ndt related quantitative modeling of coupled piezoelectric and ultrasonic wave phenomena ·...

NDT Related

Quantitative Modeling of

Coupled Piezoelectric and

Ultrasonic Wave Phenomena

R. Marklein

Electromagnetic Theory, Department of Electrical Engineering,University of Kassel, D–34109 Kassel, GermanyE–mail: [email protected]–technik.uni–kassel.de

Dedicated to Professor Dr. H. Wustenbergon the Occasion of his 60th Birthday

Abstract. This paper presents an overview of the development of numerical tools for the computa-

tional modeling of acoustic (A), elastodynamic (E), and coupled piezoelectric (P) and ultrasonic wave

phenomena in the time domain related to nondestructive testing with ultrasound. For this purpose, a

numerical method, the Finite Integration Technique (FIT), is directly applied to the set of governing

equations of the underlying wave propagation in integral form. Space and time discretization of the

governing equations on a staggered grid system results in a set of discrete matrix equations of three

explicit time domain modeling codes acronymed PFIT, EFIT, and AFIT. Each of the discrete matrix

equations has a one–to–one correspondent in the original set of equations. Analytical properties of

the original set of equations are conserved. Mathematical properties like consistency and convergence

of the codes are proved and stability conditions are derived. The sparse matrices are highly suitable

for high–performance vector computers, workstations, or even for personal computers and state–of–

the–art laptops. To handle complex geometries, an IGES interface to commercial CAD systems is

implemented, and to solve arbitrary initial–boundary–value problems several kinds of boundary con-

ditions can be defined. The numerical codes PFIT, EFIT, and AFIT have widespread applications

in quantitative computational NDT, for example starting from piezoelectric transducer modeling and

optimization to ultrasonic wave propagation, diffraction, and scattering in inhomogeneous anisotropic

dissipative and even statistically heterogeneous materials. Some of the analytical and experimental

validations and applications to real life NDT problems are given in this paper illustrating the potential

of these numerical modeling tools.

1

Contents

1 Introduction 3

2 Numerical Modeling of Ultrasonic Wave Phenomena with EFIT and AFIT 4

3 Numerical Results obtained with EFIT and AFIT 12

3.1 Analytical and Numerical Validation of EFIT . . . . . . . . . . . . . . . . . . . 123.2 Experimental Validation of EFIT . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 AFIT and EFIT Modeling of Ultrasonic Pipeline Inspection . . . . . . . . . . . 173.4 EFIT Modeling of Ultrasonic NDT of Concrete . . . . . . . . . . . . . . . . . . 183.5 Transducer Radiation Modeling with EFIT . . . . . . . . . . . . . . . . . . . . . 203.6 EFIT Modeling of the Ultrasonic NDT of Fiber–Reinforced Double T–Stringer . 233.7 EFIT Modeling of Ultrasonic Wave Propagation in an Austenitic V–Butt Weld

and Notch Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8 ULIAS — EFIT Modeling Module . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Numerical Modeling of Coupled Piezoelectric and Ultrasonic Wave Phenom-

ena with the Piezoelectric Finite Integration Technique — PFIT 31

5 Experimental Validation and Numerical Results of PFIT 33

5.1 Piezoelectric Pz27 Disk Transducer on a Brass Cylinder . . . . . . . . . . . . . . 335.2 Piezoelectric Pz27 Disk Transducer on a Brass Cylinder with Backwall Breaking

Notch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Concluding Remarks 38

Acknowledgements 38

References 39

2

1 Introduction

The numerical time domain modeling of transient waves of all types in various disciplines ofscience and engineering is utilized for increasingly sophisticated applications to communication,remote sensing, medical imaging, and nondestructive testing.

In recent years ultrasonic nondestructive evaluation has been more and more character-ized by quantitative methods, which is particularly true for modern diffraction tomographicimaging algorithms. These developments and their appropriate applications to increasinglycomplex NDT problems — complex relating to materials as well as to geometries — like in-homogeneous anisotropic austenitic welds and fiber–reinforced composites, require a deep andfundamental physical understanding of ultrasonic wave propagation, diffraction and scattering.Consequently, numerical codes to model these phenomena should be as close to real life aspossible with only marginal or even no mathematical approximations. Only for canonical prob-lems, where the geometries comply with coordinate systems in which the governing equationsare separable, analytical solutions can be constructed, but, for the general case it is necessaryto turn to direct numerical methods.

This means that a direct discretization technique has to be applied to the governing equa-tions of the underlying physical phenomena. Well known numerical methods of that kindcomprise the

• Finite Difference Method (FDM) [e. g. Alterman, 1968; Bamberger et al., 1980; Chin etal., 1984; Temple, 1988; Bond et al., 1988; Harker, 1988; Harker et al., 1990, 1991],

• Finite Element Method (FEM) [e. g. Ludwig, 1986; Ludwig & Lord, 1988; Ludwig et al.,1989; Lord et al., 1990; Lerch, 1990, 1991; Stam, 1990; Stam & Hoop, 1990; You et al.,1991; Roberts, 1991], and

• Finite–Difference Time–Domain (FD–TD) method [e. g. Madariaga, 1976; Aki & Richards,1980; Virieux, 1984, 1986; Stanke & Chang, 1989].

In this paper, the well–established Finite Integration Technique (FIT) [Weiland, 1977; Fellinger,1991a; Fellinger et al., 1995; Wolter, 1995b; Marklein, 1998; Bihn, 1998] which is a generaliza-tion of the Finite Volume Method (FVM) [e. g. Versteeg & Malalaskera, 1995], is applied to thegoverning equations in integral form to derive numerical codes to simulate typical ultrasonicNDT situations as shown in Figure 1 including the piezoelectric elements and external electricalcircuit elements [Marklein, 1998]. Figure 2 relates the three explicit time domain modelingtools based on FIT [Marklein, 1998]:

• PFIT — Piezoelectric Finite Integration Technique,

• EFIT — Elastodynamic Finite Integration Technique, and

• AFIT — Acoustic Finite Integration Technique.

PFIT is particularly developed for the modeling of the excitation and reception mode of apiezoelectric transducer including an external electrical load. Furthermore, PFIT includes thefeatures of EFIT and AFIT. EFIT and AFIT have been designed to model the elastodynamicwavefields in solids and acoustic wavefields in fluids and gases. Besides that, AFIT can be alsoused to model scalar approximations of elastic wavefields in solids.

3

Emitter/Receiver Unit

E

R

(NetworkAlgorithm)

Emitter/Receiver Unit

E

R

(NetworkAlgorithm)

PiezoelectricTransducer A

PiezoelectricTransducer B

PiezoelectricElements (PFIT)

Backwall Breaking Crack

IncidentUltrasonicWavefield Diffracted/Reflected

Ultrasonic Wavefield

Diffracted/ReflectedUltrasonic Wavefield

Elastic Solid (EFIT)

Figure 1: Typical ultrasonic NDT situation: Nondestructive testing with ultrasound of a block of steel

with a backwall breaking crack

Excitation,

Reception,

Propagation,

Diffraction, and

Scattering of

Ultrasonic Wavefields

Piezoelectric (P) Materials

Elastic (E) Solid Materials

Acoustic (A) Fluid/Gaseous Materials or

Scalar Treated Elastic Solid Materials

....................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................... .......................

................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................ PFIT

............................................................................................................................................................................................................................................................................................................ EFIT

............................................................................................................................................................................................................................................................................................................ AFIT

Figure 2: Numerical ultrasonic modeling tools: PFIT, EFIT, and AFIT based on FIT

2 Numerical Modeling of Ultrasonic Wave Phenomena with EFIT

and AFIT

Historically, the Finite Integration Technique (FIT) has first been applied in electrodynamicsto Maxwell’s equations in integral form by Weiland [1977] to model electromagnetic problems[see also Weiland, 1986, 1990; Bartsch et al., 1992]. Based on these developments a commer-cial software package called MAFIA (MAFIA: solution of MAxwell’s equations using a FiniteIntegration Algorithm) is today available [CST, 1997].

In the time domain the resulting discrete equations of MAFIA are similar to the equationsderived by Yee [1966] starting from the differential form of Maxwell’s equations, which aretoday known as the Finite–Difference Time–Domain (FD–TD) method [e. g. Taflove, 1995].

Fellinger & Langenberg [1990] [see also Fellinger, 1991a; Fellinger & Marklein, 1991b] trans-fered the ideas applied in electrodynamics to the linear governing equations of ultrasonic wavesin solids, the linear elastodynamic case, i. e. Newton–Cauchy’s equation of motion and theequation of deformation rate for the homogeneous isotropic case and acronymed the numeri-cal procedure Elastodynamic Finite Integration Technique (EFIT). With EFIT, Fellinger has

4

furnished a tool for the modeling of elastodynamic waves in the time domain, which has beenfurther developed throughout the following years to handle arbitrary inhomogeneous dissipa-tive (viscid) anisotropic materials [Marklein et al., 1994a, 1994b, 1995b; Marklein, 1998] andcomplex geometries [Langenberg et al., 1996; Marklein et al., 1997]; today it is widely used asa well–accepted quantitative modeling tool in NDT.

