0:00 The following presentation is by Vanessa Cool of KU Luven on from structure to sound unlocking the potential of viro acoustic design. So I’m from the LMSD research group. It’s also shown over there. It’s a meatronic system dynamics research group. So we are looking to the design and analysis of dynamical components and a few examples are shown here on the slide. So if you want to design these components of course there are a lot of requirements that needs to be met and a first requirement that everyone knows is it needs to have certain stiffness or certain strength.

0:40 Everyone wants to sit in a safe car. Now the last years that’s not the only requirement or the only performance indicator anymore that’s important. We also want to go to more lightweight structures due to ecological requirements. And now more recently also noise is becoming a very important issue because noise pollution is the big the second biggest pollutant on earth. Now we want to already include this noise this acoustic performance into the design because nowadays this is mostly done if we already have a design and then people start thinking what about the acoustic performance of it.

1:15 Now I’m have a mathematial background and there we like to make things periodic and I just wanted to give a bit of a crash course on how we model these periodic structures because it will be used during the presentation and we call this unit cell modeling. So we start from our periodic structure here and what we will do is we will take the smallest non-repetitive part called the unit cell and now this unit cell it’s the only thing that we will model.

1:41 So we will not model the full component anymore but only this unit cell with our favorite discretization technique. So I’m using the finite element techniques. And now what we are going to do is we are going to assume this unit cell is infinitely repeated in all directions of periodicity. So how does this happen in a numerical way? We are just applying periodic boundary conditions. If we have this unit cell there are two different performance indicators that are often used to look at the performance of our meth material.

2:10 The first one are dispersion curves. So here you can see a relationship between the frequency and the wave number. But most importantly each point on this curve shows a certain wave that is propagating through the structure. Now if you want to have a very good acoustic performance, we want to have a region in the frequency where we do not have any of these points and this is called a band gap.

2:32 Another performance indicator is the sound transmission loss SDL. Where we will look at if a acoustic wave is exciting our structure what’s the transmitted pressure at the top and now we want to minimize this so the higher this curve in my presentation the better we’re doing acoustically fine now I’m saying I want to optimize things and what I’m going to do is structural optimization and more specifically topology optimization and making it even more specific density based topology optimization and also gradientbased so we start from a design domain looks a bit like this.

3:09 And this is defined with a lot of different elements or finite elements. And now each element has a design variable shifting between zero and one. Meaning if we will place no material of if we will place material. Next up, we need to define the problem that we want to solve. This will be or I will present different frameworks and it will each be the same problem formulation.

3:31 Namely, as an objective, we will look at that acoustic performance that we want to maximize. But we also have two constraints. A constraints on the stiffness of our structure to have a certain stiffness or a certain strength and then also a constraint on the volume to have that lightweight characteristic in there. And if everything goes well, we will get some topologies that look like that. So clearly defined topologies in black and white elements.

3:58 Now, if you’re from the field of topology optimization, you know that you need some regularization because if you do nothing, we get designs that look a bit like that because and as an engineer, we’re not really happy with that. So, we’re using firstly a density filtering to smoothen everything out and then a heavy side projection to really go to that black and white design anymore that we don’t have any gray values anymore.

4:17 Why I’m showing this more technical thing is because you are doing that heavy side projection three times. So we have three different three different design fields that in the end are included in the optimization and this is also called a robust topology optimization meaning that the already a slight bit of manufacturability is in there. So a slight shift in the design is already included in into the optimization as said viro acoustic design.

4:46 So we also need to include of course the viro acoustic information into our optimization. So if we have a certain design domain we will each time have a structural part given with the equation there and also an acoustic part and more importantly we are also considering the coupling between the acoustic and the structural part. Now this is the finite element formulation that we are using. So for the people that are interested, it’s a displacement pressure formulation within red decoupling matrices shown.

5:16 How this works in a topology optimization. So for the structural part, this is really a typical density based topology optimization. So we will have our real material input. So the Young’s modulus and our density in the region where we have a structure and then some virtual properties in the region where we had acoustics. Now then we can sum up all the element contributions to get our element stiffness and mass matrices.

5:43 Similarly for the acoustic part now we will have a real acoustic properties in the part of course where we have acoustics and then virtual properties in the part where we have the solids. And then for the viro acoustic coupling we are using a technique from the literature in which we have an element coupling matrix by just integrating over one element. And this allows us to very simply integrate the viro acoustic coupling.

6:05 Also in the topology optimization I promise different frameworks. So I will present three different frameworks from the unit cell level to the component level. So on the unit cell level we are looking to one cell which is repeated in two directions of periodicity and we will optimize those dispersion curves that I talked about. Then we will go to the supercell level where we will we specifically will look to sandwich panels.

6:28 So a double panel with a certain core in between that we want to optimize. And here we assume infinite periodicity in one direction while it’s finite in the other direction. And then the last one the component level where we do not have that infinite periodicity anymore into the optimization. So firstly the unit cell modeling. So this is the mathematical formulation of it. But what does it specifically mean?

