Wave propagation in homogeneous elastic half-space using the Dual Reciprocity Boundary ...

129 Pages·2015·8.86 MB·English

by  Tosecký, Andrej

Wave propagation in homogeneous elastic half-space
using the Dual Reciprocity Boundary Element Method
by
Andrej Tosecky´
Dissertation submitted to the Faculty of Civil Engineering
of the Ruhr University Bochum
in partial fulfillment of the requirements
for the degree of Doctor Engineer.
October 2005
Supervisor: Prof. Dr. Gu¨nther Schmid, Ph.D.
Co-Supervisor: Prof. Dr. rer. nat. Klaus Hackl
Abstract
The Dual Reciprocity Boundary Element Method (DRBEM) is studied thoroughly in this disser-
tation. At the current state-of-the-art this method is used to study various physical phenomena
(e.g. heat conduction, acoustic effects, fluid mechanics) in closed domains. Some pioneering at-
tempts have been made, aspiring after enhancement of the field of application of the DRBEM to
infinite 2D domains. To the best knowledge of the author, the DRBEM has never been applied to
3D half-space dynamics yet. There are some extra requirements for the mathematical formulation
of the method which make its extension to infinite and semi-infinite domains a real pain. First
part of the dissertation deals with mathematical derivation of the method. Stress is laid upon
peculiarities of the method when applied to infinite and semi-infinite domains. It has been found
out that semi-infinite domains are even more difficult to deal with, since after discretization of the
half-space surface by boundary elements, an excessive spurious wave reflection on the border of
the discretized region occurs. Mathematical derivation is followed by its numerical implementa-
tion and computer coding. The developed method has been implemented into an own computer
code XSpeed. Wave propagation due to various kinds of loading in time-domain is studied af-
terward, aimed at validation of the method. Special attention is devoted to moving loads (e.g.
high-speed trains) and study of the effect known as “critical velocity”. The main advantage of the
DRBEM over its counterparts used to model infinite and semi-infinite domains (such as classical
BEM formulation, integral transformations, Thin Layer Method) is, that it produces time- and
frequency-independent matrices (mass and stiffness matrix namely), by preserving boundary-only
discretization (no internal nodes necessary). This makes the method very attractive, since this
feature is very close to the common engineering understanding and the final equation of motion
has similar form as the one known from the Finite Element Method. Moreover, the formulation
allows for seamless incorporation of non-homogeneous initial conditions, i.e. non-zero initial dis-
placements, velocities, accelerations and surface tractions can be prescribed.
Submitted on October the 25th, 2005
ii
Acknowledgement
I owe my sincere gratitude to all the people who have made this thesis possible, directly or
indirectly.
First and foremost, I’d like to thank Professor Gu¨nther Schmid for having given me the oppor-
tunity to conduct my doctor thesis with so much freedom and open space for my own ideas and for
being the person who has always encouraged me and trusted in me especially in those moments
when my own hope of finishing this thesis was hanging by a thread. His advices, seeming very
simple at the first sight, gave rise to new ideas leading to great leaps forward. Thank you dear
Professor Schmid for having broadened my views by your words of wisdom.
Finishing this thesis would not have been possible without people dear to my hearth. Without
their constant support it would be pretty hard to stay in balance throughout the sometimes really
exhausting phases of the labor on this work. My special thank goes to my friends Michael Krug
and Bernd Neuhaus and to my colleagues Vera Feldhaus and Wolfgang Hubert.
My gratitude goes also to Professor Klaus Hackl for his interest in my work and for accepting
the role of my co-adviser.
My last thanks is for my mom who has patiently withstood the long time of my being some-
where else ...far from home. Thank you.
Bochum, October the 8th, 2005 Andrej Tosecky´
iii
“Life is a comedy for those who think ...and a tragedy for those who feel.”
Horace Walpole
iv
Contents
Notation vii
1 Motivation 1
2 Preliminaries 3
2.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Underlying mathematical theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Lam´e-Navier’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Solution to the Lam´e-Navier’s equation . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Weak form of the equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Boundary integral equation for elastodynamics . . . . . . . . . . . . . . . . . . . . 15
2.7.1 Classical BEM approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Navier’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 The Dual Reciprocity Boundary Element Method 18
3.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Transformation of the domain integral . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Inertial forces approximating functions f(x) . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 Principle of superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Solution utilizing body force’s potentials . . . . . . . . . . . . . . . . . . . 27
3.3.3 Fourier transformation of Navier’s equation . . . . . . . . . . . . . . . . . 28
