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#013: Shape Optimization vs Analytical Solution

Before setting up any numerical calculation, you should always consider if you can get similar insight with simpler means, such as analytical expressions, lumped circuits, or similar techniques. In today's blog I look at how the result of a shape optimization simulation could actually have been investigated much easier with an analytical expression.

I have several interests within the field of vibroacoustics, which hopefully shows in the different blog posts on my page. But some topics are just closer to the heart than others, and for me microacoustics has been one such topic for several years, and optimization has become of great interest to me in recent years, especially topology optimization. I am by no means an expert on the topic, I am however very aware of how such complex topics are often difficult to get into the established industry, so any example that I can get working, will be useful in convincing managers to direct resources towards these topics, which could also be of benefit for the universities/students that have knowledge of the topic in general, but lesser knowledge about how it can be applied to real-life applications. This intersection between industry and academia is somewhat of a vacuum, that I am trying to fill.

I have also recently been playing around with shape optimization, and just as with topology optimization, I am not an expert at all.

But you just have to try and see what comes out of it. So I turned to my trusty application; the viscometer: Two larger volumes conneted by a smaller tube. This example is advantageous when it comes to investigating thermoviscous effects.

I tried doing shape optimization on the smaller tube only. Since it is up to you to control the allowed shapes, you need to come up with a strategy for describing the shape. You can opt for a more freeform shape optimization, where you pretty much only have to make sure that mesh nodes don't end up in a configuration, where mesh lines cross each other, and perhaps also have some box constraints, within which all nodes have to reside. From an industry perspective, this is probably not the way to go, because you can end up with very pointy and weird shapes that no one will allow, both because of aesthetics and also manufacturability. You can also opt for having the shape being described by some mathematical function/functions, that limit how much the optimized geometry can differ from the initial. I have found a lot of inspiration of for coming up with appropriate functions. Btw. if you like the content on, you need to open a extra browser tab and start reading COMSOL's blog too :-)

I am not going to go into the details of the shape optimization, as there was no real goal, at least from an industrial perspective, but have a look at a design that I ended up with:

The geometry of the smaller tube has changed into somewhat of a starfish-shape. Lets have a look a circular cross-section and the optimized geometry:


The thermal and viscous fields will be different in the two cross-section, as can be seen in the figure. For one, the contours for the optimized geometry are not scaled down versions of the geometry perimeter, whereas for the circular cross-section they are.

Now, as mentioned in a previous post, I am currently looking into how to use results from fluid dynamics in acoustics. Having just read a lot of literature on laminar flow, I remembered having seen this exact geometry before [1]. The geometry is shown below

Since the geometry is a perturbed circle, the authors use pertubation theory and the result known for the circular cross-section to come up with an analytical expression for the new geometry. Comparing results of the so-called correction factor from fluid theory to viscous losses in acoustics, for the circular and perturbed geometry, respectively, I could see that there was a perfect correspondence. This means that if I had known in advance that the outcome would look "starfishy", it might have been time well spent to start out with the analytical expression, which covers a great varity of shapes, i.e. how many times does the perimeter oscillate around the circular perimeter, and how much does it diverges from circular. With a simple parameter study, I could have plotted various possible outcomes, without doing any simulations. Now, of course the shape optimization could have ended up with a more elaborate design, but I knew in advance that it was limited by my choice of shape functions, so I could at least have come a great deal of the way.

There is no winner here, and I am not saying that you should always lean towards analytical solutions, or numerial solutions, for that matter. All I am saying is that the more knowledge you have about analytical solutions, and special techniques such as perturbation theory, conformal mapping, variational calculus, asymptotic expansions, and whatever else that might help you with YOUR particular application, the better you will be able to decide how to progress efficiently. And try to look towards other physics such as flow theory, electromagnetics, and so on, because sometimes solutions have found their way into only some physics, but not the one you are currently investigation.

[1]: "Reexamination of Hagen-Poiseuille flow: shape-dependence of the hydralic resistance in microchannels", Mortensen, Okkels, and Bruus, Mic - Department of Micro and Nanotechnology, Technical University of Denmark

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