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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Series in Acoustics Noise and Vibration
- Sprache: Englisch
Covers the theory and practice of innovative new approaches to modelling acoustic propagation There are as many types of acoustic phenomena as there are media, from longitudinal pressure waves in a fluid to S and P waves in seismology. This text focuses on the application of computational methods to the fields of linear acoustics. Techniques for solving the linear wave equation in homogeneous medium are explored in depth, as are techniques for modelling wave propagation in inhomogeneous and anisotropic fluid medium from a source and scattering from objects. Written for both students and working engineers, this book features a unique pedagogical approach to acquainting readers with innovative numerical methods for developing computational procedures for solving problems in acoustics and for understanding linear acoustic propagation and scattering. Chapters follow a consistent format, beginning with a presentation of modelling paradigms, followed by descriptions of numerical methods appropriate to each paradigm. Along the way important implementation issues are discussed and examples are provided, as are exercises and references to suggested readings. Classic methods and approaches are explored throughout, along with comments on modern advances and novel modeling approaches. * Bridges the gap between theory and implementation, and features examples illustrating the use of the methods described * Provides complete derivations and explanations of recent research trends in order to provide readers with a deep understanding of novel techniques and methods * Features a systematic presentation appropriate for advanced students as well as working professionals * References, suggested reading and fully worked problems are provided throughout An indispensable learning tool/reference that readers will find useful throughout their academic and professional careers, this book is both a supplemental text for graduate students in physics and engineering interested in acoustics and a valuable working resource for engineers in an array of industries, including defense, medicine, architecture, civil engineering, aerospace, biotech, and more.
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Seitenzahl: 442
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Series Preface
1 Introduction
2 Computation and Related Topics
2.1 Floating‐Point Numbers
2.2 Computational Cost
2.3 Fidelity
2.4 Code Development
2.5 List of Open‐Source Tools
2.6 Exercises
References
3 Derivation of the Wave Equation
3.1 Introduction
3.2 General Properties of Waves
3.3 One‐Dimensional Waves on a String
3.4 Waves in Elastic Solids
3.5 Waves in Ideal Fluids
3.6 Thin Rods and Plates
3.7 Phonons
3.8 Tensors Lite
3.9 Exercises
References
4 Methods for Solving the Wave Equation
4.1 Introduction
4.2 Method of Characteristics
4.3 Separation of Variables
4.4 Homogeneous Solution in Separable Coordinates
4.5 Boundary Conditions
4.6 Representing Functions with the Homogeneous Solutions
4.7 Green’s Function
4.8 Method of Images
4.9 Comparison of Modes to Images
4.10 Exercises
References
5 Wave Propagation
5.1 Introduction
5.2 Fourier Decomposition and Synthesis
5.3 Dispersion
5.4 Transmission and Reflection
5.5 Attenuation
5.6 Exercises
References
6 Normal Modes
6.1 Introduction
6.2 Mode Theory
6.3 Profile Models
6.4 Analytic Examples
6.5 Perturbation Theory
6.6 Multidimensional Problems and Degeneracy
6.7 Numerical Approach to Modes
6.8 Coupled Modes and the Pekeris Waveguide
6.9 Exercises
References
7 Ray Theory
7.1 Introduction
7.2 High Frequency Expansion of the Wave Equation
7.3 Amplitude
7.4 Ray Path Integrals
7.5 Building a Field from Rays
7.6 Numerical Approach to Ray Tracing
7.7 Complete Paraxial Ray Trace
7.8 Implementation Notes
7.9 Gaussian Beam Tracing
7.10 Exercises
References
8 Finite Difference and Finite Difference Time Domain
8.1 Introduction
8.2 Finite Difference
8.3 Time Domain
8.4 FDTD Representation of the Linear Wave Equation
8.5 Exercises
References
9 Parabolic Equation
9.1 Introduction
9.2 The Paraxial Approximation
9.3 Operator Factoring
9.4 Pauli Spin Matrices
9.5 Reduction of Order
9.6 Numerical Approach
9.7 Exercises
References
10 Finite Element Method
10.1 Introduction
10.2 The Finite Element Technique
