WO2007107504A1

WO2007107504A1 - Method of determining the shape of a weld pool appearing during implementation of a welding process

Info

Publication number: WO2007107504A1
Application number: PCT/EP2007/052488
Authority: WO
Inventors: Duc Dung Doan; Franck Gabriel
Original assignee: Commissariat A L'energie Atomique
Priority date: 2006-03-17
Filing date: 2007-03-16
Publication date: 2007-09-27
Also published as: EP1996367A1; FR2898530A1; FR2898530B1

Abstract

Method of determining the shape of a weld pool. According to the method: a) a welding process is selected; b) a two-dimensional or three-dimensional virtual pool shape is selected; c) the shape of the virtual weld pool is modelled by a mathematical function that contains parameters to be identified; d) the temperature field in a virtual solid surrounding the weld pool is calculated by imposing boundary conditions; e) an actual example of the first and second parts (2, 4) is manufactured; f) they are welded by means of the selected process; g) the temperature is measured at M points in the actual solid zone; h) the temperatures measured in step g) are compared with the temperatures calculated, at the same points, in step d), by calculating the function (I) in two dimensions or the function (II) in three dimensions; i) S(Γ) is compared with a predefined value; j) if S(Γ) is greater than the predefined value, another virtual pool shape is selected and steps c) to h) are repeated until S(Γ) is smaller than the predefined value; and k) if or when S(Γ) is smaller than the predefined value, the shape of the virtual weld pool that has allowed S(Γ) to be calculated is accepted as actual value and the temperature field calculated from this shape is accepted as the actual temperature field.

Description

METHOD FOR DETERMINING THE SHAPE OF A FUSION BATH APPEARING DURING THE IMPLEMENTATION OF A

WELDING PROCESS

DESCRIPTION

TECHNICAL AREA

The invention relates to a method for determining the shape of a liquid metal melt occurring during the implementation of a method of welding a first part and a second part.

To improve productivity and reduce costs, automotive, aerospace and other industries. are moving more and more towards simulation tools. This is particularly the case with regard to welding operations.

One of the major industrial challenges of welding assembly concerns the prediction of the mechanical effects of the process (residual stresses, distortions, fatigue resistance, etc.). These effects are directly dependent on the temperature changes imposed by the welding process i.e. the thermal loading of the process.

To gain access to this thermal loading of the process, two methods are commonly used: the direct method and the so-called equivalent source method. The direct method is to attempt to solve the problem of heat transfer in the two parts of welded structure, the melted part and the solid part. To do this, we must resolve the Navier equations Stored in the liquid, the energy conservation equations taking into account the interaction between the energy arc and the part, the interaction between the liquid part and the solid part, and the equations of the transfer of heat in the solid.

The set of equations above has never been solved in its entirety. Indeed, the complexity of the problem is such that only a part of this problem has found solutions for particular effects, for example the effect of surface tension, the Marangoni effect.

The equivalent source method consists of giving a priori a form of the energy source and then solving a problem of nonlinear conduction of heat in the parts to be assembled. The parameters of this source of energy are identified by an inverse problem from the temperature measured in the solid and the macrography of the welding bath (cf ref [2] or ref [3]).

This method provides access to the thermal loading of the welding process but:

• The source model is specific to each welding process. • We impose a solution that is not necessarily representative of the physics of the processes implemented.

• Simulation of temperatures in the vicinity of the melt is difficult to achieve. The present invention relates to a method for determining the shape of a melt which overcomes these disadvantages.

This process is characterized by the following steps: a) a welding method is chosen; b) a two-dimensional or three-dimensional virtual bath shape is chosen according to the welding method chosen in step (a); c) model the shape of the virtual melt by a mathematical function that contains parameters to identify; d) calculating the temperature field in a virtual solid surrounding the melt by imposing boundary conditions; e) making a real copy of the first and second pieces; f) they are welded using the process chosen in step (a), which reveals a real melt and solid zone; g) measuring the temperature in M points of the real solid zone, M being at least equal to the number of parameters to be identified; h) comparing the temperatures measured in step (g) with the calculated temperatures at the same points in step (d) by calculating the function

Λ M ₂

S (F) = -V \ T (x _m , y _m ; F) -Y ^m \ in two dimensions

Mf

or the function in

three dimensions ; i) comparing ¹ ^ (T) to a predefined value; j) if 5 ^* (r) ^is greater than the predefined value, another form of virtual bath is chosen and steps (c) to (h) are repeated until "S ^* (r) is less than the predefined value; k) if or when S (T) is less than the predefined value, we accept the shape of the virtual melt which made it possible to calculate S (T) as real value and we accept the temperature field calculated from this form like the actual temperature field.

