Reconstruct Permeability of Heterogeneous Body using T-Ω 3-D Formulation and Gauss-Newton’s Method with Total Variation Regularization Yassine FALEH, Abdeaziz SAMET
Ammar Brahim KOUKI, Ahmed KHEBIR
Research Unit Components and Electronic Systems, Polytechnic school of Tunisia, La Marsa, Tunis, Tunisia,
[email protected],
[email protected]
Ecole de Technologie Supérieure, ETS, Montréal, LACIME, Montreal, CANADA,
[email protected],
[email protected]
Abstract— In the inverse problems several techniques are used, in particular induced currents techniques. In spite of their complexities, a diversity of fields, such as imaging, use these techniques. Owing to the fact that it is not a direct problem, several conditions must be joined together to give satisfactory results. The choice of the Model, the Numerical Technique, the Iterative Method and the choice of Regularization Method are the factors which influence the exactitude and the convergence of the results. In this paper, we are going to determine a permeability distribution of heterogeneous body, from a nondestructive evaluation signal, using the numerical approach. We will solve the forward solution with a three-Dimensional T-Ω formulation of Finite Element Method in order to determine the magnetic field, and a Gauss-Newton algorithm with Total Variation Regularization to solve and stabilize the solution resulting from the optimisation problem. The permeability of the object is assumed to be linear and isotropic. Results for the permeability imaging will be reconstructed using synthetically generated data. Keywords- T-Ω formulation, Induced Currents Techniques, Inverse Problèmes, Gauss-Newton Algorithm, Total Variation Regularization
I.
INTRODUCTION
Recently, many researchers chose the electromagnetic method to analyze and resolve the inverse problems. [2]; Indeed, the methodology of inversion in the electromagnetic problem, in particular 3D, interest several fields of which geophysics, the medical imagery and several other fields requiring nondestructive techniques (NDT) [7][10][12][15]. In this context, this work was realized. The objective is to determine the distribution of the electric permeability of a heterogeneous body by using a hybrid electromagnetic simulation technique. Based on the Eddy-current techniques, this method belongs to the Magnetic Induction Tomography and consists to determining the electric parameter of the interest object while being based to the measure of impedances between pair of coils (one for transmitting and the other for detecting) [12][13]. This type of problem is an illposed and nonlinear inverse problem.
This kind of problems needs a forward solver where we determine the magnetic field or the electric field (which contains information on the electric parameter distribution) and an inverse phase in which iterative computing is used for the determination of unknown parameter. This task can be very difficult because of non-linearity and of the sensitivity of the problem [5]. In our approach, we employ an inaccurate methodology solving the direct problem and the inverse problem simultaneously in an iterative process. This approach enabled to develop an effective algorithm; In fact, starting from the measurement of tension in the reception coil, we can determine the magnetic field by using a Three-Dimensional TΩ formulation of Finite Element Method [2][3] which had already scanned the object of interest and contains information about the distribution of the parameter previously aimed at. This is a forward solver applied in low frequency. A GaussNewton algorithm with Total Variation Regularization is used to solve and stabilize the solution of inverse solves. The choice of a low frequency is related with the domain of interest and type of exploration system, such as geophysics in which electromagnetic analysis low frequencies of the wells offers the promise to give information on permeability distribution under surfaces with better resolution of what exists before [9]. We present in this paper the application of a numerical approach based on T-Ω FEM 3D and Gauss-Newton’s algorithm with Total Variation Regularization to image the electrical permeability distribution of an Heterogeneous body from a non-destructive evaluation signal. This paper is organised as follows. In section II, we are going to present the principle of the technique and to describe the forward problem in section III. The formulation of the non-linear inverse solver will be given in section IV. The Section V will be reserved to present the model of simulation. The results, the discussions, and the conclusion will be presented at Section 4.
II.
