NDT.net • Sep 2004 • Vol. 9 No.09 
Active Flaw Detection in 3D MediumEdouard NesvijskiDepartment of Civil Engineering University of Minnesota, 500 Pillsbury Drive SE,Minneapolis, MN 55455 nesvi002@umn.edu Corresponding Author Contact: Email: nesvi002@umn.edu ABSTRACTAcoustic methods have a great potential for materials characterization in industry because they can evaluate fatigue and fracture of construction materials without their destruction using special signatures of dissipated energy as active or passive acoustic signals which are generated from materials during service period or testing procedure. The most applicable active acoustic methods such as impactecho (IE) and phasearray (PA) techniques are applied to locate voids, defects or growing cracks and microcracks in strictures. However, this task for large threedimensional volume structures has some specific difficulties connected with 3D sources location. This work is devoted to analysis of mathematic algorithms of 3D acoustic sources locations using geometric acoustic techniques.Keywords: geometric acoustics, sources locations, impactecho, phasearray. 1 IntroductionThreedimensional volume structures remain basic for contemporary structural design. Some composite materials such as thermoplastics, fiberplastics, and concrete are still holding the favorite position among other construction materials in the 21st century. Many composite structures show symptoms of deterioration before expiration of their designed service life time. Repair and replacement of composites are very expensive and laborconsuming tasks. Nondestructive testing (NDT) techniques give tremendous opportunity to prevent future failure of structures and increase service time because they allow analyzing information in time and insitu and predict structures behavior. Acoustic methods keep a leading position among NDT techniques for testing of threedimensional volume structures.2 Acoustic methods2.1 Impactecho methodImpactecho testing of materials is used for practical applications in civil engineering. Usually there are no problem with flaw coordinates calculation for 1D and 2D models, but for threedimensional volume structures flaw location arise some specific problems connected with design of measuring setup and coordinate calculations [1,2].2.2 Phasedarray methodPhasedarray method is based on multisensor technique for scanning threedimensional volume structures using time delay calculations technique. Meanwhile, the problem of coordinates calculations in threedimensional volume structures is one of the obstacles on the way to successfully apply this technique for practical tasks [35].2.3 Geometric acoustic methodsGeometrical acoustic methods are using a postulate about linear ray propagation of acoustic waves in infinite or semiinfinite medium. It is also possible to assume that this medium has homogeneous and isotropic properties. To apply this method to a real material, which is neither homogeneous nor isotropic and also may contain some inclusions, pores, flaws etc., it is possible to use MonteCarlo simulation to generate random deviation of measuring data connected to this irregularities [6,7]. Described technique allows avoiding complications with modeling of wave propagation in inhomogeneous or unisotropic media, moreover, stratified materials could be presented as combination of homogeneous layers [ 8,9 ].3 Measuring designs for acoustic sources locations3.1 Measuring algorithmThere are several designs for acoustic sources location in threedimensional volume structures: number of transducers is dictated by the task of finding 3D sources coordinates X, Y and Z. Solutions could be found by simultaneous consideration of three combined equations. Here locations of transducers in 3D medium are indicated as (a_{j},b_{j},c_{j}), where j is transducer number (Figure 1).
