by Jobin Xaviour Puthuparampil - University of Toronto T-Space · 2018-11-15 · Abstract Aeroacoustic Noise Prediction and Acoustic Optimization of Mu ers Jobin Xaviour Puthuparampil

Aeroacoustic Noise Prediction and Acoustic Optimization of Mufflers

by

Jobin Xaviour Puthuparampil

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Department of Mechanical and Industrial EngineeringUniversity of Toronto

© Copyright 2018 by Jobin Xaviour Puthuparampil

Abstract

Aeroacoustic Noise Prediction and Acoustic Optimization of Mufflers

Jobin Xaviour Puthuparampil

Master of Applied Science

Department of Mechanical and Industrial Engineering

University of Toronto

2018

Noise control of large diesel and natural gas generators is achieved through industrial mufflers. Design

of such mufflers relies heavily on general guidelines. But these guidelines are not suitable for complex

mufflers; instead, automated optimization provides an effective means of design. Optimization of a

plug flow muffler (PFM) is conducted in this work with two different approaches: 1) a relatively simple

gradient-descent algorithm (L-BFGS-B) maximizing the transmission loss of the PFM, and 2) a multi-

objective (transmission loss and pressure drop) simulation-based optimization using the Efficient Global

Optimization (EGO) algorithm. The EGO algorithm is shown to be well suited for muffler optimization,

performing vastly better than the commonly used NSGA-II algorithm. In addition, the initial steps

towards the prediction of aeroacoustic noise (self-noise) in mufflers is accomplished through the CFD

simulation of the tandem cylinder benchmark experiment.

ii

This work is dedicated with love to my parents, my siblings, and my precious little niece.

iii

Acknowledgements

I acknowledge and thank Safety Power Inc. (SPI), especially Bob Stelzer, Robert Desnoyers, and Henry

Pong, for all the support, monetary and otherwise, that has led to the successful completion of my

master’s degree. I am especially grateful for SPI’s openness to my ideas and willingness to pursue them.

I would also like to acknowledge MITACS and NSERC for monetary support, as well as CMC and

SOSCIP for providing software and computational resources.

I thank Prof. Pierre Sullivan for the years of guidance and support, through my master’s degree and

prior to it. You were able to simultaneously nurture my desire to explore new ideas while providing the

right amount of supervision to coax me onto the right path. Thank you for providing me the opportunity

to do this master’s and for introducing me to the wonderful folks at SPI. For this, and for everything

not mentioned here, I will always be grateful.

By the end of my two years of master’s, I am in ever greater and unrequitable debt to my parents

and siblings. I thank my brother for giving me the confidence to tackle grad school. Thanks to my

sister for her gracious efforts in proof-reading my work many times during the duration of my master’s.

I thank my parents for all the sacrifices they’ve made for me, including those within the last two years;

they’ve shown me how to love.

I also thank my friends who have kept me sane throughout the last two years, especially those I’ve

grown closer to in this time.

Finally and most importantly, I would like to thank God for His love and blessings. I especially thank

Him for my curious mind and drive in seeking answers. A. M. D. G.

iv

Contents

Acknowledgements iv

Table of Contents v

List of Tables vii

List of Figures viii

Nomenclature x

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background and Theory 3

2.1 Aeroacoustic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Acoustic Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Acoustic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Plug Flow Muffler (PFM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Transfer Matrix Method (TMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Perforate Impedance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Pressure Drop Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Global Optimization Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Single-Objective Optimization (SOO) . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Multi-Objective Optimization (MOO) . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.3 Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Muffler Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Methodology 24

3.1 Aeroacoustic Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Simulation Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 Receiver Noise Calculation - Acoustic Analogy . . . . . . . . . . . . . . . . . . . . 27

3.2 Single-Objective Optimization (SOO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

v

3.2.1 TMM Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 FE Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 SOO Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Multi-Objective Optimization (MOO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 FE Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 Pressure Drop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.3 Dimensionality Reduction - Feature Selection . . . . . . . . . . . . . . . . . . . . . 33

3.3.4 MOO Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Results and Discussion 37

4.1 Aeroacoustic Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Simulation Verification - Hydrodynamic Results . . . . . . . . . . . . . . . . . . . 37

4.1.2 Receiver Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Single-Objective Optimization (SOO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1 TMM & FE Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 NLO Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Multi-Objective Optimization (MOO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 Empirical Pressure Drop Model Validation . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.3 EGO’s Performance in MOO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.4 Analysis of EGO Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Conclusion and Future Work 61

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.1 Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.2 Muffler Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Appendices 63

A Mann-Whitney U (MWU) Test 63

B Tandem Cylinder Simulations - Additional Results 65

B.1 2D TKE - Line Extractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.2 Span-wise Correlation of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Bibliography 67

vi

List of Tables

2.1 Taxonomy of multi-objective EGO algorithms used in present work . . . . . . . . . . . . . 21

3.1 CFD setup overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Hydrodynamic metrics used to compare simulation to experimental data . . . . . . . . . . 26

3.3 Parameters defining the PFM and associated values for the Wu et al. muffler [89] . . . . . 28

3.4 Octave band limits and associated engine noise, noise criterion, and NLO weights for SOO 30

3.5 Bounds and constraints applied on SOO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6 Geometric parameters defining the PFM, recast for multi-objective optimization as ratios 32

3.7 1/3 octave band limits and TLO weights for MOO . . . . . . . . . . . . . . . . . . . . . . 34

3.8 Bounds and constraints applied on MOO . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Microphone A tonal peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Comparison of the simulation in the present work to a selection of those in literature [59] 45

4.3 Pearson correlation coefficient - prediction vs. experimental data . . . . . . . . . . . . . . 46

4.4 Optimized muffler dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Summary of dimensionality reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.1 IHV and Iε+ for Objective 1; bolded algorithm is the baseline in MWU test . . . . . . . . 63




vii

List of Figures

2.1 Acoustic sources present in a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Sound waves around a muffler element; the pressures used in the TL calculation (Equation

2.10 and 2.12) is highlighted in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Cross-section of the plug flow muffler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Dimensions defining the perforated expansion . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Dimensions defining the perforated contraction . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Non-zero mean flow caused by grazing flow and/or bias flow on a perforated surface . . . 11

2.7 Electrical flow resistance network depicting the two components of a plug flow muffler . . 12

2.8 Optimization taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 Pareto front in a multi-objective minimization problem . . . . . . . . . . . . . . . . . . . . 16

2.10 NSGA-II algorithm, image based on [35] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.11 EGO algorithm as presented by Jones et al. [46] . . . . . . . . . . . . . . . . . . . . . . . 18

2.12 Multi-objective EGO taxonomy, based on [40]; grayed out boxes and dotted lines represent

aspects not explored in the present work. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.13 Hypervolume calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.14 Additive epsilon indicator (Iε+) calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Tandem cylinder setup, with microphone locations (not to scale) . . . . . . . . . . . . . . 25

3.2 Tandem cylinder meshes - URANS, LES, and DES, respectively . . . . . . . . . . . . . . . 26

3.3 Cross-section of a plug flow muffler and associated dimensional parameters . . . . . . . . 27

3.4 Single-objective optimization (SOO) methodology: evaluation and optimization . . . . . . 28

3.5 Multi-objective optimization (MOO) methodology: evaluation and optimization . . . . . . 32

3.6 Dimensionality reduction through feature selection methodology . . . . . . . . . . . . . . 33

4.1 Mean x-velocity with streamlines from the DES simulation . . . . . . . . . . . . . . . . . 38

4.2 Cp along the surface of the cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Cp′ RMS along the surface of the cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 u along y = 0 in gap between the two cylinders, and after second cylinder . . . . . . . . . 39

4.5 Contours of normalized 2D TKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 PSD of surface pressure at 135° on C1 and 45° on C2 . . . . . . . . . . . . . . . . . . . . . 42

4.7 SPL at the three microphone locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.8 TMM evaluation of transmission loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.9 FEM evaluation of transmission loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.10 Optimized muffler transmission loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

viii

4.11 Original engine noise, NC-60 criterion, and the output sound pressure levels of the opti-

mized muffler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.12 Parametric study - output sound pressure levels . . . . . . . . . . . . . . . . . . . . . . . . 49

4.13 Histogram of NLO evaluations at the final result of each optimization . . . . . . . . . . . 50

4.14 Dimension-reduced visualization of one cluster of 883 unique solutions (local minima),

colored by their associated NLO evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.15 Comparison of pressure drop calculated via the empirical pressure drop model and CFD,

for 6 random PFM geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.16 Median attainment surfaces comparing the 4 EGO variants . . . . . . . . . . . . . . . . . 54

4.17 Median attainment surfaces comparing different Nevals . . . . . . . . . . . . . . . . . . . . 54

4.18 Median attainment surfaces comparing different sets of initial conditions . . . . . . . . . . 55

4.19 Median attainment surfaces comparing the ParEGO algorithm and NSGA-II . . . . . . . 56

4.20 Hypervolume metric (IHV ) comparing the ParEGO algorithm and NSGAII . . . . . . . . 57

4.21 Binary additive epsilon metric (Iε+) comparing the ParEGO algorithm and NSGAII . . . 57

4.22 Visualization of kriging model developed by the ParEGO algorithm for one set of TLO

and DP weights, with the optimal points on the Pareto front approximation overlaid as

white points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.1 2D TKE along y = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.2 2D TKE along x = 1.5 D and x = 4.45 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.3 Span-wise correlation of surface pressure at 135° on C1 and C2 . . . . . . . . . . . . . . . 66

ix

Nomenclature

∆P Pressure drop

νo Kinematic viscosity of air

ω Angular frequency

ρo Ambient density of air

σ Perforated plate area porosity

ξ Perforate impedance

co Ambient speed of sound in air

Cp Coefficient of pressure

Cp′ RMS Root-mean-square of coefficient of pressure

Dh Perforated hole diameter

H Dynamic head

Iε+ Binary additive epsilon indicator/metric

IHV Hypervolume indicator/metric

k Angular wave number

LW Acoustic power

LW Sound power

M Mach number

OAR Open area ratio

p Acoustic pressure

Q Volume flow rate

R Flow resistance

Rp Span-wise correlation of pressure

Re Reynolds number

x

S Cross-sectional surface area

To Ambient temperature of air

Tp Perforated plate thickness

U Stream-wise velocity

V Flow velocity

1D One-dimensional

2D Two-dimensional

2DA Two-dimensional axisymmetric

3D Three-dimensional

BFGS Broyden-Fletcher-Goldfarb-Shanno (algorithm)

CFD Computational fluid dynamics

CFS Correlation-based feature selection

DES Detached eddy simulation

DNC Direct noise computation

DP Pressure drop objective

EGO Efficient global algorithm

EIF Expected improvement function

FEM Finite element method

FFT Fast Fourier transform

FWH Ffowcs-Williams and Hawkings

GA Genetic algorithm

LES Large eddy simulation

LHS Latin hypercube sampling

MOO Multi-objective optimization

MWU Mann-Whitney U

NC Noise criterion

NLO Noise level objective

NSGA-II Non-dominated sorting genetic algorithm-II

NSPD Normalized stagnation pressure drop

xi

PCA Principal component analysis

PFM Plug flow muffler

PSD Power spectral density

PSO Particle swarm optimization

QFF Quiet Flow Facility

SA Simulated annealing

SOO Single-objective optimization

SPL Sound pressure level

TKE Turbulent kinetic energy

TL Transmission loss

TLO Transmission loss objective

TMM Transfer matrix method

URANS Unsteady Reynolds-averaged Navier-Stoked

xii

Chapter 1

Introduction

1.1 Motivation

Large-scale (500 kW and above) diesel and natural gas generators produce up to 105 dBA of noise at

a distance of 7 m from the engine [4]. With peak allowable noise levels based on regulations ranging

anywhere from 57 dBA to 72 dBA in suburban to industrial areas respectively, there is a clear need for

acoustic control [4]. Generators have many use cases ranging from providing backup power, to offsetting

energy demand peaks from the grid (peak shaving), to continuous power generation. This versatility

has led to generators being installed in residential zones in recent years, necessitating strict acoustic

attenuation in order to operate legally with minimal disruption to the public.

Noise control of the exhaust of an internal combustion engine is typically achieved with a muffler.

Mufflers are acoustic filters designed to reduce the magnitude of audible noise (20 Hz to 20 000 Hz).

There are two main mechanisms of passive noise attenuation used in mufflers:

1. Reactive silencing: structural elements within the muffler passively create sound waves that are out

of phase with the incoming sound, causing destructive interference that reduces the amplitude of

the sound. This mechanism is most effective in the low-frequency range (< 500 Hz). It is achieved

by tuning the physical dimensions of the components inside the muffler to change the phase of

internally reflected waves.

2. Absorptive or dissipative silencing: viscous effects dampen the sound energy through the use of

fibrous or foam-like acoustic material. Its effect is more pronounced in the mid to high frequency

range, and the use of more acoustic material generally results in better attenuation. Forcing the

flow to pass through perforated surfaces (solid surface with an array of small holes with a diameter

of a few millimeters) is another method of dissipative silencing as viscosity is quite effective at noise

attenuation at these scales. The perforation geometry can also be tuned to affect performance.

Tuning the physical dimensions of the acoustic components inside the muffler is crucial to the operation

of a muffler. General guidelines exist for such tuning, but only for basic muffler designs. In order to

truly design the best muffler for a given generator, a computer-based automated optimization process

is necessary. Such optimization aims to maximize or minimize an objective by varying a set of pa-

rameters, given a set of constraints. Thus, tuning a muffler’s geometry to produce maximum acoustic

1

Chapter 1. Introduction 2

performance is well suited for an automated optimizer. In addition to maximizing acoustic performance,

a muffler must be designed to create the least amount of back-pressure on the engine as possible, as

higher back-pressures reduce engine efficiency. To account for this, optimization which simultaneously

tries to maximize acoustic performance while minimizing back-pressure can be conducted. Additionally,

complex mufflers require computer simulation in order to accurately determine its acoustic and flow

characteristics; mathematical descriptions will generally be ineffective when a design is of industrial

complexity and size. Such simulations can take a considerable amount of computational time, but are

necessary in the optimization process to retain accuracy. Thus, it becomes crucial to use an efficient

optimization algorithm when conducting simulation-based optimization. All aforementioned ideas are

explored in the single-objective and multi-objective optimization sections of this work.

As the turbulent exhaust gasses from the engine flow through the muffler, it can produce flow-induced

noise. This phenomenon, also referred to as self-noise, usually produces broadband (across a large por-

tion of the frequency spectrum) noise. However, if the flow interacts with sharp edges or perforations

within the muffler, it has the potential of creating tonal noise as well. The first steps towards the predic-

tion of such noise is conducted in the aeroacoustic modelling sections of the thesis through a benchmark

problem.

1.2 Objectives

This work has two major objectives:

1. Accurate and computationally cost-effective means of predicting aeroacoustic noise

2. Develop a methodology for the automated optimization of large industrial mufflers, simultaneously

emphasizing speed and accuracy of the optimization

Chapter 2

Background and Theory

2.1 Aeroacoustic Modelling

Aeroacoustic modelling assists the prediction of sound produced from flow turbulence. This sound is

generally broadband, with sound energy spread over a large range of audible frequencies. However,

certain flow mechanisms, such as vortex shedding, can add a tonal component to the noise generated.

There are various geometries in a muffler that can lead to tonal flow generated noise, including sharp

edges with perpendicular flow or perforated surfaces. This is in addition to broadband aeroacoustic noise

generated by fast moving flow which can be approximated using empirical relationships, and is a strong

function of the flow velocity [1].