AFIT, the acoustic pendant to EFIT, is generally derived by applying FIT to the governingequations of acoustic waves in integral form [Marklein & Fellinger, 1990; Marklein, 1992a, 1998].

Stimulated by the developed AFIT– and EFIT–code, similar algorithms have been imple-mented in the above mentioned MAFIA package by Wolter & Weiland [1995a], Wolter [1995b]and Bihn & Weiland [1996, 1997], Bihn [1998].

A CAFIT algorithm for ultrasonic NDT problems with cylindrical symmetry has been de-veloped by Peiffer et al. [1997].

The underlying governing equations of elastodynamic waves rely on the following two fun-damental physical laws [Achenbach, 1973; Nelson, 1979; Ben–Menahem & Singh, 1981; Auld,1990; Hoop, 1995]

(Newton’s 2nd law) force = mass × acceleration ⇒ f tot(R, t) = ρ0(R)∂2

∂t2u(R, t) (1)

(Hooke’s law) stress =stiffness × deformation ⇒ T(R, t) = c(R) : S(R, t) (2)

with the total force density vector f tot, the mass density at rest ρ0, the particle displacementvector u, Cauchy’s stress tensor of second rank T, the stiffness tensor of fourth rank c, and the

deformation tensor of second rank S. All field quantities are functions of the position vector R

and time t and all material parameters are assumed as time independent. The colon denotes thedouble–scalar product with the property ab : c d = (a · d)(b · c), the centered dot representsthe scalar product.

With the linear momentum vector j and the particle velocity vector v given by

j(R, t) = ρe0(R)v(R, t) (3)

v(R, t) =∂

∂tu(R, t)

Eq. (1) reads

f tot(R, t) =∂

∂tj(R, t) .

Then, Eq. (2) and Eq. (3) are the constitutive equations of a linear inhomogeneous anisotropicnondissipative elastic solid.

Introduction of Cauchy’s equation for the total force density acting on an elastic solidparticle

f tot(R, t) = ∇ · T(R, t) + f(R, t)

with f(R, t) accounting for a given or impressed force density, yields the Newton–Cauchy’sequation of motion, which is in integral form for a volume V with the closed surface S = ∂V

given by

∫∫∫

V

ρe0(R)∂

∂tv(R, t)dV = ©

∫∫

S

n · T(R, t)dS +

∫∫∫

V

f(R, t)dV (4)

5

with n being the outward normal vector of the closed surface S.According to the law of deformation a particle flux results in a time variation of the defor-

mation volume. The equation of deformation rate reads in integral form for a volume V withthe closed surface S = ∂V∫∫∫

V

s(R) :∂

∂tT(R, t)dV = ©

∫∫

S

sym nv(R, t)dS +

∫∫∫

V

h(R, t)dV (5)

with n being the outward normal vector of the closed surface S and the compliance tensor offourth rank s being the inverse of the stiffness tensor, which yields Hooke’s law in the form

S(R, t) = s(R) : T(R, t) .

Like f , h is a given or impressed source term, the so–called source of deformation rate accom-plishing the symmetry of both equations. The operator sym nv(R, t) gives the symmetricpart of the dyadic nv(R, t).

Elementary

Material

Cell m(n)

DiscreteMaterialGrid M

........................................................................

.......................

ContinuousMaterial ........................................................................

.......................

”Continuous“ SpacePhysical World

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

......

..

....

......

......

......

......

..

....

......

......

......

.

.

.

.

.

.

”Discrete“ Space of

PFIT, EFIT, and AFIT

Figure 3: Discretization of the ma-

terial in elementary cubic material

cells (m(n)) resulting in a regular

material grid M

A given ultrasonic NDT situation is generally described by a given material distribution, forinstance a homogeneous isotropic nondissipative block of steel. Discretization of the

”continu-

ous“ material by elementary cubic material cells m(n) ∈ M with the volume V = ∆x3, n beingthe global cell or node number (see Fig. 3), yields a regular material grid M , which defines the

grid G (see Fig. 3 and Fig. 4), whereas a dual grid G is staggered to both (see Fig. 4). Thematerial parameters are assumed constant in each elementary material cell, but each materialcell can be filled with different material parameters.

Figure 5 shows the EFIT grids M , G, G, and the allocation of the discrete elastodynamicfield components assigned to the node n, allowing a consistent implementation of the continuityand boundary conditions.

For a source–free interface between two solid materials, medium 1 and medium 2, thecontinuity conditions are

v(2)(R, t) − v(1)(R, t) = 0

n ·[T(2)(R, t) − T(1)(R, t)

]= 0

6

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Figure 4: Material grid M and the staggered dual grid system G and G

with n being the interface normal pointing into medium 2. v(i) and T(i) are the field contribu-tions at the endface on side of medium i with i = 1, 2.

Replacing medium 1 by vacuum and suppressing the index 2 the (homogeneous) boundaryconditions for R ∈ boundary and ∀ t are

v(R, t) = 0 motion–free boundary condition

or

n · T(R, t) = 0 stress–free boundary condition

with n being the surface normal pointing into the medium.Generally, each governing equation is solved for each discrete field component of v = viei,

i = 1, 2, 3 and T = Tijeiej, i, j = 1, 2, 3 on the left–hand side of Eq. (4) and Eq. (5) for a cellvolume V = ∆x3. For example, the first Newton–Cauchy grid equation for the v1 componentis (see Fig. 6)

e1 ·

[∫∫∫

V

ρe0(R)∂

∂tv(R, t)dV = ©

∫∫

S

n · T(R, t)dS +

∫∫∫

V

f(R, t)dV

]

⇓

ρ(m)e0 v

(m)1 (t)(∆x)3 =[T

(r)11 (t) − T

(l)11(t) + T

(f)12 (t) − T

(b)12 (t) + T

(d)13 (t) − T

(u)13 (t)

](∆x)2 + f

(m)1 (t)(∆x)3

with m = middle, r = right, l = left, f = front, b = back, d = down, and u = up. The dotindicates the first time derivative.

7

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Figure 5: Allocation of the discrete elastic field

components assigned to the global grid node n

x x x x x x x x x x x x x x x x xx xx x xx x x xx x x xx x x xx xy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yyyyyyyyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyyyyyyyz | ~l~ x x x x x x x x x x x x x x x x xx xx x x xx x x xx x x xx x xx xy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yyyyyyyyyyyyyyyyyyyyyyyy yyyyyyyyyyyyyyyyyyyyyyy

z l~l~

z h~Q

z ~U

z ~

z 2~

) )yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy y y y y y y y y y y y y y yy y y y yy y yy y y y y y y y y y y y y y y y y y y y y y y

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx ~

\

Figure 6: Elastodynamic case (EFIT): elemen-

tary integration cell of the v1 component

In a global node formulation this grid equation reads

ρ(n)e0 + ρ

(n+M1)e0

2v

(n)1 (t)(∆x)3 =

[T

(n+M1)11 (t) − T

(n)11 (t) + T

(n)12 (t) − T

(n−M2)12 (t) + T

(n)13 (t) − T

(n−M3)13 (t)

](∆x)2 + f

(n)1 (t)(∆x)3

with the global grid node n

n = nG(n1, n2, n3) = 1 + M1(n1 − 1) + M2(n2 − 1) + M3(n3 − 1) , n ∈ 1, 2, . . . , N ,

N = N1N2N3, ni ∈ 1, 2, . . . , Ni, and the step counters

x1 direction: M1 = 1

x2 direction: M2 = N1

x3 direction: M3 = N1N2 .

Performing this procedure for all components of v and T and formulating the grid equationsin matrix form yields the following Newton–Cauchy’s grid equation of motion and the gridequation of stress rate, the discrete Elastic Grid Equations (EGE) of EFIT

v(z− 1

2) = [ρe0]

−1[DIV][RTv ]−1[AT

v ]T(z− 1

2) + [ρe0]

−1f(z− 1

2)

T(z) = [c][RvT]−1[GRAD][Av

T]v(z) + g(z)

with the algebraic field vectors f, g, v, T, the material matrices [ρe0], [c], the topo-

logical operators [DIV], [GRAD] containing only the values −1, 0, and 1, the matrices [RTv ],

[RvT] containing the line elements, and the averaging matrices [AT

v ], [AvT].