10.3 Discretization of the Domain
10.4 Defining Basis Elements
10.5 Expressing the Helmholtz Equation in the FEM Basis
10.6 Numerical Integration over Triangular and Tetrahedral Domains
10.7 Implementation Notes
10.8 Exercises
References
11 Boundary Element Method
11.1 Introduction
11.2 The Boundary Integral Equations
11.3 Discretization of the BIE
11.4 Basis Elements and Test Functions
11.5 Coupling Integrals
11.6 Scattering from Closed Surfaces
11.7 Implementation Notes
11.8 Comments on Additional Techniques
11.9 Exercises
References
Index
End User License Agreement
List of Tables
Chapter 02
Table 2.1 Binary representation of integers
Table 2.2 Binary representations of fractions
Table 2.3 Four alternate representations of the first 16 (base 10) integers
Table 2.4 Exponent and mantissa widths and exponent bias for floating‐point numbers
Table 2.5 Machine epsilon for floating‐point types
Table 2.6 Size of various numbers in bytes
Table 2.7 Open‐source alternatives to MATLAB, MAPLE, or Mathematica
Table 2.8 List of C/C++ numeric libraries and other items
Table 2.9 List of tools related to finite and boundary element analysis
Chapter 03
Table 3.1 Three representations of the second‐order wave equation
Chapter 04
Table 4.1 Associated Legendre polynomials
Table 4.2 List of eigenfunctions and eigenvalues for three ideal cases
Table 4.3 Modified Green’s functions for a single boundary
Table 4.4 Locations and relative phase of images for three cases
Chapter 06
Table 6.1 Hermite polynomials
Table 6.2 Mode degeneracy for 2‐dim and 3‐dim examples
Table 6.3 Boundary conditions for relaxation
Chapter 08
Table 8.1 Indices for neighboring points contributing to Laplacian in 3 dim
Table 8.2 Stencils for spatial derivative
Table 8.3 Index sets for boundary points
Table 8.4 Stencils for finite differences in time
Table 8.5 Example estimates of the number of sample points required for setting up an FD grid
Chapter 09
Table 9.1 Common choices for operator splitting used to develop the parabolic equation
List of Illustrations
Chapter 03
Figure 3.1 Description of the tangent plane at a point on a curved manifold
Chapter 04
Figure 4.1 Sample mode,
, for two soft (left), two hard (center), and mixed (right) boundary conditions
Figure 4.2 Image method defined
Figure 4.3 Multiple images for satisfying two boundary conditions
Figure 4.4 Images for a source in a right‐angle corner
Figure 4.5 Images for a point source in a rectangular enclosure
Figure 4.6 Contour plot of |
p
| built from images
Figure 4.7 Contour plot of |
p
| built from modes
Chapter 05
Figure 5.1 Group velocity and phase velocity curves for a normal mode problem
Figure 5.2 Incident, reflected, and refracted waves at an interface between two media
Chapter 06
Figure 6.1 The Garrettt–Munk sound speed profile (left) and its effective potentials (right)
Figure 6.2 Effect potentials for the linear sound speed profile at various strengths
Figure 6.3 Effective potentials for an ideal refractive waveguide
Figure 6.4 First three modes for the quadratic potential
Figure 6.5 Modes of the linear potential for pressure release (top) and hard (bottom) boundary conditions
Figure 6.6 Perturbation applied too linear sound speed and soft boundary conditions
Figure 6.7 Definition of relaxation variables and lattice points
Figure 6.8 Relaxation applied to linear sound speed and soft boundary conditions
Figure 6.9 Setup for the Pekeris waveguide
Figure 6.10 Waveguide with irregular boundary
Figure 6.11 Setup for coupled mode
Chapter 07
Figure 7.1 Illustration showing quantities defined in the paraxial ray trace
Figure 7.2 Ray trace for the linear sound speed profile
Figure 7.3 Sample ray trace for the linear wind profile
Figure 7.4 Sample ray trace for the Kormilitsin profile
Figure 7.5 Ray trace for the refractive waveguide
Figure 7.6 Ray trace for a linear refractive sound speed profile and hard reflective ground as
Figure 7.7 Three‐dimensional ray trace for the Garrett–Munk profile
Chapter 08
Figure 8.1 Sample 2‐dim grid with rectangular boundary
Figure 8.2 Example of irregular boundary illustrating interior, boundary, and near points
Figure 8.3 Cutting of a cubic point lattice by a sphere
Figure 8.4 Local normal derived from edge vectors of a triangular patch
Figure 8.5 Identification of interior and boundary points and local normal vectors for implementing Neumann boundary conditions in 2 dim
