Ultrasonic System Measurement Model - II
Learning Objectives
Definition of a Measurement Model
Three Types of Measurement Models
Auld Model
Thompson-Gray
Model for 2D scatterers
Verifying the Modeling Assumptions
and Measurements Needed
Ultrasonic Measurement Models
What is an Ultrasonic Measurement Model? A model that
expresses the output voltage, vR(t) ,received from a flaw in a
ultrasonic testing configuration in a form that explicitly states the
modeling or measurement parameters needed to obtain that
voltage (immersion shown only as a specific example)
Pulser
Receiver
n
B
A
component
being inspected
Sf
vR ( t )
surface of the flaw which has
an outward normal, n
All the measurement models discussed use VR(ω), the frequency components of vR(t)
FFT
VR (ω ) ↔ vR ( t )
Ultrasonic Measurement Models
Electroacoustic Measurement Model - 2002
Dang, C. J., Schmerr, L.W., and A. Sedov," Modeling and measuring all the elements of an ultrasonic nondestructive
evaluation system. I: Modeling Foundations," Research in Nondestructive Evaluation, 14, 141- 176, 2002.
VR (ω ) = tG (ω ) t R (ω ) Vi (ω ) t A (ω )
driving voltage
= s (ω ) t A (ω )
output voltage
s (ω )
Vi (ω )
VR (ω )
n
Sf
pulser
cabling
tG ( ω )
sending
transducer
t A (ω ) = ?
propagation,
flaw scattering
receiving cabling receiver
transducer
t R (ω )
Ultrasonic Measurement Models
Auld Electromechanical Reciprocity Relation
Auld (1979) , using reciprocity relations and the solutions to two specific
problems, developed an explicit relation that has been widely used
B. A. Auld, Wave Motion,, Vol. 1, (1979), pp. 3-10.
The two problems:
problem (1)
problem (2)
n
B
A
A
B
Sf
transducer A firing,
flaw present
transducer B firing,
flaw absent
Ultrasonic Measurement Models
Auld Electromechanical Reciprocity Relation
Γ
n
A
B
P
Sf
(
)
1
(1)
( 2)
( 2)
( 2)
δΓ =
t
⋅
v
−
t
⋅
v
dS
∫
4P S f
δΓ… change in cable transmission coefficient due to the presence of the flaw
P… power delivered by the transducer
t (1) , v (1)
stress vector and velocity for problem (1) : A firing and flaw present
t ( 2) , v ( 2)
stress vector and velocity for problem (2): B firing and flaw absent
Ultrasonic Measurement Models
Auld Electromechanical Reciprocity Relation
Γ
n
A
B
P
Sf
(
)
1
(1)
( 2)
( 2)
( 2)
δΓ =
t
⋅
v
−
t
⋅
v
dS
∫
4P S f
Auld's assumptions:
1) linear, reciprocal electromechanical system (harmonic disturbances)
2) fundamental TEM mode propagating in the cable
Since we expect VR (ω ) ∝ δ Γ (ω )
an ultrasonic measurement model
Auld's relation "essentially" is
Ultrasonic Measurement Models
An Alternative Reciprocity Relation
In 1998, Dang, Schmerr, and Sedov obtained a completely explicit model
relationship using a purely mechanical reciprocity relationship plus
several other relatively weak assumptions
problem (1)
problem (2)
n
A
A
B
B
(1)
(2)
vA
vB
Sf
VR (ω ) =
s (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
Sf
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Ultrasonic Measurement Models
VR (ω ) =
s (ω )
s (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Sf
… system function (pulser, cabling, receiver)
v (A1) (ω )
… average velocity over the face of transducer A in problem (1)
vB( 2) (ω )
… average velocity over the face of transducer B in problem (2)
Z rA (ω )
… radiation impedance of transducer A in problem (1)
Ultrasonic Measurement Models
VR (ω ) =
s (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Sf
assumptions used to obtain this explicit model
1) linear, reciprocal acoustic and elastic media (harmonic disturbances)
Ultrasonic Measurement Models
2) we also assumed the output velocity for the transducers can be written
in a separable form. For transducer A, for example
ρ1 , c1
density, wave speed of the fluid
v (A1) ( x, ω ) = v(A1) (ω ) f ( x )
A
v(A ) (ω )
1
average velocity
Here, the "forces" are defined in terms of weighted pressure integrals
over the transducer face, using the same functions used to describe
the velocity. For example, for transducer A
Ft (ω ) =
∫ p ( x, ω ) f ( x ) dS ( x )
SA
For a piston transducer
f =1
These two assumptions lead to an explicit
expression for the acoustic/elastic transfer function:
flaw
A
Ft
Transmitted
force
B
FB
Blocked
force
FB (ω )
1
(1)
( 2)
( 2)
(1)
t A (ω ) =
dS
= A
t
⋅
v
−
t
⋅
v
∫
(1)
( 2)
Ft (ω ) Z r (ω ) v A (ω ) vB (ω ) S f
(
)
Ultrasonic Measurement Models
3) we also assume that the pulser, receiver, transducers and cabling
can be replaced by equivalent linear, time-shift invariant (LTI) systems,
i.e.
