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Published: 2020-12-28

# Better understanding of computer-based design optimization and sound propagation in the wind

NEWS Linus Hägg introduces a framework for filters that is used to prevent the appearance of small details in computer-based design optimization. He also analyses mathematical models for the propagation of sound. Linus defended his dissertation on October 22 at Umeå University.

Text: Ingrid Söderbergh

The goal of topology optimization is to determine the best design of a given material. A classic example is to design as rigid a console beam as possible from a given amount of steel. By dividing the area into a finite number of squares and using a computer to decide which squares should contain steel and which should be empty, the design of the console beam can be optimized in a completely virtual design process. For grids with relatively few squares, it is in principle possible to evaluate all possible designs with the help of a computer. In practice, however, much finer grids are used which require efficient calculation and optimization algorithms that systematically propose improving design changes.

The problem is that the devil is lurking in the details. An ever finer grid leads to ever finer structures in the optimized designs, which in turn places ever higher demands on the manufacturing process. This shortcoming is an effect of the fact that the original problem, which in some sense corresponds to a grid with an infinite number of infinitely small squares, lacks solutions.

By introducing non-linear filters in the problem formulation that mimic morphological image processing operators, the minimum allowable size of the details can be controlled independently of the size of the squares. The effect of these filters can be compared to that achieved if the designs are drawn with pencils of a certain predetermined size - coarse pencils can not draw fine details.

In his dissertation, Linus Hägg introduces the framework for generalized fW average filters that enables uniform analysis of both new and a majority of existing filters for topology optimization, among other filters that mimic morphological operators.

“I present a mathematical proof that shows that the console beam optimization problem and similar problems in topology optimization with the help of generalized fW average filters become solvable. The filters are thus not only a cosmetic fix that prevents the appearance of too small details in the optimized designs when the grid is refined, but also handles the underlying lack of solutions,” he says.

The dissertation shows that the smallest size of a design can be characterized with the help of morphological operators.

“The interesting thing about such a characterization is that it leads to a side condition that can be used to limit the minimum size of the holes while limiting the minimum size of the structures in the optimized designs. This is in contrast to morphology-mimicking filters which generally cannot do both and simultaneously.”

Examines model for sound propagation

In the second part of his dissertation, Linus Hägg studies the Friedrich system that models the propagation of sound. It is often assumed that the sound propagation takes place in stagnant air, but those who have listened to an outdoor concert in gusty winds have not failed to note the wind's effect on the sound.

In the early 1930s, the Frenchman Henri Galbrun formulated an equation that models sound propagation with regard to wind. Despite the majority of relatively terrestrial applications, Galbrun's equation is also used as a model for oscillations in a star's plasma.

The equation actually consists of a system of three equations, one for each component of the so-called Lagrangian displacement field from which the entire sound field can be calculated. Galbrun's three equations can be compared with linearized Euler's equations, which are a system of six equations that are normally used to model sound propagation with regard to the wind. Halving the number of equations makes Galbrun's equation attractive from a computational perspective.

Although Galbrun's equation was formulated almost 100 years ago, there are still largely no universal answers to fundamental questions about the existence, unambiguity and sensitivity of solutions. These three properties, which greatly affect the ability to calculate reliable solutions using computers, are usually summarized in the concept of correctness.

In his dissertation, Linus Hägg analyses the correctness of Galbrun's equation and shows an alternative derivation of Galbrun's equation in which the Lagrangian displacement field is defined via a solution to linearized Euler's equations.

“In this way, the derivation highlights the possibility of constructing solutions to Galbrun's equation from solutions to linearized Euler's equations. In the special case where the wind does not cross the boundary areas of the calculation area, I show that linearized Euler's equations are corrected and that my construction of the Lagrangian displacement field is well defined.”

Linus Hägg was born and raised in Skellefteå. He has a master's degree in technical physics and a degree from Umeå University.

Press photo. Credit: Victoria Skeidsvoll