Time of wave propagation between flaw with coordinates (X,Y,Z) and transducers locations (a_{j},b_{j},c_{j}) and j=1,2,3 produced a nonlinear system of equations
where V  velocity of bulk wave propagation, and taking into account that one of transducers (with code J = 1 ) is bifunctional, i.e. is working as transmitter and receiver in turn. Time of propagation from this transducer to a flaw and back T_{1} is described as formulae
and time propagation from transmitting transducer to others receiving transducers is
and substitute t_{j} from (2) and (3) at (1) we receive final nonlinear system of equations regarding flaw (X,Y,Z) coordinates:
3.2 Nonlinear equations solutionSolution for (4) could be find as some algorithms for nonlinear equations where 3D coordinate (X,Y,Z) are considered as unknown x_{1},x_{2} and x_{3} accordingly [10].It possible to assume that function is at least continuous and that _{ } for all unknown x_{j} where j =1,2,3 and some algorithm assumption, for instance differentiability, should be added to calculations. General solution is presented below [11]. Nonlinear system of equations with n unknown variable could be presented as
To simplify expressions (5) it is possible to describe it as:
System of nonlinear equations may be solved by an iterative methods by computing a sequence of points x^{(0)} with tendency and . The vector is a starting point of calculations, and algorithm is done when _{ } , where is special tolerance of calculations. There are several techniques for calculation of nonlinear equations and one of them is Newton's method [12]. Newton's method is providing a solution for system of nonlinear equations using assumption that functions are differentiable and smooth:
where  is some function estimation; and  the derivative matrix,  iterate assumptions.
The calculation algorithm starts with initial value and iterations by repeating
where k=0,1,2,... Then after evaluation of function f(x^{(k)}) and f'(x^{(k)}) derivative it is possible to constrain the firstorder Taylor approximation of the functions and to look for solution as f(x^{(k)}) + f'(x^{(k)})(yx(x^{(k)})=0 getting final forms y = x^{(k)}  Df(x^{(k)})^{1} f(x^{(k)}) for each next iterate x^{(k+1)} where x^{(k+1)} = x^{(k)}  Df(x ^{(k)})^{1} f(x^{(k)}) and k=0,1,2,3,... . Example of algorithm testing for 3D flaw coordinates calculation iterating by Newton's method presented in the Figure 2.
3.3 Computing of flaw coordinatesFor threetransducer design of measurement initial time could be evaluated for impactecho and phased arrays methods using time t_{equip} of generated pulses by testing equipment as initial time t_{0} . The main problem is measuring precision of initial time t_{0} and time differences Dt_{jj} between echopulse responses to each of transducers.Precision of calculations using presented algorithm depends on real conditions of measurements which are common for practical applications and give measuring errors connected with:
Blockscheme of MonteCarlo simulation of measuring process and estimation of algorithm precision are presented in the Figure 3:
MonteCarlo method is a powerful instrument for simulating of flaw coordinates in 3D medium. There are a lot possibilities to provide simulations of flaw locations, for example using combination of deterministic and stochastic functions and cyclic measurements modeling k = 1,2,..,n:
Where (X_{0},Y_{0},Z_{0})  first random flaw coordinates in 3D space. Then it is possible to determine positions for transducers with locations (a_{m},b_{m},c_{m}) with m  transducers codes as constants with some deviations of positions
There are (Da_{m,k},Db_{m,k},Dc_{m,k})  values of transducer position deviations on each cyclic measurements simulation k . After that calculating algorithms for threetransducer design using formulae (4)(9) the following data (X*_{k},Y*_{k},Z*_{k}) are received. Statistical analysis of differences between generated data (X_{k},Y_{k},Z_{k}) and computing data (X*_{k},Y*_{k},Z*_{k}) allow getting estimations of reliability of these algorithms and evaluate influence of different factors. The estimating criterion is normalized metrics DIF_{XYZ} of these differences as an average of m cycles of simulation and calculated using formulae [13]:
where m cycles of MonteCarlo simulations. Computing of flaw coordinates of was provided for slow growing flaw. MonteCarlo simulations of time measuring errors helps to analyzed possibilities of proposed algorithm for practical tasks. Results of moving source coordinate calculations are shown in the Figure 4.
Statistical analysis of MonteCarlo simulations of flaw locations in 3D medium based on m= 50 cycles. Measuring error data calculated by formulae (12) are presented in the Table below:
ConclusionDetermination of flaw coordinates in 3D medium meets some problems connected with measuring design and materials properties. Positioning of transducers affects accuracy of measurements. Inhomogeneity and anisotropy of materials are considerably influencing measurement results. Threetransducer design allows to measure flaw coordinates if arriving time is unknown. This design could open new avenues for flaw coordinates determination.References

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