Direct noise computation (DNC) can simulate the pressure variation as a function of time by solving

the compressible flow equations. The pressure field is then decomposed into a hydrodynamic component

and an acoustic component, following which the acoustic component can be analyzed in the frequency

domain to reveal the sound pressure level (SPL) at various frequencies. DNC proves to be extremely

challenging because of the implementation of:

1. high order spatial discretization schemes to account for the large mismatch in energy scales of the

hydrodynamic and acoustic components (acoustic energy is smaller than the hydrodynamic energy

by an order O(M4) where M is the Mach number) [88]

2. large domains, required to capture the relatively large lengths of acoustic waves, compared to

hydrodynamic length scales; computational requirement is proportional to Re3t/M

4 where Ret is

the turbulent Reynolds number [74]

3. Absorbent boundary conditions, which produce minimal spurious reflections into the domain [88]

2.1.1 Acoustic Analogy

Acoustic analogies, originally proposed by Lighthill [57, 58] and developed by Curle [24] and Ffowcs-

Williams & Hawkings (FWH) [32], aim to rectify the challenges seen with DNC by decoupling the

hydrodynamic and acoustic fields. Traditional time-resolved flow simulations are used to estimate the

turbulent properties of the flow. Then, the strengths and locations of acoustic source terms such as

3

Chapter 2. Background and Theory 4

Figure 2.1: Acoustic sources present in a flow

monopoles, dipoles, and quadrupoles are determined based on the turbulence in the flow. These three

sources radiate acoustic energy with different directivity and efficiencies (Figure 2.1). The contribution

of these acoustic sources are assessed to estimate the acoustic pressure fluctuations at far-field acoustic

listeners locations, which can subsequently be converted to the frequency domain to show the SPL at

various frequencies.

Lighthill manipulated basic flow equations to convert them into a form similar to the inhomogeneous

wave equation, creating the aeroacoustic analogy [57, 58]. It is derived beginning with the conservation

of mass and momentum

∂ρ

∂t+∂ (ρui)

∂xi= m (2.1)

∂ (ρui)

∂t+∂ (ρuiuj)

∂xj= fi −

∂p

∂xi+∂τij∂xj

(2.2)

where ρ is the density and u is the velocity, m is a mass source term, p is the hydrostatic pressure, and

τij is the shear stress tensor. Subtracting the divergence of Equation 2.2 from the time derivative of

Equation 2.1 to eliminate the term with ρui yields

∂2ρ

∂t2=∂m

∂t− ∂fi∂xi

+∂2 (ρuiuj)

∂xixj+∂2p

∂x2i

− ∂2τij∂xixj

(2.3)

∂2ρ

∂t2=∂m

∂t− ∂fi∂xi

+∂2

∂xixj(ρuiuj + pδij − τij) (2.4)

Subtracting c2o∇2ρ = c2o∂2

∂x2i

(ρ) = ∂2

∂xixj

(c2oρδij

)from both sides, where co is the ambient speed of sound,

gives∂2ρ

∂t2− c2o∇2ρ =

∂m

∂t− ∂fi∂xi

+∂2

∂xixj

(ρuiuj +

(p− c2oρ

)δij − τij

)(2.5)

which has a similar form to the inhomogeneous wave equation. The right hand side of Equation 2.5

consists of the source term (S), which has three terms, the latter of which is simplified by Lighthill by


an argument of practicality [57]:

S = Smonopole + Sdipole + Squadrupole (2.6)

Smonopole =∂m

∂t(2.7)

Sdipole = − ∂fi∂xi

(2.8)

Squadrupole =∂2

∂xixj

(ρuiuj +

(p− c2oρ

)δij − τij

)≈ ∂2

∂xixj(ρuiuj) (2.9)

Lighthill’s original formulations focused on quadrupoles only, such that S = Squadrupole [57, 58]. Lighthill

provided an exact solution to Equation 2.5 using Green’s functions with the simplified Equation 2.9.

Curle expanded Lighthill’s exact solution to Equation 2.5 while considering S = Sdipole + Squadrupole

(i.e., sum of Equations 2.8 and simplified 2.9) where the dipole sources, distributed on the surface of a

stationary object in flow, contain the noise created by the force exerted by the fluid on the solid [24].

The exact solution provided with the FWH analogy [32] includes monopoles and thus uses the complete

source term (Equation 2.6, with simplified Equation 2.9). Monopoles are generated through the physical

displacement of fluid volumes by solid surfaces.

The relative importance of each source varies based on the type of flow encountered; the dipole is

dominant when periodic flow separation occurs [37], and will be the focus of the aeroacoustic modelling

in the present work (Sections 3.1 and 4.1).

2.2 Acoustic Modelling

Transmission loss (TL) is the difference in acoustic power between the sound travelling toward the muffler

on the inlet side (LWi) and sound travelling away from the muffler on the outlet side (LWo); direction of

sound travel is important as any reflections will transport sound power in the opposite direction (Figure

2.2).

TL = LWi − LWo (2.10)

Sound power (LW ) normal to a surface, such as the inlet or outlet cross-section of a muffler, is

LW =Sp2

ρoco(2.11)

where S is the cross-sectional area of the surface, p is the acoustic pressure, and ρo and co are the density

and speed of sound of the medium. Expressing TL as a decibel (dB) using Equations 2.10 and 2.11,

with the direction specified, results in

TL = 10 log10

∣∣∣∣ Sip2i+

Sop2o+

∣∣∣∣ (2.12)


Figure 2.2: Sound waves around a muffler element; the pressures used in the TL calculation (Equation2.10 and 2.12) is highlighted in red.

Figure 2.3: Cross-section of the plug flow muffler

where pi+ is the acoustic pressure of the incoming incident wave, po+ is the acoustic pressure of the

outgoing transmitted wave, and Si and So are the inlet and outlet cross-sectional areas, respectively.

Various analytical and numerical methods exist to predict the TL of a given muffler. Analytical meth-

ods include the transfer matrix method (TMM) [63], modal meshing approach [62], and exact 2D/3D

analytical solutions [e.g., 42, 75], while numerical alternatives are typically the finite element method

(FEM) and boundary element method (BEM). The present work utilizes TMM and FEM for analytical

and numerical predictions, respectively.

2.2.1 Plug Flow Muffler (PFM)

This work focuses on the plug flow muffler (PFM) (Figure 2.3), a muffler consisting of a perforated

expansion followed by a perforated contraction. Engine exhaust gas enters the first section of the inner

tube, is forced out through the perforations on the inner tube to the surrounding annular chamber due

to the presence of the flow plug, rejoins in the second portion of the inner tube after passing through the

second perforated surface, and finally exits the muffler. This type of muffler has a relatively high pressure

drop due to the presence of the plug as well as the perforated surfaces, but yields high transmission losses

in a small form factor. It utilizes both reactive and dissipative mechanisms of noise control through the

various tubes and perforated surfaces, respectively. The PFM can be used as a component within a larger

muffler, as it is well suited for attenuating the mid frequencies (500 Hz to 2000 Hz); this frequency range

is generally not addressed easily with a traditional industrial muffler’s reactive or dissipative sections.


2.2.2 Transfer Matrix Method (TMM)

The transfer matrix method (TMM) requires the calculation of a 2 x 2 transfer matrix, also known as

the “four-pole parameters”. Sullivan and Crocker proposed the first mathematical model of a perforated

muffler and analyzed a non-plug concentric tube muffler [85]. Jayaraman and Yam modified Sullivan and

Crocker’s model with a decoupled solution applied to the two cross-flow elements found in a plug flow

muffler: a perforated expansion (Figure 2.4) and a perforated contraction (Figure 2.5) [43, 86]. The key

improvement was that each muffler element only required one set of calculations. However, a limitation

in their derivation defined the mean flow velocity in the inner tube and outer annulus be the same. It

is clear that the change in cross-sectional area from the inner to the outer section results in a change in

mean flow velocity, except for the zero mean flow case. Munjal et al. [65] addressed this limitation and

produced an analytical decoupling method which accounted for the difference in mean flow velocity in the

interior and exterior annulus. This approach is well suited for the analysis of the plug flow muffler shown

in Section 2.2.1. TMM splits the muffler into a sequence of its sub-components, analyzed individually,

and then combined to reveal their total effect. For a given frequency, each sub-component results in a

transfer matrix (T ), (p1

ρocou1

)= T

(p2

ρocou2

)=

[TA TB

TC TD

](p2

ρocou2

)(2.13)

The transfer matrix relates the acoustic pressure (p) and the velocity (u) at the inlet of the sub-

component, subscript 1, to the outlet condition, subscript 2; ρo and co are the ambient density and

speed of sound. The muffler’s complete transfer matrix is

Tmuffler = T1T2 . . . Tn =

[TA TB

TC TD

]1

[TA TB

TC TD

]2

. . .

[TA TB

TC TD

]n

(2.14)

where the subscript represents the sub-component under consideration. In the case of a plug flow muffler,

two acoustics sub-components need to be considered (n = 2): a perforated expansion and a perforated

contraction. Munjal shows that the transfer matrix for a perforated expansion chamber (Figure 2.4) is

[64]

TA = P12 +A1A2 TB = P14 +B1A2 (2.15)

TC = P32 +A1B2 TD = P34 +B1B2

A1 = (X1P22 − P42) /F1 B1 = (X1P24 − P44) /F1

A2 = P11 +X2P13 B2 = P31 +X2P33

F1 = P41 +X2P43 −X1 (P21 +X2P23)

X1 = −i tan (koLa1) X2 = i tan (koLb1)

where La1 and Lb1 refer to the lengths of the solid sections of tube before and after the expansion’s

perforated section. The transfer matrix for a perforated contraction chamber (Figure 2.5) is [64]


Figure 2.4: Dimensions defining the perforated expansion

Figure 2.5: Dimensions defining the perforated contraction

TA = P21 +A1A2 TB = P23 +B1A2 (2.16)

TC = P41 +A1B2 TD = P43 +B1B2

A1 = (X1P11 − P31) /F1 B1 = (X1P13 − P33) /F1

A2 = P22 +X2P24 B2 = P42 +X2P44

F1 = P32 +X2P34 −X1 (P12 +X2P14)

X1 = −i tan (koLa2) X2 = i tan (koLb2)


where La2 and Lb2 refer to the lengths of the solid sections of tube before and after the contraction’s

perforated section. In Equations 2.15 & 2.16, the matrix P is

P =

P11 P12 P13 P14

P21 P22 P23 P24

P31 P32 P33 P34

P41 P42 P43 P44

= [A(0)][A(1)]−1 (2.17)

where the A(x) matrix’s elements are (for i = 1, 2, 3, 4)

A1,i = ψ3,ieβix A2,i = ψ4,ie

βix (2.18)

A3,i = − eβix

iko +M1βiA4,i = − ψ2,ie

βix

iko +M2βi

ψ and β are the eigenmatrix and eigenvector, respectively, of the following matrix−α1 −α3 −α2 −α4

−α5 −α7 −α6 −α8

1 0 0 0

0 1 0 0

(2.19)

α1 = − iM1

1−M21

(k2a + k2

o

ko

)α2 =

k2a

1−M21

α3 = − iM1

1−M21

(k2a − k2

o

ko

)α4 = −

(k2a − k2

o

1−M21

)α5 = − iM2

1−M22

(k2b − k2

o

ko

)α6 = −

(k2b − k2

o

1−M22

)α7 = − iM2

1−M22

(k2b + k2

o

ko

)α8 =

k2b

1−M22

ko = ω/co

M1 = V1/co M2 = V2/co

k2a = k2

o −4ikoD1ξ

k2b = k2

o −4ikoD1

(D22 −D2

1) ξ

where ω is the angular frequency, ξ is the specific perforate impedance (Section 2.2.4), and V1 and V2

are the mean flow velocities in the inner tube and outer annulus, respectively.

Once the complete transfer matrix is calculated (Equation 2.14), the transmission loss is [65]

TL = 10 log10

[SiSo

(1 +M1

1 +M2

)2 ∣∣∣∣TA + TB + TC + TD2

∣∣∣∣]

(2.20)

TMM is a 1D method. It only considers the propagation of plane waves, and so, is only valid below the


cutoff frequency (fc) for the first circumferential mode [11, 31], above which higher wave modes occur:

fc =1.84coπD2

(2.21)

2.2.3 Finite Element Method (FEM)

An alternate method to predict the TL of a muffler is the finite element numerical discretization of the

Helmholtz equation. This equation can be written with pressure as the scalar of interest as

∇ ·[− 1

ρo(∇p− qd)

]− ω2p

ρoc2o= Qm (2.22)

where ρo is the density, co is the speed of sound, ω is the angular frequency, Qm is the monopole source

term, and qd is the dipole source term, and the pressure p is a time-harmonic wave of the form

p (x, t) = p (x) eiωt (2.23)

Equation 2.22 can be parametrically solved for a range of frequencies to determine the pressure dis-

tribution within a domain at each frequency. Following this, the transmission loss is calculated as a

decibel:

TL = 10 log10

(LWi

LWo

)(2.24)

where LWi and LWo correspond to the acoustic power at the muffler inlet and outlet respectively.

Acoustic power is calculated using the pressure field solution integrated over the surface (S) of interest

(cross-sectional surface of the inlet or outlet):

LWs = 2

∫S

|p|2

2ρoco(2.25)

Note that accuracy is not limited to a cutoff frequency as with TMM; it is limited by the size of the

mesh used. However, the computational cost of FE is at least 120 times TMM, in the case of the plug

flow muffler, discouraging the use of FE in muffler optimization.

2.2.4 Perforate Impedance Model

An important feature of a plug flow muffler is the perforated surfaces that the flow moves through.

Accurate descriptions of the acoustic effects of perforated surfaces exist primarily in empirical work.

In general, acoustic impedance of perforated surfaces have been classified in (a) zero mean flow and

(b) non-zero mean flow. Zero mean flow, allowing only for the oscillatory acoustic flow, is a physically

unrealistic condition as mufflers are generally always subject to some amount of exhaust flow. However,

it is still a valuable tool in developing the understanding behind perforated surfaces. Non-zero mean

flow can be caused due to one or both of the following flows: bias flow or grazing flow. Bias flow involves

flow passing perpendicularly through the perforations, whereas grazing flow considers flow parallel to

the perforated plate (Figure 2.6).


Grazing FlowBias Flow

Figure 2.6: Non-zero mean flow caused by grazing flow and/or bias flow on a perforated surface

Zero Mean Flow

Three models for zero mean flow perforate impedance (ξ) are considered in this work:

1. Sullivan [84]

ξ =p

ρocou=

6× 10−3

σ+ik (Tp + 0.75Dh)

σ(2.26)

2. Bento Coelho [25]

ξ =p

ρocou=

1

ρoco(R+ iX) (2.27)

R =1

σ

[ρo

(d′

d′′

)√8νoω +

(ρo8co

)(ωDh)

2

]X =

(ωρoσ

)(d′′ +

(d′

Dh

√8νoω

))d′ = Tp +Dh

d′′ = Tp +

(8

3π

)Dh

(1− 0.7

√σ)

3. Bauer [9]

ξ =p

ρocou=

√8νokco

(1 +

Tp

Dh

)σ

+ik (Tp + 0.25Dh)

σ(2.28)

where k is the angular wave number, Tp is the thickness of the perforated plate, Dh is the diameter of

the perforated holes, σ is the area porosity of the perforated sections, ρo is the ambient air density, co

is the speed of sound in ambient conditions, νo is the ambient kinematic viscosity, and ω is the angular

frequency.