Figure 8.6 Example 5 × 5 grid
Figure 8.7 Leapfrog method on 1 + 1 dim lattice
Figure 8.8 Location on lattice where variables are evaluated
Figure 8.9 Time steps for a 2 + 1 dim implementation
Chapter 09
Figure 9.1 Computational stencil for the Crank–Nicolson method
Chapter 10
Figure 10.1 One‐dimensional patches and basis elements defined
Figure 10.2 Sample tiling of a square with right triangle
Figure 10.3 Sample tiling of a square with right triangles
Figure 10.4 Example of a random tiling of a square using Gmsh
Figure 10.5 Cube cut into five tetrahedra
Figure 10.6 A cube in 3‐dim tiled with tetrahedra using Gmsh
Figure 10.7 Two‐dimensional rooftop basis defined
Figure 10.8 Mapping of a generic triangle (left) into the standard triangle (center) and to a square (right)
Figure 10.9 Example of Dunavant quadrature points in simplex space, order = 12
Chapter 11
Figure 11.1 Definition of terms for the exterior problem
Figure 11.2 Real part of Green’s function
Figure 11.3 Variables used for evaluating singular integrals
Figure 11.4 Evaluation of
Figure 11.5 Contour plot of Figure 11.4
Figure 11.6 Example of far field bi‐static scattering from a sphere
Guide
Cover
Table of Contents
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Wiley Series in Acoustics, Noise and Vibration
Computational Acoustics
Bergman
January 2018
Wind Farm Noise
Hansen
February 2017
The Effects of Sound on People
Cowan
May 2016
Engineering Vibroacoustic Analysis:
Methods and Applications
Hambric et al.,
April 2016
Formulas for Dynamics, Vibration and Acoustics
Blevins
November 2015
COMPUTATIONAL ACOUSTICS
THEORY AND IMPLEMENTATION
David R. Bergman
Exact Solution Scientific Consulting LLC, Morristown, NJ, USA
This edition first published 2018© 2018 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of David R. Bergman to be identified as the author of this work has been asserted in accordance with law.
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MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
Library of Congress Cataloging-in-Publication Data
Names: Bergman, David R., author.Title: Computational acoustics : theory and implementation / David R. Bergman.Description: Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |Identifiers: LCCN 2017036469 (print) | LCCN 2017046745 (ebook) | ISBN 9781119277330 (pdf) | ISBN 9781119277279 (epub) | ISBN 9781119277286 (cloth)Subjects: LCSH: Sound‐waves–Measurement. | Sound‐waves–Computer simulation. | Sound‐waves–Mathematical models.Classification: LCC QC243 (ebook) | LCC QC243 .B38 2018 (print) | DDC 534.0285–dc23LC record available at https://lccn.loc.gov/2017036469
Cover Design: WileyCover Image: © Vik_Y/Gettyimages
Series Preface
This book series will embrace a wide spectrum of acoustics, noise, and vibration topics from theoretical foundations to real‐world applications. Individual volumes will range from specialist works of science to advanced undergraduate and graduate student texts. Books in the series will review scientific principles of acoustics, describe special research studies, and discuss solutions for noise and vibration problems in communities, industry, and transportation.
The first books in the series include those on Biomedical Ultrasound; Effects of Sound on People, Engineering Acoustics, Noise and Vibration Control, Environmental Noise Management; and Sound Intensity and Windfarm Noise. Books on a wide variety of related topics.
The books I edited for Wiley—Encyclopedia of Acoustics (1997), The Handbook of Acoustics (1998), and Handbook of Noise and Vibration Control (2007)—included over 400 chapters written by different authors. Each author had to restrict their chapter length on their special topics to no more than about 10 pages. The books in the current series will allow authors to provide much more in‐depth coverage of their topic.
The series will be of interest to senior undergraduate and graduate students, consultants, and researchers in acoustics, noise, and vibration and in particular those involved in engineering and scientific fields, including aerospace, automotive, biomedical, civil/structural, electrical, environmental, industrial, materials, naval architecture, and mechanical systems. In addition the books will be of interest to practitioners and researchers in fields such as audiology, architecture, the environment, physics, signal processing, and speech.
