The Pennsylvania State University
The Graduate School
Graduate Program in Acoustics
A WAVE SUPERPOSITION METHOD
FORMULATED IN DIGITAL ACOUSTIC SPACE
A Dissertation in
Acoustics
by
Yong-Sin Hwang
© 2009 Yong-Sin Hwang
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
May 2009
The dissertation of Yong-Sin Hwang was reviewed and approved* by the following:
Gary H. Koopmann Distinguished Professor of Mechanical Engineering Dissertation Co-Advisor Co-Chair of Committee
Victor W. Sparrow Professor of Acoustics Dissertation Co-Advisor Co-Chair of Committee
Martin W. Trethewey Professor of Mechanical Engineering
John B. Fahnline Assistant Professor of Acoustics Research Associate, Applied Research Laboratory
Anthony A. Atchley Professor of Acoustics Head of the Graduate Program in Acoustics
*Signatures are on file in the Graduate School
ii
ABSTRACT
In this thesis, a new formulation of the Wave Superposition method is proposed
wherein the conventional mesh approach is replaced by a simple 3-D digital work space
that easily accommodates shape optimization for minimizing or maximizing radiation
efficiency. As sound quality is in demand in almost all product designs and also because
of fierce competition between product manufacturers, faster and accurate computational
method for shape optimization is always desired. Because the conventional Wave
Superposition method relies solely on mesh geometry, it cannot accommodate fast shape
changes in the design stage of a consumer product or machinery, where many iterations
of shape changes are required. Since the use of a mesh hinders easy shape changes, a new
approach for representing geometry is introduced by constructing a uniform lattice in a 3-
D digital work space. A voxel (a portmanteau, a new word made from combining the
sound and meaning, of the words, volumetric and pixel) is essentially a volume element
defined by the uniform lattice, and does not require separate connectivity information as a
mesh element does. In the presented method, geometry is represented with voxels that
can easily adapt to shape changes, therefore it is more suitable for shape optimization.
The new method was validated by computing radiated sound power of structures of
simple and complex geometries and complex mode shapes. It was shown that matching
volume velocity is a key component to an accurate analysis. A sensitivity study showed
that it required at least 6 elements per acoustic wavelength, and a complexity study
showed a minimal reduction in computational time.
iii
TABLE OF CONTENTS
LIST OF FIGURES .....................................................................................................v
LIST OF TABLES.......................................................................................................vii
ACKNOWLEDGEMENTS.........................................................................................viii
Chapter 1 Introduction ................................................................................................1
1.1 Motivation.......................................................................................................1 1.2 Background.....................................................................................................5 1.3 Proposed work ................................................................................................7
Chapter 2 Meshfree Method and Voxelization...........................................................16
2.1 Introduction to the Meshfree Method .............................................................16 2.2 Representation of a solid in 3D space.............................................................18 2.3 Voxelization....................................................................................................20
Chapter 3 Method of Wave Superposition and Interior Points...................................29
3.1 Method of Wave Superposition .....................................................................29 3.2 Singularity in the Method of Wave Superposition and Self-Terms ...............33 3.3 Non-Uniqueness Problem...............................................................................36
Chapter 4 Digital Space Method.................................................................................39
4.1 Voxelization of Solid Surface.........................................................................39 4.2 Volume Velocity Approximation ...................................................................41
Chapter 5 Validation of the new method: Test Cases and Results .............................48
5.1 Voxelized Geometry From a Mesh Sphere ....................................................48 5.2 Voxelized Sphere From an Analytical Description ........................................55 5.3 Voxelized Cylindrical Tank, Cylinder With Two Hemi Spherical Caps .......59 5.4 Complexity Study ...........................................................................................67 5.5 Sensitivity Study.............................................................................................70
Chapter 6 Conclusions and Future Work....................................................................76
6.1 Conclusion ......................................................................................................76 6.2 Assessment of the Accuracy of the Digital Space Method.............................79 6.3 Future Work....................................................................................................80
Bibliography ................................................................................................................82
iv
Appendix A Sample Matlab code for the Digital Acoustic Space Method ...............85
v
LIST OF FIGURES
Figure 1-1: Easy Adaptation of Shape Change............................................................4
Figure 1-2: Flowchart of processes in the conventional Wave Superposition method ..................................................................................................................10
Figure 1-3: Representation of line mesh in 2D digital space......................................12
Figure 1-4: Flowchart of processes in the proposed Digital Acoustic Space method ..................................................................................................................13
Figure 2-1: Boundary, and Volumetric representation of a sphere..............................19
Figure 2-2: Use of Voxels in Various Disciplines .......................................................21
Figure 2-3: Example of Voxelization Threshold for Difference Geometry ................23
Figure 2-4: Application of Threshold ..........................................................................24
Figure 2-5: Example of Voxelization Threshold for Connected Geometry ...............25
Figure 2-6: Voxelization by sweeping along an axis...................................................26
Figure 2-7: Mesh vs. Voxel .........................................................................................27
Figure 3-1: Arbitrarily shaped vibrating structure. s’s represent sources and σ ’s represent discretized surface elements on the surface ..........................................32
Figure 4-1: Activating Voxels in Digital Space...........................................................40
Figure 4-2: Voxelization of the first octant of a sphere ..............................................42
Figure 4-3: 2D voxelization of the first octant of a sphere ..........................................43
Figure 4-4: 2D voxelization of the first octant of a sphere, Digital Space method version...................................................................................................................44
Figure 4-5: Adjustment of Volume Velocity for Voxelized Sphere............................46
Figure 5-1: Mesh Representation of the Modified Dodecahedron ..............................49
Figure 5-2: Voxelization of the Sphere Mesh..............................................................51
Figure 5-3: Voxel Representation of Modified Dodecahedron ...................................52
Figure 5-4: Sound Power from voxelized sphere ........................................................53
vi
Figure 5-5: Sound Power from voxelized sphere with internal points ........................54
Figure 5-6: Setting a Threshold for a Curved Plate .....................................................56
Figure 5-7: Voxelized Pulsating Sphere; 1 m/ssv = , Threshold, t = 0.2 m ..............57
Figure 5-8: Voxelized Pulsating Sphere; 1 m/ssv = , Threshold, t = 0.1 m ..............58
Figure 5-9: Voxelized Pulsating Sphere; 1 m/ssv = , Threshold, t = 0.05 m .............58
Figure 5-10: Digital Representation of Cylindrical Tank ............................................59
Figure 5-11: Breathing mode for Cylindrical Tank with vs = 1 m/s ............................60
Figure 5-12: Displacement plot of Mode (1,8) ............................................................61
Figure 5-13: Displacement plot of Mode (1,10) .........................................................62
Figure 5-14: Displacement plot of Mode (2, 10) ........................................................62
Figure 5-15: Radiated Power from cylindrical model vibrating in (1,8) mode ...........64
Figure 5-16: Radiated Power from cylindrical model vibrating in (1,10) mode ........64
Figure 5-17: Radiated Power from cylindrical model vibrating in (2,10) mode ........65
Figure 5-18: Radiated Power from finer cylindrical model vibrating in (1,8) mode..66
Figure 5-19: Radiated Power from finer cylindrical model vibrating in (2,10) mode .....................................................................................................................67
Figure 5-20: Complexity Study for Computation Cost................................................69
Figure 5-21: Simulated execution time difference for Digital Space method vs. Wave Superposition method.................................................................................70
Figure 5-22: Cubic Box with a Spherical Source 71
Figure 5-23: Radiated Power from a Pulsating Box 73
Figure 5-24: Radiated Power from a Pulsating Box, magnified at ka = 3.7 to 4 74
vii
LIST OF TABLES
Table 3-1: Summary of Greens functions for Pressure and Velocity monopole self term.......................................................................................................................36
viii
ACKNOWLEDGEMENTS
I’d like to express my sincerest appreciation to all who helped me to finish my
doctoral thesis and made my time in State College a much enjoyable experience.
Especially Dr. Gary H. Koopmann and Dr. Victor W. Sparrow who served as co-chair
persons, I’d like to thank them for their tremendous support and unlimited thesis ideas,
direction and supervision. I’d like to also thank Dr. John Fahnline and Dr. Martin
Trethewey for their time and advices to complete my thesis.
To my family, I give them a million thanks. I thank my sister Hye-Won Hwang,
her husband Soo-Hyuk Choi and my cute nephew Myung-Joon for their encouragement
and love from afar. To my mother, Sang-Eun An and my father Doo-Hwa Hwang, I am
forever indebted. They bore me, raised me, taught me and supported me in a loving
environment. Without their continuous encouragement and support, my thesis would not
have been possible.
I am deeply grateful to my fiancée Kelly Jung. Her love, emotional support, and
encouragement helped me get through difficult times and complete this thesis. And to my
friends, here in State College and other places, I thank them for their friendship that
helped me to become a better friend and a better person.
Last, but not least, I’d like to extend special thanks to the faculty and staff
members of the Graduate Program in Acoustics for their friendship and support.
Chapter 1
Introduction
1.1 Motivation
With improvements in today’s quality of living, sound quality has become a
major part of well-being. Examples include buildings with sound proof windows in urban
areas or perhaps, better sounding stereo speakers. On the other hand, it is an established
fact that noisiness is one of the main discriminators for consumers. According to a recent
article in the Wall Street Journal, orders for Raytheon’s new executive jet, the Premier I,
were cancelled because of perceived noise inside of the cabin. Clearly, sound quality
issues are driving industries to meet the expectation of consumers. Therefore, if the shape
of a machine or product is to be designed optimally from a sound radiation perspective,
computational acoustics must be an integral component in the overall design process at
the earliest stages. Naturally, many design tools and simulation methods are already
available to help industries design and manufacture optimal products. However, finding
an optimal product design is not a trivial task, and it takes many iterations of design
changes. In our competitive time, companies understand that the impact of accurate and
fast processes at the design stage is huge and this fact motivates improvements in design
tools such as finite element and boundary element methods. This thesis focuses on a
fundamental change in the meshing routines common to these two methods.
2
In conventional Finite Element Methods (FEM) and Boundary Element Methods
(BEM), to build a mesh, one has to divide the domain of the problem into well-
proportioned, non-overlapping elements that fit the geometry perfectly. Moreover, the
connectivity of nodes, which are the vertices of the elements hence defining the elements,
is also needed at all times. Having to make a mesh and do a new analysis, each and every
time, is indeed a tedious process. Furthermore, for building complex models, meshing is
not a trivial task. Also, one needs to ensure correctness of the mesh even with
commercialized programs and, to check and maintain the integrity of a mesh that require
thorough inspection by a human and much human effort. For instance, in order to obtain
an acceptable computational solution, one has to ensure that;
1. All elements are well-proportioned. The ratio of the largest and the smallest
dimensions of an element should be close to one, ensuring that there are about
equal number of elements per wavelength in all directions.