Thanks to these characteristics, we ignore the complex thermal phenomena that occur in the liquid, which simplifies the problem to be solved very significantly by reducing many of the parameters to be determined. In addition, by accessing the temperature field and S (F), the amount of energy passing through the melting front is also accessed.

Advantageously, but not necessarily, the mathematical function that contains parameters to be identified is a Bezier curve or surface.

In the case of a two-dimensional fusion front, the mathematical function that contains parameters to identify is a Bézier curve defined by the equation:

where P (t) passes through all the points of the curve of Bezier, where P ₁ is the control point, n the degree of the Bézier curve, N = n + 1 the number of points of n control P ₁ , te [0,1] and B (t) are polynomials of i

Bernstein.

In the case of a three-dimensional fusion front, the mathematical function that contains parameters to be identified is a Bézier surface defined by the equation: ^mn TYl Tl

P (u, v) = ΣΣB. (u) B. (v) P, with u e [0, l], v e [0,1]

I = O j = O ^l J

B ^m (u) = - - u (1 - u) "ii \ (m - i) \

B ^H (v) = v ^J (1 - v) "- ^}

J JKn - J) I where P (u, v) passes through all points of the Bézier surface, P ₁₃ is the control point, mxn is the degree of the Bézier surface there is (m + l) x (n + 1) control points including (m + 1) points in the direction "i" and (n + 1) points in the direction "j"; mn B (u), B (y) are Bernstein polynomials. Advantageously, in step (j), the Levenberg-Marquardt algorithm is used to choose another form of virtual bath.

In the case of a two-dimensional bath, in step (d), the temperature field is calculated in a virtual solid surrounding the melt by solving the following equation system: dT _s (x, y) r -, hp

P _s c _s u ^ = V _x [Z ₃ VT ₁ (x, y)] - γ (T _s - T _ιmp) (x, y): Ω: D

- - = 0 in y = L; - - = 0 in x = - L ₁ and - - = 0 in y = 0 (2) dy dx dy

T = T _imp in x = + L ₂ ; T (x, y) = T _f (x, y) eF (3)

with: λ _s Conductivity of the solid, T _s Temperature in the Wm ^-1 K ^"1 solid, ⁰ C p _s Density of the solid, T _e Outside temperature, kg m ^{" 3 0} C c Thermal capacity T _{imp Required} temperature, mass , J kg ^"1 K ^{" 1} ° ç u Speed of the medium of T _f Melting temperature, welding, ms ^"1 ° C

F Liquid-Solid Interface Ω _s Solid domain h Coefficient of exchange _α emissivity

In the case of a three-dimensional bath, in step (d), the temperature field is calculated in a virtual solid surrounding the melt by solving the following equation system:

Pfi ju ^dTΛX _dχ ' ^y ' ^Z) - V [AV T _s (x, y, z)] (x, y, z) eΩ _s (1)

- - = 0 in y = L _Λ> ; - - = 0 in x = -L, and - - = 0 in y = 0 ιo \ dy ^y dx dy dT (x, v, z _n ) -λ _s - g ^ + hT (x, y, z _Q ) = hT _e enz = z _Q = 0 (3) dT (x, y, z)

- + hT {x, y, z * _e ep _D )> = hT _e in z = z " _e ep _D = e: 4) dz

T = T _imp in x = + L ₂ ; T (x, y, z) = T _f [x, y, z) eF: 5)

Most often the welding process is chosen from the group comprising TIG, MIG / MAG, electron beam, plasma, laser, hybrid welding processes.

For most purposes, the degree of the Bézier curve is equal to 3 or 4 in 2D and the degree of the Bézier surface is between 2x3 and 4x3 in 3D.

Other features and advantages of the invention will become apparent upon reading the following description of embodiments with reference to the appended figures. In these figures: Figure 1 illustrates a Bezier curve with four control points; Figure 2 is a macrography of a non-debonding melt; FIG. 3 is a view of a curve of

Bezier of degree 3 corresponding to the macrography of FIG. 2; Figure 4 is a macrography of a melting bath opening; FIG. 5 is a view of a curve of

Bezier of degree 3 corresponding to the macrography of FIG. 4; Fig. 6 is a diagram illustrating an example of a two-dimensional melt; Figures 7 to 9 are three Bézier curves of degree 4 (of order 5); Fig. 10 is a diagram of the welding process; FIG. 11 is a diagram of the spatial domain in 2D case; Figure 12 is a diagram of the spatial domain in 3D case; Figure 13 illustrates the Bezier surface of degree 3x2 with its control points; Figure 14 illustrates the temperature field calculated in 3D case.