8

The staggered grid allocation of the discrete field components also requires a staggeredallocation in time in full and half time steps ∆t according to the integration schemes (seeFig. 7)

v(z) = v(z−1) + ∆tv(z− 1

2)

T(z+ 1

2) = T(z− 1

2) + ∆tT(z) .

Scalarization of the elastodynamic case yields the acoustic case, therefore the hydrodynamic

$ ¡ ¡ ¡ ¡

¢

¢

£"¤4¥¦ §¨©Qª £>«¤>¥ ¦ §¨M¬ ª £"¤4¥¦ §Uª

£®w¥ ¦ §¨ ¬ ª £ «®¥ ¦ §ª £®w¥ ¦ §U¯ ¬ ªFigure 7: Staggered time grid of EFIT

pressure p is introduced

T(R, t) = −p(R, t) I

with the unit dyadic I = δijeiej; δij is the Kronecker symbol being equal to 1 for i = j andequal 0 for i 6= j.

Furthermore, Hooke’s law (2) reduces to a scalar formulation according to

(Hooke’s law) stress = stiffness × deformation ⇒ −p(R, t) =1

κ(R)S(R, t)

with the compressibility κ and the scalar deformation S.Generally, AFIT is derived by applying FIT to the governing equations of acoustic waves

in integral form [Marklein & Fellinger, 1990; Marklein, 1992a, 1998].The governing equations of linear acoustic waves for nondissipative materials are Newton’s

equation of motion and the scalar deformation rate equation

∫∫∫

V

ρa0(R)∂

∂tv(R, t)dV = −©

∫∫

S

n p(R, t)dS +

∫∫∫

V

f(R, t)dV

∫∫∫

V

κ(R)∂

∂tp(R, t)dV = −©

∫∫

S

n · v(R, t)dS −∫∫∫

S

h(R, t)dV

with the mass density at rest ρa0 and the given or impressed scalar source of deformation rateh.

Figure 8 illustrates the AFIT grids M , G, G, and the allocation of the discrete acousticfield components assigned to the node n, allowing a consistent implementation of the continuityconditions, which are for a source–free interface between two acoustic media, medium 1 andmedium 2, for R ∈ interface and ∀ t,

n ·[v(2)(R, t) − v(1)(R, t)

]= 0

p(2)(R, t) − p(1)(R, t) = 0

with n being the interface normal pointing into medium 2. v(i) and p(i) are the field contribu-tions at the endface on side of medium i with i = 1, 2.

9

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Figure 8: Allocation of the discrete acoustic field

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ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò

ñ ñ

ññ

ññ

ññ

Figure 9: Acoustic case (AFIT): elementary inte-

gration cell of the v1 component

Replacing medium 1 by vacuum and suppressing the index 2 the (homogeneous) boundaryconditions are for R ∈ interface and ∀ t

n · v(R, t) = 0 motion–free boundary condition

or

p(R, t) = 0 pressure–free boundary condition

with n being the surface normal pointing into the medium.For instance, the first Newton grid equation for the v1 component is (see Fig. 9)

e1 ·

[∫∫∫

V

ρa0(R)∂

∂tv(R, t)dV = −©

∫∫

S

n p(R, t)dS +

∫∫∫

V

f(R, t)dV

]

⇓ρ

(m)a0 v

(m)1 (t)(∆x)3 = −

[p(r)(t) − p(l)(t)

](∆x)2 + f

(m)1 (t)(∆x)3 .

The further application of FIT yields the discrete Acoustic Grid Equations (AGE) of AFIT,Newton’s grid equation of motion, and the scalar grid equation of pressure rate in the form

v(z− 1

2) = −[ρa0]

−1[R]−1[grad]p(z− 1

2) + [ρa0]

−1f(z− 1

2)

p(z) = −[κ]−1[div][R]−1v(z) − [κ]−1h(z)

with the algebraic field vectors f, h, v, p, the material matrices [ρa0], [κ], the matrices

containing the line elements [R], [R], and the topological operators [grad], [div] containingonly the values −1, 0, and 1.

Then, the time integration scheme (see Fig. 10)

v(z) = v(z−1) + ∆tv(z− 1

2)

p(z+ 1

2) = p(z− 1

2) + ∆tp(z)

10

ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö öö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö öööööööööööööööööööööööö ööööööööööööööööööööööö÷$øùú÷û û û ûööööööööööööööööööööööööööööööööööööööööööööööööööö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö

ööööööööööööööööööööööööööööööööööööööööööööööööööü ööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööö ö öööö öö öö öö öö ö öö ö öö ö ö ö ö

ööööööööööööööööööööööööööööööööööööööööööööööööööö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö ö öööööööööööööööööööööööööööööööööööööööööööööööööööü öööööööööööööööööööööööööööööööö

öööööööööööööööööööööö ööö öö öö ö öö öö ö ö öö ö ö öööööööööööööööööööööööö

ý"þ4ÿ ýþ>ÿ ý"þ4ÿ ý4ÿ ý4ÿ ý4ÿ

Figure 10: Staggered time grid of AFIT

completes the AFIT algorithm.The EFIT– and AFIT–codes are explicit time domain algorithms of leapfrog–type and of

2nd order in space and time. Consistency has been proven for both codes [Marklein, 1998].

Stability for a regular grid (M , G, G) is given by the multidimensional Courant–Friedrichs–Levy(CFL) condition in n–dimensions

∆t ≤ ∆tmax =1√n

∆x

cmax

with the spatial dimension n and the maximal energy velocity cmax. If consistency of the gridequations has been proven and the stability condition is satisfied, then convergence is ensuredby the Lax equivalence theorem [Richtmyer & Morton, 1990].

To model an ideal scatterer or an infinite modeling domain several boundary conditionshave been implemented (see Table 1).

Boundary condition EFIT AFITStress–free •Pressure–free •Motion–free • •Open • •Plane wave • •

Table 1: Implemented boundary

condition in EFIT and AFIT

In the following, some EFIT and AFITmodeling examples are presented [see alsoFellinger & Langenberg, 1990; Langenberg etal., 1990, 1993, 1994, 1995, 1996, 1997a,1997b; Fellinger 1991a; Fellinger & Marklein,1991b; Marklein, 1991a, 1992a; Marklein& Fellinger, 1991b; Marklein et al., 1992b,1994a, 1994b, 1995a, 1995b, 1996a, 1996b,1996c, 1996d, 1997a, 1997c; Hofmann, 1994;

Walte et al., 1993, 1996, 1997; Klaholz et al., 1995; Fellinger et al., 1995; Rudolph et al., 1995;Schmitz et al., 1995; Bergmann, 1996; Kostka et al., 1997; Hannemann et al., 1998; Marklein,1998].

11

3 Numerical Results obtained with EFIT and AFIT

3.1 Analytical and Numerical Validation of EFIT

Once a numerical method is applied to compute a physical solution the results, and there-with the numerical code, have to be validated either against analytical solutions or numericalreferences.

For a first analytical and numerical validation of the 2–D EFIT–code a strip–like pistontransducer radiating into a 2–D homogeneous nondissipative isotropic half–space has beenchosen (see Fig. 11). Figure 12 shows the comparison of the time history of the horizontal

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......

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n · T = T33(t)e3

n · T = 0 n · T = 0

Figure 11: Strip–like piston transducer with

uniform normal stress load on a homogeneous

isotropic stress–free elastic half–space: 2–D EFIT

grid with two selected observation points: P1 =

(0, 12) mm and P2 = (12, 12) mm. For the

EFIT modeling a stress–free boundary condition

has been applied on all four boundaries, for the

EFEM modeling a stress–free boundary condition

has been applied on the top, left, and right side,

and a displacement–free boundary condition has

been chosen on the bottom side.

Figure 12: EFIT (—), EFEM (- - -), and SZEW (– – –) comparison for a strip–like piston transducer:

time history of the particle displacement components u1(t) and u3(t) at P1 (left) and P2 (right)

12

and vertical particle displacement with the spectral decomposition method (SZEW: SpektraleZerlegung nach ebenen Wellen) [Fellinger & Langenberg, 1990; Fellinger, 1991a] and an elasto-dynamic FEM–code (EFEM) developed by Ludwig [1986, 1988, 1989] [Marklein, 1991; Fellinger& Marklein, 1991b]. Both results are documenting a very good agreement between all threemethods.

