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Forward Solution Speed-Up for 3D Eddy Current Inversion Zsolt Badics, Senior Member, IEEE, József Pávó, Yoshihiro Matsumoto, and Hidenobu Komatsu
Abstract—The forward solver is the bottleneck in eddy current flaw reconstruction because the solution of a dense system matrix is required. Two methods are developed to increase the solution speed. The cell merging technique gives impressive speed-up for cracks but a fully adaptive version for volumetric flaws has not been found yet. The other algorithm based on sparse representation of rank deficient matrix blocks and the Woodbury procedure is slower than the cell merging technique but it is robust and still significantly faster than the LU factorization. Creating a hierarchical partitioning tree ensures the stability of the latter algorithm. Index Terms—Eddy current inversion, flaw reconstruction, matrix decomposition, sparse representation.
I. INTRODUCTION
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N RECENT papers, we have introduced an efficient inversion strategy for defect shape reconstruction from eddy current signals [1], [2]. The strategy is based on the minimization of a nonlinear least-squares error functional, and incorporates a forward solver whose performance is tuned to calculate eddy current signals rapidly. If we want to use the forward solver for a wide variety of defect types including cracks, the forward solution may slow down significantly for large volumetric flaws. Since our ultimate goal is to develop a reconstruction procedure that can be utilized in a real-time eddy current inspection tool, further speed-up of the forward solution is necessary. Numerical experience with crack-type defects shows that the current dipole density, which is our state variable, shows a very characteristic spatial distribution [3], [4]. The spatial variation of the current dipole density is smooth inside a flaw while it changes rapidly approaching the flaw boundaries. This observation suggests that we can “merge” flaw cells in the large inner zone of a flaw while keeping the necessary resolution near the flaw boundaries. Merging means that we introduce only a few unknowns for a group of connected cells. In this paper, we illustrate through a set of examples that manually merging cells of crack-type defects provides impressive speed improvement without loss of accuracy. Unfortunately, we have not been able to come up with an adaptive merging algorithm Manuscript received October 25, 1999. The work of J. Pávó was supported by the EU INCO-COPERNICUS Project ERBIC-15-CT-960703, and by the Hungarian Scientific Research Fund through Grant T-023559. Z. Badics is with the Ansoft Corporation, 4 Station Square, Suite 200, Pittsburgh, PA 15219 USA (e-mail:
[email protected]). J. Pávó is with the Department of Electromagnetic Theory, Technical University of Budapest, Budapest 1521 Hungary (e-mail:
[email protected]). Y. Matsumoto and H. Komatsu are with the Nuclear Fuel Industries, Ltd., 950 Ohaza-noda, Kumatori-cho, Sennan-gun, Osaka-fu 590-0451 Japan (e-mail: {y-matsu; komatsu}@nfi.co.jp). Publisher Item Identifier S 0018-9464(00)06846-1.
