Geodetic Iterative Methods for Nonlinear Acoustic Sources Localization: Application to Cracks Eects Detected with Acoustic Emission

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1 6th NDT in Progress 2011 International Workshop of NDT Experts, Prague, Oct 2011 Geodetic Iterative Methods for Nonlinear Acoustic Sources Localization: Application to Cracks Eects Detected with Acoustic Emission Michal ZÁVESKÝ 1, Václav K S 1 1 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Trojanova 13, Prague , EU - Czech Republic; zaveskymichal@seznam.cz, v.kus@centrum.cz Abstract This paper is related to Acoustic Emission (AE) principle, which is a Nondestructive Testing method (NDT) for structural health monitoring and many other applications. We deal with the geodetic curve based localization of the defects on surfaces of solid bodies mathematically described as a conjunction of several simpler parametrized shapes like cylinders, toroids, spheres, cones,... with intersections. We demonstrate a numerical solution of geodesic equations by the algorithm Newton- Raphson method. For faster computations we propose a few improvements using Sequential algorithm applied to bisected points of the intersections, which is compared with trivial Fundamental method. The whole algorithm for nding geodesics on combined surfaces is applied to virtual testing body which was designed according to the solid watering can. Keywords: Acoustic emission, geodetic curve, geodesic equations, Newton-Raphson method, Sequential algorithm 1. Introduction and AE source localization principle The main goal of our work is the precise mathematical model, which detects positions of acoustic emission sources (cracks) in various materials. Acoustic emission (AE) diagnostics belongs to nondestructive defectoscopy methods and the AE localization principle is based on geodesic curves on combined surfaces. The geodesics will be described by geodesic equations, which include information about the surface shape. Using a numerical iterative solution of this geodetic equations we obtain a procedure determining geodesics on elementary surfaces, whose compositions give us the real surfaces in engineering industry. Thus it is quite necessary to face the problem how to nd the exact geodesic curves for a number of intersected elementary surfaces. The created algorithm will allow us to detect the crack eect on real solids in the future. We focus on the crack emissions, i.e. discrete AE sources. The localization task consists in obtaining quantity depending on the distance between emission event and the AE sensors.

2 For determination of the real AE source position of elastic waves in practice, we use time dierences of arrivals of AE signal to AE sensors placed on the material surface. In the case of three sensors we have the following three time dierences measured throughout the experiment t 12 = t 1 t 2, t 23 = t 2 t 3, t 13 = t 1 t 3, where t 1, t 2, t 3 are individual signal time arrivals to the sensors 1,2,3. Notice that t 12, t 23, t 13 are linearly dependent as t 23 = t 12 t 13. These time dierences can be detected by elastic waves analysis method, see [1]. Assume that we know the velocity of elastic waves propagating in a given material and let us denote it by c. If we consider the localization of emission event on the plane and we use three sensors then the measured time dierences t 12, t 23, t 13 correspond to the distances between the source and AE sensors l 1, l 2, l 3 by the following relations c t 12 = l 1 l 2 = l 12, c t 23 = l 2 l 3 = l 23, c t 13 = l 1 l 3 = l 13. As well-known, it is possible to obtain the position of AE source by means of two arms of hyperbolas corresponding to distance dierences. The above principle of AE sources localization indicates that the key challenge is to nd the shortest distance between two surface points: the imaginary AE source position and the sensor position. Consequently, the computed distances (lengths) can be compared with the real values measured. 2. Mathematical background for geodesics Dierential geometry provides us with the necessary tools to solve the task of nding shortest line on the body surface between two given points, see e.g. [2]. Denition 1. A regular curve C on regular surface S is called geodetic curve (or geodesics) if the normal vector of C is parallel to the normal vector of S at all surface points p C. It can by shown that the geodesic curve between two points of a surface has locally minimal length. This is the favorable property which is essential for us. Thus the geodetic curve is a generalization of the straight line in linear plane to curved surfaces and every straight line is geodesics as well. Proposition 1. Let S R 3 be a regular surface with parametrization x : U R 2 R 3 and α(t) = x(u(t), v(t)), t I R, is the representation of curve by means of given parametrization x. Then the curve α(t) is geodetic curve if the functions u(t) and v(t) satisfy the following system of two second order dierential equations ü + Γ 1 11 u 2 + 2Γ 1 12 u v + Γ 1 22 v 2 = 0, v + Γ 2 11 u 2 + 2Γ 2 12 u v + Γ 2 22 v 2 = 0, (1) where Γ k ij are so called Christoel symbols corresponding to the regular surface S in parametrization x.

