Direct model optimisation for data inversion. Application to ultrasonic characterisation of heterogeneous welds

ARTICLE IN PRESS NDT&E International 42 (2009) 47–55

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Direct model optimisation for data inversion. Application to ultrasonic characterisation of heterogeneous welds Ce´cile Gueudre a,, Loı¨c Le Marrec b,1, Joseph Moysan a, Bertrand Chassignole c,2 a b c

LCND, Laboratoire de Caracte´risation Non Destructive, Universite´ de la Me´diterrane´e, 13625 Aix-en-Provence, France IRMAR, Institut de Recherche Mathe´matique de Rennes, UMR 6625 du CNRS, Campus de Beaulieu, 35042 Rennes, France `res, De ´partement MMC, Avenue des Renardie`res, 77818 Moret sur Loing, France EDF R&D, Les Renardie

a r t i c l e in fo

abstract

Article history: Received 2 October 2007 Received in revised form 30 May 2008 Accepted 10 July 2008 Available online 22 July 2008

Numerous multipass welds in austenitic stainless steel are made on the primary circuits of nuclear power stations. The heterogeneous anisotropic nature of these welds causes disturbance to ultrasonic propagation. Simulation is a useful tool when attempting to understand physical phenomena. With this objective, a finite element code called ATHENA was developed. Sufficiently realistic modelling of the material is proposed by the Modelling anIsotropy from Notebook of Arc welding (MINA) model, which determines the orientation of the weld grains. Optimisation by inversion of the MINA model is proposed in this study. The results validate the strategy and open perspectives regarding use on real data. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Ultrasonics Inverse problem Optimization Weldment Anisotropy

1. Introduction Numerous multipass welds in austenitic stainless steel are made on the primary circuits of nuclear power stations. Fig. 1 shows that the grain structure of these welds is highly complex and difficult to predict. An austenitic weld is both anisotropic (grain elongation parallel to the lines of heat dissipation and according to a privileged crystallographic direction) and heterogeneous (change of grain orientation in the welded volume). Ultrasonic nondestructive testing of this type of weld, to detect and characterise potential defects, is difficult to interpret due to the deviation and splitting of the ultrasonic beam caused by the structure. Use of an ultrasonic propagation simulation code is recommended to obtain a greater understanding of the propagation phenomena. This paper points up a precise material modelling as it is the key point before using a wave propagation code in such complex welds. The input data required for simulation is a sufficiently realistic description of the weld. Modelling of the material was proposed by the Modelling anIsotropy from Notebook of Arc welding (MINA) model developed at

 Corresponding author. Tel.: +33 442 939 034; fax: +33 442 939 084.

E-mail addresses: [email protected] (C. Gueudre), [email protected] (L. Le Marrec), [email protected] (J. Moysan), [email protected] (B. Chassignole). 1 Tel.: +33 223 23 66 83; fax: +33 223 23 67 90. 2 Tel.: +33 160 73 68 25; fax: +33 160 73 68 89. 0963-8695/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2008.07.003

LCND3 [1]. This model is currently applied to the welding method using coated electrode in flat position. It makes it possible to have an accurate prediction of grain orientation based on information recorded in the welding book, parameters specific to the welding method and rules derived from crystalline growth. Elasticity constants are then deduced from these orientations. Ultrasonic propagation modelling is carried out using a finite element code called ATHENA, developed by EDF4 and INRIA.5 This combination of two models to simulate ultrasonic inspection in multipass welds has been validated by comparisons with experimental measurements [2]. MINA is a robust model which can be used to perform parametric studies, in particular to evaluate the effect of grain structure dispersion on the inspection results. If the input parameters of the model are not exactly known, this creates differences between modelling and experimental measurements. Inversion is then a highly promising direction with the perspective of using ultrasonic data to determine the real structure of the weld. This is a key point when trying to interpret the results of an inspection, since it has been shown that the propagation phenomena could vary significantly from one structure to another. To obtain a model as close as possible to reality and demonstrate that model inversion is possible, optimisation of 3 4 5

Laboratoire de Caracte´risation Non Destructive. Electricite´ De France. Institut National de Recherche en Informatique et Automatique.

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c θB

a

θC

zoom

1 cm b Fig. 1. Cross-section macrograph of a multipass weld in austenitic stainless steel made using a method with coated electrode. In a zoom the pass inclination parameters.