for general volumetric flaws yet. Therefore, we have tried to find an alternative way of utilizing the hidden redundancy in the system matrix. It has been long recognized that system matrices derived by using Method of Moments (MoM) have rank deficient sub-matrices that express relationships between groups of well-separated source and field points. The most popular technique that exploits this feature is the Fast Multipole Method [5], [6]. Other techniques are the Impedance Matrix Localization [7] and the Reduced Representation Technique [8]. Since our system matrix shows characteristics similar to a MoM matrix, we can also identify rank deficient blocks and derive a sparse—or reduced—representation of the system matrix. We have to solve an equation system for huge number of right-hand sides because we need a solution for every observation point. Therefore, our goal is to find a direct technique to solve the equation system. An attempt to derive a direct solver for such matrices is made in [9] but in all other references, iterative solvers are considered. The basic idea of a direct solver based on the utilization of the Woodbury formula is hinted at in [9] but it is not explored because it is found to be numerically unstable. In this paper, we develop further the idea based on the Woodbury formula and show that stable solution can be obtained if we number the flaw cells using a hierarchical tree-like data structure. Note that the Woodbury formula was successfully applied in the field of eddy current NDE for a different purpose in [10]. II. SYSTEM MATRIX AND SIGNAL Fig. 1 depicts a typical arrangement and shows the anomalous region that is the largest possible region where flaws are expected. (Detailed description of how to form an anomalous region can be found in [1]–[4].) A flaw is described by the flaw , where and are the conductivity of function, the flawed and unflawed arrangements, respectively. The flaw function can be expressed as
(1) is 1 in cell and 0 elsewhere, and where the cell function denotes the number of cells [1], [2]. Maxwell’s equation can be reformulated where current dipole density as and is the current density of the eddy current probe. We introduce an extra source in order to represent flaws. Suppose
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region where the mentioned cells are located, we can linearly interpolate at the cell in the middle and then (2) becomes
(4)
Fig. 1. Eddy current testing arrangement. An impedance probe is scanned parallel to the axis of a tube specimen along an anomalous region. The outer diameter of the tube is 11.2 mm and the wall thickness is 1.25 mm. The outer and inner diameters of the probe coil are 3.4 mm and 0.6 mm, and its height is 1.0 mm. The frequency of the excitation is 400 kHz. The cell size is 0.125 mm 0.024 radian 0.25 mm where the dimensions are in r , and z directions, respectively.
2
Note that the number of unknowns in (4) are reduced by 3, that , are eliminated. Similarly, can be is, interpolated linearly in a region composed of more than three cells, thus, eliminating the inner unknowns in the region. The idea can easily be extended to 3D cell arrangements. The advantage of cell merging is that the new system matrix can be assembled using the elements of the original equation system (3) while the number of unknowns is reduced considerably. On the other hand, the problem with cell merging is that we have not been able to develop an automatic adaptive algorithm yet.
we have observation points. Approximate the current dipole density at observation point as
IV. SPARSE DIRECT SOLVER A. Sparse Representation
(2) are three mutually orthogonal vector where functions whose supports are cell . As described in [2], we can by solving the multiple obtain the unknown coefficients right-hand-side problem
Consider two well separated groups of cells in a flaw. There are four blocks that correspond to the two groups as one can see below
(5)
(3) consists of reactions Symmetric complex matrix . Matrices between the current dipole density pulses , where are reactions between the pulses and the currents of the probe located at observation point . After solving (3), we obtain the signal
and contain reactions Matrices among cells in cell groups 1 and 2. Let us concentrate on the rank that expresses the relationship deficient block between the two cell groups. Without loss of generality, suppose . Then generate the Singular Value Decomposition (SVD) as
(3a)
(6)
, and denotes the transpose of . The physwhere ical meaning of a signal is usually a set of impedance changes of the probe coil at the observation points. In the next section, we describe how we can exploit the sparse nature of the system matrix and solve it significantly faster than by LU factorization.
where U and V are unitary matrices, and indicates the Hermiand tian conjugate. Further, are the singular values of . to a precision is such that The numerical rank of [11] (7)
III. BASIC CONCEPT OF CELL MERGING For the sake of brevity, we are going to illustrate cell merging here by discussing cell merging only in one coordinate direction. Suppose that three cells of the dipole density function (2) are neighbors along one of the coordinate directions. Let they . If the varibe cells ation of the vector function is predicted to be smooth in the
Then
can be approximated as (8)
where , in the sense of and . Since we need to represent . significantly fewer elements than
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In practice, we are better off if we use the modified Gram–Schmidt algorithm [11] to obtain a sparse representation (9) is a unitary and is an upper where triangular matrix [11]. The algorithm creates the columns of and one by one and it is easy to check when the numerical rank with respect to is achieved [11]. B. Direct Sparse Solution Partition matrix into four blocks and generate the sparse representation of the off-diagonal elements as (10) (11) Then (3) becomes (12)
Fig. 2. Example of forming a partitioning tree in a 2D anamalous region.