3.2 Experimental Validation of EFIT

The first experimental validation of EFIT has been performed for the insonification of a backwallbreaking crack by a 45 shear wave probe (see Fig. 13). The numerical results obtained withthe 2–D EFIT–code are compared to those of an accompanying experiment. This problem alsohas been modeled with 2–D AFIT in order point out explicitly the difference between acousticand elastodynamic waves. The time history of the displacement as well as the amplitude ofthe normal traction load within the finite transducer aperture was carefully measured with anelectrodynamic probe. The elastic material steel is characterized by the velocities cP = 5.9mm/µs and cS = 3.2 mm/µs. In order to compare the acoustic and elastodynamic case forthe shear wave an acoustic material with cP = 3.2 mm/µs has been chosen. The backwallbreaking crack is modeled as an ideal crack with plane surface and stress–free/pressure–freeboundary condition. The crack is illuminated with a 45 shear wave angle probe within aprescribed synthetic aperture. Within the transducer aperture — the black bar in Fig. 13

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MWB45–2 (45 SV–Wave Transducer)Synthetic Aperture

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................................................................. Stress–free Boundary Condition for 2–D EFIT andPressure–free Boundary Condition for 2–D AFIT Modeling

..................................... Open Boundary Condition for 2–D EFIT and 2–D AFIT Modeling

..................

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Backwall BreakingCrack

........................................................

........................................................105

...................................52.5

...................................

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27

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x0=13.74

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14

Figure 13: Steel

block with a

backwall breaking

crack

— the elastic/acoustic surface is excited by a normal traction distribution weighted by themeasured sole function. The time history of the normal traction is prescribed as a 4 cycleraised cosine pulse with a center frequency of 2 MHz (see Fig. 14).

Time domain snapshots of EFIT and AFIT are given in Figure 14, which are showingexplicitly the difference between the elastodynamic case and the scalar approximation.

Figure 15 compares experimental and simulation results for a single A–scan, where receptionhas been modeled via averaging over the transducer aperture with the same amplitude weightingas applied for transmission. The coincidence is nearly complete, except for the late pulses beingassociated with Rayleigh waves traveling down and up along the crack faces and being convertedinto shear waves. The somewhat different amplitudes and the nonresolved time separation ofthe latter in the experiment may be due to the fact that the realization of the crack was a fatiguecrack with a surface roughness, whereas the model crack was perfectly plane and stress–free.Nevertheless, the modeling result obtained with EFIT complies very well with the experiment,giving us some confidence for further applications of this code.

13

2–D EFITSnapshot

2–D AFITSnapshot

2–D EFITSnapshot

2–D AFITSnapshot

Figure 14: Time domain snapshots of 2–D EFIT and 2–D AFIT

Another way to compare the modeling results with the experiment is to use the output rf–data field as a testbed for an imaging algorithm like the Fourier–Transform Synthetic ApertureFocusing Technique (FT–SAFT) [Mayer et al., 1990], which is the Fourier–Transform–Versionof SAFT [Schmitz et al., 1995]. The rf–data fields are produced in a pulse–echo experimentwith the proper amplitude tapering of the transducer in the transmitting as well as in thereceiving mode. Since FT–SAFT “knows” nothing about mode conversion and resonances,each time of flight curve is attributed to a distinct defect, resulting in the image of Fig. 16: thecorner between the crack and the backwall is properly imaged from the most pronounced timeof flight curve, a weaker image of the crack tip is obtained, and the resonances are focused to aghost defect. Figure 16 not only presents simulations, but also experimental results, and, as weanticipated, our model is well reproduced. When investigated within the framework of inversescattering, the derivation of SAFT exhibits that a linearization is involved and, therefore, inthe present version SAFT is not bound to handle resonances and mode conversions properly;neither is it designed for inhomogeneous anisotropic materials. A theory of inverse scatteringis available which identifies these drawbacks quantitatively [Langenberg, 1987; Langenberg, etal., 1993, 1997a].

14

Experiment

2–D EFITModeling

Figure 15: Experimental and 2–D EFIT modeled A–scan for a selected transducer position, where

the crack tip echo is more prominent than the corner reflection. The single pulses can be inter-

preted as follows: SS: Shear–Shear–Reflection (crack tip echo); SPS: Shear–Pressure–Shear–Mode–

Conversion; SSS: Shear-Shear–Shear–Reflection (corner reflection); SRS: Shear–Rayleigh–Shear–

Mode–Conversion; SRRS: Shear–Rayleigh–Rayleigh–Shear–Mode–Conversion

15

a)

b) x1 in mm ⇒

x3

in mm⇓

Figure 16: FT–SAFT image obtained from the simulated rf–data (a) and SAFT image from the

measured experimental rf–data (b)

16

3.3 AFIT and EFIT Modeling of Ultrasonic Pipeline Inspection

The principle geometry for another example of 2–D EFIT modeling is given in Figure 17. Here,an interface crack at x = 40 mm is considered. Figure 18 shows two 2–D EFIT |v|–

0 67x1 in mm ⇒0

10

33.5

⇑x3in

mm

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BoundaryOpen

Boundary

np = 0

n · T = 0

........................................

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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................•

A=22

D=15

W=10

E

Transducer

.........x=40

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...........α = 18Water

Steel

Figure 17: Ultra-

sonic pipeline in-

spection: geome-

try for 2–D EFIT

modeling

Pressure Wave Pressure Wave

ShearWave

Figure 18: 2–D EFIT |v|–snapshots of the ultrasonic wavefield

24 28 32 36 40 44 48 52 56 60 64 68 72 76 80

t in µs ⇒

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-50

-40

-30

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4

Figure 19: Top: EFIT modeled A–scan; bottom: comparison between modeled (solid line) and exper-

imental ALOK-scan (dashed line). 1: P–P; 2: P–4S–2S–P; 3: P–4S–4S–P; 4: P–4S–6S–P

snapshots for an insonification angle of α=18. The modeled A–scan is displayed at the topof Fig. 19, below, a comparison between the modeled and experimental ALOK–scan is given.Convincing coincidence has been found. The four strongest echo signals can be interpreted viasnapshots analysis.

17

3.4 EFIT Modeling of Ultrasonic NDT of Concrete

Figure 20 shows three typical concrete samples with different parameters. Due to the additivesand air inclusions which may occur concrete is a statistically heterogeneous material. Usuallyconcrete consists of cement and several additives like basalt, plaster, and biatitgranite. Differentsituations can be modeled: concrete without and with air inclusions. Figure 21 displays one ofthe concrete models under concern. The base material is cement and the additives are basalt,plaster, and biatitgranite with a max. aggregate size of 8 mm: a grading curve B8 is used.The additives are modeled by approximately 60,000 ellipses which are varying statistically insize, orientation, and additives (see Fig. 21). Details can be found in Marklein [1998]; firstapplications of the EFIT–code and the application of an inverse scattering algorithm to theEFIT–data can be found in Marklein et al. [1996b, 1996c]. Each material cell of the EFITgrid can be filled with different material parameters. From a numerical point of view the totalnumber of the ellipses is only limited by the number of material cells. In this example the cellsize is ∆x = 250 µm and the resulting mesh size is 2,000 × 2,000 with 4 × 106 material cells. Thedetail drawings at the right side of Figure 21 clearly indicate the statistical distribution of theellipses. A normal pressure probe is applied in the pulse–echo mode on the top surface which hasa diameter of D = 5 cm and a center frequency of fc = 80 kHz. The time history of the probe ismodeled by a raised cosine with two cycles. For EFIT modeling a stress–free boundary conditionis applied on the top and bottom surface and an open boundary condition [Higdon, 1991, 1992]at the left and right boundary in order to model an infinite concrete structure. Snapshots of theultrasonic wavefield are shown in Figure 22. With increasing time, the wavefield becomes totallydistorted because of multiple reflections and mode conversion effects at the air inclusions. Dueto the open boundary condition reflections and mode conversions are suppressed at the verticalboundaries. The backwall echo of the pressure wave shows up in Figure 23 at the point t = 240µs. The traveling time of the backwall echo can then be further analyzed, e. g., for thicknessdetermination.

Similar results can be found in Schubert & Kohler [1997] and Burr et al. [1997].

Figure 20: Three typical concrete samples with different grain size, grading curve, and water/air

concentration

18

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0 250 500x in mm ⇒

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0 37.5 75x in mm ⇒

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Figure 21: Concrete model for 2–D EFIT modeling with 6% air concentration; 3% in the grain class

1 and grain class 2

a) t = t1 = 35.65 µs b) t = t2 = 95.08 µs c) t = t3 = 154.50 µs d) t = t4 = 213.93 µs

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Figure 22: 2–D EFIT |v|–snapshots of the ultrasonic wave propagation, diffraction, and scattering:

a) bis d) with 6% air; 3% each in the grain class 1 and 2

EP

BE

0 30 60 90 120 150 180 210 240 270 300 3300 356,55

t in µs ⇒

-1.5

-0.75

0

0.75

1.5

⇑v3

Figure 23: 2–D EFIT modeled A–scan: EP = excitation pulse and BE = backwall echo

19

3.5 Transducer Radiation Modeling with EFIT

This example illustrates what happens if a 45 shear wave transducer from the shelf, designedfor conventional applications in isotropic materials like ferritic steel, is applied to the surfaceof a transversely isotropic material like austenitic steel characterized by a grain orientation a.

a)

b)

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Figure 24: Superimposed 2–D EFIT |Sel|–snapshots of a 2 MHz 45 shear wave transducer identify

the geometric ultrasonic (GU) beam, according to the geometric optical (GO) beam in electromagnet-

ics: a) isotropic ferritic steel; b) transversely isotropic austenitic steel with a = cos 60e1 + sin 60e3

20

Figure 24 shows EFIT snapshots of the elastic Poynting vector displaying the rather dra-matic effect in the anisotropic case, where the main beam is skewed with regard to the originallyintended energy direction. A weld in austenitic steel might not even be

”hit“ by the ultrasound.