PRINCIPLE OF THE TECHNIQUE
Our technique falls within the framework of Tomography based on Magnetic Induction Tomography (MIT) [2][6] and consists in determining the electric parameter of the object of interest _ the distribution of conductivity is a case in point. Here, we shall follow the measures of impedances between a pair of coils including one for excitation and the other for detection. Starting from the measurement of voltage in the reception coil, Authors wishing to emphasize a term may use bold or italic but never underscore. We can determine the electromagnetic field which had already scanned the object of interest. By using T formulation of FEM 3D as electromagnetic analysis method, we can determine the magnetic field that contains information about the distribution of the parameter previously aimed at. This field is propagated while interacting with the entourage including the body to be explored, which causes the creation of a magnetic field in the detection coil called secondary field. This last is used to calculate the predicted induced voltage in the detection coil. The algorithm of the method of adopted resolution (Figure1) is summarized at the following stages [4]. In an iterative process, we start by injecting an initial value of the conductivity, which we use to calculate first the secondary magnetic field H by the technique T .
fields which uses a magnetic vector potential (A-formulation) [8]. This method consists to expressing the magnetic field intensity H according to current potential vectors, T and T0 and the magnetic scalar potential . The analysed region R can be divided into two regions: the current carrying region, RC , surrounded by the current free region (nonconducting region), R0 . The free region is delimited with the surface 0 and RC with C . The quantities of the fields derived from potentials are expressed by:
H T0 T grad
(1)
J rotT0 rotT
(2)
The curl T0 ( rotT0 ) is the density of excitation current in all study area of the problem and the curl T ( rotT ) is the density of eddy current in the region of interest; By using the Maxwell equations and the complementary equations where we replace E and B by their expressions according to H and J , then these two lasts by their expressions of the equations (1) and (2), we obtain:
rot rotT jT j grad rot rotT0 jT0
(3)
in Rc
j div T grad j div( T0 )
in Rc
(4)
Here, is the permeability and is the resistivity. Equations (5) and (6) show the approximation of the current vector potential T , by the Edge Basis Functions and the magnetic scalar potential , by the Nodal Basis Functions along the edges. T T
ne
(n)
tk N k
(5)
k 1 nn
(n)
Figure 1. Algorithme of permeability determination
Then, we calculate the predicted voltage which will be compared with the measured induced voltage. We apply the Method of Gauss-Newton with Total Variation regularization to calculate which will be used to update . Thus, the iterative process will be to repeat until convergence between the calculated voltage and the measured voltage. The unicity of the result and the stabilization of the solution are ensured by the regularization. III.
FORWARD SOLVER
T is one of the various formulations of FEM 3D . It’s a simple and economical method for calculating 3D eddy currents and require memory and CPU time more less compared with other formulation specially the 3D analysis
N k
(6) k
k 1
Here, t k indicates the integral lines of the current vector n
n
potential T along the edges, and k is the values of to the nodes. We apply the boundary conditions of Dirichlets and of Newmann on the potentials vectors and scalars, we find then:
T n 0 ou rotT n rotT0 n
(7)
0 cons tan t ou n.( T grad )
(8)
n.T0
n is the outer normal unit vector. By introducing the equations (5) and (6) into (3) and (4) whole, while holding account of the boundary conditions objects of the equations (7) and (8) and by using basic
functions Ni as weighting functions, in the Galerkin method, we obtain: n curlNi .. curlT dR j Ni .T ( n ) dR j Ni .grad ( n ) dR Rc
Rc
Rc
curlNi . curlT0 dR j Ni .T0 dR, Rc
(9)
i 1, 2,...., ne
Rc
n j .gradN i .T dR j gradN i .grad ( n ) dR Rc
Rc
j gradN .T dR, i
0
(10)
i 1, 2,...., nn
process is repeated until the measurement data agree with the calculated voltages. B. Computing Jacobian Matrix The Jacobian matrix (Jn) is the matrix of discretization of f in the step n. Each column of the Jacobian matrix is the calculated sensitivity term for an element for all measurements. The general form of the sensitivity formula [1][11][14] when coil “i” is an excited and coil “j” is a sensing coil, by ignoring second order terms is:
Rc
We model the current potential vector T by the Edge Basis Functions. The magnetic scalar potential and the static current potential vector T0 are modelled by Nodal Basic Functions. The use of the two types of basis functions (Edge and Nodal) at the same time in only one hybrid formulation makes it possible to effectively determine the magnetic field and consequently the voltage of the induction coil. IV.