Non-Zero Mean Flow

A unified model for perforate impedance is used in this work to account for non-zero mean flow [30]:

ξ = θ + iχ (2.29)

θ = Re

{jk

σCD

[Tp

F (µ′o)+

δreF (µo)

fint

]}+

1

σ

[1− 2J1 (kDh)

kDh

]+

0.3

σMg +

1.15

σCDMb

χ = Im

{jk

σCD

[Tp

F (µ′o)+

0.5Dh

F (µo)fint

]}


Figure 2.7: Electrical flow resistance network depicting the two components of a plug flow muffler

F (KDh) = 1− 4J1 (KDh/2)

KDhJ0 (KDh/2)K =

√− iωνo

=

√− iωρo

µoK ′ =

√− iων′o

=

√− iωρo

µ′o

δre = 0.2Dh + 200D2h + 16000D3

h fint = 1− 1.47√σ + 0.47

√σ3

where Mg is the grazing flow Mach number, Mb is the bias flow Mach number, J0 and J1 are Bessel

functions of the first kind (first and second order respectively), CD is the orifice discharge coefficient as

given by Elnady [30], and µ′o = 2.179µo.

2.3 Pressure Drop Modelling

Flow networks can be modelled using using the circuit and resistance network analogy. The plug flow

muffler is modelled as a simple resistance network (Figure 2.7). The flow resistance (R) of any component

in a flow path is

R =∆P

Q|Q|(2.30)

where ∆P is the pressure drop across the component and Q is the volume flow rate. Equation 2.30 can

be simplified for a unidirectional flow to be

R =∆P

Q2(2.31)

The total equivalent resistance (Req) of the network in Figure 2.7 is given as

Req = Rexp +Rcon (2.32)

Therefore, the pressure drop across the plug flow muffler (PFM) is

(∆P )PFM = ReqQ2 = (Rexp +Rcon)Q2 (2.33)

Munjal et al. [66] experimentally developed relationships to describe the flow resistance of both per-

forated expansions (Rexp) and perforated contractions (Rcon) based on geometrical parameters. The

flow resistances are represented by normalized stagnation pressure drops (NSPD = ∆PH ), where H is the

dynamic head of the incoming flow,

R = (NSPD)H

Q2(2.34)


Combining equations 2.33 and 2.34, where Q is constant throughout the muffler:

(∆P )PFM =

[NSPDexp

(H

Q2

)+ NSPDcon

(H

Q2

)]Q2

= (NSPDexp + NSPDcon)H

= (NSPDexp + NSPDcon)

(1

2ρoV

2inlet

)(2.35)

where ρ is the density of air or combustion gas passing through the muffler and Vinlet is the area-averaged

velocity at the inlet. Munjal provides the following relationships for calculating the NSPD [66]

NSPDexp =

(∆P

H

)exp

= 3.136(OARexp)−1.391 0.3 < OARexp < 2.2 (2.36)

NSPDcon =

(∆P

H

)con

= 2.208(OARcon)−1.5818 0.31 < OARcon < 1.66 (2.37)

Open area ratio (OAR) compares the open area present on the surface of the perforated tube to the

cross-sectional area of the perforated tube. OAR is

OAR =πD1Lσπ4D

21

=4Lσ

D1(2.38)

where D1 is the diameter of the perforated tube, L is the length of the perforated section of tube, and

σ is the open area porosity of the perforated tube.

2.4 Global Optimization Methodology

Sections 2.2 and 2.3 provide a means of analyzing the plug flow muffler’s two main characteristics: trans-

mission loss and pressure drop. Aside from general design trends (e.g., increased OAR leads to lower

pressure drop), a specific design methodology is not directly discernible from the analysis above, due

to the complex nature of the transmission loss prediction. In order to design the best possible muffler

for a given set of operating conditions and constraints, automated design via optimization is the solution.

Optimization is the process of finding the best performing state of an objective function, applied in

this work to find designs that maximize acoustic performance. Formally, it is

minimizex

f(x)

subject to gi(x) ≤ 0, k = 0, . . . ,m,

hj(x) = 0, k = 0, . . . , p

(2.39)

where f(x) is the objective function to be optimized (traditionally, a minimization) with an n-variable

long vector x, gj(x) and hi(x) are sets of inequality and equality constraints respectively, and m ≥ 0

and p ≥ 0 [13]. There are multiple ways to look at the taxonomy of optimization problems as the field

is very large and there exists many interrelated ideas, but Figure 2.8 provides an overview from the

perspective relevant to this work. The five major attributes of an optimization problem are:


1. Variables: The goal of any optimization problem is to determine the variable value(s) that provide

the optimal conditions for f(x). There can be just one variable to optimize (n = 1), in the

uni-variate case, or as more often seen in engineering, multiple variables (n > 1), leading to multi-

variate optimization. These variables can be continuous or discrete or a combination of both.

2. Objectives: In single-objective optimization, there is only one objective function to optimize,

leading to a clearly defined optimality criterion. However, in multi-objective optimization, the

objectives may conflict; there may not be a clear optimality criterion leading to one ideal x where

all objectives are optimized. Instead, there is a set of well performing solutions that satisfy each

objective to a different degree - i.e., a Pareto front.

3. Constraints: If constraints exist, they come in two forms, inequality (gi) or equality (hj), that

can be applied as linear constraints or non-linear constraints. Linear constraints are imposed on

the variable x while non-linear constraints are applied on the evaluation of the function, f(x). If

m = 0 and p = 0, it becomes an unconstrained optimization problem. In addition to linear and

non-linear constraints, upper and lower bounds on the variables (x) can be established to limit the

optimization within a predefined boundary; in an engineering context, these may be boundaries

dictated by physical limitations or other costs.

4. Linearity: A linear optimization problem requires both the objective function and the constraints

to be linear functions. If this condition is not met, the problem is termed non-linear. In a linear

problem, any local optima discovered is always the global optima as well, allowing for simpler and

more robust optimization algorithms, and the ability to solve large problems.

5. Convexity: A function is convex when any line drawn between two points on the function (chord)

lies above the function [13]. A convex minimization problem requires all objective function as well

as the constraints to be convex. In a convex minimization problem, any local minima discovered is

also the global minima. However, in a non-convex problem, there is no such guarantee, requiring

the search of all local minima to truly guarantee the global minima. Thus, if a problem has multiple

local minima, it can be classified as non-convex.

The present problem is bounded, multi-variate (continuous variables), non-linear, and non-convex. Thus,

a global optimization strategy able to balance local (narrowing down on optima) and global (exploration)

searches is required. Both single-objective and multi-objective formulations were considered in this work,

each achieving separate goals.

2.4.1 Single-Objective Optimization (SOO)

A single-objective formulation of the present problem involves the optimization of transmission loss alone.

Many algorithms exist for non-linear single-objective optimization, including the Broyden-Fletcher-

Goldfarb-Shanno (BFGS) algorithm [14, 33, 36, 77], non-linear conjugate gradient methods such as

Fletcher-Reeves [34], Levenberg-Marquardt algorithm [56, 61], and the Nelder-Mead method [67]. The

present work utilized the BFGS algorithm to solve the single-objective optimization problem.


Figure 2.8: Optimization taxonomy

Broyden-Fletcher-Goldfarb-Shanno (BFGS)

The BFGS algorithm was discovered independently by Broyden [14], Fletcher [33], Goldfarb [36], and

Shanno [77]. It is a second-order iterative quasi-Newton method that calculates the gradient of a func-

tion and estimates the inverse of the Hessian to guide itself towards a local minima.

A function f(x) can be estimated by its second-order Taylor approximation

f (xk + ∆x) ≈ f(xk) +∇f(xk)T∆x+1

2∆xT∇2f(xk)∆x (2.40)

where xk is the current approximation of the minima, ∆x is a small step to explore the function at a

nearby value, ∇f(xk) is the gradient at xk, and ∇2f(xk) is the Hessian. The gradient of Equation 2.40

is

∇f (xk + ∆x) ≈ ∇f(xk) +∇2f(xk)∆x (2.41)

To determine the minima, the gradient (Equation 2.41) is set to 0 and rearranged to solve for ∆x,

resulting in

∆x = −(∇2f(xk)

)−1∇f(xk) (2.42)

In the case of a quasi-Newton method, the Hessian is approximated as B and Equation 2.42 is written

as

∆x = −B−1∇f(xk) (2.43)

Iterative updates of B (or B−1) to reveal better approximations of the true Hessian (or inverse Hessian)

is the key aspect of BFGS, as is achieved via

Bk+1 = Bk +yky

Tk

yTk ∆xk− Bk∆xk (Bk∆xk)

T

∆xTkBk∆xk(2.44)

where ∆xk = αkB−1k ∇f(xk) with αk determined from the Armijo rule [5] and yk = ∇f(xk+1)−∇f(xk).


Figure 2.9: Pareto front in a multi-objective minimization problem

Memory efficient versions of BFGS have been developed, known as limited-memory BFGS (L-BFGS),

which limit the amount of data used in the Hessian approximation [68, 16]. Additionally, BFGS was

originally formulated as an unconstrained optimization algorithm; variable bounds constraints have been

implemented on the limited memory BFGS in an algorithm referred to as L-BFGS-B [15]. It is important

to note that the BFGS algorithm does not guarantee global minimums, but a local one. To increase the

probability of discovering the global minima, a common technique is to employ a multi-start framework,

where multiple independent optimizations are run in parallel, with each starting at a different random

initial condition [28].

2.4.2 Multi-Objective Optimization (MOO)

Attempting to optimize both transmission loss and the pressure drop through the muffler leads to a

multi-objective problem with competing objectives. The optimal solution to a multi-objective problem

is a Pareto front when nobj = 2 or Pareto surface for nobj > 2. The Pareto front consists of a set of

Pareto efficient solutions to the optimization problem; Pareto efficient (equivalently, non-dominated)

solutions are those where any improvement in one objective must be met with a deterioration in another

objective (Figure 2.9). In the engineering context, a Pareto front is a trade-off curve, representing the

interplay between competing objectives.

Genetic Algorithm

Evolutionary algorithms such as the genetic algorithm are well suited for the discovery of Pareto fronts

as multiple solution candidates are considered and improved through the optimization process. Non-

dominated sorting genetic algorithm-II (NSGA-II) has been shown to be an effective variant of the genetic

algorithm in multi-objective optimization [27], outperforming other established evolutionary algorithms

such as Pareto archived evolution strategy (PAES) [52] and strength pareto evolutionary algorithm

(SPEA) [98]

NSGA-II’s algorithm flowchart is seen in Figure 2.10. It begins with a randomly generated population of

size npop whose objective functions are first evaluated. This population is ranked via the non-dominated

sort; the population is organized into multiple levels of non-domination, resulting in multiple fronts.

After the initial sorting, the child population is created (i.e., next generation) using standard genetic

algorithm processes such as selection, crossover, and mutation, which are described extensively in other


Figure 2.10: NSGA-II algorithm, image based on [35]

sources [98]. Additionally, to promote exploration of the entire Pareto front, the crowding of solutions

is discouraged via calculation of a crowding distance metric.

The elitist aspect of NSGA-II combines the parent and children populations together and applies the

non-dominated ranking procedure and crowding distance calculations on the combined set, from which

the best performing npop individuals are chosen, with higher preference to non-domination - if two so-

lutions share the same non-domination rank, the less crowded solution is chosen. The process loops

back to the creation of children, resulting in he next generation, and continues until a stopping criterion

is satisfied, typically maximum number of generations ngen. This results in Nevals = npop · ngen total

evaluations of the objective function.

Efficient Global Optimization

Efficient global optimization (EGO) is a surrogate model based optimization technique first proposed as

a single-objective optimization algorithm by Jones et al. [46]. EGO uses kriging, the process of fitting

a Gaussian process model, to create a surrogate model to estimate where the optima lies, along with

an expected improvement function (EIF) to determine where the next evaluation point must be. The

algorithm for EGO as developed by Jones et al. is shown in Figure 2.11; the single objective EGO will

be described before moving onto the multi-objective formulation. EGO begins with the generation of a

set of initial points using Latin hypercube sampling (LHS) and subsequent calculation of the objective

function value at each point. This information is used to create the Gaussian process-based surrogate

model, providing both an estimate of the objective value at a given location along with the uncertainty

of the estimate. The process only continues if the magnitude of the cross-validated standardized error

residuals of the surrogate model (or the surrogate model of the log of the objective space) is less than 3

- a sufficient range for the error as stated by Jones et al. The next step is to determine the location for

the next objective function evaluation.

EGO determines the location of the next evaluation by balancing local and global searches through


Figure 2.11: EGO algorithm as presented by Jones et al. [46]

the use of an expected improvement function (EIF). EIF is defined as:

EIF(x) = (fmin − y)D

(fmin − y

s

)+ sP

(fmin − y

s

)(2.45)

where fmin is the lowest value of the objectives that are evaluated directly thus far, y is the surrogate

model’s prediction of the objective value, s is the standard error (uncertainty) of the surrogate model’s

estimate, D is the normal distribution function, and P is the normal density function. EGO achieves

a natural balance between local and global search since small values of (fmin − y) and large values of s

promote local and global searches, respectively. Maximizing the EIF will reveal the next evaluation point

which would maximize information gain about the actual minima and the objective surface, f(x). In

EGO, if the maximized EIF is above 1%, this point is evaluated and the algorithm iterates, fitting a new

surrogate model with the new information. If the EIF < 1%, it signals an accuracy based termination

condition.

EGO has been extended to a multi-objective optimization algorithm by multiple researchers in recent

years; Horn et al. provides a framework to understand the variations that exist [40]. A visual repre-

sentation of the taxonomy of multi-objective EGO algorithms developed in [40] is shown in Figure 2.12.

The initial design, as with single-objective EGO is typically generated using LHS. There are two options

available when fitting a kriging model: 1) each objective is fit with a separate surrogate model, or 2) the

multiple objectives are scalarized (weighted sum of each objective), after which one surrogate model is

fit over the scalarized objective space. Much work has been done in utilizing a variety of infill criteria for

the generation of an evaluation point (candidate generation), such as hypervolume contribution [69] and

additive ε-indicator [87], in addition to the EIF. In general, these methods involve 1) multi-objective op-

timization of a given criterion when individual models have been fit for each objective, 2) single-objective

optimization of a criterion, or 3) multiple single-objective optimizations of a criterion. If required, mul-

tiple candidates can be generated using either 1) or 3). Following the generation of multiple candidates,

a selection strategy can be used to narrow down the number of candidates to a desired value based on

computational budget or other requirements. After the evaluation of the candidate(s), the process can

return to the fitting of the surrogate model and iterate, typically until a computational budget is met.


Figure 2.12: Multi-objective EGO taxonomy, based on [40]; grayed out boxes and dotted lines representaspects not explored in the present work.

The present work explores 4 multi-objective EGO variants, with the taxonomy of each presented in

Table 2.1:

1. ParEGO [51]: This is a scalarization based method which randomizes the weights used for each

objective at each iteration. Thus, the EIF is calculated and optimized a single surrogate model,

while still having a general idea of all objectives involved without focusing down on any single

objective.

2. SMS-EGO [69]: After developing individual surrogate models for each objective, SMS-EGO con-

verts the MOO into a SOO by calculation of the hypervolume (S-metric). Hypervolume is the

volume (or area in a bi-objective optimization) encompassed by the Pareto front (or Pareto front

approximation), with reference to a prescribed point [6]. In practice, the hypervolume is calculated

through rectangular approximations as shown in Figure 2.13. Candidate points are generated and

chosen based on highest hypervolume contribution potential as this would push out the Pareto

front.

3. ε-EGO [87]: Similar to SMS-EGO, ε-EGO fits individual surrogate models to each objective and

then converts the MOO into a SOO by calculating the additive epsilon indicator Iε+. This is the

minimum distance that a solution set needs to be translated to just weakly dominate another set,

with positive numbers signifying a shift towards the ideal point (Figure 2.14) [97] (a more detailed

explanation of Iε+ is given in Section 3.3.4). By minimizing Iε+, a candidate point which pushes

out the Pareto front can be selected for evaluation.