3 The set (1) is called the system of geodesic equations for the surface S considered and the coecients Γ k ij are given as follows. First, for q = (u, v) U we dene the matrix G q = dx T q dx q, dx q = ( x u, x ) = v Proposition 2. If we denote the elements of matrix G q by g ij and the elements of inverse matrix G 1 q by g ij then the Christoel symbols (corresponding to regular surface S with parametrization x) are dened for i, j, k = 1, 2 by the relations x u y u z u x v y v z v. Γ k ij = g kl ( i g jl + j g li l g ij ), (2) l=1 where 1, 2, denote u, v, respectively. Moreover, it holds that Γ1 12 = Γ 1 21 and Γ 2 12 = Γ We assume a regular surface S with the parametrization x : U R 2 R 3, further we select two points p, q U and an interval I = (a, b) R. The task is to nd geodetic curve on the surface S passing through the two corresponding surface points x(p) = x(u(a), v(a)) and x(q) = x(u(b), v(b)). This is equivalent to nding the curve γ : I = (a, b) U, γ(t) = (u(t), v(t)), whose coordinates satisfy the equations (1). We solve the system (1) numerically by the method of nite dierences, especially by the iterative Newton-Raphson procedure (NR), see in [3], while the Christoel symbols for a given regular surface S R 3 are computed analytically in advance. Using Newton-Raphson method we nd a sequence of (N + 1) tuples in U, which converges to a solution of (1). This numerical solution requires an initial path which was considered here to be a straight line linking together the two boundary points in the corresponding parameter space of the surface S. The numerical solution of (1) was also found alternatively by the algorithm called Functional iteration procedure (FI) but we found that the convergence of FI algorithm is still accurate but very slow and, moreover, it is unstable in a neighborhoods of some critical surface points (e.g. the poles of a sphere). 3. Finding geodesics on surfaces The algorithms NR and FI were implemented in Matlab version (R2010b) and calculations were running on notebook ASUS K50IJ with processor Intel Core Duo T5900. Input values are: (I1) the boundary points in the space of parameters p, q U R 2, (I2) large positive N, (I3) number of NR iterations allowed or any other stopping condition. Output values are: (O1) graphic visualization of the surface, the initial path and the resulting geodesics, (O2) length of geodesics between boundary points on the surface,

4 Figure 1: NR geodesics on elementary surfaces (red line inital path, black line geodesics) (O3) total time of running the program (including visualizations), (O4) total number of iterations of NR in the case of combined surfaces. It is necessary to realize that the NR procedure works on space of parameters U, depends on an initial path chosen and also on the parametrization of surface used. First, we illustrate the geodetic curve computations for the elementary surfaces such as the plane, cylinder, cone, sphere, and torus which can be seen in Figure 1. Our choice of the initial path allows us to nd geodesics on the plane and the cylinder without using NR procedure because the corresponding Christoel symbols are all zeros for these two simple surfaces. It means that the direct image of straight line in parametric space results in the geodesics in plane or cylinder. Another non-numerical improvement is connected with the sphere since the length of geodesics can be computed straightforwardly. Let us assume we have a sphere with radius r, centered at the point s = (x s, y s, z s ), and two dierent points on the sphere x(p) = (x p, y p, z p ) and x(q) = (x q, y q, z q ) are at disposal. We compute the angle between the vectors s p := (x s x p, y s y p, z s z p ) and s q := (x s x q, y s y q, z s z q ) by the formula β = arccos s p s q s p s q. The length of geodesics on sphere joining the surface points x(p) and x(q) is equal to L = rβ. In spite of that fact, we used the NR procedure for determining the geodesic curves on spheres due to visualization purposes. 4. Geodetic curves for combined surfaces We created many various types of combined surfaces by translations and/or rotations of elementary surfaces. It rephrases the question of nding the geodesic curves on such combined surfaces since now we have to minimize the sum of all geodesic lengths on the elementary surfaces which the combined surface consists of. Evidently, it must be computed

5 over an appropriately chosen points of intersections of these simple surfaces. In other words, in case of two surfaces with the only one set of points of intersection, we want to nd min { L p i + Li q i = 1, 2,..., M }, where M is the number of points of intersection and {L p i i = 1, 2,..., M}, or { L i q i = 1, 2,..., M }, are the sets of geodesic lengths on elementary surfaces from boundary points p, or q, to the points of intersection, respectively. This process can be intuitively generalized to greater number of intersection sets of points considered. Fundamental method (FM). We assume that we have already computed the points of intersections (e.g. equidistantly distributed) before nding geodesics. The Fundamental method tests all the points of all intersections without any feedback on its positions. It is clear that we surely nd the shortest curve connecting the two boundary given points on the combined surface, but the procedure is very time-consuming. In the case of the only one intersection with M points, this algorithm nds M dierent curves connecting boundary points. However, if the surface is composed of more components and the geodetic curve goes through m intersections, then FM must nd m M k curves to pick up the right solution. Here, k=1 M k denotes the number of points of k-th intersection. The following Table 1, corresponding to Figure 2, shows how the accuracy of the FM algorithm decreases if we reduce the number of points of intersections to achieve a reasonably short computation time. Figure 2: Fundamental method applied to combined surfaces (see Table 1) Combined surface M m M k Comput. time (s) Length GC L 41 L 201 k=1 Cupola m = Cylinder+planes m = Sphere+cylinders m = Table 1: Fundamental method (M = M 1 = M 2 in case of two intersections) We can say that an advantage of FM is only an infallibility, but the computational times rise up exponentially with increasing number of intersections. Hence, below we propose a new algorithm depending on positions of points of intersections which are passing through during the computation process.