2. A mesh does not contain a gap or a hole. Otherwise they may create an
erroneous boundary condition.
3. Size of neighboring elements may not change drastically ensuring gradual
transition from larger to smaller elements or vice versa.
4. Element distortion is not desired. Distortions must be monitored and
minimized to ensure the accuracy of the solution.
In order to use mesh geometry in a shape optimization, it is necessary to rebuild
the mesh or use highly complex adaptive remeshing algorithms to accommodate any
change in shape. This process is repeated every time one needs to modify geometry, even
for a small change. Moreover, the use of mesh geometry can be quite difficult to use in a
3
shape optimization process because of the reasons mentioned above. Thus, having to
remesh or use complex remeshing algorithm definitely reduces computational efficiency
since many iterations of shape changes are necessary for obtaining an optimal design.
Therefore, considering the difficulties that mesh geometry imposes on changing
geometry of a structure, the use of mesh elements are not suitable for a shape
optimization process.
There are a number of techniques and methods that address these issues in
computational methods, namely meshless methods [3-6]. These methods, although they
alleviate the burden of meshing, still demand heavy computation in the analysis phase.
This is especially true for BEM since it has a full matrix to invert, in which each element
of the matrix is a surface integral of a Greens Function describing the relationship
between pressure and particle velocity.
If one can adopt a new way to represent a shape, free from the restrictions that
meshing imposes – a necessity to remesh or use complex remeshing algorithms, it is safe
to say the new approach would be more suitable for a shape optimization. This thesis
proposes the following computational methodology. An arbitrary shape is represented by
uniformly spread digital nodes similar to the idea of pixels. In order to accommodate a
change in the shape, one only has to switch on or off the digital nodes for the new shape.
As shown Figure 1-1, nodes are distributed uniformly in a 2-D digital space (Blue nodes).
For this example, let the blue nodes be inactive or switched off and the red nodes be
active or switched on. Draw an arbitrary shape in the digital space and turn on the
appropriate nodes to represent the shape (Red nodes). When a change in shape is made by
moving the top upward, the new shape can be easily represented by turning on a few
4
more appropriate nodes to represent the change. In this way, there is no need to neither
rebuild the entire space nor use a complex algorithm to accommodate the change. It is
also intuitively a desirable model to be used in a shape optimization.
One minor benefit comes from the uniformly spread digital work space. In
conventional BEM methods, a Greens Function where the source point and the field point
that are collocated becomes singular and requires appropriate integration methods to
integrate over the element [7, 8]. If the element areas are different from one another, and
generally so, it is necessary to integrate over each and every collocated element to
calculate the surface integral of the problem domain to populate the system matrices.
However, with uniformly distributed nodes, all nodal areas are kept equal therefore;
integrations for the singular Greens functions can be reused for all nodes. In other words,
because the integration over a collocated term only depends on the element size and
shape, the elements of the same size and shape result in the same integration value.
Figure 1-1: Easy Adaptation of Shape Change
(a.) before (b.) after
5
Therefore one can expect a reduction in computational time for constructing the system
matrices by reusing integration value for matrix elements.
In this thesis, an improvement to the meshing procedure used in the present
conventional Wave Superposition method is reformulated by describing the surface
geometry of an acoustic radiator in digital work space. First, a digital space, filled with
uniformly distributed points or nodes is constructed. The uniformly digitized space
allows equal nodal area for each and every node throughout the entire space. Therefore,
all nodes are described in relation to its location for forming a contiguous shape and there
is no need for keeping connectivity information.
1.2 Background
In general, FEM and BEM have been used widely for various problems including
acoustic problems. FEM has also been used to solve interior acoustic problems. These
problems are characterized by bound geometry and a linear set of equations solved by
various methods. BEM has been used to solve radiation problems. These problems are
characterized by shell-like or hollow geometries that require numerical integration over
geometry surfaces. In these numerical methods, generating and analyzing a complex
structure have always been time consuming tasks. In conventional techniques, one needs
to generate a grid, to define a model shape in the grid, to generate mesh, to define node
connectivity of the mesh and then to solve for a system of equations constructed from the
nodes. The major issue is that every time there’s a change in the mesh, the entire meshing
process has to be repeated.
6
Two major methods, the Wave Superposition method and Meshfree methods,
answer to the issue in question. The Wave Superposition method has the potential to
adopt a different – free of mesh - approach of representing a geometry [9]. Meshfree
methods are developed by many researchers to respond to the need to better model a
crack propagation, fractures in a structure and large deformation and have many different
names; Diffuse Element [10], Element Free Galerkin [11], Principle of Unity [12, 13],
hp-Clouds [14] to name a few. In conventional finite element methods, with large
deformation of geometry, repeated meshing is required to avoid the breakdown of
calculation due to excessive distortion. All meshless methods only require a set of
scattered nodal points that represent the geometry of interest. Hence there is no need to
undertake multiple steps to generate meshes; hence, it takes much less time to generate a
complex model than it would for conventional finite element methods and boundary
element methods. Many of them, however, appeared from approximation techniques such
as Moving Least Squares method (MLS) to approximate a true surface of a structure with
unorganized nodal points and do not interpolate nodal values. Consequently, they do not
exactly reproduce essential boundary conditions[4, 15]. Moving Least Squares method is
an approximation method for reconstructing continuous functions from a set of nodes, to
obtain a weight function [16]. The biggest weakness of this method is that it takes a set of
equations to solve for each node; therefore it is computationally more costly in practice.
The Superposition integration method arose from calibration procedure often used
in boundary element studies and is equivalent to the Helmholtz-Kirchhoff integral[6, 12].
In this method, one defines a set of nodal points spread on the surface of the geometry
and sets them as simple sources. This method inherently is similar to meshless methods
7
in modeling of complex shapes and is simpler to describe their acoustical fields than
existing Boundary Element Methods[17, 18]. Two problems can be tackled for
improvement; internal modes of a model due to the non-uniqueness problem and the
inherent singularity problem of the Greens Function used in the Superposition integration
method. For the non-uniqueness problem, one can use internal points or surfaces that give
a boundary condition inside the geometry, or instead of placing sources inside of the
geometry, one can place them outside[19-21]. For the singularity problem, Eric Constans
showed an analytical expression for the Green’s Function response where the source
location and the observation location coincide. He used a coordinate transform to
overcome the inherent singularity problem in the Green’s Function[22]. Zellers et al.
followed up on this idea and completed analytical expressions for monopole pressure and
velocity [9].
1.3 Proposed work
This thesis will focus on building a discrete digital space in an attempt to develop
a more suitable computational method to be used in a shape optimization, therefore to
give an answer to the industrial need of a fast and reliable tool for product design. This
new method is largely based on the method of Wave Superposition. Naturally, in order
to show the improvements of the Digital Acoustic Space method, it is mainly compared
to the method of Wave Superposition.
8
To begin, it is of value to mention that the method of Wave Superposition
comprises some of the advantages of meshless methods. The two main characteristics of
the most current meshless methods are;
1. use of nodes, free from mesh elements.
2. localized approximations to system’s solution.
First, instead of using a mesh, one can freely populate an area with nodal points to
investigate the system without heavy overhead due to the connectivity of mesh elements.
Second, the solution of the system is obtained locally. This second characteristic
emphasizes the usefulness of the first. By approximating the solution of the system
locally, adding or removing nodes won’t affect the overall accuracy of the solution.
Rather it enhances the accuracy of the solution for a structure that features large
deformations, discontinuities or parts of irregular and complex shapes that were
conventionally difficult to model and analyze properly.
The method of Wave Superposition does not have all the enhancements that
meshless methods have as mentioned above. In order to obtain a solution, unlike
meshless methods, all elements have to be solved simultaneously. Similarly to meshless
methods, however, the method of Wave Superposition permits using only points in space
for constructing system matrices, when surface velocities are given as boundary
conditions. This feature allows the method of Wave Superposition to describe
geometries in a “meshless” fashion and is exploited in the Digital Acoustic Space method.
To understand the new method, it is imperative to understand the method of
Wave Superposition. Figure 1-2 shows the logical flow of the Superposition method. In
the Superposition method, the matrix inversion process is the most time consuming part
9
in the analysis phase. Evidently, developing a faster inversion algorithm is definitely a
way to improve the computation speed, and there certainly are researchers dedicated to
finding faster and more optimal matrix inversion algorithm. For this thesis, however, a
general matrix inversion function, inv(M), where M is square matrix, supplied by Matlab,
is sufficient as the main objective of this thesis is to develop a new approach to represent
a geometry, suitable for a digital geometric work space in conjunction with the Wave
Superposition method.
One can also find another place to minimize computation efficiency other than a
faster inversion algorithm. It is a numerical integration of uniform elements. Typically a
numerical integration takes many operations to complete and, it also needs to be done on
every element. If one can reduce the number of integrations to perform, that translates to
faster computation. The reduction in the number of integrations is possible if there is an
analytical expression that can replace the integral and if one integral over an element can
be used repeatedly for other elements.
10
Figure 1-2: Flowchart of processes in the conventional Wave Superposition method
No
Yes
Define or import geometry mesh
Calculate acoustic element location from geometry
Calculate surface normals
Analysis
Construct system matrices with Greens functions G and G∇
Define surface velocity, v
Solve for source strength, s from given velocity, v
Wave Superposition Method
Geometry
Is Design requirement
met?
End
[ ][ ]G
G∇
{ } [ ] { }1s G v−= ∇ ×
Calculate press-ure, p and radia-tion power, ∏
{ } [ ] { }p G s= ×
{ } { }*1 Re[ ]2
p vΠ = ×
Calculate Singular Greens function (diagonal terms) and regular Greens function (off-diagonal )
Use Gaussian quadrature,
1( )
n
i ii
Gds G sω=
≈∑∫
when 0r = , for all acoustic elements
Modify mesh geometry
11
In building the system matrices, [ ]G and its gradient, [ ]G∇ the field-source
collocated Greens function requires special attention when integrated over the element
surface, since the integrand is singular. Singular functions are usually not possible to
integrate unless solved analytically or numerically avoiding the singular point.[23] These
singular integrals in the Wave Superposition method are independent of the surface
normal because the position vector between the source and the field point are always the
same. This leads to the main focus of the thesis. If the limits of the integral are kept the
same, these singular integrals can be calculated only once and applied to all collocated
elements. The limits are determined by the area of an element; hence, by keeping the area
congruent, one can achieve the necessary condition. There are not many platonic solids
that guaranties equal element area on the surface. This implies that it is not trivial to
make a platonic solid that guaranties equal area for all of its surface elements let alone
make an arbitrarily-shaped volume. The most feasible shape of lattice in this case is the
cube. A cube is one of the platonic surfaces and also can be divided into infinitesimally
small and congruent cubes. Therefore, a cube can support any size mesh and give an
equal surface area to all nodes.