Figure 2.13: Hypervolume calculation

Figure 2.14: Additive epsilon indicator (Iε+) calculation

4. MSPOT [94]: This conducts an NSGA-II multi-objective optimization directly on the model pre-

diction using multiple surrogate models fit to each objective, resulting in a Pareto set of viable

points for evaluation in the next iteration. The single point for evaluation is chosen based on it’s

hypervolume contribution.

2.4.3 Dimensionality Reduction

Dimensionality reduction is the task of reducing the the degrees of freedom (number of dimensional

parameters being optimized) an optimization algorithm needs to consider in an effort to mitigate the

curse of dimensionality [70]. In addition to optimization speedup, dimensionality reduction can result in

better solutions as unimportant variables that add noise to the optimizer’s model of the objective space

(i.e., over-fitting) are removed. In general, it can be achieved through: 1) feature extraction, where m

new variables are generated from the original n variables with m < n, or 2) feature selection, where a

subset of m variables are chosen from the original n variables, or 3) a combination of feature extraction

and selection. This work utilizes feature selection, as the physical meaning of the variables are retained

- a fact that proves useful in an engineering context.


Table 2.1: Taxonomy of multi-objective EGO algorithms used in present work

Algorithm Initial Design Model FittingCandidateGeneration

CandidateSelection

StoppingDecision

ParEGO

LHS

Model ofscalarization

SOO of EIF1 point,generation criterion Budget

SMS-EGO Individualmodels foreach objective

SOO ofhypervolume

ε-EGOSOO ofε indicator

MSPOTMOO of modelpredictions

1 point,hypervolume contribution

There are three types of feature selection methods:

1. Filter: Ranks the variables or subsets of variables by a directly calculated metric, and is very quick

[78].

2. Wrapper: This method is more applicable in the machine learning context. The metric used

to evaluate a given subset is determined by the accuracy of a trained model, generally leading to

effective dimensionality reduction at the cost of the resulting subset’s performance being specifically

tied to the machine learning algorithm used [78].

3. Hybrid: Combination of filter and wrapper methods [78].

Filter methods are well suited for dimensionality reduction in optimization as they are not tied to any

specific algorithm’s ability to learn underlying structures. Correlation-based feature selection (CFS) is

a well-established metric for the filter method that aims to choose variables with a high correlation to

the desired output (transmission loss of the muffler, in this case) but have low correlation with other

variables, reducing redundancy in the input variable subset [39]. A variety of methods and heuristics

exist for the selection of the m-variable subset including best first [72], greedy forward selection [49], and

genetic algorithms. However, as the number of variables are relatively low in the present case (n = 11),

an exhaustive search was conducted, looking through every possible subset combination (211 = 2048

combinations). In summary, dimensionality reduction prior to optimization was conducted using a

filter-based feature selection using CFS to evaluate an exhaustive list of all possible variable subsets.

2.5 Muffler Optimization

Engineering design optimization generally falls under two categories: topology optimization and shape

(parameter) optimization. The former assumes very little about the design and the process is tasked

with generating the details of the design - in the case of a muffler, the inputs would just be the bounding

dimensions of the muffler and the algorithm would be responsible for filling in all the internal compo-

nents such as baffles and tubes. On the other hand, shape optimization involves the tuning of a set

of parameters that control the dimensions or properties of components within a predetermined design.

Topology optimization has the potential of generating radically new designs, but its prevalence is limited


in the field of muffler optimization. Due to the general nature of this method, its designs are limited to

simple mufflers such as simple expansion chambers [90], expansion chambers with offset inlet and outlet

[54], and dual expansion chambers [55, 26, 53].

The bulk of the existing work on muffler optimization involves shape optimization. Yeh et al. initiated

this work on a simple expansion chamber with an offset inlet and outlet configuration [93], followed by a

multi-chamber expansion chamber [92, 91], all using 1D TMM models. While [93] used algorithms such

as exterior penalty, interior penalty, and method of feasible directions with success, [92] and [91] used

genetic algorithms (GA). Barbieri and Barbieri focused on 2D FEM-based optimization of an expansion

chamber with extended inlet and outlet using the method of feasible directions [7]; this algorithm doesn’t

see much use in later literature. Chiu and Chang used GA in the optimization of a multi-chambered

muffler with perforated tubes, reverting back to 1D TMM [22]. They expanded the work by consider-

ing pressure drop as a constraint (calculated via empirical relationships) while employing the simulated

annealing (SA) algorithm in the optimization of 1D TMM models of perforated mufflers [23]. SA based

optimization of a plug flow muffler using 1D TMM without considering pressure drop is conducted in

[17]. Much of the work following this included perforated elements in the mufflers, modelled with 1D

techniques, relying on GA [19, 81] or SA [18, 20]. Huang et al.’s work was the first to consider pres-

sure drop in a truly multi-objective fashion, using optimization by design of experiments (DOE) [41].

Siano et al. continued using pressure drop in a multi-objective setting, utilizing genetic algorithms for

optimization [81]. Additionally, single-objective muffler optimization has incorporated pressure drop via

scalarization (weighted sum of acoustic and pressure drop objectives) [80, 79]. Barbieri et al. introduced

the use of particle swarm optimization (PSO) using a 2D FEM model of a simple expansion chamber

with extended inlet and outlet [8].

Analyzing recent literature at large (92 papers with ‘muffler optimization’ since 2003) shows a few

striking observations:

1. Majority of the work (70.7%) used SA or GA. These methods use anywhere from 2000 function

evaluations [23] to 64000 function evaluations [22] for SA and GA, respectively.

2. Majority of the work (81.5%) used 1D TMM to model the muffler [e.g., 96, 71, 48, 21]. This is

tied to observation #1, as SA and GA require a large number of function evaluations in order to

be successful, necessitating low-fidelity 1D models for the sake of computational time.

3. Multi-objective work considering TL and pressure drop as two distinct objectives is scarce (3.3%).

This is a crucial consideration as high back-pressure on engines lead to loss of performance and

efficiency.

Sections 3.2 & 4.2 address the first observation by using a much simpler algorithm (L-BFGS-B) in order

to optimize TL, removing the complexity associated with implementing algorithms such as GA or SA.

The latter two observations are addressed in Sections 3.3 & 4.3 by using a truly multi-objective method

capable of optimization using a relatively small number of function evaluations (EGO).


2.6 Chapter Summary

This chapter provided the theoretical foundations for various aspects of the design of a muffler. Section

2.1 detailed the prediction of turbulence-induced noise in a muffler. Section 2.2 covered the analysis of the

plug flow muffler’s acoustic performance and Section 2.3 presented a means of characterizing the pressure

drop across the muffler. Finally, Sections 2.4 and 2.5 addressed methods related to the automated design

of an optimal muffler and surveyed the current muffler optimization literature, respectively.

Chapter 3

Methodology

3.1 Aeroacoustic Predictions

The present work was an introductory investigation into using the acoustic analogy (Section 2.1.1) to

predict the SPL caused by turbulence-induced noise at specific receiver locations. NASA’s tandem

cylinder benchmark for aeroacoustic noise [60, 59] was replicated using incompressible computational

fluid dynamics (CFD) using three established methods for time-resolved simulation: unsteady Reynolds-

averaged Navier-Stokes (URANS), large eddy simulation (LES), and detached eddy simulation (DES).

The tandem cylinder experiment was conducted at NASA’s Quiet Flow Facility (QFF) - an open jet

wind tunnel in an anechoic room. Two cylinders (C1 and C2) of equal diameter (D = 0.057 15 m) with

a 16 D span were placed in oncoming flow with a uniform flow velocity (Uo = 43.4 m/s;ReD = 166, 000),

one downstream of the other, separated by a center-to-center distance L = 3.7 D (Figure 3.1). The aeroa-

coustic noise created through turbulent vortex shedding was measured by an array of microphones in the

far-field. Three microphones, A, B, and C, were considered in this work, located at (−8.33 D, 27.815 D),

(9.11 D, 32.49 D), and (26.55 D, 27.815 D), respectively, from the centroid of C1.

In the present work, the tandem cylinder experiment was simulated using ANSYS Fluent 17.1 on an

8-core computer with three methods, URANS, LES, and DES, in order to determine the accuracy of each

method, while considering the computational cost of each. The setup parameters are shown in Table

3.1. The LES was a low-fidelity simulation due to the coarse span-wise resolution. The three meshes

used are pictured in Figure 3.2. The DES mesh has RANS-based meshing (high aspect ratio) near the

walls and transitions into an LES-based mesh in the ”focus region” as defined by Spalart [83].

3.1.1 Simulation Verification

The accuracy of the simulations was verified through the comparison of various hydrodynamic values

and statistics. Table 3.2 shows the list of flow parameters that were compared against experimental

values for URANS, LES, and DES.

Section 3.2 was previously published in J. Puthuparampil, H. Pong, and P. Sullivan, Modelling and Optimization of PlugFlow Mufflers in Emission Control Systems, in SAE 2017 Noise and Vibration Conference and Exhibition, 2017. [71].

24

Chapter 3. Methodology 25

Figure 3.1: Tandem cylinder setup, with microphone locations (not to scale)

Table 3.1: CFD setup overview

Mesh Setup URANS LES DES

Mesh Type Planar, extruded Planar, extruded Planar, extrudedFirst Layer Thickness 100 y+ 57 y+ 1 y+Num. of Elements (Plane) 15,292 81,609 88,842Span 16 D (33 elements) 16 D (120 elements) 3 D (120 elements) †

Span Resolution [mm] 27.7 7.62 1.42875Total Num. of Elements 504,636 9,793,080 10,661,040

Modelling Setup

Turbulence Model k-ε Dynamic Smagorinsky-Lilly SST k-ωP-V Coupling SIMPLE Fractional Step (NITA) Fractional step (NITA)Momentum Discretization 2nd order upwind Bounded central difference Bounded central differencek Discretization 2nd order upwind – 2nd order upwindε Discretization 2nd order upwind – –ω Discretization – – 2nd order upwindPressure Discretization 2nd order PRESTO 2nd orderTransient Discretization 2nd order implicit 2nd order implicit Bounded 2nd order implicitTime Step [s] 2.5× 10−5 1.5× 10−5 1.5× 10−5

Time Sampled [s] 0.30 0.15 0.2658Num. of Flow-Through 9.47 4.75 8.06

Boundary Conditions

Inlet Velocity [m/s] (43.4, 0, 0) (43.4, 0, 0) (43.4, 0, 0)Pressure Outlet [gauge Pa] 0 0 0Cylinder Surfaces No-slip No-slip No-slipTop and Bottom Free-slip Free-slip Free-slipSides (Span-wise) No-slip No-slip Periodic

† Full-span noise contribution estimated using long-span correction [76]


Figure 3.2: Tandem cylinder meshes - URANS, LES, and DES, respectively

Table 3.2: Hydrodynamic metrics used to compare simulation to experimental data

Parameter Description URANS LES DES

Cp Coefficient of pressure (Cp) on the surface of cylinders Yes Yes YesCp′ RMS Root-mean-square of Cp on the surface of cylinders No Yes Yesu Mean stream-wise velocity along y = 0 Yes Yes Yes2D TKE Contour plots of 2D turbulent kinetic energy (TKE) No Yes YesPSD Power spectral density of pressure at 135° on C1 and 45° on C2 No No YesRp Span-wise correlation of pressure at 135° on both cylinders † No No Yes

† Span-wise correlation results in Appendix B.2


Figure 3.3: Cross-section of a plug flow muffler and associated dimensional parameters

3.1.2 Receiver Noise Calculation - Acoustic Analogy

Fluent implements the full FWH analogy, but since the cylinders were stationary in flow, the FWH

analogy reduced to the Curle analogy. Furthermore, dipole sources were dominant as periodic flow sepa-

ration through vortex shedding was the primary flow characteristic (Section 2.1.1). Thus, the quadrupole

contributions were not considered, simplifying the computation.

The cylinder surfaces were used as noise sources to estimate the time-resolved sound pressures at mics A,

B, and C. The sound pressure at each mic was transformed into the frequency domain with a fast Fourier

transform (FFT) using a Hamming window with a 0.50 overlap (10 Hz resolution for URANS, 15 Hz for

LES, and 10 Hz for DES). The pressure levels were then converted to decibels using the standard pressure

reference value of pref = 2× 10−5 Pa, arriving at SPL (dB) at a range of frequencies.

3.2 Single-Objective Optimization (SOO)

In general, the plug flow muffler can be fully defined by 14 design parameters, shown in Figure 3.3

and Table 3.3. Prior to optimization, the TMM and FE models were evaluated against a baseline plug

flow muffler geometry presented by Wu et al. [89]; its dimensions are seen in Table 3.3. TMM and FE

were conducted with all three zero mean flow impedance models (Section 2.2.4), and the accuracy of

each was evaluated against the experimental data [89]. The best performing TMM impedance model

was used in the TMM-based SOO, conducted at room temperature (To = 22 ◦C, ρo = 1.1965 kg/m3,

co = 344.82 m/s, νo = 1.5295× 10−5 m2/s). The final optimized design’s TL prediction was then verified

using the FE model. The general methodology for SOO is seen in Figure 3.4.


Table 3.3: Parameters defining the PFM and associated values for the Wu et al. muffler [89]

Parameter Description Wu et al. Muffler [89]

D1 Inner diameter 0.0508 mD2 Outer diameter 0.1076 mLa1 Length of first solid tube section (expansion section) 0.0317 mL1 Length of perforated section (expansion section) 0.0952 mLb1 Length of second solid tube section (expansion section) 0.0255 mLa2 Length of first solid tube section (contraction section) 0.0063 mL2 Length of perforated section (contraction section) 0.0952 mLb2 Length of second solid tube section (contraction section) 0.0953 mσ1 Area porosity of perforated expansion tube 21.68 %σ2 Area porosity of perforated contraction tube 13.54 %Tp1 Thickness of perforated expansion tube 1.1938 mmTp2 Thickness of perforated contraction tube 1.1938 mmDh1 Perforate hole diameter of expansion tube 6.35 mmDh2 Perforate hole diameter of contraction tube 6.35 mm

Figure 3.4: Single-objective optimization (SOO) methodology: evaluation and optimization


3.2.1 TMM Setup

The TMM for a plug flow muffler (Section 2.2.1) was implemented in Python 2.7 utilizing the numpy

module for matrix operations. The Sullivan, Bento Coelho, and Bauer zero mean flow impedance models

(Section 2.2.4) were implemented on the perforated surfaces. TMM is only accurate below the cutoff

frequency (Equation 2.21); D2 was held as a constant in the optimization process at the same value as

the Wu et al. muffler, leading to a cutoff frequency of 1877 Hz for all mufflers.

3.2.2 FE Setup

The FE simulation was conducted through COMSOL Multiphysics 5.2a using the Acoustics Module with

a parametric frequency sweep study. The mufflers were implemented with both two-dimensional axisym-

metric (2DA) and three-dimensional (3D) geometries. First, the muffler was parametrically modelled.

Plane wave boundary conditions were applied at the inlet and outlet, along with hard wall boundary

conditions at all relevant surfaces. The perforated boundaries were defined via custom functions for the

Sullivan, Bento Coelho, and Bauer impedance models (Section 2.2.4). An incident pressure wave of 1

Pa amplitude was applied normal to the inlet as the initial condition. A free triangular mesh (for 2DA)

and a tetrahedral mesh (for 3D) were used, ensuring the maximum element size was limited by

Max. Size =co

10 · fmax(3.1)

where fmax is the maximum frequency of interest in the simulation. Equation 3.1 ensured that the

model always had enough spatial resolution to resolve the smallest wavelength under consideration [3].

TL solutions were obtained for a parametric frequency sweep from 10 Hz to 4000 Hz (fmax = 4000 Hz),

with a resolution of 10 Hz.

In the SOO process, FE simulations are only used to check the accuracy of the TL predictions pro-

duced by TMM; FE is not used in the actual optimization.

