Numerical Evaluation of Acoustic Green's Functions

259 Pages·2014·5.66 MB·English

NUMERICAL EVALUATION OF
ACOUSTIC GREEN’S FUNCTIONS
A thesis submitted to the University of Manchester
for the degree of Doctor of Philosophy
in the Faculty of Engineering and Physical Sciences
The University of Manchester
2014
Adrian R. G. Harwood
School of Mechanical, Aerospace & Civil Engineering
Contents
Abstract 11
Declaration 12
Copyright Statement 13
Acknowledgements 14
Abbreviations 15
Nomenclature 16
1 Introduction 18
2 Noise Prediction Schemes 21
2.1 Computational Aero-Acoustics . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Types of Computational Approaches . . . . . . . . . . . . . . . 22
2.1.2 Exclusively Numerical Approaches . . . . . . . . . . . . . . . . 24
2.1.3 Exclusively Analytical Approaches . . . . . . . . . . . . . . . . 31
2.1.4 Hybrid Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.5 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.7 Applications of CAA . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Green’s Functions in Acoustics 58
3.1 Green’s Function and Hybrid Schemes . . . . . . . . . . . . . . . . . . 60
3.2 Acoustic Application of Green’s Functions . . . . . . . . . . . . . . . . 61
3.3 Methods of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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3.3.1 Analytical Green’s Functions . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Series Representations . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.3 Problem-Specific Construction Techniques . . . . . . . . . . . . 64
3.3.4 Other Methods of Construction . . . . . . . . . . . . . . . . . . 66
3.4 Green’s Functions in other contexts . . . . . . . . . . . . . . . . . . . . 67
4 Choice of Numerical Methods 69
5 Compact Green’s Function Method 73
5.1 The Compact Green’s Function . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Constructing the Compact Green’s Functions . . . . . . . . . . . . . . 75
5.2.1 Potential Flow Solutions . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Application to 2D Problems . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.1 Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.2 Backward-Facing Step . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Influence of Grid and Map Configuration . . . . . . . . . . . . . . . . . 86
5.4.1 Backward-facing Step . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.2 Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4.3 Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Application to 3D Problems . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 Compact Green’s Function for a Throttle . . . . . . . . . . . . . . . . . 106
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Boundary Element Method 110
6.1 Introduction to Boundary Methods . . . . . . . . . . . . . . . . . . . . 111
6.1.1 Boundary Element Method . . . . . . . . . . . . . . . . . . . . 112
6.1.2 Issues with Implementing BEM . . . . . . . . . . . . . . . . . . 112
6.1.3 Hybrid BEM/BIE Schemes . . . . . . . . . . . . . . . . . . . . . 115
6.2 Green’s Function for the Helmholtz Equation . . . . . . . . . . . . . . . 116
6.3 Direct Collocation Boundary Element Formulation . . . . . . . . . . . 117
6.4 Dirichlet-to-Neumann Operator . . . . . . . . . . . . . . . . . . . . . . 120
6.5 BEM Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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6.7 1D Waves in a Rigid Channel . . . . . . . . . . . . . . . . . . . . . . . 126
6.8 2D Green’s Function for a Rigid Channel . . . . . . . . . . . . . . . . . 129
6.8.1 Range Selections . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.8.2 Accuracy variation with Frequency . . . . . . . . . . . . . . . . 134
6.8.3 Execution Time variation with Frequency . . . . . . . . . . . . 135
6.8.4 Accuracy variation with Mesh Density . . . . . . . . . . . . . . 137
6.8.5 Execution Time variation with Mesh Density . . . . . . . . . . . 137
6.8.6 Influence of Cross-Channel Mesh Density . . . . . . . . . . . . . 139
6.9 Throttle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.10 2D Green’s Functions with other boundaries . . . . . . . . . . . . . . . 144
6.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7 Sinc-Galerkin Method 147
7.1 Sinc Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Green’s Function for the Helmholtz Equation . . . . . . . . . . . . . . . 153
7.3 1D Sinc Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4 2D Sinc-Galerkin for the Helmholtz Equation . . . . . . . . . . . . . . 157
7.5 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.5.1 Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5.2 Removal of the Singularity . . . . . . . . . . . . . . . . . . . . . 166
7.6 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.6.1 Source Specification . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.6.2 Green’s Function for the 2D Helmholtz equation . . . . . . . . . 168
7.6.3 Centred Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.6.4 Near-Boundary Source . . . . . . . . . . . . . . . . . . . . . . . 173
7.7 2D Sinc-Galerkin Helmholtz Neumann BVP . . . . . . . . . . . . . . . 179