Ft (ω ) = tG (ω ) Vin (ω )
VR (ω ) = t R (ω ) FB (ω )
Then, we can write the reciprocity relationship in the form
VR (ω ) =
t R (ω ) tG (ω ) Vin (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Sf
or, finally
VR (ω ) =
s (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
Sf
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Ultrasonic Measurement Models
A General Ultrasonic Measurement Model
VR (ω ) =
s (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Sf
assumptions
1) linear, reciprocal acoustic and elastic media (harmonic disturbances)
2) the output velocity for the transducers can be written in a separable
form
3) the pulser, receiver, transducers and cabling can be replaced by
equivalent linear, time-shift invariant (LTI) systems.
Ultrasonic Measurement Models
4) if we also assume that the transducers behave as piston radiators at
high frequency then
Z rA (ω ) = ρ1c1S A
and our explicit model becomes, finally
VR (ω ) =
s (ω )
ρ1c1S Av A (ω ) vB
(1)
( 2)
t
(
∫
(ω )
Sf
(1)
⋅v
( 2)
−t
( 2)
⋅v
(1)
) dS
Ultrasonic Measurement Models
5) now, assume the incident waves are quasi-plane waves
in the vicinity of the flaw, e.g.
v (j1)
inc
(
= V (1) ( x1 , x2 , x3 , ω ) d (j1) exp ik2e(1) ⋅ x
(
v (j2) = V ( 2) ( x1 , x2 , x3 , ω ) d (j 2) exp ik2e( 2) ⋅ x
)
)
and also define normalized amplitudes of these quasi-plane waves as
(1)
V
1
Vˆ ( ) = (1)
vA (ω )
Vˆ ( 2) =
V(
2)
vB( ) (ω )
2
Ultrasonic Measurement Models
Then the form of the measurement model becomes
⎡ 4πρ 2 c2 ⎤ ˆ (1) ˆ ( 2)
(1) (1)
%
% dS
A
VR (ω ) = s (ω ) ⎢
V
V
v
j ,τ ij
⎥∫
⎣ −ik2 ρ1c1S A ⎦ S f
(
velocity fields
in problems (1),(2)
due to unit average
velocities on the
transducer faces
)
scattered wave fields
from the flaw
The flaw scattering term is rather complex:
(
)
A v% (j1) ,τ%ij(1) =
τ ij(1) , v(j1)
e(j ) , d (j
2
2)
nj
1
4πρ c
2
2 2
(
(
) )
(
τ%ij(1) di( 2) − Cijkl d k( 2) el( 2) / c2 v%i(1) n j exp ik2e( 2) ⋅ x
stresses and velocity components in problem (1)
components of the incident wave direction and polarization
in problem (2)
components of the unit outward normal to the flaw surface
Cijkl
elastic constants tensor
1
1
v% (j ) ,τ%ij( )
the normalized fields
are defined as velocity and stress in
problem (1) due to an incident wave of unit displacement amplitude
v% (j1) =
(1)
τ%ij =
−iω v (j )
1
V( )
−iωτ ij(1)
1
V( )
1
U (1) = 1
τ%ij(1) , v% (j1)
)
Ultrasonic Measurement Models
3-D scattering amplitude
u scatt ( y , ω ) = U 0