Different welding processes (MI G, MA G, Laser, electron beam, hybrid ...) give different forms of molten bath. In order to have access to very free forms with the minimum of parameters in accordance with the invention, the shape of the bath is parameterized with a Bézier curve of degree n (or a Bézier surface of degree m × n). The classical definition of Bézier curves and surfaces is based on the family of Bernstein polynomials. These polynomials were used by Bernstein for the polynomial approximation of functions. They are used in the description of the Bézier model by the definition points named also Bézier points or "points of control". The essential properties of these polynomials come from their intervention within the binomial formula when developing (t + (lt)) ⁿ with n a nonzero natural integer. To have the basic functions of Bernstein, we write the number 1 in the form: l = (t + (lt)) ⁿ .

Thus, in the case 2D: The value of n defines the degree of the Bernstein polynomial and also the number of control points (N = n + 1), therefore, the type of the Bezier curve.

For example, for n = 3 we have:

I = I ³ = (t + (1-t)) ³ = (lt) ³ + ³ (lt) ² t + 3 (1-t) t ² + t ³ therefore, the basic functions are:

B0 (t) = (lt) ³

Bl (t) = 3 (lt) ² t

B2 (t) = 3 (1-t) t ²

B3 (t) = t ³ and of course we have

BO (t) + B1 (t) + B2 (t) + B3 (t) = l, Vt

To build a curve, you then need 4 control points. In the plane (x, y) for example, describes the four control points P ₀ (x 0, y), P (xl, yl), _P2 (x2, y2), _P3 (x3, y3) ( see Figure 1).

The Bezier curve with four control points is then formulated as follows:

P (t) = B0 (t) P ₀ + B (t) Pi + B2 (t) P ₂ + B3 (t) P ₃ , te [0, 1] or in vector form

(xOBO (t) + xlBl (t) + x2B2 (t) + x3xB3 (t), ylB1 (t) + y2B2 (t) + y3B3 (t))

This curve passes through the points Po (x0, y0) and P3 (x3, y3). Its tangent to these points is the line connecting Po (xO, yO) to Pi (xl, yl) and the connecting line

P ₃ (x ₃ , y 3) to P ₂ (x ₂ , y ₂ ). The shape of the curve depends on the position of Pi (xl, yl) and P ₂ (x2, y2). We can generally formulate the equation of the Bezier curve of degree n (order n + 1) in the following way: n

P (t) = ΣB ⁿ (t) P _ι

not

B (t) are Bernstein polynomials. i

Several welding studies give us possible forms of melting zone, therefore, of fusion front of several welding processes. This forehead is characterized by certain shapes: parabola, conic, triangle or more complicated shapes like bulge or nail head. By using the Bézier curve of degree 1 to 4, most of these forms can be interpreted. We will now describe the Bezier curves of degree 3 and 4, whose application is very wide.

The Bezier curve of degree 3 (n = 3) is defined by the following equations: P (t) = (lt) ³ P ₀ + ³ (lt) ² tPi + 3 (1-t) t ² P ₂ + t ³ P ₃ O≤t≤l P '(t) = - 3 (lt) ² P ₀ + 3 (1-t) (1 -3t) Pi + 3t (2-3t) P ₃ + 3t ² P ₃

By studying the possible shapes of the Bezier curve of degree three (Figure 1), we show that with four control points we can interpret most of the forms of fusion front. By way of illustration, FIG. 2 shows a macrography of the fusion front of a non-emergent weld and in FIG. 3 the parametrization of this fusion front by a Bezier curve of degree 3. FIG. 4 shows the macrography of the melting front of an open weld and in FIG. 5 the parametrization of this fusion front by another Bezier curve of degree 3. In the case of FIG. 3, as in FIG. In the case of Figure 5, only half of the fusion front has been shown. We thus see that with Bezier curve shapes of degree 3 we will be able to interpret most of the possible forms of the fusion front. Note that for the four control points, the two points P ₀ and P ₃ are imposed by the macrography because they are visible directly on the macrography and it is clear that they remain the same throughout the process ( see Figure 6). The question is therefore to determine Pi and P ₂ such that the shape of the macrography is found.

With the Bézier curves of degree 3 (of order 4) one can parameterize:

• Most forms of fusion front of the electron beam process except the case of bulge.

• Simple shapes of the fusion front in case of laser welding.

• All forms of fusion front in the case of open and non-through TIG welding and in the case of MIG / MAG welding.

