Genetic algorithm for the optimization of damping material

Jain Chacko 1

University of Applied Sciences Würzburg-Schweinfurt Ignaz-Schoen-Str. 11, 97421 Schweinfurt

Stefanie Retka 2

University of Applied Sciences Würzburg-Schweinfurt Ignaz-Schoen-Str. 11, 97421 Schweinfurt

ABSTRACT A main research topic in acoustics is the reduction of indoor noise. Application of damping in the reduction of noise is the most commonly used form of solution. This paper focuses mainly on the numerical optimization of damping structures in interior problems thereby employing a genetic algorithm in determining an approximated solution on damping parameters for the simulation. This algorithm is intended for time and cost saving, specifically linked to numerical simulations influencing the early development process. As part of the genetic algorithm implementation, initially the algorithm is used to determine the approximation of a mathematical constant. Numerical optimization of damping structures aims at fulfilling three goals in contrast to the common approach of applying masses to the structure over a wide area in order to achieve the desired acoustic component properties. First, the optimal damping structure for the required frequency range, and second, the optimal position of the damping material have to be found . Third, the use of damping material has to be minimized with the focus on lightweight design. The numerical approach used for the study is the finite element method (FEM) and the numerical model is generated using the open-source software ’Gmsh’. FEniCS is used as the open-source library for the FEM interface.

Keywords: genetic algorithm, damping parameters, numerical optimization.

1. INTRODUCTION

The general theoretical description based on the optimization of damping mechanics are briefly discussed in this section. Main topics include damping mechanism, impedance and di ff erential equations used for modeling acoustic systems. One of the widely used methods in reducing vibration and noise in mechanical systems involves the implementation of a damping mechanism [1]. Based on the system of application and complexity, di ff erent types of damping mechanisms are used. Damping is easily defined as the removal of energy from a vibrating system. A small damping force magnitude, compared to the elastic and inertial forces, can have a great e ff ect on the stability of dynamical systems under certain circumstances such

1 jain.chacko@fhws.de

2 stefanie.retka@fhws.de

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

as shaft whirl or pipeline flutter [2]. Thus damping material optimization plays an important role for the behaviour of mechanical systems. A damped free vibration of a single degree-of-freedom (SDOF) mass-damper system is given by

m ¨ u + b ˙ u + ju = 0 , (1)

where m is the mass suspended from the spring, j is the sti ff ness of the spring, u is the displacement of the mass from mean position and b is the damping coe ffi cient [3]. Solving the above equation by finding the general solution results in the following equations

ζ = b

b c b c = 2 p

jm = 2 m ω, (2)

with j = ω 2 m , where ω is natural frequency, ζ is the damping ratio or fraction of critical damping, b c is the critical damping coe ffi cient. Following equation (2) shows the relation between critical damping coe ffi cient b c and natural frequency ω . Low level of acoustic noise is often a requirement in various environments such as factories or o ffi ces. To solve most noise control problems, a thorough understanding of impedance is necessary. Acoustic impedance is described as the resistance opposing the propagation of sound. Impedance has both, real and imaginary part. The real part is called resistivity and the imaginary part is called reactance. Structural resisting behaviour in response to external forces or moments can be measured using mechanical impedance [4]. The impedance due to a harmonic force F = ˆ Fe j ω t with the velocity response v = ˆ ve j ω t and the impedance due to harmonic moment ˆ M with angular velocity ˆ θ are defined as

Z F = ˆ F ˆ v Z M = ˆ M ˆ θ , (3)

where Z F is force impedance and Z M is moment impedance. An analogous formulation is used for the acoustic impedance Z , which corresponds to the ratio of the acoustic pressure P to its volume velocity U

Z = P

U . (4)

Acoustics generally deals with modeling sound propagation in terms of pressure distribution. The wave equation is used to describe systems in the time domain and respectively the Helmholtz equation is used for frequency domain. Both result in partial di ff erential equations (PDE). The Helmholtz PDE is derived from the wave equation assuming a harmonic time dependency describing the acoustic behavior of an incompressible and ideal gas. The three dimensional wave equation [5] with time t , sound velocity c and sound pressure p is

∂ 2 p ∂ x 2 + ∂ 2 p

∂ y 2 + ∂ 2 p

c 2 ∂ 2 p

∂ z 2 = 1

∂ t 2 , (5)

and its respective Helmholtz equation obtained using Fast-Fourier-Transformation (FFT) is

∂ 2 p ∂ x 2 + ∂ 2 p

∂ y 2 + ∂ 2 p

∂ z 2 + k 2 p = 0 , (6)

with k as the wave number, k = ω / c where ω is the natural frequency and c being speed of sound.