12
The proposed Digital Acoustic Space method allows uniform nodal areas; hence,
one integration for all collocated nodes is sufficient. In essence, this is a work space in
computer memory filled with nodes that are uniformly distributed much like graph paper.
When a geometry is given, only the nodal elements that describe the geometry will
remain and be assigned with velocities and surface normals. Figure 1-3 shows a simple,
arbitrarily shaped line mesh and how it is represented in 2D digital space. In this case, the
elements on the line mesh are denoted with black dots and arrows for normal vectors. The
locations of the elements are then translated by finding the closest grid point for the
Figure 1-3: Representation of line mesh in 2D digital space
13
actual locations. Normals that are calculated from the mesh are directly imported and
used in building the system matrices, [G] and its gradient.
Figure 1-4: Flowchart of processes in the proposed Digital Acoustic Space method
Modify Geometry
Yes
Import geometry
Activate/ Deactivate voxels
Import surface normals and
surface velocity, v
Analysis
Construct Greens function matrices of Active nodes [ ] [ ],G G∇
Calculate Singular Greens functions (diagonal) and reg-ular Greens func-tions (off-diagonal)
Solve for source strength, s
Digital Acoustic Space Method
Digital Space & Geometry
Is Design requirement
met?
End
[ ] [ ],G G∇
{ } [ ] { }1s G v−= ∇ ×
Calculate pressure, p { } [ ] { }p G s= ×
Calculate sound power, ∏ from p and v
{ } { }*1 Re[ ]2
p vΠ = ×
Define Digital Acoustic Space
Use Gaussian quadrature,
1( )
n
i ii
Gds G sω=
≈∑∫
when 0r = , for all acoustic elements
No
14
Figure 1-4 shows the logic flow of the proposed the Digital Acoustic Space
method. First, a digital space with uniformly distributed grid points is built. Each grid
point is the center of a cube representing a single voxel. Then the element locations from
the mesh are imported to the digital work space. When the geometry information is
imported, then one can activate or deactivate the voxels. Activation means that a part of
the structure’s surface occupies, or passes through a voxel. Therefore, that voxel is
included in the system matrices and the expression deactivate simply means it is
excluded from it. The final step of the geometry importation is the importation of the
surface normals and the volume velocity distribution that are obtained from the mesh and
usually a FEM analysis, respectively. The analysis part of the Digital Acoustic Space
method follows much of the conventional Superposition method. The main difference
from the Superposition method is that, in building system matrices, computation time is
reduced by using one single singular integral for all field-source collocated elements.
Also, when the design requirements is not met and structural modification is required,
instead of re-meshing the entire geometry and repeating the whole geometry portion of
the processes, one can topologically activate or deactivate grid points to achieve
necessary changes which tremendously helps to reduce overall computation time.
The new method, unfortunately, requires certain overheads that were not present
for the conventional Wave Superposition method. First, importing a geometry requires
calculation of new element locations on grid points of the digital work space. At the
worst case, if the mesh sizes are drastically different from one element to another, the
mesh needs to be re-meshed or a clever voxelization scheme is needed to remedy the
problem. For example, a larger mesh must be divided into a smaller mesh, or it needs to
15
be represented by more than one voxel. If the mesh is too small, one voxel needs to
include more than one patch from the mesh and appropriate geometric information, by
means of averaging or interpolation, must be assigned to the voxel. On the other hand, if
the mesh is conditioned to even mesh size or the geometry is described directly in the
Digital Space, the overhead is minimized or eliminated completely.
The remainder of this thesis is organized as follows. Chapter 2 describes current
meshless methods and the voxelization techniques. The relevance between the two
methods is briefly discussed. Chapter 3 introduces the reader to the method of Wave
Superposition. This chapter also explains the rational for choosing the method of Wave
Superposition for the Digital Acoustic Space method. Chapter 4 gives an explanation on
the implementation procedure of a voxelized geometry, combined with the method of
Wave Superposition method in the Digital Acoustic Space. A validation of the proposed
method by computing radiated sound power from structures of simple and complex
geometry is provided in Chapter 5. A sensitivity study and a complexity study are also
presented in chapter 5. Finally, in Chapter 6, some conclusions and points toward future
work are provided.
Chapter 2
Meshfree Method and Voxelization
2.1 Introduction to the Meshfree Method
In order to lighten the burden of meshing a geometry, several numerical methods
have been developed. These are Diffuse Element method (DEM) [3], Element-Free
Galerkin method (EFG) [4], hp-clouds (HP) [5, 6], and Natural Element method (NEM)
[7], to name a few. All of these methods utilize irregularly distributed particles or nodes
in a problem domain combined with weight function or kernel function to approximate
the solution only locally confined by the domain of influence, hence solving the problem
piecewise, although the means of approximating solutions are different. In this section,
some of the Meshless methods are reviewed and their relevance to the Digital Space
Method is explored.
Probably the most avant-garde Meshless method known to numerical method
community is Smoothed Particle Hydrodynamics(SPH) [8]. SPH first used the concept of
using discrete particles and kernel function that govern the amount of influence by
particles determined by distance between them, within the domain of influence. While
SPH was conceived in fluid dynamics, another Meshless method came to be known in
solid mechanics. Developed by Nayroles et al. in 1992, DEM was introduced as a
generalized FEM with improvements such as removing limitations of the regularity of
approximated functions and most notably, removing the mesh generation requirement [3].
17
DEM uses a method for function approximation from a set of points which is found out to
be Moving Least Squares (MLS) technique to locally approximate the solution to the
system’s governing partial differential equation.
Belytschko et al. followed the work of Nayroles et al. and introduced their
Element-Free Galerkin method (EFG) in 1994 [4]. The EFG method, much similar to
DEM, does not require mesh elements and also employs a function approximation
technique (MLS) that they credited to Lancaster and Salkauskas.[9] It was also
Belytschko et al. who identified that Nayroles et al. employed MLS that provides a
function approximation for an arbitrary set of nodes.
In 1998 Atluri and Zhu developed a true meshless Local Petrov-Galerkin method
(MLPG) as this did not require element mesh for neither the interpolation of solution nor
for Gaussian integration.[10] Among other notable meshless methods, the H-p clouds
method developed by C. A. Duarte and J. T. Oden in 1996 also uses MLS to build a
Partition of Unity (PoU). [5, 6]
All of the above meshless methods share one common feature. The most notable
feature is that they all use freely distributed nodes in 3D space, hence “meshless”.
Moreover, they no longer require mesh elements to approximate a solution, but rather a
cluster of nodes to solve the system’s governing partial differential equation only locally.
Since the solution of a system is obtained from summing local solutions, nodes can be
easily added or removed without affecting the overall solution and without having to
mesh a geometry anew. Considering these advantages, meshless methods are much more
attractive alternatives to FEM which is more difficult and computationally more
expensive at the mesh generation stage.
18
Unlike meshless methods, the method of Wave Superposition and the Digital
Space Method do not a use domain of influence to limit coupling between nodes and
obtain solutions locally. In this regard, the meshless methods and the two latter methods
are different. However, in spite of the discrete formula used in the method of Wave
Superposition, hence the Digital Space Method, one can set up the system into nodal
points as acoustic elements. The ability to formulate the entire problem domain in terms
of digital node point of the Wave Superposition method allows it to be an excellent
candidate for a shape optimization scheme. Moreover, this is the first successful case
where the method of Wave Superposition is incorporated with a meshless feature in a
digital topology – an on or off switch on a grid of workspace. While there was an attempt
to incorporate meshless feature, also in digital topology, to the method of Wave
Superposition by Zellers[11], the surface normal calculation was not robust enough to
handle various surfaces.
2.2 Representation of a solid in 3D space
In this section, two different digital representations of a solid is briefly discussed.
As the word digital may suggest, in computer graphics, all images are represented with a
discrete picture element (pixel) without exception whether the information about the
actual shape is either numerical or analytical. This implies that, in order to represent a
solid in a computer domain for the purpose of the numerical method posed in this thesis,
one must convert a solid into a discrete format.
19
There are mainly 2 ways to digitize and represent a solid in 3D digital space. One
is boundary representation (B-Rep) where only surface is defined by small, connected
planar patches. The other is volumetric representation where the volume of the solid is
represented by volume elements (voxel), i.e., an extension of the 2D pixel concept to a
3D volume. Figure 2-1 shows examples of boundary representation and volumetric
representation of sphere.
In B-Rep, nodes are distributed on the solid’s surface and by connecting the nodes,
polygons, typically triangles or quadrilaterals that represent the surface, are formed. B-
Rep can be more efficient and easier than volumetric representation to work with.
Naturally, most commercialized boundary element methods use a shell mesh or surface
mesh to construct their system of matrices.
Volumetric representation, on the other hand, uses voxels to build the volume of a
solid. The most common building mechanism is to put the center of voxels on the vertices
Figure 2-1: Boundary, and Volumetric representation of a sphere
20
of a cubic lattice that are comprised within the volume of the solid to evenly fill its
volume.[12] Since only the center of a voxel is considered for determining the position of
a voxel that will represent the volume, often a volumetric threshold is given when
determining the position of voxels in order to give a better representation of the actual
shape.[13] For instance, when the surface of a solid passes through a space occupied by a
voxel, and yet, the center of that voxel is marginally outside the solid, then the voxel in
question may not be accounted for building the solid according to the simple method
described above, while it should be included. A detailed scheme of voxelization is given
in the following section.
2.3 Voxelization
Use of voxels is fairly new in the domain of BEM [11]. However, in other
disciplines, voxels are popular for their ability to represent 3D data sets. For instance, it’s
been used widely in the computer gaming community. With the emergence of 3D games
in the mid 1990’s, voxels started to replace crude pixels since they could represent a more
realistic 3D world that can form and reform on a player’s whim such as 3D terrain and
buildings [14]. In the scientific community, voxels are popular for their ability to present
3D data sets. A most notable use is in magnetic resonance imaging (MRI) [15]. Once a
3D scan of a human or another object is scanned, the data is built in a 3D body. Since the
data is built in 3D, one can easily take a cross-section of 3D data set and examine the
intended area with no difficulty. Figure 2-2 shows some examples of voxel application in
some disciplines.