The Bezier curve of degree 4 (n = 4) is defined by the following equation:

P (t) = (lt) ⁴ P ₀ + 4 (lt) ³ tPi + 6 (lt) ² t ² P ₂ + 4 (1- t) t ³ P ₃ + t ⁴ P ₄ O ≤ t ≤l Aside from the hybrid welding process that creates more complex melting zones, most of the edge shapes created during welding can be set. Obviously, it is then more difficult to parameterize the fronts because here we have a curve of order 5 having 5 control points of which we know generally only two in advance. In addition to all forms accessible to Bezier curves of order 1 to 4, FIGS. 7 to 9 show Bezier curves of order 5.

With the Bézier curve of degree 4 (of order 5) one can parameterize:

• All simplified forms of the fusion front obtained by welding (electron beams) and laser while obscuring the effects of instability.

• Simple forms of fusion front in the case of hybrid welding.

In the case of a three-dimensional fusion bath:

The definition of a parametric surface is the extension of the parametric curve in which we have two parameters u and v which each vary in [0,1].

To have the basic functions of Bernstein, we write the number 1 in the form: l = (u + (1-u)) ^m and l = (v + (1-v)) ⁿ . The value mxn defines the degree of the Bézier surface. Generally, the Bézier surface is formulated as follows: ^mn mn

P (u, v) = ΣΣB \ u) B Xv) P _y with [0, \], ve [0, \] ι = 0 j = 0

m

B. (") = - - ^K1 O- ^K ) 'i ι \ (mi) \

n "ι

B (v) = v ^J (lv) ^{n ~ J}

J JKn-J) I

Where P (u, v) passes through all points on the Bézier surface. P ₁₃ is the control point, mxn is the degree of the Bézier surface, there are (m + l) x (n + l) control points including (m + 1) points in the direction

"I" and (n + 1) points in the direction "j"; m n

B (u), B (v) are Bernstein polynomials. ¹ J

We have general properties for surfaces of degree mxn (order (m + 1) x (n + 1)):

• The surface passes through the end points P _o o, Pon, P _m o, Pmn-

• The surface P (u, v) is continuous and has continuous derivatives at all orders. "Each control point exerts an attraction on the portion of the surface proportional to the bi-varied coefficients of the Bernstein polynomial.

• The surface is continuous in the convex envelope of polygon P ₀₀ , P _0n , P _m0 , P _mn . "The control of the surface is global. • The Bezier surface is independent of the axis system to which it is attached.

With Bezier surfaces of degree 3x3:

Piu, v) fΣ ^{3 |} B _R " ^m V (, u Λ) B _R " ^π ω (y) P, _igj

I = Oj = O ' ^J

= P ₀₀ (lu) ³ (lv) ³ + P ₀₁ (lu) ³ 3v (lv) ² + P ₀₂ (lu) ³ 3v ² (lv) + P ₀₃ (lu) V + P ₁₀ 3u (1-u) ) ² (lv) ³ + P _u 3u (lu) ² 3v (lv) ² + P ₁₂ 3u (lu) ² 3v ² (1-v) + P ₁₃ 3u (1-u) ² v ³ + P ₂₀ 3u ² (1-u) (lv) ³ + P ₂₁ 3u ² (1-u) 3v (1v) ² + P ₂₂ 3u ² (1-u) 3v ² (lv) + P ₂₃ 3u ² (1-u) v ³ + P ₃₀ U ³ (1-v) ³ + P ₃ iU ³ 3v (lv) ² + P ₃₂ u ³ 3v ² (1-v) + P ₃₃ U ³ V ³ or even better of degree 4x3:

= P ₀₀ (1-u) ⁴ (1-v) ³ + P ₀₁ (1-u) ⁴ 3v (1-v) ² + P ₀₂ (1-u) ⁴ 3v ² (1-v) + P ₀₃ (1-u) V + P ₁₀ 4u (1-u) ³ (1v) ³ + P ₁₁ 4u (1-u) ³ 3v (1v) ² + P ₁₂ 4u (1-u) ³ 3v ² (1-u) v) + P ₁₃ 4u (1-u) ³ V ³ + P ₂₀ 6u ² (1-u) ² (1-v) ³ + P ₂₁ 6u ² (1-u) ² 3v (1-v) ² + P ₂₂ 6u ² (1-u) ² 3v ² (1-v) + P ₂₃ 6u ² (lu) ² v ³ + P ₃₀ 4u ³ (1-u) (lv) ³ + P ₃₁ 4u ³ (1-u) u) 3v (lv) ² + P ₀₂ (1-u) ² + P ₃₂ 4u ³ (1-u) 3v ² (lv) + P ₃₃ 4u ³ (1- u) v ³

+ P ₄₀ U ⁴ (1-v) ³ + P ₄₁ u ⁴ 3v (1-v) ² + P ₄₂ u ⁴ 3v ² (1-v) + P ₄₃ U ⁴ V ³

Most melts can be configured in three dimensions.

