0:14 I think you can hear me. So, good morning. My name is Raul Llamas and I am presenting today my PhD topic on on GPU enabled large-scale structural topology optimization. About me, I I am a student, a PhD student at the ETS, then I will wait. I have also another two jobs, a full-time job as an aeronautical engineer in a well-known company, and also a part-time lecturer at the University European University of Madrid.

0:52 The PhD is under the University of Seville. Before I started my PhD, I had hobbies, as you can see there. I’ll come back to them. As a lecturer at the European University, I teach two main groups of subjects, flight mechanics and aircraft design. In aircraft design, I I don’t miss any opportunity to mention about topology optimization and the students have to do exercises and get acquainted with the technology.

1:20 As an aircraft designer, I’ve done a few big things. The biggest in many ways is the the wing covers of the Airbus A380 wing. This is a picture which is almost to the date 25 years old. I produced the picture when I was working on this for an internal presentation. It was shown it was used for a paper later on. Everything I will present about my work in industry is open domain from conferences and other sources.

So, this this was a very interesting job. The algorithm used here, I cannot be too explicit, but it’s not very different to the stuff I’ll I’ll talk about later on. This was to my knowledge the first application of genetic algorithms in a significant structure. In parallel the year 2000 2001 I was the guy who introduced the structural topology optimization in Airbus. This is a well-known very well-published example.

2:21 In my view today we have an overabundance of topology optimization. Many people are abusing the tool in my opinion. 25 years ago it was exactly the opposite. It was quite hard to convince people that there was value and and and effectively to explain what it was about. I think every all all of you understand the technology now and and we have different challenges today that I will talk about later.

2:49 Even before introducing that tool I was interested in the topic from real papers. This is a video of my computer at the time. They are leads my first topology optimization example which was the Eiffel Tower. It had a 1,000 elements. I used the student version of Nastran and QuickBASIC. At some point I will recover it and I’ll be able to show it. But around the same time I started playing with the algorithm the early versions of the algorithm I’ll talk in a second and I used the Nastran and Lisp to get the little bridge.

3:26 So I feel interested in the topic for a long time. Oops. Some public works that I can show a patent from 2007. This is a fuselage of an aircraft with propellers at the back to simplify things. And we have to consider the the failure of the propeller that can cut the structure. So this is a structural topology optimization of the structure taking into account the possible trajectories of the propellers.

3:55 So, it’s a a very early case using in-house methods, my actually my algorithm of failure topology optimization under failure conditions. The The last example is on 2020. This is quite a a quite complex example where we have aeroelastic loads in the loop. So, at every iteration loop, we are computing aeroelastic loads. The loads become internal loads, and then we do the structural sizing of the composite structure with a compatible laminates.

4:26 And you see in these bulkheads a topology optimization. So, this is topology optimization with laminate sizing under aeroelastic loads fully consistent. Part of the the job that I can show today from a conference in 2012, this is an aircraft design modeling tool. The state of the art, we still use it. In 2012, it was usable, but not at the state that it is today, but I cannot show much more than this.

4:59 The tool lift or became a hydrofoil design tool for America’s Cup boats. And in fact, the main developer now is working full time on that. You can check his company, Take Off Technologies. You can play there with the state of the art the state of the art aerofoil or hydrofoil modeling tool and analysis. So, the main topic, GPU-enabled large-scale topology optimization. Super simple algorithm, three lines. In fact, the Julia I use Julia as a language code to do that sequence of optimization is 42 lines of code.

5:40 Most of you will be familiar with the 88 lines of Python code for TopOpt. You can do it in in in much less lines of codes with a comparable code structure. So, it’s the algorithm belongs to the family of fully stressed design. It’s even simpler than fully stressed design. So, we assign a density based on the allowable. The secret sauce here is what stress we use. And And that’s a part of the publications that I will publish hopefully soon.

6:12 So, you assign the density based on the stress divided by the allowable. Then you average the densities. It’s a convolution. It’s a filtering process. And then you threshold the you cull elements which are not working very hard. This is a well-known algorithm. Has been discredited during many years by academia because sometimes it’s optimum, sometimes you can prove that it’s not optimum. And most of the research has been focused on showing that it’s not optimum.