3.2.3 SOO Setup

Objective Function

The objective function used, noise level objective (NLO), was defined as

NLO =∑i

|ESLi − TLi|wi (3.2)

for i = 63, 125, 250, 500, 1000, 2000, 4000 Hz

where i denotes the set of octave band center frequencies of interest, ESLi is the excess sound level

at a given octave band, TLi is the averaged muffler transmission loss over the octave band under

consideration, and wi is a weight applied to each octave band that determines the band’s contribution

to the NLO. ESLi is

ESLi = ENi −NCi (3.3)

where ENi is the measured engine noise (sound pressure level at 1 m) in decibels and NCi is the noise

criterion standard required for the application [10]. The difference between the two quantities provides


Table 3.4: Octave band limits and associated engine noise, noise criterion, and NLO weights for SOO

Octave Band [Hz] Lower Limit [Hz] Upper Limit [Hz] EN [dB] NC-60 [dB] wi

63 40 90 98.3 77 0.2125 90 180 107.1 71 0.2250 180 360 102.5 67 0.2500 360 710 102.5 63 0.18751000 710 1420 104.0 61 0.15252000 1420 2840 100.8 59 0.0554000 2840 4000 96.5 58 0.005

a suitable TL target for the design. This work used the NC-60 limits as the desired noise criterion; the

maximum permissible output noise level for each octave band is provided in Table 3.4, along with the

measured sound pressure level (ENi) of a CAT C27 750 kW generator at 1 meter. It is important to

note that Equation 3.2 uses transmission loss where insertion loss would be a more technically correct

value. However, insertion loss requires knowledge of the engine (source) impedance, muffler outlet (ter-

mination) impedance, and the impedance of any ducting used between the engine and the muffler. Since

transmission loss and insertion loss are correlated, the use of transmission loss in Equation 3.2 has no

bearing on the optimization methodology described.

The muffler’s average transmission loss over an octave band (TLi) is

TLi = 10 log10

Upper Limit of i∑

j=Lower Limit of i

10TLj/10

ni

(3.4)

where ni is the number of elements summed for a given octave band. The upper and lower frequency

limits for a given octave band is listed in Table 3.4. The frequency limits are rounded to the nearest

10 Hz, as the solution is calculated with a 10 Hz resolution over the frequency range from 10 Hz to

4000 Hz. Note that Equation 3.4 simply calculates an average of the decibel values of TL.

The NLO weights (wi) in Table 3.4 were chosen (a) to prioritize the frequency ranges where a reactive

muffler is traditionally used as an effective method of noise attenuation and (b) to account for the

inherent inaccuracies that occur above the cutoff frequency (fc = 1877 Hz). The higher order modes

that occur above the cutoff frequency are not included in TMM, but the TL contribution of the plane

wave mode that occurs in the high frequency regime is included; thus, the overall TL predicted in the

high-frequency range is useful even though not entirely accurate, leading to non-zero weights in these

octave bands.

Implementation

The L-BFGS-B algorithm (Section 2.4.1) from the scipy.optimize module [47] was used for the SOO, aim-

ing to minimize Equation 3.2. The algorithm was implemented in a randomized multi-start framework

[28] with parallel computation accomplished using the pp module. The dimensional variables defining the

PFM were optimized subject to the bounds and constraints seen in Table 3.5. The limits were informed


Table 3.5: Bounds and constraints applied on SOO

Parameter Lower Bound Upper Bound

rD = D1/D2 0.4 0.8La1 [m] 0.0 0.15L1 [m] 0.05 0.25Lb1 [m] 0.0 0.15σ1 0.14 (14 %) 0.25 (25 %)

Constraint Equality

La2 La1

L2 L1

Lb2 Lb1σ2 σ1

by physical and manufacturing constraints. The perforated expansion and perforated contraction were

defined as symmetrical using equality constraints, to simplify the manufacturing process. The variables

D2, Tp1, Tp2, Dh1, and Dh1 were held constant as they were seen to have a comparatively low effect

on the TL in a sensitivity study. Therefore, these variables were set to their respective values from the

baseline Wu et al. muffler [89].

The optimization was performed 2000 times in parallel on a 4-core Intel i7-4790 processor. Each ini-

tialization of the optimization algorithm began at random values between the prescribed limits in Table

3.5, striving towards a minima, terminating when

NLOk−1 −NLOk

max(NLOk,NLOk−1, 1.0)< 1× 10−4 (3.5)

where NLO (Equation 3.2) is evaluated at the current iteration, k, and the previous iteration, k − 1.

Research Objective

The SOO work primarily aimed to answer the following objective: can L-BFGS-B, a relatively sim-

ple algorithm, be used in place of complex stochastic algorithms such genetic algorithm or simulated

annealing to successfully conduct muffler optimization?

3.3 Multi-Objective Optimization (MOO)

In the MOO process, an FE simulation was used for the TL prediction, removing any limitation as-

sociated with the TMM cutoff frequency. Firstly, the 14 variables defining the plug flow muffler were

recast to those presented in Table 3.6 (Figure 3.3 for visual reference). Lt and D2 were held con-

stant at 17 inches and 12 inches respectively, as these bounding dimensions are generally prescribed by

space limitations. Since larger mufflers are generally more effective, a good muffler design will always

take advantage of all the space available. The importance of the remaining 12 variables in predicting

the acoustic performance of a PFM was discovered using feature selection; the most important subset

of variables were used in the MOO process, with the other variables held constant at their nominal values.


Table 3.6: Geometric parameters defining the PFM, recast for multi-objective optimization as ratios

Paremeter Description

Lt Total lengthD2 Outer diameterrD = D1/D2 Diameter ratiorexp = Lexp/Lt Expansion chamber length ratiorL1 = L1/Lexp Expansion perforated tube length ratiorL2 = L2/(Lt − Lexp) = L2/Lcon Contraction perforated tube length ratior1 = La1/(Lexp − L1) Expansion solid tube length ratior2 = La2/Lcon − L2 Contraction solid tube length ratioσ1 Area porosity of perforated expansion tubeσ2 Area porosity of perforated contraction tubeTp1 Thickness of perforated expansion tubeTp2 Thickness of perforated contraction tubeDh1 Perforate hole diameter of expansion tubeDh2 Perforate hole diameter of contraction tube

Figure 3.5: Multi-objective optimization (MOO) methodology: evaluation and optimization

The pressure drop of the muffler was estimated to provide a competing objective to TL in MOO pro-

cess. The empirical model was validated through CFD before its use. The optimization was conducted

at typical operating temperatures for a muffler (To = 400 ◦C, ρo = 0.5245 kg/m3, co = 520.07 m/s,

νo = 6.1961× 10−5 m2/s) and a typical operating flow rate (Q = 800 ACFM). The MOO algorithm

aimed to simultaneously optimize two aforementioned objectives, resulting in a Pareto front showing the

trade-off between TLO and DP for the PFM. The MOO methodology is depicted in Figure 3.5.

3.3.1 FE Setup

The FE simulation was set up in a similar manner as in Section 3.2. A few key differences are:

• the simulation’s frequency range was from 80 Hz to 4470 Hz with a resolution of 10 Hz

• only a 2DA model was implemented, as any inaccuracies caused by neglecting 3D modes are

minimal in the frequency range considered


Figure 3.6: Dimensionality reduction through feature selection methodology

• only the Elnady perforate impedance model (Section 2.2.4, Equation 2.29) was used, as it is able

to account for the effects of flow through the muffler

3.3.2 Pressure Drop Model

The empirical pressure drop model (Section 2.3) for the PFM was implemented in in Python 2.7. The

output of the calculation was converted from Pa to inH2O as it is a more industrially relevant unit of

measure. A model, rather than a CFD simulation, was used to calculate pressure drop due to the large

savings in computational time. However, the model was first checked against a 2DA steady state RANS

CFD conducted in Fluent for 6 random PFM geometries. The perforated plate was implemented in

Fluent as a porous surface using tabulated discharge coefficients [82]. Once validated, the model was

used in the MOO.

3.3.3 Dimensionality Reduction - Feature Selection

To determine the importance of each of the 12 variables from Table 3.6 (excluding Lt and D2 as they are

held constant), dimensionality reduction through feature selection was performed. 2400 (= 200·nvariables)

randomly generated mufflers (i.e., 2400 randomized sets of the 12 geometric parameters) were evaluated

via a 2DA FE simulation. The TL curve for each muffler was transformed into a single-number rep-

resentation, TLO (Equation 3.6). 10 different TLO were calculated for each muffler using 10 different

randomly generated sets of weighting factors (wi) from Equation 3.6 to generalize the findings of the

feature selection process.

The feature selection process (Section 2.4.3) was conducted using an exhaustive search of all subsets

of the 12 variables ranked through the correlation-based feature selection (CFS) metric, implemented in

Weka 3.8 [38]. For a given wi, 10-fold cross-validated feature selection was performed, which resulted

in a set of percentages representing how often each variable was chosen in the 10-fold cross-validation;

these values are associated with the importance of each variable. This was repeated for all 10 wi and

the results were averaged, to arrive at a generalized and statistically robust subset of most important

variables to be used in optimization. The dimensionality reduction methodology is pictured in Figure

3.6.


Table 3.7: 1/3 octave band limits and TLO weights for MOO

1/3 Octave BandTLO Weight (wi)

Band Number Lower Limit [Hz] Center Frequency [Hz] Upper Limit [Hz]

9 89 100 112 0.00759387310 112 125 141 0.00759387311 141 160 178 0.01012516412 178 200 224 0.01518774613 224 250 282 0.02278161914 282 315 355 0.03290678415 355 400 447 0.04556323916 447 500 562 0.06075098517 562 630 708 0.07847002318 708 800 891 0.09872035119 891 1000 1122 0.13668971720 1122 1250 1413 0.15187746321 1413 1600 1778 0.14276481522 1778 2000 2239 0.10033809823 2239 2500 2818 0.05807544424 2818 3150 3548 0.02296693325 3548 4000 4467 0.007593873

3.3.4 MOO Setup

Objective Functions

The acoustic objective function used, transmission loss objective (TLO), was defined as

TLO = −25∑i=9

TLi · wi (3.6)

where i denotes the 1/3 octave band number from 100 Hz (band number 9) to 4000 Hz (band number

25), TLi is the averaged transmission loss over the 1/3 octave band, and wi is the weight applied to each

1/3 octave band. The negation allows the minimization of Equation 3.6 to be associated with higher

acoustic performance. TLi is calculated using Equation 3.4, but with 1/3 octave band resolution using

the upper and lower frequency limits shown in Table 3.7 (TL calculated by FE at a resolution of 10 Hz

is interpolated to a 1 Hz resolution to match the 1/3 octave band limits closely). The weights used are

also listed in Table 3.7. The weights were chosen to maximize the PFM’s performance between 891 Hz

and 1778 Hz (bands 19 through 21); however, any set of arbitrary weights summing to 1.0 is sufficient.

The pressure drop objective function (DP) was calculated in units of inH2O. As the pressure drop

model is empirically derived, it should not be considered valid outside the range of experimental pa-

rameters. Thus, linear constraints were introduced to limit the OAR as per Equations 2.36 and 2.37:

0.3 < OARexp < 2.2 and 0.31 < OARcon < 1.66.

The complete set of variable bounds and constraints used in MOO is seen in Table 3.8. Based on

the results of feature selection, the less important variables were be held constant at their nominal

values during the MOO.


Table 3.8: Bounds and constraints applied on MOO

Parameter Lower Bound Nominal Value Upper Bound

Lt [in] – 17.0 –D2 [in] – 12.0 –rD 0.1 0.5 0.9rexp 0.1 0.5 0.9rL1 0.1 0.5 0.9rL2 0.1 0.5 0.9r1 0.1 0.6 † 0.9r2 0.1 0.4 † 0.9σ1‡ 0.1 (10 %) 0.5 0.9 (40 %)

σ2‡ 0.1 (10 %) 0.5 0.9 (40 %)

Tp1 [mm] 0.5 1.1938 2.0Tp2 [mm] 0.5 1.1938 2.0Dh1 [mm] 2.0 5.0 8.0Dh2 [mm] 2.0 5.0 8.0

Constraint Lower Bound Nominal Value Upper Bound

OARexp 0.3 – 2.2OARcon 0.31 – 1.66

† Asymmetric r1 and r2 shown to boost PFM performancein a sensitivity study; peak values chosen as nominal‡ 10 % to 40 % porosity linearly mapped from 0.1 to 0.9

Implementation

The EGO algorithm (Section 2.4.2) from the mlrMBO package for the statistical programming language

R was used in MOO [12]. All 4 EGO variants (SMS-EGO, ε-EGO, MSPOT, and ParEGO) are available

in mlrMBO. In addition, the NSGA-II algorithm (Section 2.4.2) from the mco package was used for GA-

based MOO (MSPOT and NSGA-II). A constant set of initial points (Ninit = 40) was used for all EGO

runs to remove any difference in performance associated with initial conditions. A maximum evaluation

budget Neval was used as the stopping criterion for all MOO; the baseline value for Neval = 240, which

is in addition to Ninit. Each method was repeated n = 8 times to produce statistics for the results.

Research Objectives & Metrics Used

The MOO work aimed to address the following objectives:

1. Which EGO variant performs best for multi-objective muffler optimization?

• Comparison of ParEGO, SMS-EGO, ε-EGO, and MSPOT

2. How sensitive is the EGO algorithm to Nevals?

• Comparison of Nevals = 240, 300, 360

3. How sensitive is the EGO algorithm to different initial conditions?

• Comparison of four randomly generated LHS of size (Ninit = 40) labeled A, B, C, and D

4. How does EGO compare to the NSGA-II algorithm (standard algorithm for MOO)?


The Pareto front approximation resulting from each method is visualized using an attainment surface

[50]. An attainment surface is a single summary curve of the n = 8 repetitions of a given optimization

method. This work utilizes the median attainment surface for all visualizations - the 4th attainment sur-

face (p = 4) when n = 8. The median attainment surface is obtained by (1) querying the n Pareto front

approximations at regular intervals, (2) choosing the Pareto front approximation that weakly dominates

at least n− p+ 1 = 5 approximations at each queried position.

Pareto front approximations are numerically compared using two different metrics - hypervolume and

the additive binary epsilon indicator. Together, they provide a holistic look at the quality of a Pareto

front [51]:

1. Hypervolume (IHV (A)): The hypervolume is the volume (or area in bi-objective optimization)

encompassed by the Pareto front approximation, with reference to a prescribed point (Figure

2.13). A larger value indicates a front of higher quality. However, it should be noted that different

fronts can have the same hypervolume; thus, other metrics are also necessary for a more complete

understanding of the quality of a Pareto front approximation. The reference point chosen was

(TLO = 0,DP = 10).

2. Additive binary epsilon indicator (Iε+(A,B)): This metric, being a binary indicator, compares two

Pareto front approximations (A,B) and produces a value IA. IA is the minimum distance that

all the points in A must be translated such that A weakly dominates B (i.e., A is equal or better

than B across the entire front). A translation towards the ideal point is positive. Translations are

conducted independently in each objective’s direction, and the maximum value is IA (Figure 2.14).

The reverse, (Iε+(B,A)), produces IB . The lower numerical value between IA and IB indicates the

better front, as it needs to be shifted less to weakly dominate the other set. If (IA ≤ 0, IB > 0),

then A fully dominates B (i.e., A is strictly better).

To answer the objectives in a statistical fashion, IHV and Iε+ from multiple runs of one method was

compared against another method using the Mann-Whitney U (MWU) test. Details are seen in Appendix

A.