The explanation of Figure 24 is given in Figure 27 using the characteristic slowness and groupvelocity diagrams (see Fig. 25 and 26).

A geometric construction of the above skewing effects can be obtained via Huygens’ princi-ple (envelope construction) applied to group velocity diagrams; this is done in Fig. 27 for theconsidered 2–D case. Figure 27a shows the properly adjusted linear time delay resulting in the

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3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 333333333333333333333333 333333333333333333333334 B C 73 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 333333333333333333333333 333333333333333333333334 B C 89

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D D D D D D D D D D D D D DD D D D D D DD DD

D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D 5 67D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D 5 689D D D D D D D D D D D $5 6E F G89

D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D B C 7D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D B C 89D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D B 8 HD D D D D D D D D D D $5 6E F G89

Figure 25: Slowness diagrams: a): isotropic fer-

ritic case, b): anisotropic austenitic case

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jjjjjjjjjjj jj j jj j j j j j jj j j j j j j j j j j jj j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j jj j j j j j j j j j jj j j j j j jj j jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj jj j j j j jj j j j j j j j j jj j j j j j j j j j j j jj j j j j j j j j j j j j j j j jjj j j j j j j j j j j jj j j j j j jj j jj j j j jj j j j jjj j j j jj j j j jjj j j jj j j jj j j jj j j jj j j j jj j j jjj j jj j jj j jj j jj j jj j jj j jj j jj j jj j jjj j jjjj j jj jj j jj jj j jj jj j jj jj j jj jjj jj j jj jj jj jj jj jj jj jj jj j jj jj jj jj jjj jj jjj jjjj jj jjj jj jj jj jj jjj jj jj jj jjjj jjj jj jjj jjj jj jjj jjj jj jjj jjj jjjj jjj jjjjjjj jjjj jjj jjjj jjj jjjjj jjjjj jjjjj jjjj jjjjj jjjjjjjj jjjjjjjjjjj jjjjjjjjjjj jjjjjjjjjjj jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

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j j jj j jj j jj j jj jj jjj jj jj jjj jjj jjjj j jjjjj jjjjjjjjjjjjjjjjjjjjjjjjj

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj jj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jj j j j j j j j j j j j j jj j j jj

j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j T d eU V W oj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j T d eU V W fgj j j j j j j j j j j T d ep q rU V W fg

j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j T mU V W n oj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j T mU V W n fgj j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j T mU V W f sj j j j j j j j j j j T d ep q rU V W fg

Figure 26: Group velocity diagrams: a):

isotropic ferritic case, b): anisotropic austenitic

case

tilted beam as desired in the isotropic case. Figure 27b displays the isotropic case and Figure27c the anisotropic case, where two time frames of the wavefronts, accounting for a broadbandsignal, emanating from the left and right corner of the aperture are superimposed. The geomet-ric ultrasonic (GU) beam, according to the geometric optical (GO) beam in electromagnetics, isthe tangent of the left and right wavefront representing all radiating line sources —

”wavelets“

— inside the finite transducer aperture (indicated as a black bar). Tracing the amplitude alonga wavefront an EFIT snapshot can also be utilized to calculate an appropriate radiation pattern

21

(see Fig. 28).

t;u

v u

w*u

xzy^ | xzy~| y^ |

y~|& x ux

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Figure 27: Huygens construction of transducer soundfields: a) linear time retardation as a function

of the local transducer aperture coordinate; b) Huygens construction for isotropic ferritic steel; c)

Huygens construction for transversely isotropic austenitic steel

22

a) b)

Excitation Pulse

t ⇒

⇑A

0 30 60

x in mm ⇒

0

30

60

⇑zin

mm

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RadiationPattern ...........................................................................................................................................................

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t ⇒

⇑v1

t ⇒

⇑v3

Excitation Pulse

t ⇒

⇑A

0 30 60

x in mm ⇒

0

30

60

⇑zin

mm

u

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t ⇒

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⇑v3

Figure 28: 2–D

EFIT modeling of

transducer radia-

tion: a) isotropic

aluminum and

b) transversely

isotropic austen-

ite with a hor-

izontal grain

orientation

(a = ex)

3.6 EFIT Modeling of the Ultrasonic NDT of Fiber–Reinforced Double T–Stringer

In the construction of aircraft wings fiber–reinforced graphite epoxy material is a new–fangledmaterial. It has a transverse isotropy characterized by the fiber direction a.

For instance, the interior of an aircraft wing consists of multiple double T–stringers which aremade of fiber–reinforced composites and which form a honeycomb structure. At the I–sectioninterface the double T–stringer has a very complicated variation of the fiber direction, but inthe EFIT model an orthogonal fiber direction at this intersection is presumed. Therefore, in theupper and lower part we have a horizontal and in the I–section a vertical fiber direction. SelectedEFIT–|v|–snapshots of the elastic wavefield of a 5 MHz normal pressure probe are given inFigure 29b. At the sole of the transmitter and receiver we applied an absorbing boundarycondition (ABC) [Higdon, 1991, 1992] in order to take into account the energy absorption ofthe mounted probe. The transmitter signal is shown in Figure 29a.

The time history of the probe is modeled by an RC2 pulse. The receiver signal is displayedin Figure 29d. This signal represents an echo sequence which is due to the wave guide behaviorof the I–section. In a second EFIT modeling we half surrounded the I–section with water inorder to model a half filled tank. Figure 29c shows the according 2–D EFIT |v|–snapshots.

Obviously, when the Lamb wave arrives at the water/I-section interface leaky waves appear.Further, when the Lamb wave hits the lower corners of the I–section, Scholte waves are generatedwhich are traveling up along the water/I-section interface. The received signal is given in Figure29e. Because of the leakage effect the amplitude of the received echo decreases and the waveguiding effect appears weaker. Due to this fact only one single transmitted echo, not a wholeecho sequence as shown in Figure 29d, is received.

23

a) Transmitter A–scan

-4000

0

4000

0.00 2.10 4.20 6.30 8.40 10.50 12.60 14.70 16.80 18.90 21.00

⇑A

t in µs ⇒b) 2–D EFIT |v|–snapshots (without water) c) 2–D EFIT |v|–snapshots (with water)

d) Receiver A–scan (without water)

-1.5

0

1.5

0.00 2.73 5.46 8.19 10.92 13.65 16.38 19.11 21.84 24.57 27.30

⇑A

t in µs ⇒

e) Receiver A–scan (half surrounded with water)

-0.4

0

0.4

0.00 2.73 5.46 8.19 10.92 13.65 16.38 19.11 21.84 24.57 27.30

⇑A

t in µs ⇒Figure 29: 2–D EFIT modeling of a fiber–reinforced double T–stringer of transversely isotropic

graphite epoxy

24

3.7 EFIT Modeling of Ultrasonic Wave Propagation in an Austenitic V–Butt

Weld and Notch Scattering

Figure 30 shows another example geometry of an inhomogeneous anisotropic NDT situation:an austenitic V–butt weld with a backwall breaking notch modeled with stress–free boundarycondition. A sketch of the two grain orientations under concern is given in Figure 31.

Simulated A–scans and 2–D EFIT v–snapshots obtained in pulse–echo mode are given forboth grain orientations in Figure 32 and Figure 33, respectively. The time domain snapshots aredisplaying the excitation and propagation of the ultrasonic wavefield inside the weld comprisingthe notch scattering, and the reception of the characteristic echo signals. It has to be pointedout, that in the case of perpendicular grain structure, no notch tip echo is received (see Fig. 32,left). For an interpretation of the obtained results using slowness and group velocity diagramsthe reader is refered to Hannemann et al. [1998].

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MWB45–2

Steel Steel

∆x

10

1

Austenitic

V–Butt Weld

30

15

......