2. GENETIC ALGORITHM

The genetic algorithm is defined as a heuristic search and optimization technique inspired by natural evolution. It comes under the class of evolutionary algorithms and has been used to find an approximate solution for wide range of complex applications. However, the genetic algorithm has its own disadvantages, as in some cases it results in low convergence rate which results in inaccurate approximation solution [6]. Genetic algorithm was first introduced by Holland [7] which was further explored by Goldberg [8]. The algorithm was developed based on Charles Darwin’s evolution theory. An example of the algorithm phases are shown as flowchart in Figure 1 below. As part of the genetic algorithm, selection, crossover and mutation are repeatedly implemented until the end criteria is fulfilled.

Figure 1: Genetic algorithm flowchart.

Di ff erent phases involved in a general form of genetic algorithm in relation to the evolution theory are briefly described below:

Population generation: The genetic algorithm operates on a population of artificial chromosomes which are usually represented in binary form. Each chromosome represents a solution to a problem and has a fitness, a real number which is a measure of how good a solution is to the particular problem [9]. Population refers to the collection of chromosomes that contains several genes representing property of an individual.

Fitness evaluation: The fitness function involves the computation for evaluating the quality of the chromosome. After the evaluation, chromosomes are arranged with best fitness placed in the top position. The fitness evaluation is repeatedly done and therefore a fast computation is the main requirement for genetic algorithm.

End criteria: When the necessary target value is reached according to a certain criteria, the computation is disrupted and the results are finalized.

Selection: Plays an important part in population evolution as it is a key aspect on identifying individuals. The procedure is designed to use fitness to guide the evolution of chromosomes.

Initial Population i - LAL Frness evaluation Mutation crossover No na F criteria Selection Yes —

Chromosomes are therefore selected for recombination. Those with highest fitness have a greater chance to be selected.

Crossover: Involves the process of modification of chromosomes. The crossover takes place after the selection of a new population. The basic idea is the mixing of genetic material from parent chromosomes to produce new chromosomes. A commonly used crossover known as one-point crossover has been implemented as part of the research.

Mutation: This is another part of the modification process involving alteration within individual genes of chromosomes. In the current algorithm a random number is generated for each iteration within a specific range that is applied to each chromosome of the selected population.

2.1. Application 1: Mathematical constant In this section a modified and improved version of the genetic algorithm based on a previously created algorithm code [10] for the approximation of ‘ π ′ using John Machin formula and the final comparison on convergence are discussed. According to John Machin’s formula for approximation,

π = (4 · arctan 1

5 − arctan 1

239 ) · 4 . (7)

For the determination of the approximation, the known values from the approximation equation ( 5 and 239 ) are implemented as variables. Therefore ‘5’ has been replaced with ‘gen1’ and ‘239’ with ‘gen2’ , both being random values. A preview of the equation with variables is shown below,

π ≈ (4 · arctan 1 gen 1[ i ] − arctan 1 gen 2[ i ] ) · 4 . (8)

With the modified algorithm, significant factors which act as boundary condition for the algorithm are mentioned in Table 1 below. Here the population is generated randomly within the given range. With each iteration, the crossover is followed with a random mutation factor in the given range which is added to each individual resulting in convergence.

Table 1: Boundary condition factors of genetic algorithm. Nr. Maximum ‘gen1’ ‘gen2’ Crossover Mutation

individual iteration range range factor

1000 10000 (1, 10000) (2, 30000) 1- point (0.99,1.01)

In the genetic algorithm, previously mentioned Equation (8) with variables is used as the fitness function for evaluation. The fitness function is implemented in python using functions in terms of easy access. Another significant feature included in the modified algorithm is the application of tuples. Tuples are immutable, and usually contain a heterogeneous sequence of elements that are accessed by unpacking or indexing [11]. As tuples are immutable, no extra space is required to store new objects which leads to faster computation. Through each iteration after the fitness evaluation, 100 best solutions of the generated population are extracted to be grouped in pairs. Each pair has two adjacent elements of ’gen1’ and ’gen2’ and its respective two elements of ’gen2’ are swapped as part of the 1-point crossover feature. An example of the crossover type used for the algorithm shown in the Figure 2 below. After the crossover, each elements are multiplied by randomly generated mutation factor.

Figure 2: An example of the crossover.