MINA parameters is being considered. This optimisation concerns in particular the pass solidification parameters which, currently, correspond to mean values obtained further to macrographic analyses of welds. The aim of inversion is therefore to improve the description of the weld studied, by comparing experimental data produced from ultrasonic inspection of a weld with corresponding simulated data. It will therefore allow the MINA model to be applied in configurations where little information is available about the model parameters (no welding book, change of procedure or operator, etc.). In the first part of this paper, the fundamentals of the MINA model are presented, the function and the influence of each parameter are thus explained. The inverse methodology is then presented. The optimisation of the inversion process is introduced afterwards. We indicate how it enables the modification of the direct model to improve efficiency of the whole inversion strategy. Conclusion presents the large prospects opened up by this study.

2. Direct model The global direct model, using the combination of the two models MINA and ATHENA, is presented first. An overall algorithm is given at the end of the paragraph to synthesise the input and output data for the two models. 2.1. The MINA model A heterogeneous anisotropic weld is described by a set of homogeneous, anisotropic media. To solve the Christoffel equation in each medium, two parameters must be determined:

 The elasticity constants of the medium. Specific studies have 

been conducted on this aspect for austenitic stainless steel welds [3]. The orientation of the coordinate systems used to express the elasticity constants, which corresponds to the mean grain orientation in the defined domain.

The MINA model creates the grain orientation of a weld macroscopically. The welds investigated are made using a method with coated electrode in flat position (Fig. 1). There is a V-shaped chamfer and the material is 316L austenitic stainless steel. The welds consist of 20–40 passes. The welded zone is about 40 mm thick. The electrode diameter varies between 2.5 and 5 mm depending on the layers. The global structure of the weld is obtained by iteratively describing (layer by layer and pass by pass) the growth of the austenite grains. This model is based on the solidification mechanisms. And it uses two separate sources of information: the welding

book and parameters of influence extracted from welds macrographs affecting the direction of grain growth.

2.1.1. Solidification mechanisms The welds exhibit an anisotropic and heterogeneous columnar grain structure, with orthotropic crystallographic symmetry. The grain elongation directions follow those of the crystallographic axes /1 0 0S. The columnar grain structures observed on the macrographs of multipass welds result from the solidification of each pass successively deposited. The physical phenomena affecting grain growth are the local direction of the temperature gradient, the epitaxy and the competition between grains (selective growth). Many thermal studies allow the temperature gradient and isotherms modelling in a welding pass [4]. Our weld pool geometry model is given by a concave description (see Fig. 2), which is justified by macrographs that reveal the shape of passes and by the fact there is no Marangoni convection effect [5]. The temperature gradient is perpendicular to the isotherms and directed from the lowest temperatures (bottom of pass) to the highest temperatures (top of pass, close to the heat source, i.e. the electrode). The grains develop along the direction of the temperature gradient. The austenite grains grow in an epitaxial mode: a grain grows from the grain of the lower pass then tends to line up in the direction of the temperature gradient [6–9]. The phenomenon of selective growth implies that the columnar grains whose crystallographic direction /1 0 0S is close to the local direction of the temperature gradient grow faster than the neighbouring grains. The growth of the latter is therefore stopped [10,11].

2.1.2. The welding book The welding book indicates the chamfer geometry (height a, base width b and top width c, see Fig. 1), the number of layers deposited, the number of passes per layer, the diameter of the coated electrode depending on the layers and the sequencing order of the passes. This order is sometimes written but is not imposed: since welding is carried out manually, the welders are not obliged to respect a specific order. It may therefore vary considerably from one welder to another or from one weld to another.

2.1.3. Signatures of weld structure and geometry When the welder makes a pass, he naturally inclines the electrode when this pass is located against the edge of a chamfer or against a previously solidified lateral pass. The axis of symmetry of the pass is therefore inclined of an angle y. The macrographs show that the angle of inclination of a pass is larger when it has been deposited against the edge of a chamfer

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49

D

Isotherms

Direction of temperature gradient C

L

H' G

h H

Hk

O Lk Fig. 2. Description of the weld passes for the MINA model. Left: geometry of a pass in a layer k. Right: lateral and vertical remelting of a pass in multipass welding.