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..........α

Figure 30: Austenitic V–butt weld with backwall breaking notch embedded in isotropic ferritic steel;

α is the inclination angle of the interface between the V–butt weld and the isotropic ferritic steel

a) b)

a_

a_a_

Figure 31: Considered grain orientation characterized by a: a) perpendicular grain orientation and b)

herringbone grain orientation

0 46.2

0.3

-0.3Corner Reflection of Notch

0 46.2

Corner Reflection of Notch

0.07

-0.07 Notch Tip

Figure 32: Austenitic V–butt weld with backwall breaking notch and an inclination of the interface

of α = 15: 2–D EFIT modeled A–scans of the perpendicular grain orientation (left) and herringbone

grain orientation (right)

25

"Gap"

"Gap"

Figure 33: Austenitic V–butt weld with backwall breaking notch and an inclination of the interface

of α = 15: 2–D EFIT v–snapshots of the perpendicular grain orientation (left) and herringbone

grain orientation (right)

26

3.8 ULIAS — EFIT Modeling Module

Within a joint European research project involving several industrial partners an ultrasonicmodeling and imaging tool acronymed Ultrasonic Inspection Applying Simulation (ULIAS) hasbeen developed [Langenberg et al., 1996; Marklein et al., 1997a].

¡ ¡¢£¤ ¤¥¦§ ¨ ¨© ª« ¬ ® ®® ®¯ ¯¯ ¯°±²³´´µµ¶· ¸¹ º»

¼½¾¿À ÀÁ ÁÂÃ

ÄÅ ÆÇ ÈÉ ÊË ÌÍÎ ÎÏ ÏÐ ÐÑ ÑÒ ÒÓ ÓÔÕ Ö× Ø ØÙ Ù ÚÛ Ü ÜÝ ÝÞßà àá

â âã ã äå æ æç è èé ê êëìíîïðñò òó ó ô ôõ ö÷øù

Emitter

ú ú ú ú ú ú ú ú ú úú ú ú ú ú ú ú ú ú úú ú ú ú ú ú ú ú ú úú ú ú ú ú ú ú ú ú úú ú ú ú ú ú ú ú ú úú ú ú ú ú ú ú ú ú úú ú ú ú ú ú ú ú ú úû û û û û û û û û ûû û û û û û û û û ûû û û û û û û û û ûû û û û û û û û û ûû û û û û û û û û ûû û û û û û û û û ûû û û û û û û û û û

ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü üü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü ü

ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ýý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý ý

þ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þþ þ þ þ þ þ þ

ÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ ÿ

Block

Flaw

Channel 2

Emitter/Receiver

Channel 1

Frame 15

Frame 1

Receiver

Figure 34: A typical

NDT situation: block

with backwall breaking

flaw and a scan path with

two channels, a multi-

static channel (Channel 1)

and a monostatic channel

(Channel 2)

ULIAS is a module based modeling, imaging, and visualization system, which also comprisesa selection of data processing modules. The data exchange between different modules relies ona unified ultrasonic data file in Ultrasonic Data Exchange Format (UDEF). This paper focuseson the EFIT modeling module, the UDEF File, the UDEF Editor, the FT–SAFT imagingmodule, and 1–D/2–D/3–D visualization capabilities (VISLIB). The 3–D graphic features arebased on the graphic libraries PEX, GL, or OpenGL. For instance, the VISLIB allows the 3–Dvisualization of defect images within the 3–D test piece geometry as provided by a CAD filein IGES format. ULIAS has been originally designed to run on UNIX based computers and issupported for the following computer platforms: IBM RISC System/6000, HP, SUN, and SGI.The user interface is based on X-Windows and OSF/Motif. The general programming languageis ANSI C. All ULIAS modules have read/write access to a selected UDEF File using the so–called I/O functions of the Low Level Library (LLL). The UDEF File is the

”center“ of the

ULIAS, it contains all parameters needed to build a realistic computer model of the underlyingNDT situation. It is a binary file containing several NDT related structures, e. g., Block,Flaw(s), Channel(s), Frame(s), Material(s), Transducer(s) (see Fig. 34). The structures Blockand Flaw have a reference to a CAD file in IGES format containing the 2–D/3–D geometry ofthe test specimen and the flaw. The EFIT Modeling Module requires a voxel grid representationof the simulation domain with material information. In order to generate such a voxel grid fora general geometry from a CAD file in IGES format, a Voxel Grid Generator (VGG) has beendeveloped, which is written in C++. Presently, we are working on a LINUX version of ULIAS.

Fig. 35 illustrates a basic configuration of the ULIAS package, which comprises the followingparts:

• the UDEF File containing NDT related structures, e. g., Block, Flaw(s), Channel(s),Frame(s), Material(s), Transducer(s);

• the EFIT Modeling Module simulating the given NDT situation exactly, i. e., arbitraryNDT situations and flaw geometries including mode conversion effects and surface wavephenomena [Marklein, 1998]. Primary results are snapshots of the ultrasonic wavefield for

27

selected time frames and recorded A–scans, which are written to a selected UDEF File.EFIT generates also 2–D/3–D ultrasonic rf–datafields, which can be used as a testbed forthe ULIAS Imaging Modules;

• the FT-SAFT (Fourier Transform-SAFT) Imaging Module, which is a scalar algorithmicversion of Synthetic Aperture Focusing Technique (SAFT) using only Fourier transforms[Mayer et al., 1990]. This results in a fast reconstruction scheme especially in 3–D. A2–D/3–D algorithm is implemented, which supports a linear/plane measurement surface.Furthermore, signal analysis and signal processing techniques like filtering, correlation,convolution, and deconvolution are implemented;

• the UDEF Editor which is a special editor for UDEF Files.

VISLIBVisualization

Library

FT-SAFTFourier Transform

SAFT

Low Level Library

LLL

VISLIBVisualization

Library

Low Level Library

LLL

EFIT Code VGG

GeneratorVoxel Grid

VISLIB

LibraryVisualization Elastodynamic Finite

Integration Technique

EFIT Modeling Module

FT-SAFT Imaging Module UDEF Editor

in IGES Format CAD Files UDEF File

Low Level Library

LLL

UDEF Editor

Figure 35: A basic configuration of ULIAS com-

prising the EFIT Modeling Module, the FT–SAFT

Imaging Module, a UDEF File, and the UDEF Edi-

tor

Preexisting Voxel Grid

.vgg.out

.vgg.err

.vgg.cfg

ANSI C igesc2l.lut

*.vox

Loop-Up TableColor/Layer

CAD SystemCAD System

IGES ConverterIGES Converter

Layer Material Coding Color Material CodingCADGeneral

System

VGG Configuration File

VGG Stdout File

VGG Stderr File

VGG Voxel Grid

EFIT Module

Voxel Grid GenerationIGES File Handling, and

CAD System,

Color to Layer

IGESC2L

Converter

VISLIB

ANSI C

LibraryVisualization

VGG

GNU C++

GeneratorVoxel Grid

ANSI C

EFIT Code

Integration TechniqueElastodynamic Finite

CADI-DEAS

System

*.igs

IGES File

*.vox

Figure 36: ULIAS: EFIT Modeling Module,

CAD file in IGES format, and the Voxel Grid

Generator (VGG)

All ULIAS modules use the 1–D/2–D/3–D capabilities of the Visualization Library (VISLIB)to display 1–D Plots, 2–D Raster Images, or to display the ultrasonic data within the 3–Dgeometry given by the CAD file in IGES format. The 3–D window allows for example lighttransformation, object transformation, volume and surface rendering, and voxel grid slicing.Fig. 36 summarizes the implemented CAD file handling and material coding strategies withthe EFIT Modeling Module. The Voxel Grid Generator (VGG) requires a layer (IGES: level)based material coding philosophy. If the used CAD system does not support layers, like theCAD system I-DEAS, the material coding can alternatively be based on colors. Then the

28

supplied Color–to–Layer Converter must be applied in order to generate a CAD file with layerbased material coding. The VGG is controled by a configuration file and is called via a UNIXsystem call. Fig. 37 shows the main window of the EFIT Modeling Module displaying the testspecimen (Block), the transducer (wedge probe), and the linear scanning path (black line) insolid mode. The specimen consists of two parts, which are screwed together. A 2–D xz–planecentered in y–direction has been selected for a 2–D EFIT modeling session in the backgroundmode. Fig. 38 displays a selected snapshot of the ultrasonic wavefield within the 3–D geometry.

Figure 37: EFIT Modeling Module: Main window with CAD file in IGES format

Here, a cross-section plane has been moved through the 3–D geometry in order to visualize theinterior of the block, the ultrasonic wavefield. The recorded A–scan is displayed in the 1–DA–Scan Window. The 3–D Main Window and 1–D A–Scan Window are synchronized in twoways: (1) the displayed A–scan is synchronized with the given receiver position, and (2) thetime point of the displayed snapshot is marked by a cross in the 1–D A–Scan Window. Thisexample gives you a first idea of the potential of ULIAS.