3.4 Chapter Summary

This chapter provided the methodology pertaining to the acoustic analysis and optimization of a plug

flow muffler. Section 3.1 presented a means of predicting turbulence-induced noise using the aeroacoustic

analogy. Section 3.2 detailed the single-objective optimization process where a relatively simple gradient-

based algorithm was used in a multi-start framework to optimize the acoustic performance of a plug

flow muffler. Lastly, Section 3.3 covers an efficient multi-objective optimization of acoustics and pressure

drop across a muffler, utilizing an FE simulation to calculate the acoustic objective function.

Chapter 4

Results and Discussion

4.1 Aeroacoustic Predictions

The tandem cylinder experiment (Section 3.1) was recreated using CFD as the first steps in exploring

turbulence-induced noise produced within mufflers. This section first evaluates the ability of different

low computational-cost CFD methods to accurately depict the flow observed in the experiment, followed

by their ability to recreate the measured noise. The acoustic predictions are then compared against

those from literature.

4.1.1 Simulation Verification - Hydrodynamic Results

The mean stream-wise velocity and flow streamlines around the tandem cylinders are seen in Figure

4.1. Like a single cylinder in cross-flow, vortex shedding occurs off C1 (front cylinder). The turbulent

structures associated with vortex shedding are then convected downstream and impinge on C2 (rear

cylinder). Two pairs of counter-rotating recirculation cells are observed, one behind each cylinder. The

recirculation cells behind C1 are larger and promote a significant amount of flow in the y-direction as

shown by the streamlines, leading to a high level of turbulent energy between the two cylinders.

The accuracy of each simulation was verified using a set of hydrodynamic parameters (Table 3.2). The

results for URANS are span-averaged, while the results for LES and DES are both span and time-

averaged, when applicable. All results were captured after initial transience in the flow exited the

domain and a periodic behaviour was observed in the coefficients of lift and drag of the cylinders.

Coefficient of Pressure

Figure 4.2 shows the mean coefficient of pressure (Cp) on the surface of each cylinder. The QFF data

shows excellent symmetry along the y = 0 plane. On C1, a stagnation point is seen at 0°, and Cp reduces

as the flow speeds up around the curvature of the cylinder. Figure 4.2 shows a region of flow separation

on C1 between 110° and 250°. The separated flow impinges on C2, reattaching at 0°. The Cp profile

on C2 is similar to C1, but with lower magnitudes of pressure. The separation region on C2 extends

between 130° and 230°. The smaller angular span of the separation region is associated with the smaller

pair of counter-rotating recirculation cells seen behind C2, as compared to C1 (Figure 4.1).

37

Chapter 4. Results and Discussion 38

−1

−0.5

0

0.5

1

0 1 2 3 4 5 6

y/D

x/D

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

u/Uo

Figure 4.1: Mean x-velocity with streamlines from the DES simulation

In general, the experimental trends are captured by all methods. There is excellent agreement ob-

served between the LES and DES predictions for both cylinders. The LES and DES Cp match the QFF

experiment for both C1 and C2, except for the separated zone behind C2 - LES and DES over-predict

the amount of suction. URANS over-predicts the peak suction location and magnitude, and under-

predicts the Cp in the separation region on C1. However, it is able to quite accurately determine the Cp

on C2. The asymmetry seen in the URANS results for C2 is associated with a lack of time-averaged data.

Figure 4.3 shows the root mean square of the pressure fluctuations (Cp′) on the surface of each cylinder.

As with Cp, Cp′ RMS is also symmetric about y = 0. On C1, two small peaks are seen in the QFF data

at 90° and 270°, and an increased level is seen in between these angles. These fluctuations are associated

with the unsteadiness in the separation points, as well as interactions between C1’s surface and the

vortex shedding that occurs [44]. The front surface of C2 sees the highest levels of Cp′ RMS due to the

impingement of the separated turbulent shear layer from C1. The fluctuation magnitude reduces on the

downstream side of C2, and reaches approximately the same levels as C1 at 180°, within the separated

region.

DES successfully predicts the Cp′ RMS on C1, except for the over-predictions at the 90° and 270°peaks. LES shows significant over-prediction for all of C1. The C2 profile is much more complex, but

LES and DES are able to predict the overall levels well and capture most of the variation on the surface

of C2 with reasonable accuracy, well within expected engineering needs.

Mean Velocity

Figure 4.4 shows the mean velocity in the x-direction (u) normalized by the Uo, along the y = 0 plane.

The negative velocity in the QFF data from 0.5 < x/D < 1.6 occurs due to two counter-rotating recircu-

lation cells visible in Figure 4.1. The stream-wise velocity then recovers to approximately 0.4Uo before

impinging onto the second cylinder at x/D = 3.2. The smaller pair of counter-rotative recirculation cells

lead to the the negative stream-wise between 4.2 < x/D < 4.6. The flow then regains momentum as it

exits the domain.


−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

0 45 90 135 180 225 270 315 360

Cp

Cylinder 1, θ [°]

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

0 45 90 135 180 225 270 315 360

Cp

Cylinder 2, θ [°]

QFFURANS

LESDES

Figure 4.2: Cp along the surface of the cylinders

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 45 90 135 180 225 270 315 360

Cp′

RM

S

Cylinder 1, θ [°]

QFFLESDES

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 45 90 135 180 225 270 315 360

Cp′

RM

S

Cylinder 2, θ [°]

Figure 4.3: Cp′ RMS along the surface of the cylinders

−0.4

−0.2

0

0.2

0.4

0.6

0.5 1 1.5 2 2.5 3

u/U

o

x/D

QFFURANS

LESDES

−0.4

−0.2

0

0.2

0.4

0.6

4.5 5 5.5 6 6.5

u/U

o

x/D

Figure 4.4: u along y = 0 in gap between the two cylinders, and after second cylinder


The general trend of the velocity in the gap region is captured by all three methods. However, URANS

significantly under-predicts the amplitude of the velocity. LES shows inaccuracies in predicting the

recirculation region in the gap, but is successful in the recovery zone prior to the second cylinder. In

contrast, the DES simulation predicts the entire gap region’s stream-wise velocity with high accuracy.

LES is the only method that predicts the velocity after the second cylinder with good accuracy; however,

it over-predicts the velocity recovery after x/D = 4.8. Both URANS and DES are able to predict the

general trends, but not the absolute values.

2D Turbulent Kinetic Energy (TKE)

The 2D turbulent kinetic energy (TKE) is calculated as

2D TKE =1

2

(u′u′ + v′v′

)/U2

o (4.1)

where u′ and v′ are the fluctuating components of the x and y velocities respectively (U = u + u′ and

V = v + v′ ), and Uo is the inlet velocity. TKE represents the energy of turbulent fluctuations per

unit mass. Figure 4.5a is the experimentally measured 2D TKE. It shows a maxima between the two

cylinders, centered approximately where the two counter-rotating cells meet (x/D ≈ 1.7 as seen from

Figure 4.1). Smaller peaks of TKE are seen just upstream and downstream of C2; the upstream TKE is

associated with turbulent structures from C1 passing around C2 [45], and the downstream peak is most

probably a combination of the convected structures from C1 and any additional structures created at C2.

Figures 4.5b and 4.5c show the 2D TKE predictions from the LES and DES simulations respectively.

Both simulations are able to capture the general trends. However, LES struggles with the TKE around

C2; it under-predicts the upstream section of C2 and over-predicts the wake TKE. In contrast, DES

over-predicts the TKE maxima between C1 and C2 while having much higher accuracy upstream and

downstream of C2. These trends are further examined by the line extractions presented in Appendix

B.1.

Power Spectral Density (PSD) of Pressure

The power spectral density (PSD) of surface pressure at 135° on C1 and 45° on C2 are shown in Figure

4.6. A sharp peak is present at 178 Hz on C1 due to the vortex shedding. This peak is also seen on

C2, as the point of measurement at 45° lies in the path of the impinging vortex. Additionally, a smaller

peak is seen at the second harmonic (356 Hz) on on C1 and very faintly on C2. Furthermore, it should

be noted that the broadband “hump” seen between 500 Hz < f < 2000 Hz on C1 is attributed to the

trip wire used in the experiment to induce turbulent behavior [59]. Additionally, the energy on C2 is

significantly higher than C1 due to the separated turbulent shear layer from C1 impinging on C2 [44].

The DES simulation is able to accurately predict the PSD on both cylinders with a high accuracy

(Figure 4.6). It captures all features of the experimental data, except for the “hump”, as no trip wire

was used in the simulation. The only other deviation is that DES predicts a quicker drop-off in energy

toward the higher frequency region (f > 2000 Hz).


−1

−0.5

0

0.5

1

0 1 2 3 4 5 6

y/D

x/D

0.00

0.05

0.10

0.15

0.20

2D

TK

E/U

2 o

(a) QFF experiment

−1

−0.5

0

0.5

1

0 1 2 3 4 5 6

y/D

x/D

0.00

0.05

0.10

0.15

0.20

2DT

KE/U

2 o

(b) LES simulation

−1

−0.5

0

0.5

1

0 1 2 3 4 5 6

y/D

x/D

0.00

0.05

0.10

0.15

0.20

2DT

KE/U

2 o

(c) DES simulation

Figure 4.5: Contours of normalized 2D TKE


60

70

80

90

100

110

120

130

50 200 500 2000 5000100 1000

PS

D[d

B/H

z]

Frequency [Hz]

QFFDES

80

90

100

110

120

130

140

150

50 200 500 2000 5000100 1000

PS

D[d

B/H

z]

Frequency [Hz]

Figure 4.6: PSD of surface pressure at 135° on C1 and 45° on C2

Summary of Hydrodynamic Results

URANS shows the ability to capture general trends well, but lacks accuracy on the magnitude of the

hydrodynamic quantities. The LES conducted here, being of relatively low-fidelity, has difficulty sim-

ulating unsteady metrics such as Cp′ RMS and TKE. However, it still shows higher accuracy than the

URANS simulations. DES most closely predicts the QFF experiment and shows a high level of accuracy

in the pressure PSD. The PSD correlates to the unsteady forces applied onto the cylinders, which is used

for the acoustic source strength calculations in the aeroacoustic analogy.

4.1.2 Receiver Noise

The noise produced by the tandem cylinders was a combination of broadband noise and tonal peaks, as

seen in Figure 4.7. The oscillatory vortex shedding phenomenon at 178 Hz, seen in Figure 4.6 produced

a strong tonal noise at 178 Hz, and less prominent peaks at its harmonics (mainly 356 and 536 Hz). This

noise was observed at all microphones in the QFF experiment (Figure 4.7), with the relative strength of

the harmonics varying between microphones while the fundamental tone remained loudest. Variations in

the SPL of the harmonics can be attributed to varying directivity at different frequencies. The following

analysis will focus primarily on microphone A, as the other SPL predictions follow similar trends.

The URANS predictions show significant under-prediction of the overall SPL and it over-emphasizes

the relative importance of the peaks above the third harmonic. However, it is able to predict the fre-

quency of the 3 major peaks relatively well with a maximum error of 8 %, as seen in Table 4.1. URANS

is only capable of capturing the major structures in the flow as the high turbulent viscosity used in

RANS models dissipate the finer details of the flow. Thus, primarily tonal information is recovered, with

very little broadband noise in the prediction.

In contrast, SPL predictions via LES shows excellent agreement of broadband noise levels till approxi-

mately 2000 Hz; the time-step used is too large to fully resolve the high-frequency content beyond this.

Since LES resolves (as opposed to models, in the case of URANS) the turbulent flow field and energy

cascade up till the limit of mesh size, detail can be recovered throughout the entire resolved range of


Table 4.1: Microphone A tonal peaks

Harmonic QFF (3.125 Hz Resolution) URANS (10 Hz Res.) LES (15 Hz Res.) DES (10 Hz Res.)

1 178 Hz, 94.1 dB 192 Hz, 82.2 dB 153 Hz, 83.3 dB 173 Hz, 94.7 dB2 356 Hz, 69.6 dB 384 Hz, 44.0 dB – 356 Hz, 70.3 dB3 536 Hz, 63.7 dB 571 Hz, 37.0 dB – 539 Hz, 65.5 dB

frequencies. However, the three tonal peaks are not reproduced by LES - there is only one spread-out

peak visible near the first harmonic. This peak’s frequency and amplitude are under-predicted by 14 %

and 11.5 %, respectively (Table 4.1). Neither the URANS or LES meshes are detailed enough in the

span-wise direction, as the entire 16 D span of the experiment was modelled while aiming to maintain

a reasonable level of computational cost (<10 million elements). Authors reproducing this experiment

have noted the importance of a fine span-wise resolution [60].

Increased span-wise resolution was achieved without significantly higher computational cost by using

periodic boundary conditions in the DES simulation. The SPL predicted by DES is in excellent agree-

ment with the experimental measurements up till approximately 2000 Hz; like LES, the time-step used

was not sufficient to resolve higher frequency. The three peak frequency predictions have a maximum

error of 2.8 %. The significance of this error reduces even further when the 10 Hz FFT resolution for the

DES data is considered; it is very likely that a finer FFT resolution, achieved simply through a longer

simulation period, would reduce this error. In addition to the excellent peak frequency prediction, the

SPL amplitude error is only 0.6 % at the first harmonic and increases to 2.8 % at higher harmonics.

Table 4.2 compares the maximum errors seen in the sound level predictions of the three models used

in the present work to a selection of published work recreating this experiment, using a wide range of

turbulence models and simulation size. It is clear that it is not necessary to model the complete span in

order to reach high accuracy - both ELAN S-A DDES and the SST k-ω DES conducted in the present

work achieve some of the lowest errors observed using a span of only 3 D. Simulations using above

11 million elements rarely see any substantial increase in accuracy. However, Table 4.2 only considers

the capability of aeroacoustic noise prediction; the higher fidelity models may be able to predict the

hydrodynamic aspects of the flow to higher accuracies.

In summary, the three simulation methods URANS, LES, and DES yielded increasing levels of success,

respectively, in recreating the complex noise produced in the tandem cylinder experiment. URANS,

designed to recreate the most dominant features of the flow, significantly under-predicts the broadband

noise levels. This is rectified by LES, where much more turbulence is resolved rather than modelled.

However, LES still showed difficulty in predicting the tonal portion of the noise. This may be due

to the coarse span-wise resolution used. DES, with a much finer span-wise resolution, produces SPL

with excellent agreement to the QFF experiment. In addition, the accuracy of the DES conducted in

this work is comparable to the the better-performing results in literature, generally achieved with much

higher computational cost. The learnings from simulating this benchmark study can be applied to more

complex geometries in the future, such as the components used in a muffler (e.g., perforated plates and

sharp edges in flow).


0

20

40

60

80

100

50 200 500 2000 5000100 1000

SP

L[d

B/H

z]

Frequency [Hz]

QFFURANS

LESDES

(a) Mic A

0

20

40

60

80

100

50 200 500 2000 5000100 1000

SP

L[d

B/H

z]

Frequency [Hz]

(b) Mic B

0

20

40

60

80

100

50 200 500 2000 5000100 1000

SP

L[d

B/H

z]

Frequency [Hz]

(c) Mic C

Figure 4.7: SPL at the three microphone locations


Table 4.2: Comparison of the simulation in the present work to a selection of those in literature [59]

Turbulence ModelNum. Elements(millions)

Span / DPeak FrequencyMax. Error

Peak AmplitudeMax. Error

PowerFLOW k-ε DES 66 16 2.0 % 5.5 %CEDRE SST DDES 17 4 14.3 % 7.0 %UPACS-LES Zonal S-A 70 18 1.6 % 1.3 %CFL3D Zonal SST 60 18 7.6 % 4.0 %ELAN S-A DDES 10 3 1.1 % 3.8 %

Present Work

Fluent k-ε RANS 0.5 16 7.9 % 41.9 %Fluent LES 9.8 16 14.0 % 11.5 %Fluent SST k-ω DES 10.7 3 2.8 % 2.8 %

4.2 Single-Objective Optimization (SOO)

The primary goal when designing a muffler is to maximize its transmission loss in the desired portions

of the audible frequency spectrum. This was conducted through single-objective optimization (SOO)

of the plug flow muffler as detailed in Section 3.2. Prior to the optimization process, the prediction

methods and perforate impedance models used were validated using experimental data from literature.