6:41 But in fact, for most practical cases, it is a a very good solution. I’ll show you a little animation. On the left, we have a a picture from the original Michell paper on topology optimization, even though he didn’t call it that way. This is a continuum solution. I guess many people overlook what what Michell’s theorem says is a the the optimum solution is a almost a continuum.

7:07 It’s not a continuum. It’s a infinitely dense lattice of infinitesimal elements in compression and tension, pure tension and compression. When we do topology optimization, what we what we apply in practice is a degenerate version of that. So, we need to remove complexity. So, we find a heuristic method. And every topology optimization method has heuristics. So, it’s not optimum by any means, regardless of any claims, because we are reducing the the the the mesh density in a way until we are happy with the final solution.

7:42 In this animation we you see that we start from a fully stressed design. They are embedded are all these infinitesimal load paths and these are being cut by the algorithm and all of these topologies belong to different families and this is a key a key point in in my research. So, this evolution that we have seen in the animation effectively we are traversing several manifolds that I will mention in a second with different topologies all of them valid or most of them valid except for the transitions.

8:16 If you look at the green curve, this is the evolution of the compliance, so the flexibility of the structure. As we increase you know, as as we progress in the topology optimization, there are little spikes. Those spikes are transitions from valid solutions where the structure becomes a mechanism. There is a very big one at the end because we are now breaking breaking significant structural members and the structure becomes a mechanism until it readjusts.

8:46 And here the diffusion the convolution that we do the the averaging of the material is what allows the structure to redistribute material and become isostatic again. Normally people talk about the the filtering method as a way to remove checkerboard artifacts and so on. It’s much more fundamental than that. So, a large part of my research consists in in exploring the theoretical underpinnings of the algorithm and these are very deep and very surprising even for this three-line algorithm method.

9:24 What we’ve discovered is that each of these solutions this isostatic solutions belong to a manifold a topological manifold with a different number of members. You can imagine that we can take this structure, which is a finite element model made of shell elements, but you can very easily interpret it as a lattice structure. And there is trivial to assign it a dimension. How many bars, how many nodes.

9:49 So, there is an intrinsic dimensionality. As we progress in the optimization, we cross manifolds with different intrinsic dimensionality. So, we go This is a bit difficult to understand in in a short time, but here we have the most basic structure with three elements, and the gray level is the the density of the elements. We can go to a full the the same color or same gray level in in all elements.

10:19 This would be in the diagonal of design space. And if this was a truss structure, and only two elements were necessary in the plane, we would end up in one manifold in one of the planes. So, one of the structures one of the elements becomes fully white or fully black, depend depending on how you call it. So, this evolution in the design space from the gray structure to a topological manifold is actually driven by the the fully stressed design method, which is very parallel or very close to the diffusion methods that people are using now for image generation and the generation of other devices that we have seen in the previous presentation.

11:07 So, the mathematics are quite daunting, but the meaning is quite simple. We have a force that pushes the structure, in this case, to a to a certain solution. The structure wants to become isostatic, and this is the gradient there, plus the the isostatic attractor. This is what Michell’s theorem says, and the diffusion is a diffusion. So, we are spreading the point in the design space to allow it to move to the attractor.

11:36 Then the cutting is just forcing the dimensionality reduction to to get the black and white. The cutting is a control parameter as well in the in the process. So, we see different solutions. All of them are valid. All of them belong to different manifolds, and we are just moving between manifolds. And the decision of engineer is where do we stop? The The most complex solution is the stiffest, but the most expensive to make or impossible to make.

12:03 In which manifold do we stop? And I see very often that people reach the final solution, and they don’t care about the intermediate solutions in other methods. Which is fine, but we are missing there a lot of richness in the in the design space. Like this. So, certainly the lower row, all of these are isostatic valid solutions. And the first one is what people call now variable thickness solution, which is effectively the implementation or the realization of the of the mid-cell fully stressed design.

