|NDT.net Sep 2004 Vol. 9 No.09|
Active Flaw Detection in 3D MediumEdouard Nesvijski
Department of Civil Engineering
University of Minnesota, 500 Pillsbury Drive SE,Minneapolis, MN 55455
Corresponding Author Contact:
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 impact-echo (IE) and phase-array (PA) techniques are applied to locate voids, defects or growing cracks and micro-cracks in strictures. However, this task for large three-dimensional 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, impact-echo, phase-array.
1 IntroductionThree-dimensional volume structures remain basic for contemporary structural design. Some composite materials such as thermoplastics, fiber-plastics, 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 labor-consuming 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 in-situ and predict structures behavior. Acoustic methods keep a leading position among NDT techniques for testing of three-dimensional volume structures.
2 Acoustic methods
2.1 Impact-echo methodImpact-echo 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 three-dimensional volume structures flaw location arise some specific problems connected with design of measuring set-up and coordinate calculations [1,2].
2.2 Phased-array methodPhased-array method is based on multi-sensor technique for scanning three-dimensional volume structures using time delay calculations technique. Meanwhile, the problem of coordinates calculations in three-dimensional volume structures is one of the obstacles on the way to successfully apply this technique for practical tasks [3-5].
2.3 Geometric acoustic methodsGeometrical acoustic methods are using a postulate about linear ray propagation of acoustic waves in infinite or semi-infinite 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 Monte-Carlo 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 locations
3.1 Measuring algorithmThere are several designs for acoustic sources location in three-dimensional 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 (aj,bj,cj), where j is transducer number (Figure 1).
Time of wave propagation between flaw with coordinates (X,Y,Z) and transducers locations (aj,bj,cj) 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 bi-functional, i.e. is working as transmitter and receiver in turn. Time of propagation from this transducer to a flaw and back T1 is described as formulae
and time propagation from transmitting transducer to others receiving transducers is
and substitute tj 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 x1,x2 and x3 accordingly .
It possible to assume that function is at least continuous and that for all unknown xj where j =1,2,3 and some algorithm assumption, for instance differentiability, should be added to calculations. General solution is presented below .
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 . 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 first-order Taylor approximation of the functions and to look for solution as f(x(k)) + f'(x(k))(y-x(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 three-transducer design of measurement initial time could be evaluated for impact-echo and phased arrays methods using time tequip of generated pulses by testing equipment as initial time t0 . The main problem is measuring precision of initial time t0 and time differences Dtjj between echo-pulse 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:
Block-scheme of Monte-Carlo simulation of measuring process and estimation of algorithm precision are presented in the Figure 3:
Monte-Carlo 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 (X0,Y0,Z0) - first random flaw coordinates in 3D space. Then it is possible to determine positions for transducers with locations (am,bm,cm) with m - transducers codes as constants with some deviations of positions
There are (Dam,k,Dbm,k,Dcm,k) - values of transducer position deviations on each cyclic measurements simulation k . After that calculating algorithms for three-transducer design using formulae (4)-(9) the following data (X*k,Y*k,Z*k) are received. Statistical analysis of differences between generated data (Xk,Yk,Zk) 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 DIFXYZ of these differences as an average of m cycles of simulation and calculated using formulae :
where m cycles of Monte-Carlo simulations. Computing of flaw coordinates of was provided for slow growing flaw. Monte-Carlo 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 Monte-Carlo 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. Three-transducer design allows to measure flaw coordinates if arriving time is unknown. This design could open new avenues for flaw coordinates determination.