As outlined in Figure 3.4, the optimization process used the TMM with the best performing impedance

model. This section also analyzes the limitations in the optimization process stemming from the use of

TMM. Finally, the solution space is visualized in order to reveal the underlying structure.

4.2.1 TMM & FE Model Evaluation

The first step was an evaluation of the 3 zero mean flow perforate impedance models using both TMM

and FE, by comparison against experiments conducted by Wu et al. [89]. The TL calculated via TMM

are presented in Figure 4.8 alongside the experimental data. The experimental data up to 3200 Hz shows

three major TL peaks occurring at 747 Hz, 1770 Hz, and 2062 Hz. The Sullivan and Bento Coelho models

are in qualitative agreement with the experimental data up to roughly 2200 Hz. Substantial deviations

only begin after the cutoff frequency, fc = 1877 Hz, and are associated with the plane wave assumption

inherent to the TMM. However, both the Sullivan and Bento Coelho models predict the general decrease

in TL above 2200 Hz. The Bauer model, while containing the 3 major peaks, exhibits an increasing fre-

quency shift at higher frequencies; the first peak is shifted by 42 Hz, the second peak by 220 Hz, and the

third peak by 270 Hz. This behavior is currently unexplained. Of the three impedance models used in

the TMM, the Sullivan model performs best, as corroborated by the high Pearson correlation coefficient

listed in Table 4.3.

Both 2DA and 3D FE simulation results are shown in Figure 4.9. The 3D simulations took 70 times the

computational time when compared to the 2DA model, primarily because of the 100 times increase in

the number of elements. The 2DA results lie almost coincident with the 3D results, implying that there

are no 3D modes of pressure distribution in the frequency range considered.

Section 4.2 was previously published in J. Puthuparampil, H. Pong, and P. Sullivan, Modelling and Optimization of PlugFlow Mufflers in Emission Control Systems, in SAE 2017 Noise and Vibration Conference and Exhibition, 2017. [71].


Table 4.3: Pearson correlation coefficient - prediction vs. experimental data

Impedance Model TMM 2DA 3D

Sullivan 0.8084 0.6683 0.6681Bento Coelho 0.7445 0.7880 0.7878Bauer 0.2696 0.8496 0.8495

0

10

20

30

40

50

60

0 500 1000 1500 2000 2500 3000

fc = 1877 Hz

Tra

nsm

issi

on

Loss

[dB

]

Frequency [Hz]

Experimental DataTMM - Sullivan

TMM - Bento CoelhoTMM - Bauer

Figure 4.8: TMM evaluation of transmission loss

0

10

20

30

40

50

60

0 500 1000 1500 2000 2500 3000

fc = 1877 Hz

Tra

nsm

issi

onL

oss

[dB

]

Frequency [Hz]

Experimental Data2DA FE - Sullivan

2DA FE - Bento Coelho2DA FE - Bauer3D FE - Sullivan

3D FE - Bento Coelho3D FE - Bauer

Figure 4.9: FEM evaluation of transmission loss


Table 4.4: Optimized muffler dimensions

Parameter Optimized Value

rD 0.4000La1 = La2 0.1498 mL1 = L2 0.1144 mLb1 = Lb2 0.060 03 mσ1 = σ2 18.24 %Objective Function (NLO) 12.86 dB

Of the three impedance models used in the FE simulations, the Bauer model shows the highest level

of qualitative accuracy, contrary to the results from the TMM. These findings are further verified by

the high correlation coefficient for the FE Bauer model seen in Table 4.3. Both the Sullivan and Bento

Coelho models show significant deviation from approximately 1400 Hz onwards. However, certain sim-

ilarities exist across TMM and FE - the order in which each impedance model peaks stays consistent.

For example, when considering the second peak in the experimental data (1770 Hz), the Sullivan model

peaks first, followed by the Bento Coelho, and finally the Bauer model. This trend is seen for all TL

peaks in both TMM and FE.

The observations in this section lead to adopting the Sullivan impedance model for TMM and the

Bauer model for 2DA FE simulations, as they provide the highest correlation and qualitative agreement

with experimental data for their respective modelling methods. In the SOO work, TMM was used in

the optimization process and the FE simulation was used only to check the final optimized result.

4.2.2 NLO Optimization

Optimized Solution

The optimization process was conducted to design a plug flow muffler that reduced the noise output of

a generator to the NC-60 standard. 2000 random initializations of the optimization were conducted in 5

hours and 40 minutes on a 4-core Intel i7-4790 processor, with only 3 instances of the optimizer failing

(due to exceeding the 15,000 iterations limit). The dimensions of the best performing design along with

its objective function value is given in Table 4.4. Figure 4.10 shows the associated TMM and 2DA FE

transmission loss predictions. Based on the results from Section 4.2.1, the FE Bauer model is expected

to predict the true TL with a much higher accuracy than the TMM Sullivan model. Thus, the current

section compares the TMM results against FE simulations, in lieu of experimental data.

All 7 major TL peaks, especially the 4 up to 2550 Hz, are predicted equally well by both the TMM and

FE. Additionally, general TL levels agree across both modelling methods, with a region of 3 peaks up to

approximately 1570 Hz, after which there is relatively low TL to 2400 Hz, following which 4 peaks occur.

The accuracy is reflected in the high correlation coefficient of 0.7520 between the two prediction meth-

ods. Due to the high level of agreement between TMM and FE, one can expect the true experimental

TL to be very similar.


0

10

20

30

40

50

60

70

0 500 1000 1500 2000 2500 3000 3500 4000

fc = 1877 HzT

ran

smis

sion

Loss

[dB

]

Frequency [Hz]

TMM - SullivanFE - Bauer

Figure 4.10: Optimized muffler transmission loss

Figure 4.11 presents the final noise output from the engine and muffler system based on the TMM

and FE predictions for the optimized muffler. The output noise sound pressure level (SPL) is defined as

SPLi = ENi − TLi (4.2)

and both the TMM and FE results agree extremely well. This is because the Noise Criterion standard

only considers an average over an octave band calculated through Equation 3.4, disregarding small

frequency mismatches.

However, a large discrepancy is observed between the acceptable output SPL determined by NC-60

and the optimized muffler systems output SPL in the 63 Hz, 125 Hz, and 250 Hz octave bands (the

low-frequency range). In fact, this difference is the major contributor to the non-zero evaluation of

the objective function seen in Table 4.4. The inability for the optimized muffler to meet the low-

frequency targets is related to the rD variable and its chosen lower limit of 0.4; TMM exhibits prediction

inaccuracies below this value. However, a strong inverse proportionality between rD and low-frequency

muffler TL was found with an FE parametric study where rD was varied from 0.1 to 0.8, keeping all

other parameters constant at their values from Table 4.4. Figure 4.12 shows the output SPL for the

parametric study; a muffler with rD = 0.1 easily outperforms the NC-60 criterion. However, choosing

such a small diameter for the inner tube introduces manufacturing challenges, lack of structural integrity

if porosity is held constant, and increased pressure drop if porosity is reduced to maintain structural

integrity [66].


0

20

40

60

80

100

120

63 125 250 500 1000 2000 4000

Ou

tpu

tS

ou

nd

Pre

ssu

reL

evel

[dB

]

Octave Band [Hz]

Engine NoiseNC-60

TMM - SullivanFE - Bauer

Figure 4.11: Original engine noise, NC-60 criterion, and the output sound pressure levels of the optimizedmuffler

0

20

40

60

80

100

120

63 125 250 500 1000 2000 4000

Ou

tpu

tS

oun

dP

ress

ure

Lev

el[d

B]

Octave Band [Hz]

NC-60rD = 0.1rD = 0.2rD = 0.3rD = 0.4rD = 0.5rD = 0.6rD = 0.7rD = 0.8

Figure 4.12: Parametric study - output sound pressure levels


0

200

400

600

800

1000

(12.0

,14

.5]

(14.

5,17.

0]

(17.0

,19

.5]

(19.5

,22.0

]

(22.0

,24.5

]

(24.5

,27.0

]

(27.0

,29.5

]

(29.

5,32.

0]

(32.

0,34.

5]

Cou

nt

NLO

36

325

900

399

179116

21 20 1

Figure 4.13: Histogram of NLO evaluations at the final result of each optimization

Optimization Analysis

A histogram of NLO evaluations of the final result of all 1997 successful optimizations is seen in Figure

4.13. It reveals that only 1.8 % of optimizations reached the lowest recorded bin, while 45.1 % of op-

timizations terminated between NLO = (17, 19.5]. Thus, the use of a multi-start algorithm is justified

and necessary to find the best performing local minima.

Of the 1997 solutions discovered, 1756 of them were unique local minima. This 5-dimensional solution

space can be visualized through dimensionality reduction. Principal component analysis (PCA) was

conducted on a z-score standardized dataset of the unique solutions to map the space onto 2 indepen-

dent variables which maximized the variance of the projection [73]. Due to the correlation between

the La and Lb values, two major clusters with very similar behavior are seen. Thus, focusing on one

of these clusters in the PCA provides a clearer visualization of underlying trends, as shown in Figure

4.14. A region of well-performing solutions is observed surrounding the point (0, -1), as marked by the

circle labelled Optimal Solution Cluster. The majority of the low-NLO solutions lie within this region,

including the previously discussed best-performing design.

Identifying a region of significance such as this provides an effective method to check whether a given

set of muffler dimensions will perform well, simply based on the X1 and X2 values it maps on to. This

check can be executed using the PCA transformation coefficients on the z-score standardized muffler


−4

−3

−2

−1

0

1

2

3

4

−3 −2 −1 0 1 2 3 4

X2

X1

12.0

14.5

17.0

19.5

22.0

24.5

NL

O

Optimal Solution Cluster

Figure 4.14: Dimension-reduced visualization of one cluster of 883 unique solutions (local minima),colored by their associated NLO evaluation

dimension values, given as

X1 =

0.6440

−0.08345

−0.3862

0.6166

−0.2213

·

z-score(rD)

z-score(La)

z-score(L)

z-score(Lb)

z-score(σ)

(4.3)

X2 =

0.3263

−0.6727

−0.1524

−0.3273

0.5573

·

z-score(rD)

z-score(La)

z-score(L)

z-score(Lb)

z-score(σ)

(4.4)

4.3 Multi-Objective Optimization (MOO)

For a more holistic design of mufflers, the pressure drop across a muffler must be considered in conjunction

with its acoustic properties. Thus, multi-objective optimization (MOO) as described in Section 3.3

was conducted. As per Figure 3.5, two steps precede the optimization process, 1) validation of the

empirical pressure drop model, and 2) dimensionality reduction in order to identify the variables of

higher importance. Following this, the optimization was conducted with the EGO algorithm and the

Pareto front approximation was compared against results from the standard NSGA-II algorithm.


0

2

4

6

8

10

A B C D E F

Pre

ssu

reD

rop

[in

H2O

]

Muffler ID

Empirical ModelCFD

Figure 4.15: Comparison of pressure drop calculated via the empirical pressure drop model and CFD,for 6 random PFM geometries

4.3.1 Empirical Pressure Drop Model Validation

The accuracy of the empirical pressure drop model was evaluated against 2DA steady state RANS

CFD using 6 randomly chosen PFM geometries (Figure 4.15). The empirical model is able to predict

the pressure drop relatively well, with a mean absolute percent error of 12 %. An average error of

this magnitude is acceptable for industrial applications. Importantly, there is a high correlation (0.97)

between the empirical and CFD-based pressure drops, meaning that the multi-objective optimization

would behave similarly regardless of the method of pressure drop estimation.

4.3.2 Dimensionality Reduction

10-fold cross-validated feature selection was performed on the 12 variables defining the PFM, for 10

different sets of TLO weights, as described in Section 3.3.3. The average frequency of a variable being

part of the reduced subset is presented in Table 4.5. rD, rexp, rL1, rL2, σ1, and σ2 are chosen in all

cross-validations for all weights. Only two other variables were chosen at all - r2 and Tp2 at an average of

18 % and 33 % of all cross-validations, respectively. Given the dominating importance of the 6 variables

at a 100 % selection rate, those were selected as the subset to be used in MOO.

4.3.3 EGO’s Performance in MOO

EGO’s effectiveness as a MOO algorithm for muffler optimization is evaluated through the four objectives

outlined in Section 3.3.4. The following naming convention is used to differentiate various optimizations:

Algorithm – Nevals – Initial Condition Set.


Table 4.5: Summary of dimensionality reduction

Parameter Average Selection Rate Across Different wi Part of Reduced Subset

rD 100 % ± 0 % Yesrexp 100 % ± 0 % YesrL1 100 % ± 0 % YesrL2 100 % ± 0 % Yesr1 0 % ± 0 % Nor2 18 % ± 7.9 % Noσ1 100 % ± 0 % Yesσ2 100 % ± 0 % YesTp1 0 % ± 0 % NoTp2 33 % ± 24.5 % NoDh1 0 % ± 0 % NoDh2 0 % ± 0 % No

Objective 1 - Comparison of EGO Variants

Figure 4.16 shows the median attainment surfaces of the four EGO variants using Nevals = 240 and

initial condition set A. No significant difference is observed between the four algorithms. MWU tests on

the distribution of the hypervolume (IHV ) and binary additive epsilon (Iε+) metrics show no statistically

significant difference either (Table A.1, Appendix A). ParEGO performs equally as well as any other

variant even though it relies on a kriging model fit to a scalarized version of the objectives. Thus,

unlike the other three methods, only one model-fitting operation is conducted, leading to a nominal

2 to 4 % decrease in execution time. However, a scalarization-based method might prove less effective

if the optimization involved a large number of objectives. It has been shown that for standard multi-

objective test functions, ParEGO performs worse than the other variants [12]. For the purposes of

muffler optimization, where there are only two primary well-behaved objectives, ParEGO proves to

be an effective choice of algorithm. The rest of the present work will use ParEGO when conducting

EGO-based optimization.

Objective 2 - EGO’s Sensitivity to Nevals

The median attainment surfaces comparing ParEGO using Nevals = 240, 300, and 360 total evaluations

show no distinct differences (Figure 4.17). Additionally, the MWU tests (Table A.2, Appendix A) present

no significant statistical difference either. ParEGO’s success at relatively low Nevals provides simulation-

based optimization at industrially relevant timescales. The ParEGO-240-A simulation takes 2.9 hours on

average. If this simulation was conducted with genetic algorithms, it would require anywhere from 2000

to 64000 evaluations (Section 2.5), taking between 20 hours to 640 hours, respectively. Additionally,

GA-based muffler optimization work state that larger evaluation budgets led to better optimized results,

with convergence seen only after a large number of evaluations [22]. GA also requires the tuning of

multiple parameters including crossover probability and mutation probability, for each problem solved

[29]. Slow convergence and the need for meta-optimization (i.e., finding the best parameters required

for GA to optimize effectively) is not ideal for an optimization process - especially not for industrial use.

In contrast, ParEGO’s relative indifference to Nevals makes it well suited for such uses.