12:37 Some other interesting parallel to the algorithm is that convolutional neural networks function exactly the same as this algorithm, exactly the same. So, we have a linear process, the FEA analysis, which is the the the the normal neural network. We have a convolution. A neural network is made of matrix multiplications plus a non-linearity. So, we have the linear solver is the matrix multiplication of the neural network in its layer.

13:10 The convolution is a convolution as the diffusion operator. And then the thresholding is a non-linearity. In neural networks, we often use a ReLU, which is just clipping the negative part. Here, we have a threshold that clips values below a certain a certain threshold. Almost any non non-linearity could work in any random work. So, in the end, you end up with a reduced dimensionality manifold of representing the topology of the structure.

13:41 This is very very similar to what people are doing with neural networks when they prune the system to make it fit in a smaller device. I will not pursue this line of work, but it’s super interesting and I think the parallels makes sense certainly for for the mathematical framework that I’m developing. This source is extremely rich. So, the method being based on on stress is not a compliance optimization method.

14:12 Deals with the stresses. So, in the end, the objective objective function is to maximize the the utilization of the material. So, try to force that every element is working at its maximum capacity for at least one load case. When we have a many load cases, is when people find that this method can be non-optimal, but still is practical. So, we have the control loop that follows the trajectory of the stress state in the structure and there is a simple PID process with some some tricks, some some controls.

14:47 And it that the control parameters are the diffusion radius and the threshold. With only those two parameters, you can end up in one of the topological manifolds, which you can choose, right? Since we have a control system in the loop, we need to be quick in the iterations. So, it’s essential that the the FEA solver is very fast. In 2D, this is not too difficult. The animation that I showed before is using a simple linear solver from Julia, no nothing exotic.

15:16 When you go to 3D, things become a bit more expensive numerically. So, here is the crux of the algorithm that enables to have a control system in the 3D optimization loop. And we are using the views. I have a a tiny Nvidia GPU at home, 10 GB of RAM, and I’m dying to get access to a larger GPU. But even with a a home kind of consumer-grade GPU, if you tailor the algorithm exactly to the problem, so we have a mesh which is made of elements of the same size, so we can map almost directly the the algebraic structure of a mesh to the GPU structure.

16:08 And do this in a matrix-free approach with a conjugate gradient solver using a field of solver in the GPU that needs a preconditioner, and this is the key. The algorithm is quite smart in finding a very efficient preconditioner. This is a bit theoretically unpleasant to discuss, but it’s the key. If you find a robust preconditioner or a cascade of preconditioners, because the algorithm finds and chooses the best depending on available VRAM and other parameters, the solver is incredibly fast.

16:39 So, I’ll show you some examples. The code is written in Julia, 100%. As features, you can read domains in STL, OBJ, and 3MF. Now, thanks to Adnan and his team, there is a 3MF library for Julia, which is very interesting because it enriches the definition of the geometry, and we can add attributes to the to the mesh to refine the mesh or set the stress values or whatever.

17:10 The outputs are specific binary plus ParaView, as you would expect. We can model actuators in the structure by defining thermal expansion coefficients in different regions. We can define regions for voids or hard areas. If you want to model a bridge like this case, we can have a void for the car passage if you like. And then the deck, we can give it the attribute of being non-designable and having a certain weight and certain stiffness.

17:39 And then we don’t need necessarily to apply a point loads, just distributed weight will will work. When we have so the process considers self-weight and and gravity loads. When you consider gravity loads or body forces is the only case where the topology can have curved elements. I see very often even flying in space topology optimized structures with a curved members. That’s I mean, that’s not optimum. Unless you have body forces which you don’t have in the space.

18:15 Anyway, the method can deal with gravity and body forces. Some examples. This is the whole structure is a bridge. Was modeled using symmetry, so only that part was optimized. This is 26 million elements. This was run in my home computer in 2 hours. The complexity is significant to to to pick up the fine detail. I mean 2 hours for this is pretty much state of the art, I would say.

18:44 You see how the deck has been defined with a a non- non-designable material but flexible and potentially with a different density and thermal properties. You can have different allowables in tension and compression. This is quite difficult to to deal with in a stiffness-based optimization. Here, since we are explicit with the stress, we can control and local state stress state as we like, even having different allowable in different regions of the model or different diffusion ratio reduces in different parts of the model.

