Exploring the Design Space For the last few years I have been giving a lecture at Leeds University on design optimisation techniques, but the one thing that always bothered me was a sketch that I did of an example design space. This sketch was just that, a simple block diagram indicating what I assumed it would look like. When I had the opportunity to write the lecture up for a paper in The Structural Engineer, I realised that the time had come to do it properly.
Design space sketch
First, some background: what is it that we mean by a Design Space? It is essentially a very useful way to think about the design options, and can consider things like steel or concrete, flat slab or beam and slab, column spacing, etc. down to what beam sizes to choose. It is sometimes also called the Search Space or Phase Space. If we consider each option type is a dimension then the result is a plane (2 options), a cube (3 options), or a hypercube (4 or more options) where all the possibilities exist. The trick is to find where our economical yet functional design sits in this space.
For this example I wanted a structure that had just two dimensions of choice, so that I could draw it, yet also exhibited interesting or complex behaviour. I chose two beams of equal length joined to each other mid span, with a central point load. Obviously one design result is that the two beams are the same size and they carry the load equally. The other is that one beam does all the work while its partner just gives restraint.
The design principle was simple too: analyse the beams, determine the moments, and choose the lightest or cheapest beam that can carry the load. In design space terms, we start from one location (the two analysed beam sizes) and jump to another (the two sections from the design). You might refer to this as a Quasi-Newton heuristic, if you were so inclined.
Two solutions
The problem with this this method on this structure is one of moment interaction. The moments are proportional to the beams' relative stiffnesses, with the stiffer beam attracting more moment. When we design the beams, the stiffer beam can thus increase in size more than the lighter one, attracting more moment, and so on. The result is that it can take several steps to converge on a stable answer (the analysed sections are the same ones found in the design).
There are nearly 100 universal beam sections in the UK catalogue, giving us close on 10,000 possible locations in the design space for this simple structure. Normally we might just make a good guess at the initial sizes and take it from there, but for this exercise I wanted to map every point of the design space.
To do all 10,000 calculations by hand would be tedious, even if we halved it by taking the problem's symmetry into account. Instead I wrote a Python script to work with the model, testing each option and outputting the results to a spreadsheet-friendly csv file.
As this script was to work with a single model I could afford to take some shortcuts. For example as I know that there are only two members then I don't need the script to either count them or check for their existence when looping through the list. Neither do I need to read the design properties so that I can update them; instead I can hard code the design values into the script. Also, if this script was actually optimizing the structure we would get it to update the analysed sections with the ones found by the design, then repeat until no sections changed. So instead of a Quasi-Newton optimisation we will get a map of how the Quasi-Newton method would traverse the space.
In this model I placed all the section sizes I wanted to check, placing them into weight order. Price order was another option, combining TATA's published price modifiers with the current base price. By doing this, then always working upwards through the list during the design check, we will either get the lightest or cheapest section that can carry the load.
The first version of this script works through all the possible pairs of beams, then determines the new pair that will carry the moment. In this instance I first reduced the section list by removing any section that has a lower moment capacity (effective length 5 m) than a lighter section. This reduced the section list down to a third, and thus the design space by a factor of nine to just over 1,000. Apart from greatly speeding up the investigation, it should also remove any sections that would not be chosen anyway: a double advantage. It is almost always worth the effort to remove the options that cannot be chosen, and the advantage grows with the dimensions of the design space. With two beams the reduction was to (1/3)2, with three beams (1/3)3, and so on.
Looking at the results we can see that if the initial sections are the same then the resulting beams will also be, as we would expect. If the sections are reasonably light then the self-weight does not play a significant part in the moment and we jump straight to the final answer. If we start at the heavy end then it can take a couple of iterations to converge.
Design space calculated
Deviate a little from this symmetry takes us to a nearly symmetrical solution, which I did not expect. Make the starting point a little more uneven and we diverge off to the extremes, where one beam does all the work. Because the ratio of stiffnesses is changing so radically, it can take over half a dozen steps to converge. An especially interesting aspect is that the Quasi-Newton method would not actually converge in this instance, as we get an oscillation between two sections for the restraining beam. A robust optimisation routine will have to watch out for this and fix on the one option that can carry the load.
The next stage was to investigate the question of design fitness. Fitness, or how close the answer is to the ideal, or
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Exploring the Design Space For the last few years I have been giving a lecture at Leeds University on design optimisation techniques, but the one thing that always bothered me was a sketch that I did of an example design space. This sketch was just that, a simple block diagram indicating what I assumed it would look like. When I had the opportunity to write the lecture up for a paper in The Structural Engineer, I realised that the time had come to do it properly.
