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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 2
Ultrasonic tomography in concrete
If the readings are performed only on opposite sides of the con-
crete element, the simultaneous system of equations is undeter-
mined, regardless the number of readings performed (JACKSON
et al.
[19]). If the readings are performed on all four opposite sides,
the simultaneous system of equations is overdetermined; for ac-
tual readings with intrinsic errors associated with, this system may
be inconsistent (Figure 4-d). The underdetermined simultaneous
system of equations is beyond the scope of this paper. Hereafter,
only the determined and the overtermined systems are discussed.
Among the methods used to solve the simultaneous system of lin-
ear equations, there are direct and iterative methods. The direct
methods used are: solution by the invertion of matrix D, Cramer’s
Rule, Gaussian Elimination, and the Minimum Least Squares
Method. On the other hand, the iterative methods are Gauss-Ja-
cobi, Gauss-Seidel, Kaczmarz and Cimmino methods.
According to Perlin [11], the best method to be used in the rectan-
gular simultaneous linear equation system caractheristic of a to-
mographic process is the Cimmino iterative method. Jackson
et al.
[19] presented an improvement of this method allowing to a faster
convergence. This optimized modification of Cimmino method is
called optimized Cimmino, and is presented in Equation 7.
(7)
P
n (k )
=P
n ( k-1 )
+W
m,n T
*
[
T
m
-D
m,n
*P
n (k-1)
]
where:
k: current iteration number;
m: number of simultaneous equations;
n: number of unknowns;
P
n
(k)
: column vector that stores the values of current iterative step ;
P
n
(k–1)
: column vector that stores the values of previous iterative step;
W
m,n
: matrix given by:
(8)
w
ij
= d
ij
N
j
*
Σ
(d
ik
)
2
mk=1
;
N
j
: number of equations where j is not zero;
d
ij
: value of row i and column j of the matrix D
m,n
.
In order to better understand the solution process of the tomo-
graphic problem, one should identify that the term inside brack-
ets represents the variation
D
T
m
(k)
of the iteration pass k. While
performing the first iteration is necessary to define an initial value
for P
n
(0)
. The transposed matrix W
m,n
calculates the variation of the
ultrasound pulse velocities of each element due to the variation
of time readings
D
T
m
(k)
, yielding
D
P
n
(k)
for iteration k.
D
P
n
(k)
is then
added to P
n
(k–1)
, resulting in the best possible velocity field.
In order to solve the proposed problem it is mandatory to use a
tomographic software. In this research, it was decided to develop
and continuously optimize a computional program, called TUCon,
which comes from Ultrasonic Tomography in Concrete.
Since object dimensions are known, the grid used and the ul-
trasound readings performed follow a reference nomenclature
according to Figure 6. TUCon calculates the path of each pulse
through each element, and thus is able to build the matrix D
m,n
.
Besides, since the ultrasound readings are known, TUCon builds
the vector T
m
, which represents the travel times.
Figure 5 – Modes of measurement – (a) simple opposite faces – (b) complete opposite faces
A
B
Figure 6 – Measurement points
for two dimensional case
