only the proportions, or both ?


John

[ckelloug]

John,

For an aggregate distribution for appolonian diameter series the d_i=lambda^n*D, the particular solution is known and straightforward compared to the general case though I don't seem to get the book formula to agree with the model. For aggregates that don't follow this appolonian size sequence exactly, the solutions for a given Phi are not necessarily unique as far as I can tell. I think it is possible to get various distributions that have quite different components but similar densities. If the optimizer is forewarned to avoid expensive stuff, it might make an interesting difference.

The equations are badly behaved enough that I have no intuitive feel for what will and won't work.

--Cameron

[Quad2T]

While browsing around on youtube I find this video just posted by ITW Polymer Technologies:

[ame="http://www.youtube.com/watch?v=_XXATSCAaCg"]YouTube - ITW Polymer Tech - Casting Division - Capabilities Video[/ame]

Cheers!

[ad_bfl]

Hi Folks

That is correct, being a engineer at heart, I am always trying to keep my costs under control, especially when the cost is coming out of my hide.

Best Regards
Al

John,

I think you too ad_bfl's post int he wrong sense. He was suggesting adding a weighting function to the simulator so that it tries to produce a high density mixture without using any more expensive stuff like zeeospheres than necessary. That is an awesome idea to me.

Once you're running numerical optimizations, adding a weighting function is easy.

Regards all,

Cameron

[greybeard]

Cameron - With my last posting, I think I drifted away from the point I wanted to make.

If recipe A includes a particular size material( or sieve range), and the available component is expensive, then I can see that a re-calculation to avoid that size might be good in keeping down the overall cost. But is it not likely that the high cost of that component is as much from the processing as from the basic material cost ?
Thus to find any well graded material at the lower end of the size scale is likely to be just as expensive.

I'm not sure that even now I've put my finger on the problem, but that will have to do.
Got to go and build a mold.
John

[ckelloug]

John,

I think we still don't understand each other on the optimization point well.

Your statement below is correct but I think it's based on the idea that the proportion of the sizes is quite fixed:
<blockquote>
Thus to find any well graded material at the lower end of the size scale is likely to be just as expensive.
</blockquote>

I think you're basing your worries about needing an expensive smaller size regardless on the idea that things have to follow the d_i=lambda^i*D idea where the sizes are very well set. This is a single particular solution to an optimzation problem I don't know how to visualize. I originally thought that as de Larrard's book is more than a bit disjoint in the way it presents the material. I'm not sure it has to be this way.

From the simulation result I posted in post 2732 however, the optimum required something like 52% zeeospheres whereas the formulation above in the same post required only 20% zeeospheres. The first got 88% density and the second 87%. I rejected the first solution as a bad idea by reflecting on it it whereas ad_bfl's suggested cost function would tell the simulator to keep looking if it found an optimum with too many zeeospheres. He's proposing optimizing density vs. cost. Cost optimization makes sense if one is optimizing Phi directly because the optimizer has already shown in can produce a mixture that sounds kinda dumb to one of us.

Following your train of thought, if you optimize for minimal difference between the K_i (minimum segregation), the size ratios follow a much more predetermined path (which generally preclude massive uses of the smaller sizes anyway) than just blind optimization for Phi. Adding a cost constraint to this optimization makes the system overconstrained in the mathematical sense but numerical optimization will do the best it can anyway. There is no guarantee that it can meet all the constraints but the result may be interesting. Solutions for a given density are not unique as far as I can tell and so it seems to me that a cost function might get the optimizer to produce solutions we like more, also it might not. . . but there's no harm in trying. Even if we build our simulator with a cost function, you can set the cost to equal for all the aggregates and get the best unconstrained solution.

Also, a cost function, while it can be applied to money, is an abstract concept that can be applied to a lot of other aspects of the problem. Using a very advanced cost function, I think I could program the optimizer to output a maximum density mixture of a specific color which used the pigment as the finest aggregate.

