Parametric Optimization in Engineering - when intuition doesn’t work

I finally got myself to write about my excitement about parametric optimization in engineering simulations. I am deeply convinced that parametric optimization takes engineering simulations to the next level. Optimization brings a paradigm shift in the field of engineering simulations. It’s a shift from the trial-and-error method to the conceptual search for the best possible. I will do my best to explain this statement in the following lines.

Trial and error method

I have been personally involved in engineering simulations since 2003. When considering Computational Fluid Dynamics (CFD), since its beginnings, the field itself made astonishing progress. To name a few advancements: Simulation models switched from 2D to 3D.?Computer power has increased 50 million times since 1975. Many related routines like model cleaning or meshing got integrated into a smooth simulation process. Single-point-simulation-mindset workflow indeed got over time smoother, automated, and much more effective.?

But surprisingly the approach to engineering simulations hasn’t changed that much. For decades, the engineers keep doing the same practice. They set up the simulation, run it and then they wait (and wait). When the simulation is finally finished, they post-process the results and make their judgment. If there are enough reasons to keep the results, then the results are claimed to be good. If the results are not “in”, then the new simulation comes up and the same process repeats again. Another try-out and another result. Another trial and another error. Again and again. Until someone is happy or some other stop criteria are reached.?

That's what I mean by the trial-and-error method in engineering simulations. That has been a standard approach to engineering simulations for the past decades. Please do not take me wrong, I know well that not everyone has done it exactly this way but most of us certainly have.?

With the optimization at hand, simulations can be done on a completely new level. In contrast with the single-point simulation mindset, the optimization mindset can deliver the best possible. And that makes a tremendous difference.

Engineers think in parameters

Engineers believe that the world consists of parameters. As well as any of its parts does. Every single-point simulation workflow consists of parameters too. Even the simplest CFD simulation has more than 150 parameters (domain topology, meshing options, physical models, numerical schemes, time management, turbulence models,?boundary conditions, fluid properties, solver options, surface roughness, and many more … - each of these topics itself includes many parameters in it). On top of that, any industrial product has many extra parameters. For example, the simplest possible simulation-ready model of centrifugal fan has about 35 parameters. At the end of the day, even the simplest possible parametric space we face in engineering simulations has about 200 parameters. Yes, it’s a 200-dimensional space to solve.

Parametric space

As you can imagine, the parametric space of 200 dimensions is pretty large. Parametric optimization is a process of finding the best possible combination of 200 parameters that leads to the best results according to the optimization function. The optimization function is a formula that means the measure of how good the results are. Optimization functions can be a very simple function of the results of the simulation. It can be for example the efficiency, power, pressure drop, drag coefficient, material stress, or any quantity we can pull out of the simulation results. Optimization function can also be a very complex combination of functions with limits and logical operations of selected quantities. We typically optimize for its minimal or maximal value.

The parametric space is extremely large and extremely complex. If we want to find anything in such a large space, we have to reduce the number of parameters (dimensions). We need to pick the number of important parameters we want to play with and the rest of the parameters would remain constant.?

A couple of examples

Note that we’re typically following the optimization function in n+1 dimensional space, where n is the number of parameters.

1D

The simplest possible parametric space is 1D. One parameter means one-dimensional parametric space. Then we are looking for an optimization function extreme in the 1+1=2D plot. Such optimization function may look like this:

The real-world example may be the following: An Airfoil CFD simulation. Let’s consider a simple fluid flow around an airfoil and we are interested in the airfoil lift force (the lift coefficient). Specifically, we are interested in the Angle of Attack at which the lift coefficient meets its maximum value. When we plot the optimization function (the lift coefficient in this case) against the only one parametric variable (the angle of attack in this case) we receive a simple 2D? - XY plot. There is no big issue to find the maximum value either manually or in an optimization loop.

Such a 1D case as is described in the example above looks cute, nice, and simple. But that's just a 1D optimization task.