3.3.4 Papkovich-Neuber potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.5 Inverse procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.1 Singular integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2 Indirect determination of strongly singular integrands and
integration-free terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.3 Understanding the matrix assemblage . . . . . . . . . . . . . . . . . . . . . 45
3.4.4 Inclusion of internal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.5 Formulation without matrix inversion . . . . . . . . . . . . . . . . . . . . . 50
3.5 Computational aspects of the DRBEM . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Does it work after all? 55
4.1 Wave propagation in elastic half-space . . . . . . . . . . . . . . . . . . . . . . . . 55
vi CONTENTS
4.1.1 Wave amplitude attenuation and radiation damping . . . . . . . . . . . . . 57
4.2 Inertial force approximating functions f(r) and amplitude decay . . . . . . . . . . 59
4.3 Wave reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Full-space vs. half-space fundamental solution . . . . . . . . . . . . . . . . . . . . 74
4.5 Singular behavior of the function f(r) = 1/r . . . . . . . . . . . . . . . . . . . . . 75
4.6 Temporal discretization and choice of time-step . . . . . . . . . . . . . . . . . . . 76
4.7 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7.1 Decay rate of Rayleigh waves excited by harmonic source . . . . . . . . . . 81
4.7.2 Spatial discretization criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.8 Concluding remarks to this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Moving loads on elastic half-space 87
5.1 Introduction and short overview of used techniques . . . . . . . . . . . . . . . . . 87
5.2 The Dual Reciprocity Boundary Element Method and moving loads . . . . . . . . 89
5.2.1 Model of moving load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.2 Further investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Concluding remarks to this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 Conclusions 103
A Investigation of time-integration stability criteria 107
B Analytical solutions to point load moving on elastic half-space 110
B.1 Barber’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.2 Payton’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Notation
General symbols
i Imaginary unit
N Number of nodes
N Number of elements
e
q Abscissa in the Gauss’s numerical integration scheme
w Weight in the Gauss’s numerical integration scheme
β, γ Newmark factors
δ() Dirac function
δ Kronecker symbol
ij
ω Coordinates in Fourier space
i
Geometry
i, j, k Unit basis vectors
J Jacobi determinant
L Distance, number of internal nodes
n Unit outward normal
n Component of unit outward normal
i
n Outward normal
∗
r Position vector
r Component of position vector
i
r Derivation of position vector with respect to ith coordinate
,i
∆x Element edge length
viii Notation
β , β Transformed coordinates by singular integration
1 2
Γ Boundary
Γ Area of single boundary element
e
Γ Part of boundary with known surface traction
t
Γ Part of boundary with known displacement
u
Γ Boundary at infinity
∞
η , η Natural coordinates
1 2
Ω Domain
ξ Source node
Material constants
c Dilatational wave propagation velocity
p
c Rayleigh wave propagation velocity
R
c Shear wave propagation velocity
s
E Young’s modulus
G Shear modulus
λ Lam´e’s constant
ν Poisson’s ratio (minor)
ρ Mass density
Physical quantities
f Frequency
t Time variable
∆t Time-step length
τ Time variable, time delay
T Positive time-domain (t > 0)
⊕
T Negative time-domain (t < 0)
⊖
u(x,t) Displacement vector
u˙(x,t) Velocity vector
u¨(x,t) Acceleration vector
v Load’s traveling velocity
ix
κ Decay rate exponent
λ Wavelength of dilatational wave
p
λ Wavelength of Rayleigh wave
R
λ Wavelength of shear wave
s
Theory of elasticity
b Body force vector (force per unit mass)
b Component of body force vector
i
B(ω) Fourier-transformed body force vector
div Divergence of vector function
e Strain tensor component
ij
grad Gradient of scalar function
tˆ Fund. sol. of elastostatics for surface tractions
ij
u Displacement vector
U(ω) Fourier-transformed displacement vector
uˆ Fund. sol. of elastostatics for displacements
ij
V (x,y,z) Vector function
∆ Laplace operator
ǫ Engineering normal strain
ǫˆ Strains of fundamental solution of elastostatics
ij
γ Engineering shear strain
σ Normal stress
σ Stress tensor component
ij
Π(x,y,z), φ(x,y,z) Scalar potential
τ Tangential stress, shear stress
ϕ(x,y,z) Vector potential
BEM and Dual Reciprocity BEM
c Integration-free terms (boundary coefficients)
ij
f(r) Radial basis inertial forces approximating function
F Matrix storing f(r) coefficients
F 1 Inverse of F matrix
−
x Notation
f(t) Vector of external forces
c Damping matrix
C Matrix storing integration-free terms
g Integrated values of displacement fundamental solution
ij
G Boundary matrix storing g coefficients
ij
h Integrated values of surface traction fundamental solution
ij
H Boundary matrix storing h coefficients
ij
k, K Stiffness matrix
m, M Mass matrix
α(t) Time-dependent factor
α(t) Vector storing α(t) factors
η Particular solution to the Navier’s equation for surface tractions
ij
η Matrix storing η coefficients
ij
ψ Particular solution to the Navier’s equation for displacements
ij
ψ Matrix storing ψ coefficients
ij