(zoom in Fig. 1). Two parameters have been defined to represent this phenomenon:

The coefficient b is then calculated as a function of the total height a of the weld, the electrode diameter Fk and the height H0 satisfying the relation (1) [12]:

 yB is the angle of inclination of the axis of symmetry of a pass 

to the vertical, when the pass is against the edge of a chamfer, yC is the angle of inclination of the axis of symmetry of a pass to the vertical, when the pass is against another pass.

These angles can be positive or negative according to the direction of inclination (respectively to the right or left) of the pass. In multipass welding, the welder waits until one pass has completely solidified and cooled before making the next pass beside it or on the next layer. When a new molten pass is deposited, the previous passes on which it has been deposited are partially remelted. Consequently, the grain orientations in this zone are modified according to the temperature gradient of the new pass. Two parameters have been defined to represent this phenomenon (Fig. 2 right):

 The lateral remelting rate, 0oRLo1, is the ratio between the 

width remelted ‘ with the next pass and the total width L of the pass deposited just beside: RL ¼ ‘/L. The vertical remelting rate, 0oRVo1, is the ratio between the height remelted h with the pass deposited just below and the total height H of the pass: RV ¼ h/H.

2.1.4. MINA modelling principle The MINA model uses parameters defined previously (welding book and signatures of weld structure and geometry). Experimentally, the lateral RL and vertical RV remelting rates differ for each pass. To reduce the number of parameters in MINA, the remelting parameters are considered to be constant for all passes. Passes have a fixed shape, with a contour consisting of two inverted parabolas. The top parabola has a fixed height H0 (Fig. 2 left). Passes in a given layer k are described by the same width Lk and the same height (Hk+H0 ) since they have been deposited with an electrode of identical diameter. A proportionality coefficient b is therefore defined between the height of a pass Hk and the electrode diameter Fk of layer k: Hk ¼ bFk. If NL is the total number of layers in the weld, and under the hypothesis that the weld must fill the height a of the chamfer: NL1 X

ðHk  hk Þ þ HNL þ H0 ¼ a

k¼1

(1)

b¼

a  H0 PNL ð1  RV Þð k¼1 Fk Þ þ RV FNL

(2)

This presents the main advantage of avoiding the introduction of a new parameter to evaluate each Hk. In a layer k having pk passes, the total width Wk of the layer is a function of the lateral remelting rate RL and the total width of the pass Lk: W k ¼ Lk ðð1  RL Þpk þ RL Þ Requiring that this width must fill the chamfer: 0 1 k1 c  b @X ðHj  hj Þ þ Hk A Wk ¼ b þ a j¼1

(3)

(4)

allows us to express the width of the pass Lk at each layer as a function of the remelting parameters and the parameters of the welding book [12]: P b þ ððc  bÞ=aÞ½bðð1  RV Þ ki¼1 Fi þ RV Fk Þ Lk ¼ (5) pk ð1  RL Þ þ RL Parameters RL and RV can therefore be used to define the position of the passes in the simulated weld but also to find the size (height and width) of each pass. Note that this way of calculation is possible because the remelting parameters are considered to be the same for all passes. In the MINA model, we defined two parameters yB and yC to simulate the inclined angle of the electrode. These angles are automatically determined in relation to the location of the pass written in the welding notebook. We made the hypothesis that this phenomenon can be described simply by a rotation of the direction of the temperature gradient without changing the geometric shape of the pass. There is no influence of inclination parameters on weld pass geometry. 2.2. The ATHENA model and the inspection configurations ATHENA is a finite element code that solves the elastodynamic equations, in transient regime, for a heterogeneous and anisotropic medium [13,14]. It allows the visualisation of the propagation of the ultrasonic field and the determination of the associated amplitudes. Interaction of the beam with a defect can also be simulated to calculate the response of a defect during ultrasonic

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x 10-10 1.5

ultrasonic amplitude

transmitter

deviation

1

0.5

splitting receiver

0 -30

-20

-10 10 20 Receiver position [mm]

30

Fig. 3. Ultrasonic wave propagation modelled by ATHENA (left) and echodynamic curve (right) for various MINA mesh sizes: dx ¼ 2 mm (dashed) and dx ¼ 0.25 mm (line).