0

2

4

6

8

10

−30 −25 −20 −15 −10 −5 0

DP

[in

H2O

]

TLO [dB]

ParEGO-240-ASMS-240-A

ε-240-AMSPOT-240-A

Figure 4.16: Median attainment surfaces comparing the 4 EGO variants

0

2

4

6

8

10

−30 −25 −20 −15 −10 −5 0

DP

[in

H2O

]

TLO [dB]

ParEGO-240-AParEGO-300-AParEGO-360-A

Figure 4.17: Median attainment surfaces comparing different Nevals


0

2

4

6

8

10

−30 −25 −20 −15 −10 −5 0

DP

[in

H2O

]

TLO [dB]

ParEGO-240-AParEGO-240-BParEGO-240-CParEGO-240-D

Figure 4.18: Median attainment surfaces comparing different sets of initial conditions

Objective 3 - EGO’s Sensitivity to Initial Conditions

ParEGO is seen to be a robust algorithm due to it’s insensitivity to initial conditions. Four different

sets of random initial conditions (A, B, C, and D) resulted in median attainment surfaces that are very

similar (Figure 4.18). The only discernible difference is ParEGO-240-C’s deviation from the other curves

near (-18.5, 2.5). This was due to the presence of an uncharacteristically well-performing muffler in this

region of the attainment surface within the set of 40 initial points for set C. Although ParEGO-240-C’s

mean statistical performance of IHV and Iε+ is marginally better than the other initial condition sets, the

MWU tests show no statistically significant difference (Table A.3, Appendix A). This robust behaviour

is very advantageous in an industrial setting as repetitions of the optimization is not necessary if the

computational resource and time are not available.

Objective 4 - Comparison of EGO Against NSGA-II

The performance of the NSGA-II algorithm using Nevals = 240, 300, and 360 show significantly worse

results than ParEGO-240-A (Figure 4.19). The IHV and Iε+ metrics in Figures 4.20 and 4.21 depict the

vast difference in performance between the ParEGO and the NSGA-II algorithms. Table A.4 (Appendix

A) contains the MWU test on the two metrics and it corroborates the statistically significant differences

seen in Figures 4.20 and 4.21. In addition, due to the Nevals = npop ·ngen restriction, NSGA-II produces

Pareto fronts of very low resolution. The resolution is determined by npop; increasing it necessitates a

reduction in ngen, for a given Nevals. Increased resolution would result in solutions that are not very

different from the random initial conditions since the population would not have had time to evolve into

a better state. The EGO algorithm circumvents this issue entirely as it is free to add or remove candidate

solutions from the final Pareto front approximation at each iteration, based only on the performance of

a given solution.


0

2

4

6

8

10

−30 −25 −20 −15 −10 −5 0

DP

[in

H2O

]

TLO [dB]

ParEGO-240-ANSGAII-240NSGAII-300NSGAII-360

Figure 4.19: Median attainment surfaces comparing the ParEGO algorithm and NSGA-II

In addition, there is no correlation seen in the NSGA-II results between the quality of a Pareto front and

Nevals; others have reported a strong positive correlation between evaluation budget and the quality of

results prior to convergence when using GA [22]. The lack of correlation here is attributed to the fact

that GA is simply not designed to perform at such low Nevals; the evolutionary strategies in the algo-

rithm are fruitful only under large populations with many generations - both of which are not possible

with low Nevals.

4.3.4 Analysis of EGO Model

A major advantage of EGO (or any model-based optimization algorithm) over evolutionary methods

is that the underlying model developed by the algorithm can be visualized and analyzed, providing

an understanding of the relationships between different optimization variables and how they affect the

objective function. Figure 4.22 is the model visualization of one ParEGO-240-A run. The contours are

coloured based on the sum of the normalized values of TLO and DP

DP + TLO = wDP

(DP−DPlow

DPhigh −DPlow

)+ wTLO

(TLO− TLOlow

TLOhigh − TLOlow

)(4.5)

where the tilde represents normalized values, DPlow = 0.0, DPhigh = 40, TLOlow = −50, TLOhigh = −10,

and wDP and wTLO are the weights for DP and TLO, both set equal to 1.0. Equation 4.5 can be thought

of as a simplified version of the scalarized objective function that is minimized by ParEGO. Minimizing

Equation 4.5 leads to better performing mufflers with low DP and high TLO. Figure 4.22 has each opti-

mization variable is plotted against every other, in order to reveal the inter-relations between variables.

In addition, the final Pareto front approximation from this run is plotted as white points on each plot.


60

80

100

120

140

160

180

200

220

ParEGO-240-A NSGAII-240 NSGAII-300 NSGAII-360

I HV

Figure 4.20: Hypervolume metric (IHV ) comparing the ParEGO algorithm and NSGAII

−5

0

5

10

15

20

25

(ParEGO-240-A,

NSGAII-240)

(ParEGO-240-A,

NSGAII-300)

(ParEGO-240-A,

NSGAII-360)

I ε+

Figure 4.21: Binary additive epsilon metric (Iε+) comparing the ParEGO algorithm and NSGAII


In general, these optimized points lie closer to the lower contour values (darker colours), validating

the visualization. However, some points are seen to stray away to higher contour values. This is because

the visualization only used one set of weights (wDP = 1.0 and wTLO = 1.0); the ParEGO algorithm uses

a randomized pair of weights in each iteration to develop it’s true optimization model. Thus, points

seen to lie outside the darker zones in Figure 4.22 most probably lie within dark zones under different

visualization weights.

Further observation and analysis of Figure 4.22 provides some insight into the behaviour of each variable:

1. rD: The graphs in column A show a maxima for low rD values, regardless of what the other

variable is. This shows the high sensitivity of rD, since at low rD the pressure drop grows very

large, while TLO doesn’t increase as dramatically. We see optimized points spanning [0.3, 0.9],

showing that an effective muffler can be constructed using a variety of expansion ratios.

2. rexp: All plots associated with rexp (A-I and column B) show a relatively symmetric trend as rexp

controls the geometric symmetry of the muffler, across the plug in the PFM. Too low would be

a short expansion section leading to high DP, and too high would be a short contraction section,

also leading to high DP. Thus, all the optimized points lie within the middle range of [0.4, 0.85].

3. rL1: As with rexp, the optimized points lie in the middle range of [0.3, 0.7], where the contour

values are seen to be quite low (A-II, B-II, and column C). Only three solutions are seen to be

outside this range into high rL1 values (the three points are seen in the top-right of plot A-II). The

high rD values of these three solutions allow for a lower pressure drop, balancing out the larger

pressure drop incurred by moving to higher rL1 region.

4. rL2: It is similar to rL1 in trends, except the optimized results seem to have a slightly larger spread

(row III, D-IV, and D-V). This could indicate that the muffler performance is marginally less

sensitive to the contraction section of the PFM. Additionally, both rL1 and rL2 show a bifurcating

behaviour when compared against rD. The optimized points greater than rD = 0.7 seem to split

into two separate paths in plots A-II and A-III. This indicates that for rD > 0.7, two separate

sets of solutions exist. Furthermore, the rL1 vs rL2 graph (C-III) shows the minima of the contour

offset from the center of the plot, signifying that a slight asymmetry between the lengths of the

perforated sections is important (too asymmetric would incur significant pressure drop penalties).

5. σ1 and σ2: The optimized points exhibit a large spread across both these variables; it is an indicator

of the relatively lower sensitivity of σ1 and σ2. Plot E-V shows that a higher number of optimized

points are skewed toward higher σ1. This could be because the pressure drop associated with the

expansion section is higher than the contraction section, for a given OAR (Equations 2.36 and

2.37). Thus, the optimizer is able to achieve lower DP by using higher σ1 values, while sacrificing

some TL performance. As σ2 has lower DP for a given OAR, the optimizer has more freedom in

exploring a larger range of this variable, resulting in a relatively even distribution of optimized

points along the σ2 axis, from approximately [0.2, 0.8].


0

1

Sum of normalized TLO and DP

A B C D E

I

II

III

IV

V

0

1

0

1

0

1

0

1

0 1 0 1 0 1 0 1 0 1

rexp

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

rL1

rL2

σ1

σ2

rD rexp rL1 rL2 σ1

Figure 4.22: Visualization of kriging model developed by the ParEGO algorithm for one set of TLO andDP weights, with the optimal points on the Pareto front approximation overlaid as white points


4.4 Chapter Summary

Section 4.1 contained the aeroacoustic noise prediction of the tandem cylinder experiments. The short-

span DES work conducted produced an excellent prediction of the measured noise, at significantly

lower computational cost than most of the work in literature. The single-objective muffler optimization

in Section 4.2 showed the effectiveness of L-BFGS-B (a relatively simple gradient-based optimization

algorithm) in a multi-start framework. In addition to arriving at a successfully optimized muffler, this

section contained analysis and visualization of the solution space. Finally, Section 4.3 demonstrated

that the EGO algorithm is robust and extremely capable of multi-objective muffler optimization, and

performs significantly better than the commonly used NSGA-II algorithm. Additionally, the models of

the solution space developed by EGO were visualized and analyzed.

Chapter 5

Conclusion and Future Work

5.1 Conclusion

The ideal muffler is found at the intersection of high acoustic performance, low pressure drop through

the muffler, and a low amount of flow-induced aeroacoustic noise. This work addressed muffler design

in a holistic manned by incorporating all three of the aforementioned aspects.

As the first steps towards aeroacoustic noise predictions of a muffler, the geometrically simpler tandem

cylinder benchmark experiment was studied. Testing the ability of URANS, LES, and DES to predict

the complex noise in a computationally cost-effective manner revealed that DES is an effective tool for

aeroacoustic prediction. DES was able to reproduce the hydrodynamics of the flow with an industrially

sufficient level of accuracy. The relatively low computational-cost DES conducted in this work produced

acoustic predictions as accurate as much more computationally expensive simulations found in literature.

In order to design a muffler with the optimal acoustic performance, the general guidelines and strategies

used by muffler manufacturers is not enough; automated optimization is necessary. The single-objective

optimization portion of this work showed that relatively simple gradient-based optimization algorithms

were successful in optimizing a plug flow muffler’s transmission loss. This is in contrast to the complex

algorithms, such as genetic algorithms or simulated annealing, typically used in such work.

However, in order to consider pressure drop alongside acoustic transmission loss in large mufflers,

simulation-based multi-objective optimization was conducted using the EGO algorithm. Surrogate

model-based algorithms such as EGO are very efficient when working with real engineering problems

with computationally expensive objective functions, as the surrogate model provides an extremely cost-

effective view of the objective space. EGO was shown to outperform the commonly used NSGA-II

genetic algorithm in the multi-objective optimization of a plug flow muffler.

61

Chapter 5. Conclusion and Future Work 62

5.2 Future Work

5.2.1 Aeroacoustics

The aeroacoustic noise predictions conducted in the present work was only an introductory exploration.

Future work can expand by:

1. Porting the simulation into an open source CFD platform such as OpenFOAM in order to allow

for computation in a high performance computing cluster

2. Simulating more complex and industrially relevant 3D geometry; several such geometries exist

as part of AIAA’s Second Workshop on Benchmark problems for Airframe Noise Computations

(BANC-II) [2] (the tandem cylinder experiment used in this work is part of this database)

3. Visualization of the noise sources to better understand the effects of different geometries and noise

producing mechanisms

5.2.2 Muffler Optimization

Optimization work using EGO can be expanded to:

1. Calculate pressure drop via a CFD simulation, removing any limitations imposed by the empirical

pressure drop model and allowing the exploration of more varied designs

2. Include a third objective which aims to minimize the strength of the aeroacoustic noise sources,

calculated through a CFD simulation (low-fidelity aeroacoustic predictions based on steady-state

RANS is available in the form of Broadband Noise Source (BNS) models [95])

3. Different styles of mufflers, including the combination of reactive and absorptive silencing

Appendix A

Mann-Whitney U (MWU) Test

To answer the multi-objective optimization objectives (Section 3.3.4) in a statistical fashion, IHV and

Iε+ from multiple runs of one method was compared against another method using the Mann-Whitney U

(MWU) test. MWU aims to discern if two independent samples were selected from the same population

(null hypothesis), but without requiring a normal distribution (unlike the Student’s t-test). Proving the

null hypothesis false would show that there is a statistical difference between two sets of data.

The MWU test relies on calculating the U statistic. If U ≤ Ucrit, where Ucrit is obtained from tabulated

values based on sample size, the null hypothesis is rejected - i.e., the two sets of data are from different

populations. In this work, the MWU test was conducted with a significance threshold (p-level) of 0.05

(two-tailed); this yields Ucrit = 13.

Table A.1: IHV and Iε+ for Objective 1; bolded algorithm is the baseline in MWU test

ParEGO-240-A SMS-240-A EPS-240-A MSPOT-240-A

IHV 186.0± 5.7 185.8± 4.4 187.8± 4.3 188.0± 4.4U Value – 30 26 27Similar to Baseline? – Yes Yes Yes

Iε+ – (1.095, 1.169) (1.022, 0.975) (0.945, 1.037)U Value – 30 25 27Similar to Baseline? – Yes Yes Yes

63

Appendix A. Mann-Whitney U (MWU) Test 64


ParEGO-240-A ParEGO-300-A ParEGO-360-A

IHV 186.0± 5.7 190.3± 6.1 189.1± 3.1U Value – 22 22Similar to Baseline? – Yes Yes

Iε+ – (1.188, 0.909) (1.376, 0.869)U Value – 16 13Similar to Baseline? – Yes No (nominally)


ParEGO-240-A ParEGO-240-B ParEGO-240-C ParEGO-240-D

IHV 186.0± 5.7 187.5± 5.4 193.5± 5.8 187.6± 2.6U Value – 29 10 23Similar to Baseline? – Yes No Yes

Iε+ – (1.178, 1.578) (1.675, 1.272) (1.254, 1.508)U Value – 30 20 029Similar to Baseline? – Yes Yes Yes


ParEGO-240-A NSGAII-240 NSGAII-300 NSGAII-360

IHV 186.0± 5.7 134.9± 46.4 123.6± 31.6 108.6± 38.4U Value – 16 0 0Similar to Baseline? – Yes No No

Iε+ – (0.927, 8.609) (0.481, 8.441) (0.396, 13.246)U Value – 4 0 0Similar to Baseline? – No No No

Appendix B

Tandem Cylinder Simulations -

Additional Results

B.1 2D TKE - Line Extractions

Horizontal line extractions (y = 0 plane) of 2D TKE in the gap between the two cylinders and aft of the

second cylinder are seen in Figure B.1. Similarly, vertical line extractions at x = 1.5 D and x = 4.45 D

are presented in Figure B.2. Both extractions are created from the 2D TKE contour plots (Figure 4.5).

B.2 Span-wise Correlation of Pressure

Figure B.3 shows the span-wise correlation of pressure along both cylinders at θ = 135° for one half

of the domain (z/D ≥ 0). The correlation is calculated between the central point at z/D = 0 and

other span-wise locations. The span-wise correlation in the QFF data reduces towards 0 as one moves

closer to the no-slip condition at the walls of the wind tunnel (z/D = 8). DES data for C1 shows a

slow decrease of span-wise correlation which does not reach 0, primarily due to the periodic boundary

conditions used. The span-wise correlation on C2 is in higher agreement with experimental data, but it

too suffers from the inaccuracies of the periodic boundary condition (in reality, it is a no-slip boundary

condition at z/D = ±8).

65

Appendix B. Tandem Cylinder Simulations - Additional Results 66

0

0.05

0.1

0.15

0.2

0.25

0.3

0.5 1 1.5 2 2.5 3

2D

TK

E/U

2 o

x/D

0

0.05

0.1

0.15

0.2

0.25

0.3

4.5 5 5.5 6 6.5

2DT

KE/U

2 o

x/D

QFFLESDES

Figure B.1: 2D TKE along y = 0

0

0.05

0.1

0.15

0.2

0.25

0.3

−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8

2DT

KE/U

2 o

y/D

0

0.05

0.1

0.15

0.2

0.25

0.3

−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8

2DT

KE/U

2 o

y/D

QFFLESDES

Figure B.2: 2D TKE along x = 1.5 D and x = 4.45 D

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

Cor

rela

tion

z/D

QFFDES

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

Cor

rela

tion

z/D

Figure B.3: Span-wise correlation of surface pressure at 135° on C1 and C2

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