Malcolm J. CrockerSeries Editor
1Introduction
Computers have become an invaluable tool in science and engineering. Over time their use has evolved from a device to aid in complex lengthy calculations to a self‐contained discipline or field of study. Coursework in science and engineering often involves learning analytic techniques and exact solutions to a sizable collection of problems. Although educational, these are rarely useful beyond the classroom. On the other side of the spectrum is the practical experimental approach to investigating nature and developing engineering solutions to practical everyday problems. Most readers are familiar with the nonideal nature of things that, in many cases, prevents one from seeing the utility of the theoretical approach. In science theorist and experimentalist see nature from a different perspective in a quest for understanding its laws but agree that the facts can only be found through observation, as patterns in data acquired via well‐planned and executed experiments designed to isolate certain degrees of freedom, to replicate an ideal circumstance to the best of our ability. Experiments can be costly but are the only mechanism for determining scientific truth. While the scientist works to create ideal circumstances to verify a fundamental law or hypothesis, an engineer must design and build with nonideal conditions in mind. In many cases the only approach available is trial and error. This requires the resources to build new versions of a device or invention every time it fails, each prototype being built and used to see what will happen and how it will fail and to learn from the experience. In this regard, computational science offers a path toward testing prototypes in a virtual environment. When executed carefully this approach can save time and money, prevent human injury or loss of life, and reduce impact to the environment.
As computers became larger, faster, and more efficient, the size of the tasks that could be performed also became larger, evolving from modeling the stress on a single beam to that found throughout the structure of a building, ship, or aircraft under dynamic loading. In recent times the use of computer‐based modeling and simulation has gained a certain credibility in fields where there is no possible experimental method available and theory does not offer a suitable path forward in exploring the consequences of natural law, in particular the fields of numerical relativity (NR) and computational fluid mechanics (CFM). In these fields of study, the computer has become the laboratory, offering us the ability to experiment on systems we cannot build in the physical world.
Over the past decade or so, we are perhaps seeing the emergence of a new class of scientist or scientific specialist, along with the experimentalist and theorist, the numericist. Just as an experimentalist needs to be aware of the science behind the inner working of their probes and detectors, the numericist must understand the limitations imposed by working with finite precision, or a discrete representation of the real number system. The computer is the device we used to probe our virtual world, and discrete mathematics imposes constraints on the precision of the probe. In moving from the world of smooth operations on the continuum to discrete representations of the same, we lose some basic kernels of truth we rely on as common sense. Namely, certain operations are abelian. More precisely there are certain procedures that when carried out by hand produce the same results regardless of the order in which the steps are performed but when executed on a computer could lead to different results for different implementations. The development of computational procedures requires attention to this fact, a new burden for the numericist, and understanding of the impact of this behavior on expected results.
This text focuses on the application of computational methods to the fields of linear acoustics. Acoustics is broadly defined as the propagation of mechanical vibration in a medium. Several aspects of this make acoustics an interesting field of study. First is the need for a medium to support the acoustic phenomenon, which unlike light propagates in free space at constant speed relative to all inertial observers. Another point of interest is that there are as many types of acoustic phenomena as there are media, from longitudinal pressure waves in a fluid to S and P waves in seismology. The material properties of the medium determine the number and type of acoustic waves that may be created and observed. We typically think of acoustics as a macro phenomenon, the result of bulk movement of the medium. However, as we probe nature at smaller scales, this type of phenomenon is precisely what is creating the acoustic phenomenon in solids and similarly particle collisions in fluids. The acoustic phenomenon is seen at small scales in lattice vibrations in crystals. Here the acoustic field is quantized and the quanta are referred to as phonons. This model is the result of an attempt to understand a phenomenon that exists at scales too large to be described by the fundamental process and too small to be a purely classical phenomenon.
The goal of this text is to introduce to the reader those numerical methods associated with the development of computational procedures for solving problems in acoustics and understanding linear acoustic propagation and scattering. The intended audience are students and professionals who are interested in the ingredients needed for the development of these procedures. The presentation of the material in this text is unique in the sense that it focuses on modeling paradigms first and introduce the numerical methods appropriate to that modeling paradigm rather than offer them in a preliminary chapter or appendix. Along the way, implementation issues that readers should be aware of are discussed. Examples are provided along with suggested exercises and references. The intent is to be pedagogical in the approach to presenting information so that readers who are new to the subject can begin experimenting. Classic methods and approaches are featured throughout the text while additional comments are included that highlight modern advances and novel modeling approaches that have appeared in the literature.