6 Figure 3: Fundamental Method (left) and Sequential Algorithm (right) Sequential algorithm (SA). This algorithm adds a new points of intersection step by step and selects automatically more convenient candidates. Suppose that an interval (a 0, b 0 ) denes the range of the parameter t (see Section 2) and the starting parameter of intersection t 0 = (a 0 + b 0 )/2 together with starting step σ 0 = b 0 a 0 are determined. Then, for i 0, we dene the iterations of SA algorithm as σ i+1 = σ i /2, a i+1 = t i σ i+1, b i+1 = t i + σ i+1, t i+1 = min L ai+1,l ti,l bi+1 {a i+1, t i, b i+1 }, where L ai+1, L ti, L bi+1 are the lengths of curves connecting boundary points and passing through the computed points a i+1, t i, b i+1. In most cases we work with the intersections in the form of a closed curve which is described by means of the parameter t within the interval (0, 2π) while the both limit points coincides for t = 0 and t = 2π. In these cases, the proposed SA algorithm always converges to the true solution. That is to say, selecting a wrong point of intersection in the rst iteration of SA, it is possible to sum up all the steps used throughout the iterative process leading to the right point. Thus in the limiting case of innite number of iterations we obtain i=1 π/2i = π achieving any right point in the range of the parameter interval. If the intersected curve is not closed or if the combined surface is too complex, it can happen that SA does not converge to the true solution due to the wrong choice in the rst step of SA. We can solve this problem by approximating the probable area of the right solution using FM at rst and subsequently we apply the SA. Fundamental method and Sequential algorithm are compared in the following test in Figure 3 and Table 2, where the accuracy denotes the minimum distance between two tested xed parameters (corresponding to points of intersection) and the number of iterations is a sum of all the iterations used in NR. We preset 5 iterations for the computation of one geodesics on torus and 1 iteration on cylinders (i.e. initial path). It is clear that the Sequential algorithm works faster and with higher accuracy then the Fundamental method.

7 Accuracy Num. of iter. Comput. time (s) Length of GC FM SA (8 iter.) 1/2 8 = Table 2: Comparison of the FM and SA (see Figure 3) 5. Simulation results on the real solid can We apply the proposed methods for computing geodesics to the testing solid designed according to real metal watering can. This vessel is a composition of some elementary surfaces, namely two cones (tin and neck), two planes (bottom and top), torus and cylinder (bail). Number of NR iterations is xed on constant value 5 for each geodesic, number of points of intersected curve N is N = 100 for the cones and N = 33 for the torus. We compare both Fundamental method and Sequential algorithm. We focus on nding geodetic curve, whose one boundary point is located on the neck and the second one on the bail. Note that the geodesic can pass through both the upper intersection (cone and torus) and the lower intersection (cone and cylinder). Hence, it is necessary to compute both possible geodesics and then choose the shorter one. Therefore, we tabulated both lengths for upper and lower intersections. The performed tests are described in the following Tables 3, 4 and Figure 4. Figure 4: Geodesics on the can from the neck to the bail (FM left, SA right), (black curve upper intersec., yellow curve lower intersec.) Accuracy neck/bail Number of iter. Comput. time (s) FM 0.05/ SA (8 iter.) / Table 3: Geodesics on the can with boundary points on the neck and the bail Length GC (upper intersec.) Length GC (lower intersec.) FM SA (8 iter.) Table 4: Geodetic lengths on the can with boundary points on the neck and the bail

8 Notice, it may occur on this watering can, as can be seen in Figure 5, that the SA does not converge for some problematical boundary points. We solve this problem, as mentioned above, by combination of FM and SA. Figure 5: Comparison of FM (left) and SA (right) for problematical boundary points (black curve upper intersec., yellow curve lower intersec.) 6. Conclusions Since our model is based on several observations of the signal arrival time dierences of elastic waves detected by piezo-ceramic sensors, the crucial task was to nd a geodetic curve connecting any given two points on a surface. The theory of dierential geometry provided us with the necessary apparatus describing geodesics, namely with the system of geodesic equations. We solved the system numerically by Newton-Raphson procedure for several useful elementary surfaces. To nd the precise geodesics on combined surfaces, we developed Fundamental method and Sequential algorithm together with their potential combination. While testing these algorithms, we found that we can detect crack eects with high accuracy and more than acceptable computing time, both for real relatively complex surfaces. Thus, the exact localization procedure is possible for the combined surfaces considered. Acknowledgements This work was supported by the grant SGS10/209/OHK4/2T/14 and by the research program of the Ministry of Education of Czech Republic under the contract MSM References 1. M. Blahá ek, 'Acoustic Emission Source Location using Articial Neural Networks', Dissertation on Department of Mathematics FNSPE CTU, Prague, M. P. do Carmo, 'Dierential Geometry of Curves and Surfaces', Prentice-Hall, New Jersey, J. Beak, A. Deopurkar and K. Redld, 'Finding geodesics on surfaces', Research report, Stanford University, 2007.