6:51 We are maximizing the band gap. So if we have our dispersion curves again that region where we do not have any of those curves that’s a region where we will have very good acoustic performance and now we want to maximize the width of this band gap. We also want a certain stiffness and definitely these problems that we’re looking at they’re very prone to having solid islands of materials because that’s just the way that we can get very high band gaps but in the end we cannot manufacture that.

7:18 So we need to ensure that our final topology also has a structural connectivity everywhere. So here we’re doing it with a technique that is called the nonlinear virtual temperature method. So what we’re doing is we’re considering two unit cells. We’re putting a temperature or a heat source on all the material of the solid and then at one side we are putting a heat sink. So now the the optimization will need to try to connect everything such that the heat can escape from that one side.

7:49 So if you have solid islands of material the heat cannot escape and then the temperature will blow up which is not allowed by the optimization. And then we also have a constraint on the amount of volume and the amount minimum and maximum volume that we can use just to to have that lightweight characteristic control in there as well. The subscripts that are shown there EBD those are those three different designs of the robust optimization that I have shown a few a few slides back another thing in the unit cell optimization is what we were going to do is using a zipper method so normally what is done in those band gap optimizations is you’re considering the whole wave propagation of all directions of wave propagation that you can do however this can be a very difficult optimization to solve and you can end up quickly in a bad local optima So what we’re going to do is we’re going to look at a region where we already have an interesting start.

8:42 So a partial band gap and then gradually start zipping those curves open until we have all the wave propagation directions taken into account. Now if we run this this is what’s happening. So you can see at the bottom that yeah the the zippering method it’s very fast and there the design is appearing and more and more also that connectivity in the horizontal direction now is ensured and this is the design that we get in the end it’s fully connected in the horizontal direction what we’re also imposing but not in the vertical direction because we’re not imposing anything there.

9:14 If we look at the performance these are our band gaps. So we indeed see a region where we have or a big region in the frequency where we do not have any of those curves. So that’s the region where we expect a very high acoustic performance and also what we see is the viro acoustic curves in black and then the structural and the acoustic ones. So this is a design which both has a band gap for the structural waves as well for the acoustic waves to make the connection to the other frameworks.

9:42 You can also look to the sound transmission loss. So here we just took a few of these cells, put it in a sandwich panel and then look at the at the sound transmission loss of it. And in the region that is indicated, we indeed see a very high acoustic transmiss sound transmission loss which corresponds to our band gap. If you now have that framework, it’s also from an engineering perspective quite interesting because now you can start playing around between the trade-off between the structural performance and the acoustic performance which of course is always present.

10:11 So I just picked a random point here, but we can start shifting with that connectivity constraint or stiffness constraint and go to a design which is more structurally connected, more stiff. But of course, there’s also a trade-off in the objective. So the objective will go down. On the other side, we can also loosen up this this constraint and end up with designs which are not at all connected anymore.

10:34 And what we see here is that we get a design which is really acoustically driven. So it has a very high acoustic band gap for the acoustic waves while for the solid waves not a lot is happening because it’s just solid materials. So here the connection gets lost. That’s for the unit cell. Then we go to the next step the supercell optimization. So this is again the mathematical formulation here.

10:58 Now as an objective we take the sound transmission loss. So we have our double panel and we only consider one supercell. I call this here in which we then have infinite periodicity at the one side and we want to minimize that transmitted pressure at the top. We also have a constraint again on the stiffness but now it’s a more easy constraint is really just a constraint on the compliance that we have which is directly connected of course with the stiffness and then we have a constraint on our volume for that lightweight characteristic.

11:30 If you run this for a specific case this is a result that we can h get out of it. So the optimized design is shown there in blue and we see that we get a very high sound transmission loss in the frequency range that we are optimizing and we’re also outperforming two reference cases. A first reference case is just the mass law. So a plate with the corresponding mass of the optimization and then an equivalent decoupling design which is a design with a similar stiffness and also the same mass which has that that peak in the sound or that dip in the sound transmission loss the same as our optimized one.

12:04 As a mechanical engineer we’re also quite interested in how this design now work. So this can be explained by looking at the displacement pattern. So firstly the design will really play on that decoupling frequency. So this is a very typical phenomenon of the double panel configuration and it’s the point that the double panel really starts behaving as a double panel and not as a single plate anymore.

12:28 But on top of that it’s also playing on mode conversion because if we see in the displacement there is bending at the bottom that’s where our excitation occurs. But due to the topology of the core this is translated to a translational motion at the other side which is of course very interesting in terms of the acoustic performance. Here again with this framework we can play around between the structural performance and the acoustic performance.

12:54 So I show the design in the middle and enforcing the structural performance more gets a drop in acoustic performance and also the other way around. So this makes it just as an engineer interesting to play around a little bit because then you can get a design that’s also corresponding to your needs. What’s also interesting to look at is the sound transmission loss because if you enforce the structure more you get a drop in acoustic performance but you also see that you get a more localized performance.

13:18 This is also visible in the displacement pattern because in the design at the bottom you only see that a small part of our design is moving while at the top which is the most flexible design you get a global displacement pattern which results of course in a better performance there. I didn’t show the convergence yet and I’m going to start movie again and that’s what where the reason because we see that quite a few iterations nothing is happening until suddenly there is a sound transmission loss peak occurring and that’s where the dynamics really kicking into our optimization and then the optimization will improve the sound transmission loss and also make it more black and white.