2D

Let’s take a look at how it looks if we increase the number of parameters by 1 and go for a 2D space. That means two parameters and a 2+1=3D plot optimization function.? Well, in 2D the optimization function may suddenly look like this:

And the corresponding real-world example may be the following. Consider a simple model of an axial fan stage. Let’s pick two parameters. Parameter 1 would be the speed of rotation and parameter 2 would be the volumetric flow rate. The optimization function would be efficiency. Let’s try to find the best set of parameters corresponding to the highest efficiency. We’ve run a couple of simulations to explore the parametric space. The optimization function plot then may look like this:

BTW: The blue points correspond to the predefined DOE (Design of Experiment) simulation runs. The green points are the optimization algorithm search and the red point marks the best point found so far. And yes. Every point is a complete single-point simulation run. The following picture shows a few more basic statistics from this axial fan project.

As you can see, by adding just one new parameter the parametric space gets suddenly much more complex (by one order of magnitude).

3D+

If we go further, the three-parameter parametric space is not possible to plot in a reasonable way. Now try to imagine how complex the five or ten-parameter parametric space can be.

Wait, and what about 200D? Such complex spaces are of course unimaginable. The human mind can't imagine more than three or four dimensions.?

Intuition doesn’t work

After we did the first few real-world projects we realized that intuition doesn’t work there. Specifically, if you look at the design candidate (3D model). You never know how well it performs. All the guesses unavoidably fail until the simulation is made. BTW: That’s what I like about engineering simulations. Guessing really doesn’t work. You have to simulate it.

There are really many effects that matter in a very dynamic system. Any of those 200+ parameters plays its role. As you can imagine, any fluid flow loves certain freedom. Then it flows smoothly and happily. For example, a flow in a pipe of a larger diameter has a lower pressure drop (pressure losses) than a thin pipe. Because of the proximity of walls, because walls cause unwanted friction (among fluid layers). However, if you increase the dimensions too much the trend gets reversed. All of sudden, too much freedom causes the opposite effect - large dimensions bring too many losses of kinetic energy because of feeding the recirculation zones. And you never know when it happens unless you simulate it.?

Again, let’s take a look at a real-world example. Recently we made an optimization project to optimize the impeller of the centrifugal fan for our customer. The object of optimization was only the impeller. There were chosen five impeller parameters: Number of blades, blade angle, blade radius, blade thickness, and impeller width. All the resting model components like the spiral (volute), outlet, or suction have remained the same.

The winner was found and the project was very successful. When we look back at the design candidates, it is difficult to recognize the difference between them with our own eyes. But their performances have actually differed a lot. The following two impellers were simulated at exactly the same conditions. The same volute, same RPM, same BC, same everything. The results gave approximately the same pressure. But the efficiency differs by 5%!

That makes savings of about €2,000.00 per year on electricity! That is quite a serious issue when speaking about a single 20kW fan.

Another real-world example is the volute optimization study we did a couple of years ago together with CAESES and GridPro. The case was a centrifugal compressor and this time it was the other way around than the previous case. We have simulated only the volutes. 330 runs.?

The image above shows three examples of volutes. The model is clipped by a plane parallel to the volute outlet to show the significant difference between them. Would you guess which one of these three volutes is the best (has the lowest pressure drop)? It is the middle one!

If you make the tongue area bigger, the results are better. But if you make it bigger again then the results are suddenly worse.? It is a typical example of non-intuitiveness in CFD. You have to simulate it! Guessing doesn’t help.

Similar stories repeat again and again in every project all along the way since we deal with simulations and optimization. Searching the parametric space is certainly a multidimensional problem. And there is no change to solve it with intuition. Manual work leads nowhere too. Parametric space is too large to navigate with random actions. Due to the complexity of the parametric space, it has to be a machine (an algorithm) that searches the space and makes the decisions. That means we need to set up the optimization process and the process has to be automated. The good news is that automation avoids human error & limits. The bad news is someone has to build up the optimization loop and that certainly requires fine skills.

Paradigm shift

Due to the actual CPU power available and advanced technology, it is now (late 2021) possible what has never been possible before. I am personally convinced that optimization is a game-changer that takes engineering simulations to the next level. Optimization brings a shift from the trial-and-error method to the conceptual search for the best possible.