19:22 I started many years ago with an Eiffel Tower and and this is now where what I am. That’s how you define the designable domain. So, the bounding box is where potentially you could have material. Then there are some point forces and then the decks for the restaurants. And the foundations. When you run that, depending on which parameters you use, you can end up with different interesting structures.

19:46 I at some point I I must Eiffel Tower exactly, but I I think I I I lost the result, unfortunately. So, the topic was large-scale topology optimization. Let’s get a sense of a scale of what a scale means. That bridge overall has a 108 million elements. I have run cases like of this size, but in this case the preconditioner has to be Jacobi in my computer. And Jacobi has certain numerical effects that don’t allow to to get so fine detail.

20:23 But then again, I’m running in a 10 GB GPU. If you run in a state of the RGB, you can go to a several hundreds or billion elements today and certainly more in the future. What does that mean? Is that in aerodynamics, you can run aeroacoustic analysis with a billions of elements. In a structures, you can do what I did in a 2017 with a cluster, 8,000 CPU hours of a compute.

20:57 That’s 1 billion was the first giga scale topology optimization. You can do this today in a single GPU if you have a access to it. All right? In terms of aerodynamics, again to get a sense of scale, this is a 60 million lattice Boltzmann analysis that we did in a research project. It’s difficult to see, but you have the results between the comparison wind tunnel and and CFD, and they are spot-on.

21:21 So, 60 million, 100 million a few hundred million elements in a structures and in aerodynamics allow you to do pretty much whatever you want. The giga scale is beyond a billion. Again, to get a sense of scale, La Sagrada Familia, the cathedral in Barcelona, I did the calculation. If you use a standard bricks, European bricks, you just need 1.2 billion bricks to model the bounding box. So, all the details that you can get in Sagrada Familia could fit in a single GPU if you have access to a enough memory with today’s algorithms and infrastructure.

21:57 So, where is all this going? The The picture there is I I have good memories from it. This was done by this guy here, who’s looking for a job. He’s a bright guy. He did this as part of his bachelor thesis, his undergraduate thesis. And this is a Julia optimization fluid-structure interaction. So, you are getting topology optimization of an airfoil under aerodynamic loads, where the aerodynamic loads You get the idea.

22:29 This is where things are going. In terms of civil engineering, you need wind loads. So, I also developed a CFD code, and it’s now integrated into topology optimization code. It’s not running the the fluid solve for each optimization of a structure. This could be done, but it’s extremely expensive. But it could be done. But the thing is if if you want to design a building, if you have a facade, at least you need the facade loads, and then you get internal structure.

22:57 So, this is now possible. So, where we are going is to fluid structure interaction. The frontier is structural stability. This is what limits topology optimization in using industry, especially in aerospace. Buckling is still a a big issue for this. Final remarks. All the algorithms that I’ve shown you, all the results and topology optimization and CFD that I’ve written have been written by AI. The algorithms have some secret sauce.

23:30 But I mean, there’s nothing totally radical here. What I think is radical is that this has been done in the last 4 months using a cloud code or codex or whatever. And this is a revolution, in my opinion. GPUs feed or AI development is feeding GPU development, which feeds the simulation, which feeds the the the need to develop simulation codes using AI to run on the GPU.

23:59 So, this is spiral is accelerating and the the repercussions in industry I don’t think we have started to feel them, but this is coming. So, we need to to be ready for that. Normally, I would finish the presentation here. But I want to talk about the future. So, I don’t know what is the future future, but I know what was the future when I was a kid.

24:23 And when I was a kid, the future consisted in in a few things. Video telephones, and which we are there. Flying cars, and we could claim we are And when I was a kid, I was very fond of this movie. And attempted a live AI demo that required internet connectivity. That did not work. But if it doesn’t work, we’ll stop here. So, this is the future. I’m I’m the futureist today, basically, at least the future when I was a kid.

So, use computational design for peace and to create things, not to destroy the world. Thank you. To learn more about the CDFAM Computational Design Symposium, access the archive of previous presentations, interviews with speakers, and information about future events around the world, visit CDFAM.com.