Since the intended audience consists of upper‐level undergraduate students, graduate students, or professionals interested in this discipline, expected prerequisites to this material are:
An introductory course that covers acoustics or fluid dynamics
Familiarity with ordinary differential and partial differential equations, perhaps a course in mathematical methods for scientists and engineers
Some exposure to programming in a high‐level language such as Maple, Mathematica, and MATLAB or its open‐source counterparts, SCILAB and Octave
The key feature of the presentation contained in this text is that it serves to bridge the gap between theory and implementation. The main focus is on techniques for solving the linear wave equation in homogeneous medium as well as inhomogeneous and anisotropic fluid medium for modeling wave propagation from a source and scattering from objects. Therefore, the starting point for much of this text will be the standard wave equation or the Helmholtz equation.
The transition from equations to computer procedures is not always a straightforward path. High‐level programming languages come with easy‐to‐use interfaces for solving differential equations, matrix equations, and performing signal processing. Beyond these are professional software packages designed to allow users to build and run specific types of simulations using common modeling paradigms. Examples include ANSYS, FEMLAB, and FEKO, just to name a few. An understanding of the math, physics, and numerics is required to evaluate and interpret the results, but low‐level programming is not necessary. Why learn these techniques? Specialized software can be very expensive, in fact cost prohibitive for students or those engaging in self‐study. Many software companies offer personal or student versions of their software at a severely discounted price and with a restricted user license. If the reader is using this text for coursework in computational acoustics at a college or university, chances are student licenses for some professional software packages are made available through the campus bookstore. If not, it is easy to find this information online. Open‐source versions of professional software exist and are worth trying. The downside to this is that bugs exist and due to certain constraints a fix may not be available in a hurry. Also, some open‐source tools are not compatible with all operating systems. Readers who like programming and are amenable to the open‐source philosophy can always contribute their fixes and upgrades (read the license). Pure curiosity drives most scientists and engineers to want to know what’s going on in any system, and this is a driver for developing homegrown algorithms even when libraries are available.
A brief description of each chapter is provided. Chapter 2 introduces topics related to numerics, computers, and algorithm development. These topics include binary representation of numbers, floating‐point numbers, and O(N) analysis, to name a few. Chapter 3 contains a survey of the linear wave equation and its connection to the supporting medium, from elastic bodies to fluids. In this chapter the linear wave equation for acoustics in a moving medium is introduced and discussed in detail. Chapter 4 introduces a variety of mathematical techniques and methods for solving the wave equation and describing the general behavior of the acoustic field. Chapter 5 discusses a variety of topics related to the analysis of acoustic waves: dispersion, refraction, attenuation, and Fourier analysis. After these chapters the structure of the text focuses on specific modeling techniques. In Chapter 6 normal modes are discussed. The wave equation is solved for a variety of 1‐dimensional (1‐dim) refractive profiles using exact methods, perturbation theory, and the numerical technique of relaxation. The chapter closes with a brief description of coupled modes and their use in modeling acoustics in realistic environments. Chapter 7 provides an introduction to ray theory and ray tracing techniques. Exact solutions to 1‐dim problems are discussed along with methods of developing ray trace procedures that account for 3‐dim propagation without simplifying assumptions. Numerical techniques are also discussed and the Runge–Kutta method is introduced. In Chapter 8 the finite difference (FD) and finite difference time domain (FDTD) technique are discussed in theory and applied to the wave equation in the frequency and time domains. Following the FD method, Chapter 9 discusses the parabolic equation and its application to modeling sound in ducted environments. Chapter 10 provides an introduction to the finite element method (FEM), introducing numerical techniques required for building an FEM model of the acoustic field in the frequency domain. The last chapter, Chapter 11, is dedicated to the boundary element method (BEM). This chapter discusses the integral equation form of the Helmholtz equation and its discretization into a matrix equation. The exterior and interior problems are discussed, but attention is spent on developing models of the scattering cross section of hard bodies. This chapter introduces techniques for dealing with singular integrals.