21
There are many algorithms to meet different use and requirement of voxelization
and there are several ways to ensure geometry is voxelized properly. While voxelization
is a research topic pursued by many in computer science and related fields, in this thesis,
a simple, yet effective voxelization scheme is used. In this section, the most common
methods for voxelization are utilized.
Before one can voxelize a geometry, defining the size of voxels and of a digital
work space is required. The voxel size solely depends on the application while the size of
a digital space depends on the capacity of the computer memory size [12]. Since this
thesis will use voxel geometry in a boundary element analysis, the primary factor for
determining the voxel size is the accuracy of the analysis. As the general rule of
numerical methods states, it requires at least 6 elements per wavelength to expect an
acceptable result [16]. Hence, one must consider the frequency range for determining an
Figure 2-2: Use of Voxels in Various Disciplines
In a medical application: high resolution MRI scan [2]
In computer graphics: Model of the Earth[1]
22
appropriate voxel size. On the other hand, the memory capacity of a computer is the
limiting factor for the digital space size. Because the proposed Digital Acoustic Space
method requires building a cubic lattice that can enclose an entire geometry of interest,
initial lattice size should be just large enough to model the geometry of interest within the
computer memory limitation.
In 3D space there are three shapes that can be voxelized: a point, a line and a plate.
Since a voxel is a volume element in a lattice system, the geometry to voxelize also has to
be in 3D. Therefore, whether the geometry in question is a point, a line, or a plate, one
needs to convert them into a 3D shape. For instance, a point can be represented with a
sphere, and similarly, a line with a cylinder and a plate with a box. However, one will
need a control mechanism for determining the size of each type of geometry. Thus,
leading to the concept of threshold. Figure 2-3 shows examples of threshold that control
voxelization. For a dot, the radius of sphere is given as the threshold limit so that it can
find an appropriate voxel in 3D space; for a line, the base radius of a cylinder that wraps
the line defines the threshold limit; and finally for a plate, the two parallel plates that
encompass the plate in the middle.
23
Depending on the size of threshold, a geometry can be represented differently in a
volume representation. Figure 2-4 shows a simple example of a dot with a different
threshold limit. As one can see below, depending on what threshold limit is given to a
geometry, in this example, a dot, it can either be represented with a single voxel or a
number of voxels. There is no single method to determine the correct threshold because,
after all, the validity of the final end result is judged by its application. This is especially
so because a voxel lacks the ability to locate its center of volume freely in 3D space. The
Figure 2-3: Example of Voxelization Threshold for Difference Geometry
t
t
Line
Plate
Dot
t
24
location is pre-determined by the lattice, and since the volume is represented with its
center, it has limited resolution for the general rendering of a surface.
In a more practical voxelization process, one will have to voxelize a combination
of simple geometries. For a combination of geometries, having only a threshold may not
be enough to select the right voxels. For instance, mesh elements are put together to
define the inside and outside of a geometry. In addition to the threshold mentioned above,
one must ensure the outside voxels are not included even if they fall within the threshold
limits. Figure 2-5 shows an example in 2D for simplicity. Two lines define the inside and
the outside and given rectangular threshold. When only a threshold is used for
Figure 2-4: Application of Threshold
t
Dot
t’
25
voxelization, the voxel P falls within the threshold; therefore, it has to be included among
the active voxels. However, when the projection is drawn from the two lines, the voxel P
is outside of the geometry. Thus, voxel P does not belong to the group of active voxels
that represent the geometry, and it must be excluded.
Once the type of the geometry and its threshold are determined, one can proceed
to voxelize an arbitrary shape. In the Figure 2-6, an example of sphere mesh is shown. A
most common voxelization scheme is to sweep through a geometry along an axis. At
each slice of a uniform lattice, one selects all voxels that comes within the threshold limit
Figure 2-5: Example of Voxelization Threshold for Connected Geometry
INSIDE
OUTSIDE
P
Active Voxels
26
as shown above. In order to ensure a solid voxel, one must perform a check to see if all
voxels have a neighbor. One may also perform several passes along a different axis.
Despite all the disadvantages, voxels provide a very attractive advantage that
counters its disadvantages. In a typical meshing process, if one wants to join or cut a part
of a structure, it is very difficult to do so once the parts are already meshed. Even when
geometry is not meshed, one has to pay special attention to a joint or cut part of structure.
Using voxels, one overcomes these difficulties because one no longer needs a
connectivity of nodes.
Figure 2-6: Voxelization by sweeping along an axis
Voxelization plane sweeping along an axis
Geometry Mesh i th slice in a digital space
Completion: voxel representation
obtained
27
Figure 2-7 shows parallel comparison between meshing and voxelization of a
model that joins two beam-like structures. When two parts are joined, for meshing, one
has to make sure that the attached area does not form an inner surface. If not, it will
create a discontinuity and result in wrong solution. In this case, only two parts are joined,
and it may seem trivial. However, when one tries to build a complex model, one may face
many operations that require human intervention of verifying proper construction of
volume, such as joining or cutting, for example. On the other hand, voxels do not see
boundaries inside a volume. In other words, voxels do not require connectivity. Therefore,
not only a discontinuity is transparent, but also one can freely add or remove voxels even
Figure 2-7: Mesh vs. Voxel
This section must be inspected
Mesh Voxel
Transparent to voxels
Enlargement
Nodes are not shared by neighboring elements
28
after the voxelization process is completed. This is the most advantageous feature that
reduces human effort in a design stage.
To summarize, this chapter shows the history of meshless methods and their brief
description to help readers understand difference between meshless methods and the
Wave Superposition method. After a short introduction to meshless methods, this chapter
also introduces readers to volume representation of a digital data in conjunction to the
meshless method and further explains how a 3D shape is converted into a volume
representation. Finally the chapter draws a hypothesis that where the conventional mesh
fails in the shape optimization, the use of voxels will succeed by accommodating shape
changes well in the shape optimization.
Chapter 3
Method of Wave Superposition and Interior Points
3.1 Method of Wave Superposition
The basic idea behind the superposition method is that an acoustic field of a
radiator can be recreated using a superposition of acoustic fields generated by an array of
simple sources distributed on the surface radiator. However because of mathematical
difficulties (the non-uniqueness problem encountered during the inversion of the matrix
operations), it has been a common practice to place simple sources inside the radiator. In
this section, the method of wave superposition is explained in detail since the Digital
Space Method is heavily based on the method of Wave Superposition. Development of
the self-term that allows simple sources to be on the radiator will also be discussed in this
chapter.
Superposition method shares many similarities with BEM except for one major
difference. While BEM requires integrating a partial differential equation over the entire
surface of the problem domain, Superposition only requires sum of the solutions to the
partial differential equations for each individual element. Also the functions used in
typical BEM and the method of Wave Superposition are different as the Wave
Superposition method assumes a very specific boundary (rigid boundary that satisfy
Neumann boundary condition) since it is based on the idea of superimposing wave fields
from an array of simple acoustic sources [6, 30-32].
30
Starting with the inhomogeneous Helmholtz equation, Eq. 3.1 to include a mass
source in the wave equation,
where, ( )p r is pressure at r , k is angular wave number, ω is angular frequency, and sm
is a mass source at sr ,
we obtain the solution [5, 33]of a form
where,
0ρ is the ambient density of a medium, ( )sq r is source strength of a monopole and V is
the volume of a vibration structure. Applying Euler’s equation to Eq. 3.2, one obtains
volume velocity ( )u r at r
where 0( ) ( )s ss r ik c q rρ= × , and c is the speed of sound traveling through the
medium. Further, applying Gauss’ theorem, one can reduce the volume integral to surface
integral. The right hand side of the Eq. 3.4 becomes,
2 2 2( ) ( ) ( )s sp r k p r m r rω δ∇ + = − 3.1
0( ) ( ) ( ) ( )s s sVp r j q r g r r dV rρ ω= −∫ 3.2
( )4
sik r r
ss
eg r rr rπ
− −
− =−
3.3
0
1( ) ( ) ( ) ( )s s sVu r s r g r r dV r
ik cρ= ∇ −∫ 3.4
31
Where nσ is the surface normal on the surface sigma and re is the radial component of
outer surface that encloses the surface σ . The second term of the right hand side is an
integral over an imaginary surface 'σ that encloses the surface, σ which expands to the
infinity. Hence the second term on the right hand side becomes to zero according to the
Sommerfield radiation condition. Substituting the result of Eq. 3.5 into Eq. 3.4 one
obtains Eq. 3.6.
Eq. 3.6. is in a continuous form, therefore, in order to make the Wave
Superposition method suitable for the proposed digital work space, one needs a discrete
equivalent of the integral expression above. One can break the surface of an arbitrarily
shaped radiator into N number of small patches on the surface as shown in Figure 3-1.
Also, these patches need to act as sources that, when the appropriate source strengths are
found, will recreate the exact pressure field due to the radiator.
{ }
{ }'
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
s s sV
s
r
s r g r r dV r
s r g r r n d r
s r g r r e d r
σ σσ
σ σσ
σ
σ
∇ −
= − ∇ −
+ ∇ −
∫∫
∫
i
i
3.5
{ }1( ) ( ) ( ) ( )su r s r g r r n d rik c σ σ
σ
σρ−
= ∇ −∫ i 3.6
32
Eq. 3.6 can be rewritten in discrete summation by letting iσ∂ small enough.
From here, one can set up a system of equations for N unknown source strength.
Place Figure Here
Figure 3-1: Arbitrarily shaped vibrating structure. s’s represent sources and σ ’s represent discretized surface elements on the surface
1
1( ) ( ) ( ) ( )i i i
N
n ni
u r s r g r r rik c σ σ σσρ =
−= ∇ −∑ 3.7
1 1
2 2
. [ ] .
. .v
v
u su s
g
u s
μ
μ
⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪= ∇⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭
3.8
33
In this case source strengths are calculated if the volume velocities on the surface
are known.
In turn, the source strength is passed to a discretized version of Eq. 3.2 and
pressure on the surface is obtained.
Once pressures and velocities are obtained, one can also compute radiation power,
Π from a structure by taking real part of dot product of volume velocity and pressure.