Table 2 shows the comparison of the previously created algorithm and the modified version on the approximation of π with five trials. Compared to the previous code with approximation to three decimal places, the modified algorithm gives a better approximation with five decimal places and the time taken for computation has also been significantly reduced.

Table 2: Approximation comparison of genetic algorithm. Trial Previous algorithm Computation time Modified algorithm Computation time

π = 3.14159265 [sec] π = 3.14159265 [sec]

1 3.14119448 42.02 3,14159201 0.49

2 3.14118177 53.48 3,14159298 0.41

3 3.14118956 48.90 3,14159317 0.83

4 3.,14118789 73.34 3,14159308 0.37

5 3.141207 38.89 3,14159347 0.17

ent sen sent eed 709556 | [703676 [[rsess6_] [Bosra asm] [sare asssr_| [zoser6

Figure below shows the error convergence graph of one of the modified genetic algorithm trials from Table 2 on the approximation of π .

Figure 3: A sample plot of error convergence.

Error convergence

2.2. Application 2: Damping parameters optimization This part of research aims to optimize damping mechanisms of technical systems with focus on optimal damping structure, optimal position of damping material and lightweight construction. The vibration behaviour and sound emission of technical systems depend on di ff erent factors such as material and structure of individual system components, the type of dissipation and the interaction between fluid and structure. Reduction of these vibration and sound emission can be achieved if the damping mechanisms are considered early in the development process. This early development process includes using numerical methods along with the optimization of technical systems. Transfer of these mechanisms to technical systems leads to quieter structural behaviour and e ffi cient design. For the calculation of the sound propagation and the acoustic pressure ˆ p in a volume area Ω , the Helmholtz equation is used in the form,

( − k 2 −∇ 2 ) ˆ p = 0 (9)

with k as the wave number, k = ω / c where ω is the natural frequency and c being speed of sound [12]. Acoustic impedance describes the motion ˆ v a induced by the applied acoustic pressure ˆ p , on a surface Γ . Here application is on the boundary layer of the area. Following expression of impedance Z is obtained with x as a point on the surface Γ , with outward pointing normal vector n . :

ˆ Z ( x , ω ) = ˆ p ( x , ω ) ˆ v a ( x , ω ) ∗ n ( x ) . (10)

As an initial step, human vocal tract (VT) is chosen as the object of investigation as it is an interesting system and works e ffi ciently in terms of energy transfer achieved. In order to gain a basic understanding of the acoustic conditions in the geometrically and functionally more complex VT (Figure 4), first, a simple structure (pipe) is used (Figure 5).

Figure 4: Experimental model of VT.

Figure 5: Simplified model of VT.

The model shows a simplified representation of the human VT (Figure 5). Compared to the the real model from the experiment [13], many details are not included in the simplified model [10]. The components of a VT include the glottis, pharynx and oral cavity. The geometry of the VT was simulated in a very simplified way by neglecting the nasal cavity and the tongue during the development of the model.

Based on this, a numerical model of the VT and the damping mechanisms was implemented and validated in Python [12]. In the next step, the model is used for the determination of damping parameters, by the application of genetic algorithm, with which the damping parameters and thus the damping behavior of the neck region can be mapped numerically correctly. The genetic algorithm is used to compute the impendance boundary condition. The resultant eigenfrequencies and pressure distributions are compared to the experimental results of the vocal tract. For this purpose the acoustic program was implemented in the genetic algorithm. The whole acoustic program functions as a fitness function of the algorithm. By specifying the real or imaginary part of the impedance, the value of the sound pressure can be determined. In the course of the genetic algorithm, the real and imaginary parts are replaced by ’gen1’ and ’gen2’ . The artificial intelligence (AI) is trained and validated on the available data of the vocal tract. In this process, the AI algorithm is linked to the numerical model of the vocal tract. The goal is to use the AI to significantly speed up the determination of the attenuation parameters.

3. CONCLUSION

The first application proved that the genetic algorithm code successfully approximates the value of the mathematical constant ′ π ′ . With the successful implementation of genetic algorithm on damping parameters, the algorithm will be further designed in such a way that it can be used to determine damping parameters for any technical application. Possible applications for numerical simulation in the automotive industry are sound propagation in the vehicle interior and vibrations of the vehicle structure. Also, noise emission from engine hoods and machine covers may not only be unpleasant, but may pose a health risk. In the aviation industry, possible applications are in the area of reducing sound propagation in aircraft interiors and minimizing vibrations of individual components.

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