A ( eiβ ; e sP )
rs
scattered displacement
from a flaw
exp ( ik p rs ) + U 0
P-wave
A ( eiβ ; e Ss )
rs
exp ( ik s rs )
S-wave
incident wave with
polarization
dα in problem (2)
U0
eiβ
U0 … displacement
amplitude for wave
of type β (β = P,S) in
problem (1)
d P e sP , d S ⊥ e sS
rs
flaw
xs
n
eαs
Scattered wave of
type α (α = P,S)
Ultrasonic Measurement Models
3-D scattering amplitude
u scatt ( y , ω ) = U 0
A ( eiβ ; e sP )
rs
exp ( ik p rs ) + U 0
A ( eiβ ; e Ss )
rs
However, it can be shown that
A ( eiβ ; eαs ) ⋅ ( −dα ) =
(
)
(1) (1)
%
A
v
∫ j ,τ%ij dS
Sf
vector scattering
amplitude of waves
in problem (1)
polarization vector of incident
waves in problem (2)
exp ( iks rs )
Ultrasonic Measurement Models
Measurement Model - Quasi-Plane Waves
⎡ 4π ρ 2 c2 ⎤ ˆ (1) ˆ ( 2)
(1) (1)
%
% dS
VR (ω ) = s (ω ) ⎢
V
V
v
A
j ,τ ij
⎥∫
⎣ −ik2 S A ρ1c1 ⎦ S f
(
A ( eiβ ; eαs ) ⋅ ( −dα ) =
(
)
)
(1) (1)
%
A
v
∫ j ,τ%ij dS
Sf
The beam modeling terms and flaw scattering terms appear separately in
this model but they still must be combined before the voltage can be
obtained
6) Now, assume a small 3-D flaw, i.e. where the incident waves do not
vary significantly in amplitude over the surface of the flaw
1
1
1
Vˆ ( ) ( x1 , x2 , x3 ) = Vˆ ( ) ( 0, 0, 0 ) = Vˆ0( )
x1
2
1
2
Vˆ ( ) ( x1 , x2 , x3 ) = Vˆ ( ) ( 0, 0, 0 ) = Vˆ0( )
O
x3
x2
Then
⎡ 4π ρ 2 c2 ⎤ ˆ (1) ˆ ( 2)
(1) (1)
⎡
⎤
%
% dS
VR (ω ) = s (ω ) ⎢
V
V
v
A
j ,τ ij
⎥⎣ 0 0 ⎦ ∫
⎣ −ik2 S A ρ1c1 ⎦
Sf
(
and, recall
A ( eiβ ; eαs ) ⋅ ( −dα ) =
(
)
(1) (1)
%
A
v
∫ j ,τ%ij dS
Sf
)
Ultrasonic Measurement Models
Measurement Model (Thompson-Gray form)
R. B. Thompson and T. A. Gray, J. Acoust. Soc. Am., Vol. 74, (1983), pp. 140-146.
⎡ 4π ρ 2 c2 ⎤
1) ˆ ( 2 )
(
β
α
α
ˆ
VR (ω ) = s (ω ) ⎡⎣V0 V0 ⎤⎦ ⎡⎣ A ( ei ; e s ) ⋅ ( −d ) ⎤⎦ ⎢
⎥
ik
S
ρ
c
−
⎣ 2 A 11⎦
beam models
A ( eiβ , eαs , ω )
plane wave far field
scattering amplitude
component
⎡ 4π ρ 2 c2 ⎤
(1) ˆ ( 2 ) ⎤ ⎡
β
α
ˆ
⎡
⎤
t A (ω ) = ⎣V0 V0 ⎦ ⎣ A ( ei , e s , ω ) ⎦ ⎢
⎥
−
ik
S
ρ
c
⎣ 2 A 11⎦
Ultrasonic Measurement Models
Measurement Model (Thompson-Gray form)
⎡ 4π ρ 2 c2 ⎤
(1) ˆ ( 2 ) ⎤ ⎡
β
α
ˆ
⎡
⎤
VR (ω ) = s (ω ) ⎣V0 V0 ⎦ ⎣ A ( ei , e s , ω ) ⎦ ⎢
⎥
ik
S
ρ
c
−
⎣ 2 A 11⎦
1) linear, reciprocal acoustic and elastic media (harmonic disturbances)
2) the output velocity for the transducer(s) can be written in a separable
form
3) the pulser, receiver, transducers and cabling can be replaced by
equivalent linear, time-shift invariant (LTI) systems.