How to modify the shape of the fusion front F.

1. The fusion front F is interpreted by the coordinates of the control points. We can write F (P ₀ , Pi ..., PN) where N is the number of parameters to identify. We use M measuring points to identify N parameters, then M≥N. When F (P ₀ , Pi ..., P _N ) is known, the temperature field in the solid T (F) can be obtained by solving the heat transfer problem. It also means that the temperature field changes when we modify the shape of F (P ₀ , Pi, ..., P _N ). We now introduce the term coefficient of sensitivity ie the derivative of the temperature in relation to the parameters to be identified

The sensitivity matrix is defined as follows:

2. A test is run and the temperature is measured at well-defined measurement points Y ^m , m = 1, ..., M where M is the number of measuring points (Figure 11 or Figure 12 for 3D)).

3. To start the calculation, we give ourselves initial parameters that is to say T ₀ (P ™, P " ¹ , ..., P ™ ¹ ), with F ₀ we solve the problem of heat transfer in the solid to obtain the temperature field T (F ₀ ) The temperature value is then extracted at measurement points.

4. The function is calculated

or the function

In 3D.

5. If S (Fo)> ε we modify the shape of the melting front T ₁ = T ₀ + [(J (T ₀ ) YJ (r _o ) \\ j (r _o ) f [YT (T ₀ )] (Method of

Gauss)

We can also use the method of Levenberg Marquardt

T ₁ = T _{0 +} [(J (T ₀₎ YJ (T ₀₎ + _{(P 0} Q ₀ Y (J (T ₀₎ Y [YT (T _0)] where μo is a positive scalar appointed damping parameter and Ω _o is a diagonal matrix The objective of the term μoΩ _o is to dampen the oscillations and instabilities due to the misplaced nature of the problem.The damping parameter μo is chosen large at the beginning of the iterative procedure (the method is then close to the method of the steepest slope) then, is reduced when one approaches the solution (the method joins the method of Gauss).

6. Loop the calculation until convergence.

EXAMPLES

Example 1: two-dimensional fusion front. It is desired to weld two plates 2.4 end to end. Each of the plates has a width L _y , a length 2L _x and a thickness e. A torch 6 generates an electric arc having sufficient intensity to melt the metal and form a melt 8. It travels with a constant speed u along the x-axis on the edge of the two plates 2 and 4. The bath 8 melting zone is moved at the same speed as the torch 6. A melting zone 10 is formed behind the displacement of the melt 8. The objective is to find by the inverse method the shape of the bath and the temperature field in the solid region with the minimum of parameters to identify.

The assumptions made to define this problem are as follows:

• At torch speed u constant, the transfers are non-stationary in a reference linked to the plate. However, by using a reference linked to the torch, a quasi-steady state is obtained: the shape of the molten bath remains constant in this frame while the material enters and leaves the field of study. The mobile repository of the torch is therefore used for the analysis of the inverse problem.

• The temperature variations in the Oz direction are assumed to be negligible compared to those in the plane (Ox, Oy), and the transfer problem is thus considered two-dimensional. • The shape of the liquid-solid interface F is parameterized by a Bézier curve.

• The position of the measuring points in the solid domain is constant in the torch reference system.

In quasi-stationary mode, the thermal field in the solid is determined by considering the temperature imposed on the liquid-solid interface, equal to the melting temperature T _f . Implementation of the process

- Step 1

One chooses a form of Fo of the melt and its parametrization by a Bézier curve of order

4 (or degree 3). This curve is based on four control points Po, Pi, P2 and P3 (that is to say eight coordinates in two-dimensional case), and has the formula:

P (t) = (lt) ³ P ₀ + 3 (lt) 2tPi + 3 (1-t) t ² P ₂ + t ³ P ₃ O≤t≤l

This approach has several advantages: on the one hand the Bézier curve ensures regularity of the shape, on the other hand the size of the vector parameters

X to identify remains limited. In addition, some coordinates of the control points may be constrained a priori by the physical nature of the problem. In the case studied here, the coordinates of P3 are known, moreover thanks to the condition of symmetry with respect to the axis "Oy" P _0x -P _1x , P _2x -P _3x , the dimension of the vector parameters X, ie the number of parameters to be identified is thus limited to three: dim (X) = 3 with X = (P _0x , Pi _y , P2 _Y ) • the initial parameter vector X is: χ ^imPut = (0.5,0.5,0.5). 2nd step