3.2 Singularity in the Method of Wave Superposition and Self-Terms
The free space Green’s function and its gradient are by themselves singular if
0sr r− = , in other words, if the field point is collocated exactly on top of the source
location, numerical values of the Greens functions cannot be used in numerical methods
for the values are infinite and cannot be expressed numerically in computers. However,
one can overcome the singularity problem by finding analytical expressions. By taking
the surface integral over the nodal area to calculate a spatial average, for both pressure or
velocity terms, one can write a bound analytical expression for the free space Greens
function and its gradient when 0sr r− = . The bound expression for both pressure and
velocity terms will be named simply self-terms – from the fact that the field point is
{ } { }1
vs g uν μ μ
−⎡ ⎤= ∇⎣ ⎦ 3.9
1
( ) ( )i
N
n i si
p r s g r rσ=
= −∑ 3.10
*
1
1 Re{ }2
N
av u pμ μμ=
Π = ∑ 3.11
34
located on the source itself. An analytical expression for the pressure, pm due to a
monopole source with source strength, sm is given by the free space 3D Green’s function
as below.
where k is the wavenumber and sr r− is the distance between the source and field point.
Rewriting Eq. 3.12 in integral form, the pressure over a circular surface due to the
monopole is defined as,
where the term gm is the spatial average of monopole pressure over the circular surface
whose normal vector is parallel to sr r− . By using a variable substitution technique, one
can set 22 zru += where r is radial coordinate on the circular surface and z is sr r− .
Integrating out the axis-symmetric θ, Eq. 3.13 becomes,
and the integration yields the bounded expression
The monopole pressure self term gm is determined in the limit of Eq. 3.15 as
sik r r
m ms
ep sr r
− −
=−
3.12
2 22
2 2 20 0
a ik r zm
m m ms ep s g rdrda r z
π
θπ
− +
= =+
∫ ∫ 3.13
∫+ −
=22
2
2 az
z
iku
m uduu
ea
gππ 3.14
{ }zikzaikm ee
akig −+− −=
22
2
2ππ 3.15
{ } { }2 2
2 20
2 2lim 1ik z ik aik a zm z
i ig e e ek a k aπ ππ π
− −− +
→
⎡ ⎤= − = −⎢ ⎥⎣ ⎦ 3.16
35
The particle velocity, vm at the field point due to the pressure radiated by the
monopole source is obtained by the linear Euler equation
where ∇ is the gradient operator with respect to field point. Assuming time harmonic and
solving for normal component of velocity, vm, in the Euler equation Eq. 3.17, one obtains
By substituting Eq. 3.13 into Eq. 3.18 and carrying out gradient operator
where sR r r= − ,
again by substitution of the variable 22 zru += and first integrating with respect to θ ,
substitute R n∇ ⋅ with zu
Then,
velocity self term is determined in the limit of Eq. 3.21 as below
mm
v pt
ρ ∂= −∇
∂ 3.17
mm m
pv v n niωρ∇
= = −i i 3.18
2
20 0
a ikRm m m
ms g s ev n R nrdrd
i i a R R
π
θωρ ωρπ
−⎛ ⎞∇ ∂= − ⋅ = − ∇ ⋅⎜ ⎟∂ ⎝ ⎠
∫ ∫ 3.19
2 2
2
2 a z iku
mz
e zg n udui ck a u u u
πρ π
+ −⎛ ⎞∂∇ ⋅ = − ⎜ ⎟∂ ⎝ ⎠
∫ 3.20
2 2 2 2
2 2 2 2
2 2a z ik ziku ik a z
mz
e i e eg n z zi ck a u cka za z
πρ π ρ
+ −− − +⎧ ⎫⎪ ⎪∇ ⋅ = − = −⎨ ⎬+⎪ ⎪⎩ ⎭
3.21
2 2
2 22 20
2 2limik zik a z
m z
i e e ig n zcka z ck aa z
πρ ρ π
−− +
→
⎡ ⎤⎧ ⎫ −⎪ ⎪⎢ ⎥∇ ⋅ = − =⎨ ⎬⎢ ⎥+⎪ ⎪⎩ ⎭⎣ ⎦
3.22
36
These self-terms are no longer function of r but function of the area of the circular
surface, A. Table 3-1 summarizes the above transfer functions [9].
3.3 Non-Uniqueness Problem
The singularity problem in Greens functions is resolved by the use of self-terms.
However, one faces yet another kind of singularity problem. That is a non-uniqueness
problem. The non-uniqueness problem occurs at frequencies that correspond to the eigen-
value of internal volume of the structure. Although any solution with zero pressure on the
surface can give a numerically correct solution, interior mode results in rank of zero
value or weak rank of system matrix causing singularity or near singularity[34]. In the
Wave Superposition method, sources are put inside the volume of a structure to avoid
both type of singularities in Greens function and internal mode. While the singularity in
the Greens function is rectified by the self-terms, one still needs to address non-
uniqueness problem that is intrinsic to the geometry of the structure. There are two
common ways to resolve a non-uniqueness problem. One method is developed by
Schenck[31] in 1967. His Combined Helmholtz Equation Formulation (CHIEF) adds
Table 3-1: Summary of Greens functions for Pressure and Velocity monopole self term
Source Type Pressure Self Term Velocity Self Term
Monopole { }2 1ik ai ekAπ − −
2 ickAπ
ρ−
37
interior points that makes the system matrix an over-determined system and eliminates
internal modes from the solution. One other solution is developed by Burton and Miller
in 1971 that combines the system matrix and its derivative to solve the non-uniqueness
problem[35]. A wave function, u which comprises an incident wave, ui and reflected
wave, ur, from a point P, satisfies the Helmholtz equation of form, 2 2 0kφ φ∇ + = in an
exterior domain, E, from a boundary B, where P E∈ . The wave function, u, is then at
least twice continuously differentiable in E and satisfies following conditions,
Also, the wave function must satisfy Sommerfeld radiation condition for its outgoing
reflected wave.
Using u in the Kirchhoff-Helmholtz integral formula,
where G(P,q) is the Greens function between a field point P and a source point q.
Applying the boundary condition ( )0,u P Bn∂
= ∈∂
, ( , ) ( )kq
G P q u qn∂∂
disappears
and can be rewritten as
Once more, rewriting Eq. 3.26 where the integral is replaced with the transform operator,
( )2 2 0,u k u P E∇ + = ∈ 3.23
( )0,u P Bn∂
= ∈∂
3.24
( ) ( , ) ( , ) ( ) ,2i k k q
q qS
uu u q G P q G P q u q dS P Bn n
⎛ ⎞∂ ∂+ − = ∈⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠∫ 3.25
( ) ( , ) ,2i k q
qS
uu u q G P q dS P Bn∂
+ = ∈∂∫ 3.26
[ ]2 k iu M u u− = 3.27
38
The corresponding homogeneous equation of Eq. 3.27 is
Eq. 3.27 has a non-trivial solution if and only if Eq. 3.28 has a non-trivial solution. In this
case, its transposed equation,
will also have a non-trivial solution. [35]
In general, a solution of the interior problem, v = 0 for 2 2 0v k v∇ + = . Also the
boundary condition 0vn∂
=∂
, is satisfied as well. While it is the case in general, if k is one
member of an infinite set, K, then there exists a non-trivial solution, v such that 0vn∂
≠∂
.
Hence Eq. 3.29 will have a non-trivial solution and therefore Eq. 3.27 also has a non-
trivial solution by combining the solution of the corresponding homogeneous equation
with a non-trivial solution.
In summary, this chapter walks through the development of the Wave
Superposition method and self-terms and, explains its use of self-terms in determining
source locations and benefits in modeling particular shapes. Since the Wave
Superposition method can be applied a lumped parameter model, such as simple sources,
it can readily be written in discrete form and well suited for voxels. This Wave
Superposition method will be used extensively in the remainder of the thesis.
[ ] 02 k
Mφ φ− = 3.28
[ ] 02 k
P
Ln
ψ ψ∂− =∂
3.29
Chapter 4
Digital Space Method
4.1 Voxelization of Solid Surface
In a typical voxelization process, one would voxelize the volume of an entire solid
into voxels. However, this thesis will only focus on voxelizing the surface of a solid,
since the Digital Space Method is rooted from BEM and therefore there is no need for
voxelizing the interior of a solid.
In order to voxelize the surface of a solid, the very first item one needs is a
description of the surface. One can attempt to describe it in two ways for the purposes of
the Digital Acoustic Space Method. One way is to import directly from meshing
software, for instance, Patran or I-DEAS, and the other is to describe the surface in
analytical expressions, e.g., the radius of a sphere.
First, voxelizing a mesh from an external meshing software is discussed. While
there are three possible geometries to voxelize as mentioned in chapter 2, in this thesis,
all elements are considered as a point; therefore, a spherical volume around a point
element is used to limit the threshold. By setting an appropriate size for threshold, one
can voxelize without adding too many voxels that would cluster together or too scarce to
represent the geometry that one is trying to voxelize. In typical boundary element
analysis, only a single layer of the surface needs to be modeled; therefore, it would be
wise to use a threshold of not much more than 1 voxel length to keep a single layer of
40
voxels to represent a surface only. Types of threshold are briefly discussed in the
Chapter 2. Also, the reason one can treat a surface mesh element as a point element is
that in the method of Wave Superposition, the surface can be defined with freely
distributed points instead of interconnected patches of mesh elements. This feature
allows one to import a mesh and convert the mesh geometry into voxels by finding a
reference point for each individual mesh element and then, applying a threshold on these
reference points to determine voxels to represent the surface as depicted in Figure 4-1.
Figure 4-1: Activating Voxels in Digital Space
Mesh elements Reference Points
Import into Digital Space
t
Activate voxels within threshold t
t
41
In the case where the sizes of the mesh elements are drastically different, the
number of voxels to represent an individual mesh will depend on the size of the mesh
element. In this scenario, one can let the smallest mesh element have a single voxel and
use more voxels for a larger mesh element.
4.2 Volume Velocity Approximation
Once a geometry is voxelized, one needs to find an appropriate volume velocity
for the geometry. Shown in Figure 4-2, a first octant of a sphere is voxelized as an
example for this study. In the typical process of the Digital Space method, the sum of the
individual areas of all active voxels is not equal to the total surface area of the
represented geometry. This becomes clearer when one understands the fundamental
difference between analogue and digital representation of geometry. For better
understanding of the problem associated with finding right volume velocity, 2D examples
will be studied instead of 3D figures. Shown in Figure 4-3 is a circle as a 2D example.