4) the transducers are assumed to behave as piston radiators operating
at high frequency
5) the incident waves are quasi-plane waves in the vicinity of the flaw
6) the scatterer is a small 3-D flaw where the incident waves do not vary
significantly in amplitude over the surface of the flaw
Ultrasonic Measurement Models
6a) If instead, the scatterer is a small
cylindrical flaw (e.g. side-drilled hole),
with incident and scattered wave
directions in the plane of the crosssection then we can assume
x1
β
eαs
ei
x3
x2
L
1
1
1
Vˆ ( ) ( x1 , x2 , x3 ) = Vˆ ( ) ( 0, x2 , 0 ) ≡ Vˆ0( )
2
1
2
Vˆ ( ) ( x1 , x2 , x3 ) = Vˆ ( ) ( 0, x2 , 0 ) ≡ Vˆ0( )
(
)
(
A v% (j1) ,τ%ij(1) , x = A v%J(1) ,τ%IJ(1) , x1 , x3
)
( I , J = 1,3)
only
In this case
⎡ 4π ρ 2 c2 ⎤
ˆ (1)Vˆ ( 2) dx A v% (1) ,τ% (1) ds
VR (ω ) = s (ω ) ⎢
V
J
IJ
2 ∫
⎥ ∫ 0 0
−
ik
S
ρ
c
⎣ 2 A 1 1 ⎦ length , L
Cf
(
x1
Cf
O
x3
)
Ultrasonic Measurement Models
3-D scattering amplitude component of a 2-D, cylindrical geometry (end effects
neglected) is just
(
)
1
1
A3 D ( eiβ , eαs , ω ) = L ∫ A v%J( ) ,τ%IJ( ) ds
Cf
so that the measurement model can be expressed, finally, as
β
α
⎡
⎤
A
e
,
e
4π ρ 2 c2 ⎤
3D ( i
s ,ω ) ⎡
(1) ˆ ( 2 )
ˆ
VR (ω ) = s (ω ) ⎢ ∫ V0 V0 dx2 ⎥
⎢
⎥
−
L
ik
S
ρ
c
⎣ 2 A 11⎦
⎣⎢ length , L
⎦⎥
Schmerr, L.W., and A. Sedov, "Modeling ultrasonic problems for the 2002 benchmark session,"
Review of Progress in Quantitative Nondestructive Evaluation, D.O. Thompson and D.E. Chimenti, Eds.,
American Institute of Physics, Melville, N.Y. , 22B,1776-1783, 2003.
Ultrasonic Measurement Models
Relationship between 2-D and 3-D scattering amplitudes
β
α
A
,
e
e
2iπ 3 D ( i s , ω )
β
α
A2 D ( ei , e s , ω ) =
k
L
which gives the alternate form
⎡
⎤
⎡ 8π i ρ 2 c2 ⎤
1
2
(
)
(
)
β
α
ˆ
ˆ
VR (ω ) = s (ω ) ⎢ ∫ V0 V0 dx2 ⎥ A2 D ( ei , e s , ω ) ⎢
⎥
⎢⎣ length , L
⎥⎦
⎣ k2 ρ1c1S A ⎦
Measurement Model for 2-D scatterers
β
α
⎡
⎤
A
e
,
e
(
4π ρ 2 c2 ⎤
3D
i
s ,ω ) ⎡
1
2
(
)
(
)
ˆ
ˆ
VR (ω ) = s (ω ) ⎢ ∫ V0 V0 dx2 ⎥
⎢
⎥
−
L
ik
S
c
ρ
⎢⎣ length , L
⎥⎦
⎣ 2 A 11⎦
1) linear, reciprocal acoustic and elastic media (harmonic disturbances)
2) the output velocity for the transducer(s) can be written in a separable
form
3) the pulser, receiver, transducers and cabling can be replaced by
equivalent linear, time-shift invariant (LTI) systems.