nondestructive testing [15]. Fictitious domains method combines a regular mesh of the calculation domain with an irregular mesh to describe a defect of a complex shape (crack). In the current code version, the attenuation coefficient due to scattering of the ultrasonic waves is not taken into account. In our study, simulations are performed in transmission mode with a transmitter in fixed position on the outer wall of the weld and a receiver scanning the opposite surface to measure the ultrasonic field transmitted (Fig. 3). The transducer radiates longitudinal waves at normal incidence in an isotropic steel, at a central frequency of 2.25 MHz. The transducer diameter is 0.5 in. Fig. 3 shows a typical simulation where colours correspond to maximum values reached by particles velocity at each point of the mesh. We can perfectly visualise the ultrasonic beam splitting and the deviation phenomena because of the heterogeneous anisotropic structure. ATHENA does not allow for the simulation of a receiver in transmission but it does give an echodynamic curve (Fig. 3), which for a given transmitter position xi, represents the maximum ultrasonic amplitude received at the bottom of the weld by the receiver scanning the opposite surface with a step of 0.25 mm, and such that 30 mmoxjo30 mm (j ¼ 1,y,241). The echodynamic curve is stacked in a vector Ei ¼ (ei1,y, ei241). 2.3. MINA and ATHENA modelling scales MINA gives the mean grain orientation in square domains of side dx ¼ 2 mm (see Fig. 6). Considering the central frequency of the ultrasonic beam (2.25 MHz) and the speed of the compression wave in steel (about 5800 m/s), the wavelength in the weld is of the order of 3 mm. In practice, the choice of domain size depends on several observations.

 The domain must be small enough to illustrate the variations



in grain orientation. Since, to a first approximation, the ultrasonic beam is sensitive to spatial variations of the order of the wavelength, domains of side approximately equal to the wavelength must be used. From another point of view, the domain size must be greater than the grain size (the grain width is of the order of 0.1 mm) to retain a physical resemblance. Lastly, since the crystallographic description of the weld is introduced in the finite element propagation code ATHENA, the domain size must be greater than the size of the finite element

(which must not be greater than one-twelfth of the wavelength, i.e. 0.25 mm).

2.4. Algorithm of the global direct problem 1. The MINA model 1.1. Inputs: welding book parameters (chamfer geometry, number of layers, number of passes per layer, sequencing order of the passes, diameter of electrodes). 1.2. Inputs: remelting parameters (RL, RV) and inclination parameters (yB, yC). 1.3. Compute size and position of each pass and the scaling parameters b. 1.4. Compute the grain orientations in each pass according to the direction of the temperature gradient, the epitaxial and selective growth modes. 1.5. Output: discrete square domain where at each pixel of the weld and the chamfer corresponds a grain orientation. 2. The FEM code ATHENA 2.1. Inputs: the output from MINA (which allows the selection of the elasticity constants of the weld stacked in a data base). 2.2. Inputs: characteristics of the transmitter. 2.3. Compute the ultrasonic propagation. 2.4. Output: the echodynamic curve. 2.5. Repeat 2.3 for all transmitter positions.

3. Inversion 3.1. Introduction MINA was validated by comparing the orientations evaluated by the model with those measured on the macrograph. A mean error of 10–151 depending on the weld [1] was obtained. The MINA model therefore gives a very encouraging, but still imperfect, description of the crystalline orientation of the weld. This is due to the uncertainties concerning the values of the model’s input parameters. Parameters RL, RV, yB, yC of MINA can only be accessed using metallographic observations on a welding coupon. Mean parameters have been determined experimentally from macrographs of three welds (D717B, D717C, D717D), made by the same welder but with different pass sequencing orders (Table 1). For a given weld, the mean value of the parameter obtained on all passes and

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Table 1 Weld type D717 [1] mean parameters Weld type

Mean parameters and standard deviations

D717A D717C D717D D717: overall mean values

/RLS

sRL

/RVS

sRV

/yBS (deg)

syB (deg)

/yCS (deg)

syC (deg)

0.48 0.45 0.49 0.47

0.11 0.11 0.08

0.29 0.24 0.24 0.26

0.13 0.04 0.06

18 16 20 18

4 3 3

13 9 15 12

2 1 1

Table 2 Weld type P [1] mean parameters Weld type

P1 P2 P3 P: overall mean values

Mean parameters and standard deviations /RLS

sRL

/RVS

sRV

/yBS (deg)

syB (deg)