42
Figure 4-2: Voxelization of the first octant of a sphere
43
In the typical voxelization shown in Figure 4-3, it is clear that the total length of
lines in both representations is not equal to each other. Taking unit length, l, arc length of
the quarter of a circle, represented by a curved black line, is 2 8 12.564
l lπ while the sum
of voxel faces, represented by red straight lines, is 18l Unlike the example shown above,
the Digital Space method allows only the equal area for all voxels no matter how many
faces they are exposing. Figure 4-4 shows a different scheme used for the Digital Space
Figure 4-3: 2D voxelization of the first octant of a sphere
Unit length, l
44
method and it is evident that even in the Digital Space Method, the total lengths are not
equal. The arc length is 12.56l, while the summed length of voxel faces, again
represented by red lines, is only 12l. Since the surface is not represented accurately
enough with voxels, even with threshold limits and control algorithms, and even though a
correct particle velocity is given to a voxel, the total volume velocity from voxelization is
not going to be equal to the original volume velocity.
Figure 4-4: 2D voxelization of the first octant of a sphere, Digital Space method version
Unit length, l
45
Figure 4-5 shows the explanation of the volume velocity adjustment graphically.
The example given in the Figure 4-5 is a spherical geometry, and a sphere is chosen for
its evenness. The aim of this process is to match the total volume velocity of the
voxelized geometry to that of the original geometry. In the case of a pulsating sphere, it is
quite simple to match total volume velocity since all voxels and elements will have the
same surface velocity. Tthus, an equal surface area translates to an equal volume velocity.
In the Digital Space Method, once a geometry is voxelized in the digital work space, the
area given to each voxel is not yet adjusted to be equal to the surface area of the original
geometry when summed together. As shown in the Figure 4-5, the original voxel area is
defined by the lattice size, and it does not know how to correct volume velocity for the
imported geometry. When the geometry is imported, the total area is calculated, and it is
divided by the number of voxels to calculate the required correction factor. Then, the
correction factor is applied to all voxels such that the total summed area of voxels is
equal to that of imported geometry. Upon comparing, the pictures of “before and after”
surface area adjustment show that the coverage is more complete after surface area
correction is performed. It is rather evident that after adjustment, the diagram look closer
to a whole sphere, and it has an equal surface area to the surface area of the original
spherical geometry.
46
To summarize, this chapter has explained how a volume representation of a shape
is implemented in conjunction to the Wave Superposition method. Because of the
limitation imposed on a uniform lattice and the voxels, matching the boundary method is
found to be a key component for obtaining accurate results for pressure and radiated
power from a radiator. In the next chapter, implementation of voxels in the Wave
Figure 4-5: Adjustment of Volume Velocity for Voxelized Sphere
Mesh element
Voxel with original voxel area
Original volume velocity,
u= Av m3/s
Element area, A m2
Velocity, v m/s
Velocity, v m/s
Velocity, v m/s
Volume velocity, u’= A’v m3/s
Volume velocity, u= Av m3/s
Voxel area(red), defined by lattice = A’ m2
Voxel after volume velocity is adjusted
Voxel area(red), adjusted to elem. Area = A
47
Superposition Method is validated with simple and complex geometries with complex
mode shapes.
Chapter 5
Validation of the new method: Test Cases and Results
All newly developed numerical methods must be validated. In this thesis, results
of sound power calculation for various voxelized geometries are compared to those from
analytical solutions and to other already proven methods using the same geometry
meshes, to validate the new method.
5.1 Voxelized Geometry From a Mesh Sphere
The first geometry one may try first is a sphere for its readily available analytical
solutions for axially and radially pulsating modes. Also, while it is not trivial to make
mesh of a sphere, it is not the most difficult task either. In this section, the mesh of a
spherical geometry, with a radius of 1 m, consisting of 60 triangular elements is used as
shown in the figure 5-1. The 60 elements are made from a dodecahedron’s 12 pentagon
face elements by dividing them into 5 isosceles triangles per each face element. Hence,
one obtains 60 evenly distributed elements that conveniently have an equal area for
voxelization. The surface velocity is, then, prescribed to the sphere for the analysis. In
this example, the surface velocity of 1 m/s is given for radial oscillation, and for axial
oscillation, 1 m/s cosxR
⎛ ⎞× ⎜ ⎟
⎝ ⎠, where x is the component along the axis of oscillation and
R is the radius of the sphere, is given.
49
In general, one needs to determine which kind of threshold to use. As mentioned
in Chapter 2, one can have a dot, a line, or a plate piece for voxelization. In the case of
voxelizing a mesh geometry, either a dot or a plate for a mesh element seem good
candidates. Since the mesh in question already has evenly distributed elements, the
centers of each element are converted into dots and a spherical threshold is given for each
dot for voxelization. Centroids (center point) of mesh elements with a spherical threshold
are overlapped onto a cubic lattice, and all voxels that falls within threshold are activated.
Once all active voxels are found, they are corrected for surface area.
Figure 5-1: Mesh Representation of the Modified Dodecahedron
50
Figure 5-2 show a 5 step voxelization. The first step is to find the centroids of
mesh elements. This step is to find a point that represents a mesh element. In the Wave
Superposition method, this step is identical to finding acoustic elements from a mesh
geometry. Secondly, they are mapped onto Digital Acoustic Space. This step is an
intermediate step for finding appropriate voxels by overlapping centroids of mesh
elements onto the Digital Acoustic Space. Thirdly, the active nodes are found by
employing necessary threshold and voxelization scheme discussed in chapter 4. In the
fourth step, only active voxels are remaining. This step is necessary for two reasons. The
inactive voxels will behave as rigid boundary that does not exist and cause the analysis to
be unreliable. Also, the system matrices will be larger by including those inactive voxels
and, consequently the computational cost will increase in memory and operation time.
Lastly, active voxels are given the correct surface area. In this step, boundary condition is
applied to the voxelized geometry for an accurate analysis. A final voxelized version of
the digital representation is shown in Figure 5-3. The blue boxes represent voxels, and the
red circles represent normal surfaces. The boxes and circles are scaled down for easy
viewing of the voxel representation of the sphere.
51
Figure 5-2: Voxelization of the Sphere Mesh
Step 1: Centroids extracted from mesh elements
Step 2: Map the centroids onto Digital Space
Step 3: Active voxels are found
Step 4: Only active voxels are selected
Step 5: Surface area is corrected
Starting from a sphere mesh, find controids of
mesh elements
52
In this case, a sphere mesh with 60 elements is voxelized. The radius of a
circle of the equivalent area to a mesh element is 4 0.25860
a ππ
= = m for the self-term in
the conventional Wave Superposition method. Thus, all voxels are initially set to 0.25 m
in edge length to give comparable spacing between the voxels. However, the area is only
0.0625 m2 for the voxel area. This pre-defined voxel area is much smaller than that of a
mesh element, 4 0.20960
A π= = m2 and, it is later adjusted to match that of the mesh
element. This step guaranties that the total volume velocities of both versions are equal.
Figure 5-3: Voxel Representation of Modified Dodecahedron
53
Figure 5-4 shows the sound power results for both radially and axially oscillating
spheres compared to the analytical solutions[1]. The approximate number of elements per
wavelength is 6. Therefore, the power output up to 6ka = can be trusted to be an
accurate assessment of radiated power from a vibrating sphere. One notices that there are
several singularities occurring at integer multiples of π and the arguments of spherical
Bessel function of the first kind and of the first order. These singularities are numerical
artifacts caused by the non-uniqueness problem due to the frequency of exterior problem
coinciding with the resonance frequency of the internal modes characterized by the
geometry.
Figure 5-4: Sound Power from voxelized sphere
54
The non-uniqueness problem can be treated with the addition of interior points,
first developed by Schenck[2]. Figure 5-5 shows improvement of radiated sound power
with the use of internal points. Generally the location of interior points are difficult to
determine, however for simple geometry such as a sphere in simple vibration mode –
radial and axial oscillation - as in this example, the candidate location for interior points
can be guessed fairly easily. For instance, a pulsating sphere behaves much like a
monopole, hence placing an interior point at the center helps to remove the interior mode
for a radially pulsating sphere. Also axially oscillating sphere mimics a dipole, hence
both positions a half radius away from the center along the axis of oscillation are
strategically good places to remove the interior modes.
Figure 5-5: Sound Power from voxelized sphere with internal points
55
5.2 Voxelized Sphere From an Analytical Description
In this section, for the analytical definition of a sphere, only radius is defined and
given to the program. It is used to voxelize a sphere and then, radiation power from
pulsating sphere is calculated. In this case, an axially oscillating mode is omitted since
the results show the similar trend. The only parameter other than radius, considered in
voxelizing from an analytical description is a threshold, t. In order to voxelize a sphere
appropriately – that is, only voxelizing the surface of sphere without allowing voxels to
form a layer or layers - for the Digital Space Method, it is important to set the threshold
correctly. Although the only known information is radius, one realizes that it is a plate
since it is the only type of geometry that is capable of representing a 3D structure among
a dot, a line, and a plate. Thus a pair of plates, outer and inner limit plates, are considered
to set a threshold for activating voxels. As depicted in the Figure 5-6, all the voxels that
are located between outer and inner limits represent the surface of the sphere, and only a
single layer is formed. One can easily realize that with a different threshold value, one
might have multiple layers of voxels or too few voxels to represent the surface only.
56
In this example, all voxel sizes are set to 0.2 m and the following figures,
Figure 5-7, Figure 5-8, and Figure 5-9 show the effect of varying threshold thickness.
Figure 5-6: Setting a Threshold for a Curved Plate
Outer limit
Sphere with radius, r
Threshold, t
Inner limit
57
The significance of results shown above is not only in the dB difference but also
in time it took for each analysis. It is obvious that if threshold is too thick, the voxelized
geometry will have multiple voxels on top of itself and it will not only result in poor
agreement with actual solution, but it will also make the system matrices much larger
than necessary by adding more number of voxels in the system matrices, regaining longer
times for the program to run the analysis. At this stage of development, an algorithm for
determining and checking threshold is not yet implemented, however it is desired for an
efficient and accurate shape optimization.