The shape of the melt being given in step 1 and thus the initial vector X, we solve the system of equations describing the temperature field in the solid domain in quasi-stationary regime for this bath: p _s c _s u ^dTλ ^ ^y) = V [λ _s VT _s (x, y)] - ^ - (T _s -T _ιmp ) (x, y) _ε Ω _s (1) ox S

T ;; . 00eennxx == --LL _ιι eett - - - = 0 in y = 0

T = T _imp in x = + L ₂ , T (x, y) = T _f (x, y) eT (3)

With:

λ _s Conductivity of the solid, T _s Temperature in the

Wm ^-1 K ^"1 solid, ⁰ C p _s Density of ^τ e External temperature, solid, kg m ^

C Thermal capacity T _imp Required temperature, ⁰ C mass, J kg ^ KT ¹ u Torch speed, m T _f Melting temperature, s ^"1 ° C

p _s = 7200 kg.m, _s p c _p = 4000000 J. m ^{^".K" 1,} λ _s = 50 W m ^"1. K ^1, T _f = 1450 ° C, T _imp = 20 ° C, u = 10 mm.s ^-1 , L _y = 50 mm, L _x = 300 mm, Li = 540 mm, L ₂ = 60 mm,

α = 0.9 (emissivity).

The second term right in the equation

(1) is used to express heat losses, S is the area of the room section and p its perimeter. Adiabatic conditions (2) are considered on the edges of the computational domain without loss of generality. The temperature is imposed (3) on the melting front as well as at the edge x = + L ₂ , see Figure 11. The hatches represent the parts of the part that are sufficiently far from the molten bath. There is therefore no heat flow and temperature imposed.

Thus, the temperature field T (x, y; F) in the solid is accessed for the form of the melt of step 1.

Step 3

Then, for example using a thermocouple located at a distance y _tc from the edge of the plate or from an infrared camera, the temperature Y ^m in the solid is measured in a number of M measuring points 12 at least equal to the number of parameters to be identified (3 as we have seen above). In our example, M = 5 is chosen (ie 5 temperature measurements are made using a single thermocouple). Only one thermocouple is needed to have multiple measurement points. In quasi stationary, the different positions x _m correspond to different recording times x _m -uxt _m , therefore, one can have several measurement points using a single thermocouple. We calculate the function

The following results are then obtained after 15 iterations:

Without being interested in what happens in the liquid part, we thus arrive satisfactorily to describe the shape of the melt and thus access to the thermal loading of the process.

Example 2: 3D case opening. As in the case of the two-dimensional example, it is desired to weld two plates end to end.

Each of the plates has a width L _y , a length 2L _x and a thickness e (see Figure 12).

The problem is physically defined as follows: an electric arc of sufficient intensity moves with a constant speed along the x-axis and is applied to the edge of two metal plates. A melt is formed and moves at the same speed as the electric arc. The assumptions made to define this problem are as follows: • At torch speed u constant, the transfers are non-stationary in a reference linked to the plate. However, by using a reference linked to the torch, a quasi-steady state is obtained: the shape of the molten bath remains constant in this frame while the material enters and leaves the field of study. The mobile repository of the torch is therefore used for the analysis of the inverse problem.

• The shape of the liquid-solid interface F is parameterized by a Bézier surface.

In quasi-stationary mode, the thermal field in the solid is determined by considering the temperature imposed on the liquid-solid interface, equal to the melting temperature T _f .

Implementation of the process

Step 1 :

We choose a form F ₀ of the melt and its parameterization by a Bezier surface of order 4x3 (or degree 3x2). This surface is based on twelve control points Po, Pi, P2,, P12 (Figure 13) (that is to say thirty six coordinates in three-dimensional case), and has the formula:

= P ₀₀ (lu) ² (lv) ³ + P ₀₁ (lu) ² 3v (lv) ² + P ₀₂ (1-u) ² 3v ² (1-v) + P ₀₃ (read) V + Pi _O 2u (lu) (1-V) ³ + P ₁₁ 2U (1-U) 3V (1-V) ² + P ₁₂ 2U (1-U) 3V ² (1-V) + P ₁₃ 2U (1-U) V ³ + P ₂₀ U ² (1-v) ³ + P ₂₁ u ² (1-v) ² + P ₂₂ u ² 3v ² (1-v) + P ₂₃ U ² V ³ ue [0, 1; ve [0,1]

This approach has several advantages: on the one hand the surface of Bézier assures the regularity of the shape, on the other hand the size of the vector X parameters to be identified remains limited. In addition, some coordinates of the control points may be constrained a priori by the physical nature of the problem. In the case studied here, the coordinates of

P20, P23, Poo, P03 are known, moreover, thanks to the conditions of symmetry with respect to the "y" axis, we have:

P21x ⁼ P20x /

PlOx / Pl2x ⁼ Pl3x

We impose PiOz = PiIz = Pi2z = Pi3z / then the dimension of the parameter vector X, ie the number of parameters to be identified is thus limited to nine: dim (X) = 9 with

X = [PuIy, Pθ2y, PlOx, Pl3x, PlIy, Pl2y, PlO2,? 21y? 22y]

The initial X parameter vector is: χ ^inPut = (0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2)

2nd step

The shape of the melt being given in step 1 and thus the initial vector X, the system of equations describing the temperature field in the solid domain in quasi-stationary regime for this bath is solved:

_Ps u c _s ^dTΛx ^ ^Z) = V [VT λ _s _s (x, y, z)] (x, y, z) eΩ _s (1)

dT dT dT

- ^L = O at y = L; - '- = 0 in x = -L _x and - ^L = 0 in y = 0 (2) dy dx dy dT (x, y, z _Q )

-λ. hT (x, y, _{Z ()} ) = hT in ZZ _n -O [3) dz

dT {x, y, z _eD )

+ "'h" T \ (^ x ,, yy ,, _^ z _eepD) ) - = hT _e enz = z "e _e p = e (4) dz

T = T _imp in x = + L ₂ , T (x, y, z) = T _f in (x, y, z) eT (5)

With: λ _s Conductivity of the solid, T _s Temperature in the

Wm ^-1 K ^"1 solid, ⁰ C p _s Density of T _e Outside temperature, solid, kg m ^u

p _s = 7200 kg. m ^"3, c p _s _p = 4000000 J. ^{m" 3.} K ^"1 , λc-SOW.m ^{" 1} . K ¹ , T _f = 1450 ° C, T _imp = 20 ° C, u = 10 mm.s ^"1 .

L _^ - 300 mm, L - 60 mm, L ₁ - 540 mm, L ₂ - 60 mm, e - 3 mm, h = ¹⁶¹ W. CTC 24AxW ⁴ ^{m 2.} IC ¹ , a = 0.9 (emissivity).

S is the area of the section of the room and its perimeter Adiabatic conditions (2) are considered on the edges of the computational domain without loss of generality. The temperature is imposed (5)

(In the 3D problem, it is on equation (5) that on the fusion front and at the edge x = + L ₂ , see Figure 12.

Thus, the temperature field T (x, y, z; F) in the solid is accessed for the melt form of step 1. This estimated field is represented in FIG. 14.

Step 3

One then measures, for example using one or more thermocouples, the temperature Y ^m in the solid in a number of points M at least equal to the number of parameters to be identified (9 as has been seen above) . In our example, M = 10 is chosen (ie 10 temperature measurements are made using a single thermocouple). Only one thermocouple is needed to have multiple measurement points. In quasi stationary, the different positions xm correspond to different recording times x _m -uxt _m , therefore, one can have several measurement points using a single thermocouple.

We calculate the function

The following results are then obtained after 25 iterations: yirnput Estimated error number in (%)

(Pθly t Pθ2γ t PlOx / Pl3x / PlIy / iterations Pl2γ / PlOz / PΣly / P 22γ / IMI i) ²

Ψ \\

(0.2 O?, 0.2, 0.2, 0.2, 2.985 25 7.8

0.2, 0.2, 0.2, 0.2)

Without therefore being interested in what happens in the liquid part, we thus arrive satisfactorily in describing the shape of the melt and thus access to the thermal loading of the process.

REFERENCES

[1] Necati "Ozisik and Helcio R. B. Orlande, Inverse Heat Transfer Fundamentals and Application" pages 37-45, Taylor & Francis Publishing, New York, 2000.

[2] Virginie Hamel, Characterization of the Heat Supply during TIG Welding of a Sheet Metal

Stainless Steel (321 & 304L), Memory of DRT (Technological Research Diploma), University of Nantes, France 2003.

[3] Jialin Guo, Estimate of the

Energy distribution induced by an electron beam in a metallic material-Application to the case of welding a steel, thesis dissertation, University of Southern Brittany, France 2005.