INVERSE SOLVER
A. Formulation We denote by Vm the vector of measured voltages, and F(μ) the vector of calculated voltages with forward process. The inverse problem corresponds to an algorithm of optimization which makes it possible to calculate μ with an acceptable error. The regularization is the fact of introducing information as a preliminary to this algorithm and that amounts minimizing the following quadratic error function.
f ( ) Vm F ( ) 2G
(11)
2
Where 2G is a term of penalty. For a judicious choice of 2G ,we will have a critical point which will be a
minimum corresponding to f ( ) 0 . We solve this minimization problem by regularized Gauss-Newton method. 2 If we consider that G L (where L approximating a differential operator) the minimization leads to linear steps, each of which is the Total variation regularized solution to the linearized problem. The equation to solve in each step is: 1
n1 J nT J n 2 LT L J nT Vm F ( ) 2 LT L
(12)
Where: Jn is the Jacobian Matrix calculated with permeability μn and L is the regularized matrix. The algorithm starts with an initial permeability distribution. We compute the calculated voltages with forward solution and we compare with predicted voltages. The permeability is then updated using the Jacobian matrix. The
(13)
E H ndS 1
2
( j H
1
H 2 ( j ) E1 E2 )dV
R
where the left-hand side is representing sensing and excitation by surface integral on surface Γ and the right hand-side is the volume integral over the perturbed region R. Hi and Ei are the magnetic and electrical fields when coil “i” is excited, i=1,2. The formula relating to sensitivity of permeability is: Vij j Ii I j
(14)
R H1 H 2 dV
C. Total Variation Regularization We used the Tikhonov regularization for determining the permeability distribution of heterogeneous body and we obtained good results which have been published previously. In the literature, the Tikhonov regularization is known as being a suitable method for case that assumes the data sets to be smooth and continuous. However the Total Variation Technique is regarded as being better when we treat discontinuous data “case of sharp edges and high contrast”. The Total Variation Regularization is expressed by: G
2
(15)
x dV 2
R
Moreover it is of norm l2 . However, if we replace norm l2 by the 1-norm of the first spatial derivation we obtain the expression of the Total Variation regularization:
G
(16)
x dV
R
Consider I the number of facets, P is number of tetrahedral elements and q i the area of each facet i between two tetrahedral for i 1, 2,....., I . The kth row of matrix
S R I P
noted
Sk
is
chosen
to
be
Sk 0...01...... 10...0 where 1 and 1 in
the columns of S corresponding to the tetrahedral with common facets k . We defined then L S T .Q.S where
Q is a diagonal matrix with Qkk qk [12]. V.
RESULTS AND DISCUSSION
A. Simulation model In order to guarantee satisfactory results, we have used a simulation model formed by a heterogeneous body surrounded by 8 coils. This model was object of preceding study [16]. The model of simulation we chose (Figure2) was constituted of a cylinder A with permeability μ 1=1 Sm-1 in which we put another cylinder B with permeability μ 2=5000 Sm-1. The two cylinders to be explored, A and B, have a length of 100mm. the diameter of A is of 100mm and that of B is of 20mm. The set is surrounded with 8 induction coils of cylindrical form as indicated in figure 2. Each one has a length of 10mm and a diameter of 50mm and formed of 10 copper wire turns. The T-Ω forward solver is carried out with COSMOSDesignSTAR, and the image reconstruction software is written in C++. The overall number of tetrahedral elements in forward model was 25090 with 4846 nodes, 31190 edges and 51435 faces. The region of interest (Cylinders A and B) which was used in inverse solution included 3384 tetrahedral elements. The simulation frequency is 5 kHz and the measured tensions are replaced by the tensions calculated by the forward solution for the true permeability of the cylinders which we added 2% of their values as noise and error of measurement.
Figure 3. Visualisation Result of 1st simulation model
VI.
We applied an algorithm based on combination of a threeDimensional T-Ω formulation of Finite Element Method, and Gauss Newton method with Total Variation Regularization. This algorithm aims at determining the permeability distribution of a non-homogeneous body by applying T- Ω FEM 3D to resolve forward problem and Gauss Newton method with Total Variation Regularization for inverse phase . The obtained results are satisfactory, although this imaging tool requires a lot of improvements. REFERENCES
Coils [1]
[2]
[3]
[4]
Cylinder A
Cylinder B [5]
Figure 2. Cross section of 1st simulation model
B. Visualization Although, the program is designed with C++, the visualization of the image has been realised with MATLAB. The longest calculation in our program relates to the matrix of regularization. first, we compute this matrix. The iterations do not require a long computing time. After 5 iterations, we have satisfactory results and the induced computed voltages converge to induced measured voltages. Concerning the first model, Figure3 shows the visualization result after 5 iterations. The clear zone in the medium represents a cross section of the cylinder A and the other dark zone shows the cylinder B.
CONCLUSION
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