Figure 5-7: Voxelized Pulsating Sphere; 1 m/ssv = , Threshold, t = 0.2 m
58
Figure 5-8: Voxelized Pulsating Sphere; 1 m/ssv = , Threshold, t = 0.1 m
Figure 5-9: Voxelized Pulsating Sphere; 1 m/ssv = , Threshold, t = 0.05 m
59
5.3 Voxelized Cylindrical Tank, Cylinder With Two Hemi Spherical Caps
In this section, a cylindrical metal tank with two hemi-spherical caps is used as an
example. Unlike a sphere, this geometry portrays a real world example with more
complex vibration modes and complex displacement. Hence complex velocity and the
results from voxelized version are compared to those from Power program written by J.
Fahnline[3]
Figure 5-10 shows the mesh representation and voxel representation of a
cylindrical tank with hemispherical caps, with 1.05 m in height and 0.2 m in radius. For
both representations, two types of volume velocity are tried. First, one assumes an
artificial breathing mode for the cylinder, prescribed with 1 m/s for its entire surface.
Figure 5-10: Digital Representation of Cylindrical Tank
60
The result is shown in Figure 5-11. Up to 400 Hz, the result from the Digital
Acoustic Space method is in agreement with that of the Power program, and beyond 400
Hz the voxel version results deviates from that of Power. 400 Hz translates to a wave
length of roughly 0.9 m, and it marks the limit of “six elements per wave length” rule for
numerical methods. While Power is also a numerical method governed by the same limit,
it does extend its frequency range by integrating over a smooth surface unlike the Digital
Acoustic Space Method which only has contiguous volume elements.
Figure 5-11: Breathing mode for Cylindrical Tank with vs = 1 m/s
61
Second, a real world velocity distribution is applied on both voxel and mesh for
testing. To the cylinder mesh shown in Figure 5-10, a moment element is attached at the
bottom of cylinder, and the bottom elements of lower hemispherical cap are given a fix
constraint. A frequency response analysis was performed using NX Nastran software.
From the data obtained via frequency response analysis, a few modes are selected for a
validation study with complex modes. Displacement data of mode (1, 8), (1, 10), and (2,
10) is extracted from a FEA model of the tank and converted into a complex volume
velocity distribution. By convention, the mode shape with half wave along the length of
cylinder and, 4 full wave lengths or 8 half wavelengths along circumference is tagged
mode (1,8) and shown in Figure 5-12. Similarly, Figure 5-13 and Figure 5-14 show mode
shapes of modes (1, 10) and (2, 10) respectively.
Figure 5-12: Displacement plot of Mode (1,8)
62
Figure 5-13: Displacement plot of Mode (1,10)
Figure 5-14: Displacement plot of Mode (2, 10)
63
The results of the radiation calculation with the volume velocity distribution from
the above modes are presented in Figure 5-15, Figure 5-16, and Figure 5-17. One notices
that unlike the previous study for breathing mode of a cylinder, these results show worse
agreement at lower frequency regions. One would assume that the cylindrical tank model
having 24 elements around the circumference would be good enough for 4 or 5 full wave
lengths according to “six elements per wavelength.” Suspecting that the voxel size is too
big, hence the voxel model is too coarse for such a complex mode shape, a finer voxel
size is used. Each mesh element is divided into 4 smaller mesh elements, then voxelized
and analyzed again. The cylindrical tank is composed of 2 types of mesh elements,
namely quadrilateral and triangular. For triangular element, new nodes are formed on the
geometric center of all edges, and by connecting all new nodes, 4 new elements are built.
For quadrilateral element, new nodes are on the geometric center of all edges, and by
connecting each new node to the centroid of the element, 4 new elements are built. For
the coarser voxel geometry, 250 voxels are used and for the finer voxel geometry, 1200
voxels are used. The finer voxel model is made from the original mesh geometry by
dividing its 300 mesh elements into 1200 finer mesh elements, then voxelizing them.
64
Figure 5-15: Radiated Power from cylindrical model vibrating in (1,8) mode
Figure 5-16: Radiated Power from cylindrical model vibrating in (1,10) mode
65
Figure 5-18 and Figure 5-19 show great improvement over the entire frequency
range used in this study. Since the coarse model worked well when given “simple”
volume velocity distribution, it is natural for one to think that prescribing correct volume
velocity is a key element in Digital Acoustic Space Method.
Figure 5-17: Radiated Power from cylindrical model vibrating in (2,10) mode
66
Figure 5-18: Radiated Power from finer cylindrical model vibrating in (1,8) mode
67
5.4 Complexity Study
In the beginning of the introduction, the author motivated this work due to a
reduction in computation time. Since the Digital Acoustic Space Method is based on the
method of Wave Superposition, most of computations are similar if not the same. Hence,
it is safe to say that the cost computation is same except for one place. The only major
difference between the Digital Space method and the method of Wave Superposition is in
self-term integration. While the method of Wave Superposition requires that self terms of
all elements are to be calculated individually, the Digital Space method requires one to do
Figure 5-19: Radiated Power from finer cylindrical model vibrating in (2,10) mode
68
that only for a single instance. For a mesh with N elements, which translates to N x N
system matrices, it takes operations of order of O(N3) for inversion, O(N2) for matrix
construction, and log(N) for element area calculation. By putting all operation order
together for each method, the method of Wave Superposition takes N3 + log(N) x N2 and,
the Digital Acoustic Space method takes N3 + N2.[4] Figure 5-20 shows the percent
difference in the number of operations per frequency versus number of elements/voxels.
While the number is insignificantly small, it’s only for single frequency analysis. If one is
required to analyze over a large frequency range or perhaps if one has to prepare one-
third octave bands, he or she will have to repeatedly construct and invert the system
matrices hundreds, thousands of times. For example, % difference in computational
operations is 0.5% for 300 elements with which the cylindrical tank was meshed. On a
PC equipped with 2.4GHz Intel Core2 processor and 3GB of RAM, it takes 3 seconds to
run an analysis for a single frequency. In an actual analysis performed for the validation,
the time difference between the Power program and the Digital Acoustic Space method
was approximately 8 seconds for 65 frequencies. When one assumes that the difference is
directly proportional in time; the computational time would have been reduced by 123
seconds if analysis over a frequency range over 1000 data points, was performed.
69
In an absolute sense, the dominant factor for the difference is matrix construction.
In that regard, one will always save computation time by utilizing the Digital Space
method. Figure 5-21 shows a simulated time reduction for Digital Space method over the
method of Wave Superposition. Again, it is per single frequency.
Figure 5-20: Complexity Study for Computation Cost
70
5.5 Sensitivity Study
Although wave superposition using digitized space is based on an already proven
method, it is imperative to do a sensitivity study in order to determine the requirements
for obtaining computationally acceptable solutions. In this section, a cubic box with a
varying number of voxels is used. The surfaces of the cubic box are given a normal
velocity distribution calculated from a simple spherical source radiating from within.
Figure 5-22 shows the configuration graphically. It also includes boundary representation
0 100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Number of elements or voxels
Abs
olut
e D
iffer
ence
in s
econ
d pe
r fre
quen
cy
Figure 5-21: Simulated execution time difference for Digital Space method vs. WaveSuperposition method
71
for a better viewing and understanding since wire frames in voxel representation make it
confusing to visualize. The red circle is a simple spherical source radially pulsating in
both representations. The blue quadrilateral element and the blue dot which is the center
of the blue colored voxel, are the i th element to which a normal velocity will be given in
this study. The blue arrow indicates the surface normal direction and the black arrow
represents the position vector from the center of spherical source to the i th element.
Figure 5-22: Cubic Box with a Spherical Source
Boundary Representation Voxel Representation
72
For a simple spherical source, the particle velocity, v , at a radial distance, r, away
is expressed as,
where vs is the complex amplitude of surface velocity and as is the radius of the
sphere[1]. To find a matching normal surface velocity, the dot product of v and the
surface normal is performed. Once all elements are given normal surface velocities,
system matrices [ ]G and [ ]G∇ are constructed according to the method of Wave
Superposition and ultimately radiated power is calculated. Since the cube is given a
matching volume velocity distribution of a simple spherical source, its sound power can
readily compared to theoretical analysis.
Figure 5-23 shows the result of radiated power from the cubic box with a different
number of elements per side. It can be easily seen that, at 3.5ka = , the cube with 2
elements per side starts to deviate from the theoretical value as well as those of the cubes
with 4, 6, and 8 elements per side. Upon inspection, for a cube of the side length,
1 ma = , 3ka = translates to an acoustic wave length of 2 1.8 m3.5πλ = = . If one follows
the general rule of numerical analysis that states a minimum of 6 elements per
wavelength, at k = 3.5, the model needs at least 3 elements along the side. The cube with
only 2 elements per side obviously does not meet this requirement, and as shown in the
Figure 5-23, deviates from the groups that satisfy the requirement and the theoretical
result.
( )2( )
2 ˆ(1 )
sik r as
s
ika v kr iv e r
kr ika− −−
=+
5.1
73
To give further affirmation, the result at ka = 4 is magnified and examined. In
order to show a better comparison, results from cubes with 3 and 5 elements per side are
added. In this case, the minimum number of elements per wavelength required is 4. It can
be seen that the results from the models with a sufficient number of elements per side,
group together while those of the boxes with 2 and 3 elements per side deviates from the
group.
Figure 5-23: Radiated Power from a Pulsating Box
74
It should be noted that even with a sufficient number of elements per wavelength,
the analysis performed using voxels differs from the exact solution above ka = 1. This is
due to the source of error discussed in previous section, namely the limited ability to
replicate the exact shape of a geometry since it has limited resolution since one can only
use the center point of volume element. Below ka = 1, the actual location of acoustic
sources becomes less sensitive to approximations, since a lumped parameter model
applies.
Figure 5-24: Radiated Power from a Pulsating Box, magnified at ka = 3.7 to 4
75
In summary, this chapter shows the validity of the newly developed numerical
method. Results of sound power calculation for various voxelized geometries are
compared to those from analytical solutions and to those from the Power program when
an analytical solution was not available. To further examine the new method, a
complexity study and a sensitivity study are performed. In the complexity study, it was
found that the time saving from the construction of the system matrices and the
calculation of the solution is minimal. In the sensitivity study, it was found that the
method required at least 6 elements per acoustic wavelength as the general rule of the
numerical method states.
Chapter 6
Conclusions and Future Work
6.1 Conclusion
In this thesis, an alternate approach is presented for improving the Method of
Wave Superposition by developing a technique that incorporates a meshless method in a
digitized domain, i.e., the Digital Space Method. This alternate approach presents two
improvements. The first major improvement is integration of an intuitive and simple
digital workspace for shape optimization that alleviates much human effort and reduces
human time when compared to the conventional meshing process commonly used in
FEM. The second is reduced computational complexity from employing such a digital
workspace, ultimately allowing reduced computation time.