We can obtain the solution of (10) by applying the Woodbury procedure [14], that is, (13) (14) Unknowns
and
are obtained by solving (15)
and in (15) can be partitioned in the same Notice that way as (3) to form (12). This means that smaller and smaller linear systems have to be solved recursively but with more and more right hand sides. This defines a recursive direct solution algorithm that we call WMGS (Woodbury with Modified Gram–Schmidt) solver. C. Cell Numbering For partitioning , we use a strategy based on a hierarchical tree-like data structure [12]. The partitioning algorithm is described in [12] in detail and we only illustrate here how to create the data structure by using a 2D example. (See in Fig. 2.) Keep in mind, however, that our real anomalous region is a 3D grid as we have seen in Fig. 1 The finest grid in Fig. 2 is the whole set of cells in the anomalous region but we have to generate the tree only for the flaw cells delineated by the thick polygon. One step up in the tree means uniting at most four (in 2D) neighboring cells. (Note that we have at most 8 leaves in 3D.) Furthermore, the number of cells we unite also depends on the shape of the flaw as it is illustrated in Fig. 2. It is easy to transform this tree into a binary tree that we call a partitioning tree. Our experiments show that if we number the cells and perform partitioning (10) according to the tree structure, we obtain a stable direct solver. The basic algorithm is summarized in Fig. 3.
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Fig. 3. Algorithm to solve (3) according to partitioning tree . is the precision parameter in (7). Function CREATE–PARTITIONING generates the block structure in (10), function MODIFIED-GRAM–SCHMIDT creates QR factorization as in (9) and function ASSEMBLE-SOLUTION performs (13). The input and output parameters are separated by vertical bars in the parameter lists of the functions.
V. EXAMPLES We investigate here two sets of signals due to flaws in the for both sets in algorithm specimen in Fig. 1. We select WMGS. This value ensures that the relative error in the signal is less than 1%. The same error level in the signal is kept in the merging algorithm as well. The number of observation points is 21 in both cases. The first set of flaws, Set 1, consists of 100% through rectangular axial slots with different lengths. In Fig. 4, we compare the performance of a LDL solver, the cell merging procedure and algorithm WMGS. (The LDL technique is basically the LU factorization for symmetric matrices.) Performing curve fitting
BADICS et al.: FORWARD SOLUTION SPEED-UP FOR 3D EDDY CURRENT INVERSION
Fig. 4. Performance of different solution techniques for Set 1. Lines represent least-squares fits to the data points.
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redundancy between the elements of the system matrix. The cell merging technique gives an impressive speed improvement for crack-type defects, in which case we have performed the cell merging manually. Unfortunately, we could not come up with a good adaptive algorithm for general volumetric flaws yet. In the future, we definitely want to explore this promising possibility. The other technique, algorithm WMGS, which is described in Section III in detail, gives a general direct method for reducing the solution time of the system equations. The basic idea that one can use the Woodbury formula recursively for matrices with rank deficient blocks has been hinted at [9]. What we have discovered is that it is necessary to create a partitioning tree in order to get numerically stable solution. Though the speed improvement using WMGS is not as impressive as in the case of cell merging, it is still significant compared to the LU factorization. One big advantage of the method is that it is easy to control the accuracy of the solution by the input parameter (see in Fig. 3). This is very important if we use the forward solver in an inversion shell because we can control the accuracy of the solution adaptively depending on which state of the inversion we are in. In the initial stage when the consecutive trial flaws differ significantly, much less accuracy is allowable than in the final refining stage of the inversion. REFERENCES
Fig. 5. Performance of algorithms LDL and WMGS for Set 2. Lines represent least-squares fits to the data points.
reveals that the LDL solver is , and the cell merging and , respecand WMGS procedures give tively. The cell merging method has an impressive performance for this set. The other flaw set, Set 2, consists of shallow volumetric flaws that spread along the inner surface of the tube from very small to maximum areas. Fig. 5 compares the performance of solvers , and algoLDL and WMGS. Solver LDL behaves as . rithm WMGS provides VI. CONCLUSION We have introduced two techniques to decrease the forward solution time in inversion loops. Both methods utilize the
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