I think you're worried that trying to optimize for cost in the simulator might reduce the flexibility (and joy) of building a machine and you're taking cost in a very concrete sense. I personally see the cost function in the the numerical optimization sense http://en.wikipedia.org/wiki/Optimization_(mathematics) which just means a way of quantifying the criteria you would like your optimizer to work towards.

Finally, John,

I want you to know that I've corrected more mistakes in my understanding of this subject and in my math from answering your questions than I think I have from anyone else's questions up to this point! So thanks for arguing points like this. I hope what I said makes sense above. If it doesn't, we can talk more.

Regards all,

Cameron

P.S. I know I owe a bunch of posts on paper corrections, Beta values, aggregate matrices, sensitivity analysis etc.

[greybeard]

Thanks for your thoughts, Cameron, and that wiki link. Haven't been there yet but I'm sure it will help my naive maths.

I hope I manage to keep my role as a sort of "layman enquirer" benefiting others as much as it does for myself, and it's not seen by our readers as unnecessary interruptions to the process.

Keep up the good work
John

[BobWarfield]

The trouble with computers in general, and optimization algorithms in particular, is the darned things do what you tell them to.

This is why the way you measure "optimal" is so critical. If it doesn't know better, it will happily jack up your most expensive component to add a half percent density. If you let it, it will happily specify 50 different ingredients, most of which will be in quantities to small to matter.

So, you really want to get your arms around these kinds of thing to make sure the answers that come back make sense.

I have to reinstall my toolset, so I haven't made much programming progress, but in thinking about it, I am imagining some of the following make sense:

- A list of available materials. Cost can be optionally included in the materials. Costs would be yet another optimization value: max density, min separation, min cost.

- Parameters to indicate the minimum and maximum number of allowable components as well as the minimum and maximum allowable fraction of any single component. For example, I might want to tell the system no more than 20% Zeeospheres and ignore combinations with a component that is less than 5% of the mix.

- Cameron mentions he automatically converts a component into multiple components since they're never pure on size. That seems a good feature, but the algorithm will need to keep that conversion clear as it manipulates the combinations.

Cheers,

BW

[lerman]

As soon as I get my program running, I'll be interested in exploring sensitivity issues.

I've already concluded that the optimum solutions can't be too narrow. Sort of by definition. We are unable to specify the mix with fine tolerance. Even if we can weigh the number 2 quartz with precision, we don't really know the distribution of its components to any precision.

I suppose we can measure the beta for each component somehow, but how precise do we expect that to be?

So, there may be narrow peaks in the solution space, but even if we find them, we can't use them because when we build this stuff we can't hit the peaks with any certainty.

Knowing that there are no useful narrow optima means that we can perhaps use a more direct means of finding the optima we can use. Have I mentioned that I'm a great fan of brute force search?

I suspect that in the long run, "we" will provide a program that can be downloaded (or perhaps accessed via a server). The program will let us input a set of materials and their parameters -- size, beta, specific gravity, cost. It might also let us specify a maximum number of components to use. We might also specify the cost of epoxy and some optimization parameters.

The program will then design a mix. It will let you either accept it or ask for another mix. When you are satisfied you will be able to tell it how much you need and it will print a bill of materials and total cost.

I think we are getting close to doing that.

Ken

[ad_bfl]

It would be very interesting to try a brute force search for three sizes and see how the solution behaves, then play around a bit with the step size in the vicinity of some local solutions to see how it behaves.

I may write my solver in Ruby (need to try something besides hello world), so it will be SLOW, but easy to tweak, if I have a idea where some local solutions exist, I can easily tweak it to check out sensitivity in those regions. Would that be of interest?

Looking at the math makes me wonder if the the loosening and wall effect equations are actually applicable if the relative proportions are way skewed, like the earlier example of a 52% or zerospheres.

Since those effects are partially based on curve fitting, then those equations may be only applicable when our mixing proportions are in the same general ballpark as his test mixtures.

Bob's idea of setting component % mixture constraints is really good, it will "hopefully" us in regions where the model is valid if we keep it in the realm of the tests used to create the model