The improvements mentioned above are achieved by constructing a uniform, 3D
cubic lattice and describing a problem domain within it. The geometry of a structure is
represented with cubic voxels defined by a 3D lattice. The geometry is represented by
uniformly distributed points that correspond to the center of voxels.
Employing a uniform 3D lattice has both primary and secondary advantages. The
primary advantages are;
1. Human time and effort is reduced. Conventionally, the method of
Wave Superposition relies on mesh elements for representing the
geometry of a structure. However, the dependency on mesh elements is
77
removed with the capability to describe a geometry with voxels.
Eliminating the dependency on mesh elements signifies a huge advantage
in modeling since one no longer is bound by mesh elements that must
satisfy many requirements for obtaining an acceptable computational
solution. In this aspect, human involvement is reduced significantly, for
one does not have to ensure that the mesh elements are defined correctly.
2. Easy adaptation. All mesh elements must be planar, which disallows any
distortion whatsoever and the connectivity of all the elements needs to be
kept separately by adding more overhead. A single change in a structure
means a whole new mesh. However, in the Digital Space Method, much
like other meshless methods that use nodes with domain of influence, one
can freely activate or deactivate a number of voxels to describe a change
in geometry. One can skip the entire process of remeshing and save much
precious human time and effort in this phase.
One major difference between the Digital Space Method and other meshless
methods is that the typical meshless methods use randomly spread points in space and for
each and every point, with a certain weight function that needs to be evaluated. The
Digital Space Method on the other hand, uses a lattice to distribute points in a uniform
manner and, only requires activating and deactivating points to define or refine the
geometry of a structure, much like switches in a digital world.
One requirement of the Digital Space Method is that, while it does not require the
weight function common to meshless methods, it does requires a volume velocity
adjustment to be made at each point. During the voxelization process, the area terms used
78
in the Greens functions are adjusted relative to the area of the voxel to give the correct
volume velocity instead of adjusting the velocity.
A Secondary advantage comes from improved self-term calculations. In the
method of Wave Superposition, an integration technique that can handle singular
functions is required to integrate over the element area for each and every element. In the
Digital Space Method, however, all voxel areas are equal. This guaranties that all self-
terms are equal, and one can calculate them by integration or by readily available
analytical expressions. Thus, one can apply a single self-term to all of the diagonal terms
in the system matrices of all active voxels. Moreover, all the off-diagonal matrix
elements are calculated for average values from the point-to-point analytical expressions
of Green functions instead of integrating over the entire problem domain. Reduced
computation time is achieved by applying a single self-term computation to all diagonal
elements and using analytical expressions for the off-diagonal elements in the system
matrices for all active voxels.
The Digital Space Method is not, unfortunately, without caveat. If a geometry
mesh is imported for voxelization from a meshing software, certain overheads are added
such as relocating acoustic elements to digital space’s nodal points, i.e., center of voxels,
and adjusting volume velocities. However, this extraneous process is performed only
once at the beginning stage. Therefore, this will not add any further burden to perform
frequency operations that generally require heavy computational cost.
By the intrinsic nature of voxelization, the surface of a geometry is represented at
a discrete step distances away from the actual surface. This causes numerical error in two
ways.
79
1. Errors in Greens function computations, since the distance, R is no longer
an exact distance between field and source points. This can be easily
improved by utilizing smaller voxels but not without heavy computational
cost.
2. Errors associated with assigning volume velocities. Since the voxels are
located away from the actual surface of a geometry (although a small
distance away) the volume velocity will have to be adjusted. However,
there is no straightforward way to determine how much the surface
velocity changes as the surface is moved away from the actual surface. To
compensate for this, however, the volume velocity can be approximated
by adjusting voxel area.
6.2 Assessment of the Accuracy of the Digital Space Method
The validation studies that compared theory versus numerical results for a
vibrating sphere ( with radially and axially oscillating modes) has shown that the error in
sound power level from the Digital Space Method falls within 1 dB over a wide range of
frequencies. In a more complex case of a cylindrical tank, vibrating in its (1,8), (1,10) and
(2,10) modes, the dB difference between a wave superposition program (POWER) using
conventional meshes and the Digital Space Method has shown less than a 2 dB difference
at the resonance frequencies at each structural mode. In summary, the overall results have
shown that the new Digital Space Method presented in this thesis can be used for a highly
accurate approximation of radiated power from an arbitrarily shaped structure.
80
6.3 Future Work
The method described in this thesis has a foreseeable and promising potential
when it is used in shape –optimization studies. Since a problem domain is described in a
digital fashion, it is very intuitive to apply this method in a topological shape
optimization. While shape optimization is not new, this is a first working method where
a geometry can systematically be modified to obtain an optimal shape.
Also, for a more accurate analysis, a few improvements are suggested. First, an
improved voxelization scheme is also desired. Although finding a better voxelization
algorithm was not the main focus of this thesis, it would be profitable to employ a more
efficient voxelization scheme to reduce the error associated with representing a surface
with cubic voxels and the overhead due to importing and voxelizing a geometry.
Also, a thorough parametric study in voxelization would be beneficial to give
guidance in the voxelization process. Incorporating voxelization and the Method of
Wave Superposition is at its infancy and there has been no prior work to give guidance in
finding a best set of parameters for voxelization. For instance, in this thesis, the threshold
necessary for constructing a single layer of voxels – in order to mimic a shell mesh – is
found by trial and error.
Because of the lack of previous studies, again, a study of incorporating a different
lattice is suggested. Although a cubic lattice was chosen for its convenience and
intuitiveness for use in digital space, a cubic lattice is not the only uniform lattice. A
cubic lattice has only three axes of symmetry. Recognizing that an arbitrary shape will
more likely have a surface whose normal is not parallel to one of the three axes; it
81
becomes rather challenging to describe a surface with cubic voxels, especially when the
relative surface areas deviate substantially in the analysis phase, requiring volume
velocity adjustment.
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Appendix
A Sample Matlab code for the Digital Acoustic Space Method
% Voxelization of an analytical geometry % This program voxelizes a geometry given % in a pre-defined grid of unniformly distributed nodal points. % by "activating" the nodal points that represent the surface of % geometry. % Then, calculates the Greens function between activated points % and normal % velocity information corrected for matching boundary condition %%%%%%%%%%%%%%%%%%%%%%%% % constant declaration % %%%%%%%%%%%%%%%%%%%%%%%% % rho: density of air % c: speed of sound in air % alpha: use monopole % beta: use dipole global rho c global alpha beta rho = 1.21; c=343; alpha=1; beta=0; %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Grid generation 3D-case % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Rmax: size of the Digital Space in one octant % L: voxel size % area: initial voxel area % a: radius of equivalent area circle for self-term calculation % Ngrid: total number of voxels in the Digital Space % grid: voxel location information % R, rx, ry, rz: define uniform lattice Rmax = 1; L = 0.2; area = L^2; a = L/sqrt(pi); R = -Rmax:L:Rmax; R_leng = length(R); Ngrid = R_leng^3; grid = zeros(Ngrid,3); rx = R; ry = R; rz = R;
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counter = 1; for ii = 1:R_leng for jj = 1:R_leng for kk = 1:R_leng grid(counter,1) = rx(ii); grid(counter,2) = ry(jj); grid(counter,3) = rz(kk); counter = counter +1; end end end % End Grid generation % %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Analytic desc. of sphere % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % radius: radius of sphere % center: center of sphere radius = 1; center = [0, 0, 0]; % Analytic desc. of sphere % %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % Voxelization % %%%%%%%%%%%%%%%%% % nodes: activated voxel % normals: normal information for active voxels % sp_th:sphere threshold nodes = zeros(1,3); normals = zeros(1,3); sh_th=0.1; counter = 1; for ii = 1:Ngrid r_dist = sqrt( sum( (grid(ii,:)-center ).^2 , 2) ); if r_dist<=radius+sh_th && r_dist>=radius-sh_th nodes(counter,:) = grid(ii,:); normals(counter,:) = grid(ii,:)-center; counter = counter+1; end end n = size(nodes,1); clear grid % gets rid of excessive nodes that makes a double surface layer % remove deactive voxels % End voxelization % %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% %% CHEIF point %
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%%%%%%%%%%%%%%%% % chiefpt: interior point location % chief_norm: normals assigned to interior points chiefpt = [0 0 0; 0.1 0 0; -0.1 0 0]; chief_norm = [0 0 0; 0 0 0; 0 0 0]; %% CHEIF point ends % %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %% Volume Velocity calc % %%%%%%%%%%%%%%%%%%%%%%%%% % v: normal velocity v = ones(n,1); %% END Volume Velocity calc % %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Wave Superposition Method % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k: wave number / frequency k = [0.1:.5:10]; % calibrating factor for vv % area_sphere: surface area of original sphere % nodes_leng: number of voxels used for imported sphere % area: corrected voxel area % a: radius of equivalent area circle for self-term calculation area_sphere = 4*pi*radius^2; nodes_leng = size(nodes,1); area = area_sphere / nodes_leng; a = sqrt(area/pi); % run Wave Superpostion method for nnn = 1:length(k) alpha = 1; % turn on monopole beta = 0; % turn off dipole %construct systems matrices [g,gp] = greensfunc3(nodes,normals,k(nnn),a); % if CHIEF point is included, use the code below [g(n+1:n+size(chiefpt,1),:), gp(n+1:n+size(chiefpt,1),:)] ... = greensfunc3_CHIEF(nodes, chiefpt, normals, chief_norm ,k(nnn),a); % inverse of gp is calculated to find source strength % then matrix multiplication by g is performed to calculate % the impedance impd = gp\g; % pressure calculation from velocity p(nnn,:) = impd*v;
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% power calculation from pressure and velocity power_pv(nnn) = 1/2*real(p(nnn,:)*real(v))*area; end %% Wave Superposition Method % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
VITA
Yong-Sin Hwang
Mr. Yong-Sin Hwang was born in Seoul, South Korea on May 28, 1976. In
December 1991, his family moved to Montreal, Canada. He graduated with a B.S. in
Applied Physics, specializing in astrophysics, along with a minor in Computer Science
from the University of Waterloo in 2000.
Mr. Hwang has since been working toward his Ph.D. in Acoustics at The
Pennsylvania State University, where he has been studying numerical methods for
radiated sound power calculation under Prof. Gary H. Koopmann.
Mr. Hwang is an active member of the Acoustical Society of America and the
Audio Engineering Society.