/yCS (deg)

syC (deg)

0.28 0.26 0.27 0.27

0.08 0.09 0.06

0.52 0.47 0.50 0.49

0.09 0.11 0.06

7 8 8 8

2 4 4

4 6 5 5

2 4 3

Yexp NDT Dissemblance criterion

Initialisationv pg

J(pg)

Direct model J≈0

ysim(pg) pg+1

Yes

No

Minimisation

Fig. 4. Schematic diagram of inversion.

the standard deviation are recorded. For the four parameters, the standard deviations are small. We can conclude that these parameters are homogeneous for a given weld. If we now compare the measurements taken on the three welds, we observe once again that the results are homogeneous (same welder). These parameters are therefore uncoupled from the sequencing order of the passes. In contrast, analysis of the macrographs of three other welds (P1, P2, P3) made by a second welder shows quite different mean parameters (Table 2). The values of parameters RL, RV, yB, yC therefore depend heavily on the welder (manual welding). For a given weld, if metallographic observations are available, mean parameters can be measured from macrographs, introduced in MINA and the description of the simulated grain structure can be affined by inversion. If no metallographic observations are available, the developed inversion method is a good way to obtain these parameters.

The comparison is made on a certain number of data from the measurements and the model. This is the parametric description. The dissemblance criterion used is the estimator in the least squares sense (where n is the number of measurements taken): JðpÞ ¼ Jðp1 ; . . . ; pm Þ ¼

n X ðysim ðp1 ; . . . ; pm Þ  Y exp Þ2 i i

(6)

i¼1

This cost function J minimisation step involves optimisation theories. The main difficulty is that we are faced with an ill-posed problem, there being no guarantee as to the existence of the solution, its uniqueness or its regularity. By studying the sensitivity of the parameters, we can determine the feasibility and the conditions required to invert the system, the fact that the ill-posed problem often being due to an unsuitable choice of parameters [19]. The more the cost function is sensitive to a parameter, in fact, the greater the chance of identifying this parameter.

3.2. Inversion procedure 3.3. Cost functions The inverse problems are now frequently encountered in the field of ultrasonic nondestructive testing [16–18]. The principle consists in comparing data obtained using experimental measurements Yexp with those obtained using a mathematical model ysim known as direct model (Fig. 4). These results depend on the parameters assigned to the model (parameters of interest p), we will therefore label the simulated results ysim (p).

The parameters of interest chosen in our study are the remelting and pass inclination parameters p ¼ (RL, RV, yB, yC). All the other parameters are considered fixed and exact. The parametric description corresponds to the full echodynamic curve at the bottom of the weld (Fig. 3). In the interest of sensitivity, five transmitter positions are used so that the ultrasonic beam crosses

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most of the weld. The cost function J therefore takes the form: 0 sim 12 exp 5 X 241 ei ðR ; R ; y ; y Þ  ei X L V B C j j A @ JðRL ; RV ; yB ; yC Þ ¼ exp eij i¼1 j¼1 sim

sim

(7)

where Ei ¼ ðei1 ; :::; ei241 Þ are the simulated echodynamic curves exp exp and Ei ¼ ðei1 ; :::; ei241 Þ the experimental ones. ATHENA does not take the attenuation of the ultrasonic waves into account. As a consequence, the amplitudes of the echodynamic curves simulated by ATHENA and those of the experimental curves can vary significantly. Inversion of experimental measures is not possible at this time. For that reason we chose to ‘‘fabricate’’ experimental data, by simulation using MINA and ATHENA. Using the same direct problem to simulate measure and estimation is commonly known as inverse crime. The main advantage is that the inversion can be done without the errors, bias and imprecision inherent in measurements. In the case of inverse crime, all the parameters and particularly the target parameters are in fact completely known. Note that if the parameters of interest are set to the exact value, the cost function is exactly null (no bias) which guarantees the existence of a solution. Moreover, the sensitivity of the cost function, the presence of local minima and the coupling of parameters of interest can be studied in the case of inverse crime. This process is a useful first step before inversion with real data to demonstrate that a model is sufficiently well established to be inversible. In our case, the experimental results are fabricated with the mean parameters (RL ¼ 0.47, RV ¼ 0.26, yB ¼ 181, yC ¼ 121) which have been determined experimentally from macrographs (see Table 1). Fig. 5 shows the shape of the cost functions according to the parameters of interest (only central position of the transmitter). In our case of inverse crime: the cost functions are null for the exact value of the unknowns (the white lines). We also observe numerous local minima for the remelting parameters RL and RV. We showed the influence of these parameters on weld pass geometry (see Section 2.1.4). A small variation of RL (or RV) can induce the allocation of pixels from one pass to another, causing a significant and nonlinear variation in the grain orientation in a 2  2 mm2 area in the weld. This results in significant modification of the echodynamic curves and therefore discontinuities in the cost functions. The function is much more regular, however, for the pass inclination parameters yB and yC (no influence on weld pass geometry, see Section 2.1.4).