7.7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.7.2 Modified Solution Method . . . . . . . . . . . . . . . . . . . . . 186
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8 Comparison & Assessment of the Methods 193
8.1 Compact GF Method vs. BEM . . . . . . . . . . . . . . . . . . . . . . 194
8.2 Sinc-Galerkin vs. BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
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8.2.1 Centred Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2.2 Near-Boundary Source . . . . . . . . . . . . . . . . . . . . . . . 203
8.3 Comparison Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.3.1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.3.2 Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.3.3 Computational Resources . . . . . . . . . . . . . . . . . . . . . 212
8.3.4 Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.3.5 Ease of Use/Implementation . . . . . . . . . . . . . . . . . . . . 215
8.4 Guidelines & Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 216
9 Closing Remarks 219
9.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Bibliography 223
Word count approx. 52,400
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List of Tables
5.1 Details of the numerical mapping for the backward-facing step from the
half-plane (4 sig. fig.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Details of the numerical mapping for a simple channel with unit flow
from the half-plane (4 sig. fig.) . . . . . . . . . . . . . . . . . . . . . . 87
7.1 Block Matrices for Sub-System Type 1 . . . . . . . . . . . . . . . . . . 187
7.2 Block Matrices for Sub-System Type 2 . . . . . . . . . . . . . . . . . . 188
7.3 Block Matrices for Sub-System Type 3 . . . . . . . . . . . . . . . . . . 189
7.4 Block Matrices for Sub-System Type 4 . . . . . . . . . . . . . . . . . . 190
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List of Figures
2.1 Illustration of many of the current variety of techniques available for
the solution of acoustic problems . . . . . . . . . . . . . . . . . . . . . 23
5.1 Circular cylinder problem in both the untransformed and transformed
domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Absolute Error of potential function for half a circular cylinder. . . . . 83
5.3 Backward-facing step problem in both the untransformed and trans-
formed domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Absolute error variation in the potential for the backward-facing step.
Note: The numerical results are shown minus a constant of integration 86
5.5 Absolute Error variation over a 30×30 grid using a strip map . . . . . 93
5.6 AbsoluteErrorvariationforimprovedpotentialflowsolutionforcircular
half-cylinder. The same colour scale has been used as for Fig. 5.2 to
illustrate the decrease in error more effectively. . . . . . . . . . . . . . . 94
5.7 The vertical axis indicates at which radial value the minimisation is
largest. These results were obtained from a grid with largest radius
equal to 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.8 Variation in absolute error with changes in vertex density. . . . . . . . 96
5.9 Maximum(right-handscale)andAverage(left-handscale)relativeerror
with polygon vertex density . . . . . . . . . . . . . . . . . . . . . . . . 97
5.10 Relative error of estimation of π for polygons of increasing number of
vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.11 Circular cylinder problem in both the physical and computational strip
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7
5.12 Average(solid)andmaximum(dashed)AbsoluteError(circle)andRel-
ative Error (square) in the numerical velocity potential around the cir-
cular half-cylinder computed using a strip map. . . . . . . . . . . . . . 100
5.13 Thecomputationtime(inseconds)requiredforconstructionofthemap-
pingfunction(top)andevaluationofthemappingfunctionoverthegrid
points (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.14 Variation of computation time involving map inversion with number
of grid points. The upper and lower bounds relate to the convergence
ability of the iterative process used to invert the map. The linear plot
is selected arbitrarily for illustration of the trend. . . . . . . . . . . . . 104
5.15 Identification of the two potential problems to be solved to obtain the
Kirchhoff vector components for the 3D problem of a hemispherical
projection from a rigid half-plane. The source is located at y and the
observer and its ‘image’ are at x and x¯ respectively. These latter two lo-
cationsareillustrativepurposesonlyandwillbeallowedtovarywhereas
the source positions stays fixed. In practice, varying the source for a
fixed observer position is also a useful activity to provide data on how
the source position affects a particular observer location. . . . . . . . . 106
5.16 Two baffle geometries analysed with the array of source positions shown
as circles. Note that the two horizontal-axis scales are slightly different
for clarity but the spacing of the source locations is identical in both
cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.17 Variation of the magnitude of the gradient of the Green’s function with