4) the transducers are assumed to behave as piston radiators operating
at high frequency
5) the incident waves are quasi-plane waves in the vicinity of the flaw
6) the scatterer is cylindrical flaw with incident and scattered wave
directions in the plane of the cross-section, where the incident waves do
not vary significantly in amplitude over the small cross-section of the flaw
Ultrasonic Measurement Models
General Measurement Model (Auld form)
VR (ω ) =
s (ω )
Z
A
r
( ω ) v A ( ω ) vB
(1)
( 2)
(
∫
(ω )
)
t ( ) ⋅ v ( ) − t ( ) ⋅ v ( ) dS
1
2
2
1
Sf
3-D small flaw (Thompson-Gray form)
⎡ 4π ρ 2 c2 ⎤
(1) ˆ ( 2 ) ⎤ ⎡
β
α
ˆ
⎡
⎤
VR (ω ) = s (ω ) ⎣V0 V0 ⎦ ⎣ A ( ei , e s , ω ) ⎦ ⎢
⎥
ik
S
ρ
c
−
⎣ 2 A 11⎦
2-D small flaw (Schmerr-Sedov form)
β
α
⎡
⎤
A
e
,
e
(
4π ρ 2 c2 ⎤
3D
i
s ,ω ) ⎡
1
2
(
)
(
)
ˆ
ˆ
VR (ω ) = s (ω ) ⎢ ∫ V0 V0 dx2 ⎥
⎢
⎥
−
L
ik
S
ρ
c
⎢⎣ length , L
⎥⎦
⎣ 2 A 11⎦
Ultrasonic Measurement Models
Using Measurement Models
Is the system linear and reciprocal?
Electrical linearity checks of pulser/receiver
For a transducer
I
V
I0 = 0
A
V∞
A
V∞ I = V I s
Is
A
V0 = 0
Ultrasonic Measurement Models
Is the transducer acting as a piston?
Compare responses with theoretical predictions in a reference experiment,
e.g.
Is a quasi-plane wave assumption valid?
Compare
with
as computed with a
⎛ τ ij(1) v( 2) v(1) τ ij( 2) ⎞
i
i
beam
∫S ⎜⎜ v(1) v( 2) − v(1) v( 2) ⎟⎟ n j dS non-paraxial
model
A
B
A
B ⎠
f ⎝
⎡ 4πρ 2 c2 ⎤ ˆ (1) ˆ ( 2) (1) (1)
⎢
⎥ ∫ V V a v% j ,τ%ij dS
⎣ −ik2 ⎦ S f
(
)
Ultrasonic Measurement Models
Is a small flaw assumption valid?
Compare
(
)
ˆ (1)Vˆ ( 2) a v% (1) ,τ% (1) dS
V
j
ij
∫
Sf
with
or
⎡Vˆ0(1)Vˆ0( 2) ⎤ A3 D ( eiβ , eαs , ω )
⎣
⎦
⎡1
⎤
1
2
(
)
(
)
ˆ Vˆ dx ⎥ A ( e β , eα , ω )
V
⎢
0
0
2
3D
i
s
∫
L
⎢⎣ length , L
⎥⎦
(3-D case)
(cylindrical case)
Ultrasonic Measurement Models
What supporting measurements are needed?
System factor
Obtain in separate
experiment
Velocity, Attenuation
Obtain in separate experiments
Transducer effective radius, focal length
a
R
Ultrasonic Measurement Models
Summary
A measurement model form is available suitable for simulating most NDE
inspections
To make these models useful one needs to have
ultrasonic beam models
flaw scattering models
experimental determination of essential parameters
Once all these elements are available one has a very powerful tool for
designing tests, optimizing components, and replacing expensive
experimental procedures.