3.4. Inversion algorithm The shape of the cost functions has resulted in the use of the global method as the genetic algorithms. These optimisation algorithms are derived from the genetics of natural evolution [20]. They are developed to find the global minimum of a function J(p) of several variables. An initial population of N individuals is first randomly generated. Each individual corresponds to a set of parameters p1 ; . . . ; pm representing a possible solution of the problem posed. At each iteration g, called generation, we create a new population which must be non-homogeneous to apply to the entire search domain, with the same number of individuals. The selection, crossover, mutation and replacement operators are therefore used. From one generation to the next, these individuals become the best ‘‘adapted’’ to their environment, finally tending towards the optimum solution. In our study, we use the genetic algorithm GA-Toolbox from MATLABs. The main algorithm parameters are therefore the population size (N), the number of bits used to code parameters (Preci), the crossover rate (0oPcCroso1) and the replacement rate (0oGgapo1). The tournament selection and the double point crossover are used. The choice of such parameters determines the efficiency of the algorithm. The search domain must be well-defined. These limits have been determined for each parameter experimentally from macrographs (see Table 1). In the case of D717 welds, the following boundaries have been chosen: RLA[0.35; 0.6], RVA[0.15; 0.4], yBA[8.51; 211], yCA[51; 17.51]. According to these boundaries and for Preci ¼ 5, RL and RV are described within (0.6–0.35)/25 ¼ (0.4–0.15)/25E0.01, and yB and yC within (21–8.5)/25 ¼ (17.5–5)/ 25E0.51. This gives a realistic precision of the parameters. An in-depth study was conducted to set the algorithm parameters N, PcCros and Ggap. We briefly present the process. Four analytic functions G(RL), G(RV), G(yB) and G(yC) are defined in order to mimic—the same shape and same number of local minima—the cost function for only one parameter (the other being fixed at the exact value). Using analytic functions reduces the time required to fit the parameters comparatively to the prohibitive computation times inherent in the FEM code. The algorithm parameters are chosen by the systematic comparison of minimisation efficiency with the analytical cost function: we observed that a set (N ¼ 16, PcCros ¼ 0.7, Ggap ¼ 0.8) was a good compromise to obtain a global minimum with a reduced number of generations. The process is then applied in the real case

Fig. 5. Cost functions in grey scale according to two parameters of interest (dx ¼ 2 mm). Left: J(RV, RL), Right: J(yC, yB) (other parameters are fixed at their exact values).

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(MINA and ATHENA codes) where the four unknowns are estimated simultaneously. A convergence criterion is also used: the algorithm is stopped when the minimum of the cost functions evaluated over 9 generations is unchanged. The inversion solution was found using the algorithm before 20 generations for a maximum number of generations set at 300. According to a convergence criterion and adapted algorithm parameters, the resolution time is 7 h on a Pentium IV at 2.4 GHz (compared to 44 h before algorithm optimisation).