29

Figure 38: EFIT modeling module: Main window with CAD file in IGES format and superimposed

wavefront snapshot with the synchronized A–scan. The time point of the displayed snapshot is marked

in the A–scan by a cross, which moves if the time point of the displayed snapshot is changing

30

4 Numerical Modeling of Coupled Piezoelectric and Ultrasonic Wave

Phenomena with the Piezoelectric Finite Integration Technique

— PFIT

In order to model coupled piezoelectric and ultrasonic phenomena, like the excitation andreception of an ultrasonic wavefield by a piezoelectric transducer, which converts an electricalsignal to an ultrasonic signal and vice–versa, the piezoelectric effect has to be taken into account.For linear inhomogeneous nondissipative piezoelectric materials the constitutive equations are[Nelson, 1979; Auld, 1990]

D(E,S) = εS · E + e : S

S(T,E) = sE : T + d231 · E

with the electric flux density vector D, the electric field strength vector E, the permittivitytensor of second rank εS measured at S = const., the piezoelectric coupling tensor of third

rank e, the compliance tensor of forth rank sE measured at E = const., and the piezoelectric

coupling tensor of third rank d. The upper indicial notation 231 indicates transposition of the

tensor elements: d231 = (dijkeiejek)231 = dijkejekei.

In general, the typical dimension of the piezoelectric element is small compared to the wave-length of the electromagnetic wave inside the piezoelectric element. Therefore it is convenientto introduce the electroquasistatic (EQS) approximation in Maxwell’s equations

E(R, t) = −∇Φ(R, t)

with the electroquasistatic scalar potential Φ(R, t) [e. g., Nelson, 1979; Haus & Melcher, 1989;Auld, 1990]. For instance, this holds for a frequency f < 1 GHz for a typical dimension ofXmax = 1 cm. Within the EQS approximation all magnetic effects are neglected.

Then, the governing equations of piezoelectric waves are Newton–Cauchy’s equation ofmotion, Poisson’s equation, and the equation of deformation rate, which read in integral formfor a volume V with the closed surface S = ∂V

∫∫∫

V

ρp0(R)v(R, t)dV = ©

∫∫

S

n · T(R, t)dS +

∫∫∫

V

f(R, t)dV

©

∫∫

S

n · εS(R) · ∇Φ(R, t)dS = ©

∫∫

S

n · e(R) ·· sym ∇u(R, t)

−∫∫∫

V

%(R, t)dV −∫∫

S

η(R, t)dS

∫∫∫

V

sE(R) : T(R, t)dV = ©

∫∫

S

sym nv(R, t)dS +

∫∫∫

V

d231(R) · ∇Φ(R, t)dV

+

∫∫∫

V

h(R, t)dV

with the mass density at rest ρp0, the electrical charge density %, and the electrical surfacecharge density η, which is in general unequal to zero at a perfectly conducting boundary.

Introducing the staggered grid system (G, G) as shown in Figure 39 with the grid spacing∆x and applying FIT to the governing equation yields a set of consistent matrix equationsof an explicit elliptic–hyperbolic time domain algorithm called PFIT [Marklein et al. 1997b;Marklein, 1998]. Two versions of the PFIT–code have been developed, a voltage or charge

31

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! !! !! !! !! !! !! !! !! !! !! !! !! !! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

#$&% '(*),+ +.-/ 02143&5 )#$&% ' -6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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" "

""

""

""

Figure 39: PFIT material grid

M , grid G, and dual grid G

and the allocation of the dis-

crete piezoelectric field com-

ponents

driven and a current algorithm (U–PFIT and I–PFIT).For instance, the current driven version of the PFIT–code has in the nondissipative case to

following form

v(z− 1

2) = [ρp0]

−1[DIV][RTv ]−1[AT

v ]T(z− 1

2) + [ρp0]

−1f(z− 1

2)

v(z) = v(z−1) + ∆tv(z− 1

2)

[div][S][εS][R]−1[grad]Φ(z) = [div][S][e][RvΦ]−1[GRad][Av

Φ]v(z)

−[V%

Φ]%(z) − [Sη

Φ]η(z)

T(z) = [cE][RvT]−1[GRAD][Av

T]v(z)

+[eT][RΦT]−1[AΦ

T][grad]Φ(z) + g(z)

T(z+ 1

2) = T(z− 1

2) + ∆tT(z) .

In order to take into account an external electrical load the PFIT algorithm, the I–PFITalgorithm has been combined with a 1–D network algorithm.

Iterative algorithms like the symmetric successive overrelaxation (SSOR) method and theconjugate gradient (CG) method are implemented in the PFIT–code to solve Poisson’s gridequation. In the following examples a CG algorithm with diagonal scaling (DCG) is used.

PFIT allows the modeling of a piezoelectric transducer as a transformer of electrical quanti-ties, for example voltage and current, into elastodynamic quantities, for instance displacementand stress.

Details can be found in Marklein [1998].

32

5 Experimental Validation and Numerical Results of PFIT

5.1 Piezoelectric Pz27 Disk Transducer on a Brass Cylinder

Figure 40 shows a photograph of the sample and the 2–D geometry for the 2–D PFIT modeling.The sample consists of a disk of piezoelectric ceramics (Pz27) which is adhesively bonded ona brass cylinder. All materials are considered nondissipative in this example. A comparisonbetween results of a 1–D lattice model [Hayward & Jackson, 1986], the 1–D PFIT modeling,and the 2–D PFIT modeling against measurement is given in Figure 41.

Only the 2–D PFIT modeling result gives a comparable amplitude of the backwall echos(BEn) and the secondary echos [Krautkamer & Krautkamer, 1983; Langenberg et al., 1994], likethe first secondary echo (SE1) in Figure 41c.

2–D PFIT snapshots are shown in Figure 42 especially demonstrating the generation of thefirst and second backwall echo and the first secondary echo.

5.2 Piezoelectric Pz27 Disk Transducer on a Brass Cylinder with Backwall Break-

ing Notch

Another example, which is of considerable interest in ultrasonic NDT, is given in Figure 43.The sample has a backwall breaking notch. This problem can not be treated with a 1–D modellike the 1–D lattice model by Hayward & Jackson [1986]. A comparison between the modeledand experimental voltage observed at the Pz27 disk is given in Fig. 44, which validates thenumerical results. The dominant signals are the excitation pulse (EP), the notch echo (NE),and the 1st backwall echo (BE1). Because of the 2–D modeling the experimental backwallecho has a smaller amplitude. A sequence of 2–D PFIT v–snapshots is displayed in Fig. 45showing the excited ultrasonic waves and the generation of the notch echo (NE) and backwallecho (BE1).

Further analytical and experimental validations and examples of the PFIT–code can befound in Marklein et al. [1997b] and Marklein [1998].

33

a) b)

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................................................

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........................................................................................................................................................................................................................................................... ....................... ................................................................................................................................................................................................................................

XBr1 = 25

.........................

.........................

...............

...............

...............

...............

...............

...............

...............

...............

...............

...............

...............

....................................

.......................

................................................................................................................................................................................................................................XBr

3 = 25

..................................................

..................................................

..................................................

Adhesive Bond(Araldite)

BrassCylinder

..........................................................

...............

................

.........................

.......................

...............................................................................

XPz273 = d = 1

............................................. ....................... x1

....................................................................x3

Pz27 Disk

Figure 40: Pz27 disk on a brass cylinder: a) photograph and b) 2–D geometry for the 2–D PFIT

modeling

.0 10.0 20.0 30.0 40.0 50.0-3.0-1.5

.0

1.5

3.04.5

.0 10.0 20.0 30.0 40.0 50.0-3.0-1.5

.0

1.5

3.04.5

.0 10.0 20.0 30.0 40.0 50.0-3.0-1.5

.0

1.5

3.04.5

.0 10.0 20.0 30.0 40.0 50.0-3.0-1.5

.0

1.5

3.04.5

a) 1–D Model [Hayward & Jackson, 1986]

t in µs ⇒

upi(t)

inV

⇒

b) 1–D PFIT Modeling

t in µs ⇒

upi(t)

inV

⇒

c) 2–D PFIT Modeling

t in µs ⇒

upi(t)

inV

⇒ SE1.....................................................................

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d) Measurement

t in µs ⇒

upi(t)

inV

⇒ EP BE1 BE2 BE3 BE4SE1.....................................................................