13:57 Now these initial iterations they’re quite interesting but also quite tricky because that’s where our optimizer is just having the mass law. So that reference case and if you’re not careful so we ran a lot of simulations you see that you end up just in a lot of cases with that reference case. So you’re running an optimization for a day or two and what you end up is a reference case of just a plate which is of course something you do not want.

14:21 We saw that the lower you go in frequency the chances of ending up with this reference case is getting bigger and bigger which is also to be expected because it is harder to get a a good acoustic performance there. So recently what we tried to do is have different continuation strategy strategies during the optimization by playing with that stiffness constraint and the frequency that we are optimizing to improve the chances of getting a very good local optima shown here in with the blue dots.

14:53 Until now everything was also numerically. Maybe you were also wondering is she going to show some experiments? Well, I’m from academia but we did some experiments. It’s a very academic case that we did here but one of the supercell optimized structures we made with a laser cutting is just a near 2D sample and then with a purely structural setup we could get an estimation of the acoustic performance.

15:15 We did it like this because we didn’t want to go to the full 3D additive manufactured sample which would make the cost also a bit higher and this in-house was made in five minutes. Now the what you can see in the results that we get a very good correspondence between a numerical result and the experimental result and also that experimentally we do it better than our reference case which is the MOS law or just a simple plate.

15:42 Then the last setup which is the component level. So this is entirely the same as the supercell level. So we are optimizing the sound transmission loss as an objective. So again minimizing that pressure at the top. We have our connectivity constraint again to have that stiffness requirement in there and also our volume constraint. And if we run this now for a specific case we get a result that looks like this.

16:06 So these are three different cases that we have looked at. A first one is that we’ve just have the full component as a design domain. Secondly, we also impose imposed some symmetry in our design domain because from a manufacturing point of view, this could be of interest. And then we did a kind of hybrid thing in which the core itself is periodically but that and the at the boundaries it has full design freedom just that it can play around a bit with the bound boundaries that we imposing.

16:32 And as expected of course the design with the full design domain will have the best sound transmission loss while if we impose symmetry the performance drops a bit. Now how does this component optimization compare with the supercell optimization? So the the component optimization I took the worst case and then I ran the same supercell optimization and first it looks that the supercell optimization is doing better but of course here now it’s an infinite sound transmission loss that is shown.

17:02 So in reality we need to go back to our real component which is a finite component and then we see that our optimization drops. So four different components are shown here because there is some freedom in how you select your supercell because yeah it’s infinite in one direction. So you can make a translation of your super supercell without the infinite sound transmission loss changing. Then a bit of a recap of the different frameworks.

17:30 So we have looked at the unit cell, the supercell and the component optimization. So in terms of computational efficiency, I would say that the unit cell optimization is a bit more interesting because of course it’s a smaller component that you need to solve. So a smaller finite element analysis that you need to do. But with one note here, the dispersion curve calculation is very computationally demanding because an AEN value computation that you need to do while for the other ones it’s just a finite element analysis which is a bit in more interesting in terms of computational costs.

18:00 In terms of design complexity, it’s the other way around. Of course, if you go to a full component and you give your optimizer the full freedom that it can have, those components will be more complex than when you enforce some periodicity in there. In terms of performance generality, what I mean with that is if you look at your dispersion curves, it’s a more general thing because then then after that you can choose also the amount of super or unit cells that you need and put them between a sandwich panel for example, but you can also use them in other applications.

18:30 Well, if you go to the supercell of the component level, you’re going a bit more already to your application which makes it less general in terms of the performance. And then in terms of the boundary conditions is also a very logical one. Of course, that infinite periodicity it’s not real realistic thing. So the component optimization is closer to the realistic boundary conditions that you can see in reality.

18:54 Then everything that I have shown is 2D. What about 3D? This we tried. So we also converted the supercell optimization to 3D. So it is possible and of course it will outperform the designs that we get in 2D because you get that extra dimension of of design freedom. However one note here is because it’s fiber acoustic this isn’t really a fun problem to solve in terms of the computational costs.

19:19 So some tricks are needed just to get it solvable because you need to think for every frequency you need to solve this problem for three designs and also every iteration. So it’s just a lot of finite element analysis that needs to be done just in one iterations and then over like a thousand or 2,000 iterations. But this it is possible and we’re now trying also an experimental validation of this part.

19:45 This brings me then to the conclusion. So I’ve shown you different frameworks. So from the unit cell level to the component level on how to bring the viro acoustic performance into account into a topology optimization already. And there’s also shown that there is an intrinsic the trade-off to be made between the structural performance and the acoustic performance which is very easy to play with if you have your topology optimization.

20:06 So as such is a quite interesting engineering design tool. And all that rest me is to thank you for attention and please don’t hesitate to reach out if you have any questions. To learn more about the CDFAM computational design symposium series, to see the archives of previous presentations, and to learn about future events, visit CDFAM.com.