4. Optimisation of the effective computation cost by modification of the direct model The computation times required to obtain results after the inversion tests nevertheless remain too high in view of the industrial constraints. The problem is the number of local minima of the cost function, which seriously slows down the search for the global minimum. Two ways have been found to reduce the discontinuities of the cost function: the modelling scale and the pass geometry. 4.1. MINA modelling scale Fig. 6 shows the crystallographic description of the weld for two MINA mesh sizes: dx ¼ 2 mm and dx ¼ 0.25 mm. We observe that the grain orientations change more regularly for dx ¼ 0.25

53

mm. For dx ¼ 2 mm, the transitions are more abrupt and the description of the structure is less precise. The consequences on the direct model are only minor, since the overall shape of the echodynamic curve is preserved (see Fig. 3). This is mainly due to the size of the wave length which is greater than the spatial discretisation in both cases. In contrast, in an inversion process, the consequences are considerable. Fig. 7 shows the same cost functions as in Fig. 5 but the MINA mesh size is dx ¼ 0.25 mm. Any modification of parameters RL and RV induces modifications in the position and size of the passes (see Section 2.1.4). Consequently, comparing data derived from measurements (or fabricated) with data derived from the direct model generates skips in the cost function. These discontinuities are less pronounced if a finer mesh is used. As a result, the number of local minima is reduced and the cost functions are more regular. Note that for estimation of yB and yC, the cost functions are almost unchanged.

4.2. Pass geometry Largest differences between orientations evaluated by MINA and those measured on the macrograph are located at the weld chamfers [1]. This is mainly due to the parabolic shape of the last pass which must fill the remaining volume. For D717 weld, with RL ¼ 0.47 and RV ¼ 0.26 (Fig. 8 left), some pixels have not been allocated (chamfers and top of weld): the current version of MINA allocates now these pixels to base metal. Taking an extreme case

Fig. 6. Grain orientations in grey scale in the weld modelled by MINA (RL ¼ 0.47, RV ¼ 0.26, yB ¼ 181, yC ¼ 121) with mesh sizes dx ¼ 2 mm (left) and dx ¼ 0.25 mm (right).

Fig. 7. Cost functions in grey scale according to two parameters of interest (dx ¼ 0.25 mm). Left: J(RV, RL), Right: J(yC, yB) (other parameters are fixed at their exact values).

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Fig. 8. Non-allocated pixels (N.a.p. in black) for D717 weld with exact remelting parameters: RL ¼ 0.47, RV ¼ 0.26 (left) and extremes values: RL ¼ 0.1, RV ¼ 0.1 (right).

Fig. 9. Evolution generation after generation of the estimated parameters RL, RV, yB, yC (left) and the cost function (right).

where RL ¼ 0.1 and RV ¼ 0.1 (Fig. 8 right), the phenomenon is amplified since some pixels in the centre of the weld have not been allocated (analogy with air bubbles). As regards inversion, it is important to place correct limits on the search domain, since these unallocated pixels necessarily generate discontinuities in the ultrasonic beam and therefore in the cost functions. Then the initially unallocated pixels are allocated by taking into account an orientation gradient on neighbouring passes. This process has little impact in our case (restricted search domain for RL and RV), but can be useful for largest search domain, for example if few initial data are available.

4.3. Results in terms of effective computation cost Modifications previously presented reduce the number of local minima on the cost functions. In terms of inversion, tests have been conducted with the genetic algorithm for the four unknowns (RL, RV, yB, yC) simultaneously. Fig. 9 shows an example of the evolution, generation after generation, of the estimated parameters and the cost function is also presented. The inversion solution is found in generally 16 generations, for a computation time of 4 h on a Pentium IV at 2.4 GHz, which is extremely satisfactory.

The estimated parameters RL ¼ 0.471, RV ¼ 0.263, yB ¼ 18.181 and yC ¼ 12.251 are independent from the initialisation and are very close to the solution, according to Preci ¼ 5.

5. Conclusion Two models MINA and ATHENA are combined to describe the propagation of ultrasound waves in multipass welds. We have demonstrated that inversion of this direct problem is now possible for the four main MINA parameters. Even if inversion is done in the framework of inverse crime, this represents a major breakthrough for ultrasonic inspection of this type of weld since interpretation of the results is known to be very difficult due to the complexity of the material. The choice of an adapted minimisation procedure is essential to demonstrate that inversion can be carried out successfully. The second condition is that the two models constituting the global direct problem must be fully operational and correctly coupled. This modelling procedure is designed to help interpret the results of ultrasonic examinations. Of course, extreme configurations where little information is available about the model parameters (false or unknown order of the passes, no welding book, etc.) are still not taken into account. Nevertheless, as the

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