Finite Element and Boundary Methods in Structural Acoustics and Vibration

466 Pages·2015·8.476 MB·English

by  Noureddine Atalla, Franck Sgard

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K20584 nd
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Finite Element and
Boundary Methods in
Structural Acoustics
and Vibration

Finite Element and
Boundary Methods in
Structural Acoustics
and Vibration
Noureddine Atalla
Franck Sgard
A SPON PRESS BOOK
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does
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Thanks to all our loved ones for their support
and patience while writing this book.
Noureddine Atalla, Franck Sgard
My deepest thanks are reserved to Sonia, who provided me
with her support and moral booster during the years of book
preparation and who gave me the energy to complete this book.
Franck Sgard

Contents
Acknowledgments xvii
Authors xix
1 Introduction 1
1.1 Computational vibroacoustics 1
1.2 Overview of the book 7
References 7
2 Basic equations of structural acoustics and vibration 9
2.1 Introduction 9
2.2 Linear acoustics 9
2.3 Linear elastodynamics 12
2.4 Linear poroelasticity 14
2.5 Elasto-acoustic coupling 18
2.6 Poro-elasto-acoustic coupling 18
2.7 Conclusion 19
References 19
3 Integral formulations of the problem of structural
acoustics and vibrations 21
3.1 Introduction 21
3.2 Basic concepts 22
3.2.1 Variational statement: Stationarity of a functional 23
3.3 Strong integral formulation 24
3.4 Weak integral formulation 25
3.5 Construction of the weak integral formulation 26
3.6 Functional associated with an integral
formulation: Stationarity principle 27
vii
viii Contents
3.7 Principle of virtual work 34
3.8 Principle of minimum potential energy 36
3.9 Hamilton’s principle 37
3.10 Conclusion 41
Appendix 3A: Methods of integral approximations—Example 41
3A.1 The problem 41
3A.2 Method of weighted residuals 41
3A.2.1 Approximation by one term 42
3A.2.2 Approximation using two terms 47
3A.3 Variational method 47
3A.4 Rayleigh–Ritz’s method 51
Appendix 3B: Various integral theorems and vector identities 52
3B.1 Nabla operator 52
3B.2 Useful vector identities 53
3B.3 Divergence theorem 53
3B.4 Stoke’s theorem 54
3B.5 Green’s first identity 54
3B.6 Green’s second identity 54
Appendix 3C: Derivation of Hamilton’s principle
from the principle of virtual work 54
Appendix 3D: Lagrange’s equations (1D) 56
References 62
4 The finite element method: An introduction 63
4.1 Introduction 63
4.2 Finite element solution of the one-dimensional
acoustic wave propagation problem 65
4.2.1 Problem statement 65
4.2.2 Step 1: Weak integral form 65
4.2.3 Step 2: Meshing 67
4.2.4 Step 3: Approximation of the independent
variable and calculation of the elementary matrices 67
4.2.4.1 Nodal approximation of the variable 67
4.2.4.2 Nodal approximation of
the weak integral form 70
4.2.4.3 Evaluations of the elementary matrices 72
4.2.5 Step 4: Assembling 76
4.2.6 Step 5: Imposition of constraints 78
4.2.7 Steps 6 and 7: Solution and convergence study 80
4.3 Conclusion 83