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Figure 41: Pz27 disk on a brass cylinder: comparison between modeled and experimental voltage at

the Pz27 disk (A–scan) for Rg = 50 Ω and a sine pulse excitation with u0 = 10 V, n = 2 cycles, and

fc = 2 MHz. EP: excitation pulse; BEn: n–th backwall echo; SEn: n–th secondary echo

34

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@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @ @ @ @C@ @ @ @ @ @ @ @ @ @ @ @ @<@ @ @ @ @ @ @ @ @ @

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

D D D D D D D D D D D D DAD D D D D D D D D D D D D<D D D D D D D D D D D D DAD D D D D D D D D D D D DBDD D D D D D D D D D D DAD D D D D D D D D D D D DBDD D D D D D D D D D D DAD D D D D D D D D D D D DBDD D D D D D D D D D D DAD D D D D D D D D D D D DBD D D D D D D D D D D D DAD D D D D D D D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD




D D D D D D D D D D D D DAD D D D D D D D D D D D D<D D D D D D D D D D D D DAD D D D D D D D D D D D DBDD D D D D D D D D D D DAD D D D D D D D D D D D DBDD D D D D D D D D D D DAD D D D D D D D D D D D DBDD D D D D D D D D D D DAD D D D D D D D D D D D DBD D D D D D D D D D D D DAD D D D D D D D D D

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

D D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD




D D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D D D DCD D D D D D D D D D D D D<D D D D D D D D D D

BE1.....................................................................................................................

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SE1..............................................................................................................

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BE2..................................................................................................................................................

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.................

t = 100∆t = 555.55 ns

t = 300∆t = 1.66 µs

t = 500∆t = 2.77 µs

t = 700∆t = 3.88 µs

t = 900∆t = 4.99 µs

t = 1100∆t = 6.11 µs

t = 1300∆t = 7.22 µs

t = 1500∆t = 8.33 µs

t = 1700∆t = 9.44 µs

t = 1900∆t = 10.55 µs

t = 2100∆t = 11.66 µs

t = 2300∆t = 12.77 µs

t = 2500∆t = 13.88 µs

t = 2700∆t = 14.99 µs

t = 2900∆t = 16.11 µs

t = 3100∆t = 17.22 µs

t = 3300∆t = 18.33 µs

t = 3500∆t = 19.44 µs

t = 3700∆t = 20.55 µs

t = 3900∆t = 21.66 µs

Logarithmic Scaling:.......

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-51.2 dB -25.6 dB 0 dB

Figure 42: Pz27 disk on a brass cylinder: 2–D PFIT |v|–snapshots. BEn: n–th backwall echo; SEn:

n–th secondary echo

35

a) b)

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........................................................................................................................................................................................................................................................... ....................... ................................................................................................................................................................................................................................

XBr1 = 25

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................................................................................................................................................................................................................................XBr

3 = 25

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..................................................

Adhesive Bond(Araldite) .................................................... ....................... ...........................................................................

XNotch1 = 2...........................................................................................................

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............................................................................................................................................................................................................................................XNotch

3 = 10

BrassCylinder

..........................................................

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...............................................................................

XPz273 = d = 1

............................................. ....................... x1

....................................................................x3

Pz27 Disk

Figure 43: Pz27 disk on a brass cylinder with a backwall breaking notch: a) photograph and b) 2–D

geometry for the 2–D PFIT modeling

.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0-1.5-1.3-1.1

-.9-.7-.5-.3-.1.1.3.5.7.9

1.11.31.5

.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0-1.5-1.3-1.1

-.9-.7-.5-.3-.1.1.3.5.7.9

1.11.31.5

...........................................................................................................................

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Excitation Pulse (EP)Notch Echo (NE)................................................................................

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Backwall Echo (BE1)

a) 2–D PFIT Modeling

t in µs ⇒

upi(t)

inV

⇒

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Excitation Pulse (EP)Notch Echo (NE)................................................................................

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Backwall Echo (BE1)

b) Measurement

t in µs ⇒

upi(t)

inV

⇒

Figure 44: Pz27 disk on a brass cylinder with a backwall breaking notch: comparison between the

modeled (a) and experimental (b) voltage at the Pz27 disk (A–scan) for Rg = 50 Ω and a sine pulse

excitation with u0 = 10 V, n = 2 cycles, and fc = 2 MHz

36

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F F F F F F F F F F F F F<F F F F F F F F F F F F FCF F F F F F F F F F F F F<F F F F F F F F F F F F FCF F F F F F F F F F F F F<F F F F F F F F F F F F FCF F F F F F F F F F F F F<F F F F F F F F F F F F FCF F F F F F F F F F F F F<F F F F F F F F F F F F FCF F F F F F F F F F F F F<F F F F F F F F F Ft = 100∆t = 555.55 ns

t = 300∆t = 1.66 µs

t = 500∆t = 2.77 µs

t = 700∆t = 3.88 µs

t = 900∆t = 4.99 µs

t = 1100∆t = 6.11 µs

t = 1300∆t = 7.22 µs

t = 1500∆t = 8.33 µs

t = 1700∆t = 9.44 µs

t = 1900∆t = 10.55 µs

t = 2100∆t = 11.66 µs

t = 2300∆t = 12.77 µs

t = 2500∆t = 13.88 µs

t = 2700∆t = 14.99 µs

t = 2900∆t = 16.11 µs

t = 3100∆t = 17.22 µs

t = 3300∆t = 18.33 µs

t = 3500∆t = 19.44 µs

t = 3700∆t = 20.55 µs

t = 3900∆t = 21.66 µs

Logarithmic Scaling:.......

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-51.2 dB -25.6 dB 0 dB

Figure 45: Pz27 disk on a brass cylinder with a backwall breaking notch: 2-D PFIT |v|–snapshots

37

Concluding Remarks

In this paper an overview on the numerical modeling tools PFIT, EFIT, and AFIT has beengiven starting from the application of the Finite Integration Technique (FIT) to the governingequations of elastodynamics and acoustics, the derivation of the algorithms EFIT and AFIT,the validation of the codes with analytical and numerical references, and the application toultrasonic NDT real life situations. In the last section the recently developed PFIT algorithmhas been presented, which allows the computer simulation of a typical ultrasonic NDT problemincluding the piezoelectric element and an external electrical load. As an example, the validationof the PFIT–code with experimental data has been given. All presented results are documentingthe power of the established numerical tools and give trust for further industrial applications:

• to help to understand ultrasonic wave propagation in complex materials and complexgeometries,

• to optimize well–accepted NDT techniques, or

• to develop new and more efficient NDT techniques.

From a numerical point of view, some of the open problems, which are presently subject of thecontinuing work, are for example

• 3–D modeling of real life ultrasonic NDT situations,

• CAD/IGES interface for inhomogeneous anisotropic materials (e. g., austenitic V–buttweld)

• irregular and triangular (Voronoi grid, Delaunay tessellation), and

• angled piezoelectric excitation.

Acknowledgements

It is an honor for the author to dedicate this overview on AFIT, EFIT, and PFIT to ProfessorDr. H. Wustenberg on the occasion of his 60th birthday.

The author would like to express his gratitude to Professor Dr. K. J. Langenberg, whohas not only stimulated the development of EFIT, but his continuous interest was extremelymotivating to make the codes AFIT, EFIT, and PFIT what they are today: powerful modelingtools in quantitative ultrasonic NDT. Numerous former and present scientists of ProfessorLangenberg’s group at the University of Kassel have contributed to the evaluation of AFIT,EFIT, and PFIT as well as their application to ultrasonic imaging: Dr. P. Fellinger, S. Klaholz,Ch. Hofmann, T. Kaczorowski, Dr. K. Mayer, R. Barmann, J. Kostka, R. Hannemann, andP. Zanger. Their help in preparing the material for various examples is gratefully acknowledged.Mrs. R. Brylla has carefully read the paper.

Finally, the author would like to appreciate the support and the trustful collaborationwith the following scientists: O. Glitza, Measurement Techniques, University of Kassel, Kassel;Dr. V. Schmitz, F. Walte, W. Muller, and Dr. M. Spies, S. Schuhmacher, Fraunhofer–Institutfur zerstorungsfreie Prufverfahren (IzfP), Saarbrucken; Dr. C. Schurig and Dr. B. Kohler,Fraunhofer–Einrichtung fur Akustische Diagnostik und Qualitatssicherung (EADQ) Dresden;Dr. M. Krause and Dr. H. Wiggenhauser, Bundesanstalt fur Materialforschung und -prufung(BAM), Berlin; Professor Dr. R. Ludwig, Worcester Polytechnic Institute, Worcester, USA;

38

Professor Dr. T. Weiland, H. Wolter, and M. Bihn, Theorie elektromagnetischer Felder, Darm-stadt University of Technology, Darmstadt; Dr. A. Hecht, BASF, Ludwigshafen; H. Willems,Pipetronix, Stutensee; W. Biesle and W.–B. Klemmt, Daimler–Benz Aerospace Airbus, Bre-men; Dr. T. Schmeidl, Siemens–KWU, Erlangen.

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