source position two baffle geometries. The circles correspond to the
thinner baffle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1 General 2D BVP geometry with the boundary segmented into those of
type 1, type 2 and type 3. . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 An arbitrary 2D waveguide which is assumed to be a parallel-sided 2D
waveguide beyond truncation boundaries denoted Γ . . . . . . . . . . 121
3
6.3 Variation in Execution Time (bottom) and Relative Error (top) with
frequency for 3 different mesh densities. Solid blue line n = 5; dashed
green line n = 33, dotted red line n = 60. . . . . . . . . . . . . . . . . . 128
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6.4 Variation in Execution Time (bottom) and Relative Error (top) with
mesh density for 3 different frequencies. Solid blue line f = 5 Hz;
dashed green line f = 17.5 Hz, dotted red line f = 30 Hz. . . . . . . . . 129
6.5 Absolute error in the amplitude against the number of elements per half
wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.6 Variation in RMS relative error with frequency f and terms in the DtN
boundary expansion M for selected mesh densities n. . . . . . . . . . . 136
6.7 Variation in execution time with frequency f for n = 500. . . . . . . . . 137
6.8 Variation in RMS relative error with mesh density n and terms in the
DtN boundary expansion M for selected frequencies f. . . . . . . . . . 138
6.9 Variationinexecutiontimewithmeshdensityforf = 20Hzandf = 20
kHz for range of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.10 Variation in RMS relative error with frequency f and terms in the DtN
boundary expansion M for selected cross-channel mesh densities n. . . 140
6.11 Variation in RMS relative error with mesh density n and terms in the
DtN boundary expansion M for selected frequencies f. . . . . . . . . . 142
6.12 2D Green’s Function for the upper half of a throttle with the blue dots
representing the element centres. . . . . . . . . . . . . . . . . . . . . . 143
6.13 Qualitative comparison between BEM (left) and analytical (right) eval-
uations of the 2D Green’s function for the Helmholtz equation for a
source located above an infinite half-plane (bottom edge of image) . . . 145
7.1 3 Sinc Basis Functions (shifted to their centring points (circles) and
transformed to interval (0,2). The blue curve is k = −1, the green
k = 0 and the red k = 1.) . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Results for the analytical expression Eqn. (7.22) (top row) and Sinc-
Galerkin M=32 (bottom row) for a source centred in the unit square . . 172
7.3 Relative error (top) and execution time in seconds (bottom) for Sinc-
Galerkin for M = 12 to M = 36 (horizontal axis) for a centred source.
The limit of exponential behaviour according to Eqn. (7.23) is shown
in black for each frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.4 ResultstheanalyticalexpressionEqn.(7.22)(toprow)andSinc-Galerkin
M=32 (bottom row) for a source near the boundary of the unit square 176
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7.5 Relative error (top) and execution time in seconds (bottom) for Sinc-
Galerkin for M = 12 to M = 36 (horizontal axis) for a near-boundary
source. The limit of exponential behaviour according to Eqn. (7.23) is
shown in black for each frequency. . . . . . . . . . . . . . . . . . . . . . 178
8.1 Error in the BEM solution for 994 uniformly distributed linear elements 195
8.2 Results for for the BEM n=12 (bottom row), analytical expression
Eqn.(7.22)(middlerow)andSinc-GalerkinM=32(toprow)forasource
centred in the unit square . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.3 Relative error (top) and execution time in seconds (bottom) for Sinc-
Galerkin (squares) and BEM (circles) for the range of mesh densities
tested (horizontal axis) for a centred source. The corresponding black
lines represent the limit of exponential convergence as computed using
Eqn. (7.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.4 Absolute Error for a centre source for Sinc-Galerkin (squares) and BEM
(circles) for f=20 (solid blue), f=680 (dashed green), f=1340 (dotted
red), f=2000 (dot-dash cyan). The corresponding black lines represent
thelimitofexponentialconvergenceascomputedusingEqn.(7.23). The
horizontal axis ticks represent an increase in the mesh density compa-
rable in both methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.5 Results for for the BEM n=12 (bottom row), analytical expression
Eqn.(7.22)(middlerow)andSinc-GalerkinM=32(toprow)forasource
near the boundary of the unit square . . . . . . . . . . . . . . . . . . . 205
8.6 Relative error (top) and execution time in seconds (bottom) for Sinc-
Galerkin (squares) and BEM (circles) for the range of mesh densities
tested (horizontal axis) for a near-boundary source. . . . . . . . . . . . 206
8.7 Graphical depiction of guidelines to aid selection of the most suitable
method for calculation of the Green’s function for the given problem.
For clarity, the formal decision notation of a diamond has been changed
to a rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
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