Claims

A method for determining the shape of a liquid metal melt occurring during the implementation of a method of welding a first piece and a second piece, wherein: a) selecting a welding process; b) a two-dimensional or three-dimensional virtual bath shape is chosen according to the welding method chosen in step (a); c) model the shape of the virtual melt by a mathematical function that contains parameters to identify; d) calculating the temperature field in a virtual solid surrounding the melt by imposing boundary conditions; e) producing a real copy of the first and second pieces (2, 4); f) they are welded using the process chosen in step (a), which reveals a melt

(8) and a real solid area; g) measuring the temperature in M points of the real solid zone, M being at least equal to the number of parameters to be identified; h) comparing the temperatures measured in step (g) with the calculated temperatures at the same points in step (d) by calculating the function

S (Y) in two dimensions

1 ^M 2 or the function S (Y) = -Σ [τ (x _m , y _m , z _m ; Y) -Y ^m ] in

IV i _m _ _] three dimensions i) S (F) is compared to a predefined value; j) if S (F) is greater than the predefined value, another form of virtual bath is chosen and steps (c) to (h) are repeated until S (F) is less than the predefined value; k) if or when S (F) is less than the predefined value, we accept the shape of the virtual melt which made it possible to calculate S (F) as real value and we accept the temperature field calculated from this form like the actual temperature field.

2. Method according to claim 1, characterized in that the mathematical function which contains parameters to be identified is a curve or a Bézier surface.

3. Method according to claim 2, characterized in that the mathematical function which contains parameters to be identified is a Bezier curve defined by the equation:

P (t) = ΣB ⁿ (t) P where B ⁿ (t) = C ⁿ t '(l-tγ -', C ⁿ = " ^l with i = \, .. n; N = n + \

where P (t) passes through all points of the Bézier curve, where P ₁ is the control point, n the degree of the Bézier curve, N = n + 1 the number of points n of control P, te [0,1] and B (t) are polynomials of i

Bernstein.

4. Method according to claim 2, characterized in that the mathematical function which contains parameters to be identified is a Bézier surface defined by the equation: ^m "mn P (μ, v) = ΣΣ #. (Μ) B (v) P _g with ue [0,1], ve [0,1]

_I = O j = OIJ

B ^m {u) = - - u'Q - uY ii \ (m - i) \

not . . _n

B (v) = - n-j

TT ^ (I - V) J β (n - j) l

where P (u, v) passes through all points of the Bézier surface, P ₁₃ is the control point, (mxn) is the degree of the Bézier surface, there is (m + l) x (n + l) control points including (m + 1) points in the direction "i" and (n + 1) points in the direction mn "j"; B (u), B (y) are Bernstein polynomials. i J

5. Method according to one of claims 1 to 4, characterized in that, in step (j), the Levenberg-Marquardt algorithm is used to choose another form of virtual bath.

6. Method according to one of claims 1 to 5, characterized in that in case of two-dimensional melt, in step (d), the temperature field is calculated in a virtual solid surrounding the melt by solving the following equation system:

_Ps u c _s ^^ - VT Vμ _s _s (x, y)] - ^l f (T _s -T _ιmp) (x, y) eΩ _s (1)

T = T _imp in x = + L ₂ , T (x, y) = T _f (x, y) eF (3)

with: λ _s Conductivity of the solid, T _s Temperature in the Wm ^-1 K ^"1 solid, ⁰ C p _s Density of the solid, T _e Outside temperature, kg m ^{" 3 0} C c Thermal capacity T _{imp Required} temperature, mass , J kg ^"1 K ^{" 1} ° Q u Speed of the medium of T _f Melting temperature, welding, ms ^"1 ° ç

F Liquid-Solid Interface Ω _s Solid domain h Coefficient of exchange _α emissivity.

7. Method according to one of claims 1 to 5, characterized in that in case of three-dimensional melt, in step (d), the temperature field is calculated in a virtual solid surrounding the melt by solving the following equation system: _P M ^dTΛX ' ^Z) = V [λ, VT, (x, y, z)] (x, y, z) ≡Ω _s : D

_ ^L = O y = L - ^s - = 0 in x = -L and _x - ^ = O j = 0 (2) Oy ^y dxdy

dT (x, VZ _n ) ~ ^λ s - Q _Z ° + hT (x, y, z _Q ) = hT _e enz = z _Q = 0 (3)

dT (x, y, z _ep )

K - + hT (x, y, z _ep) = hT enz _e = e z = (4,

T = T _imp in x = + L ₂ , T (x, y, z) = T _f in (x, y, z) eT (5)

with: λ _s Conductivity of the solid, T _s Temperature in the

Wm ^-1 K ^"1 solid, ⁰ C p _s Solid density, T _e Outside temperature, kg m ^{" 3 0} C c Thermal capacity T _{imp Required} temperature, mass, J kg ^"1 K ^{" 1} ° Q u Speed of average of T _f Melting temperature, welding, ms ^"1 ° ç

8. Method according to one of claims 1 to 7, characterized in that the welding process is selected from the group comprising TIG, MIG / MAG welding, electron beam, plasma, laser, hybrid.

9. Method according to one of claims 2 to 7, characterized in that the degree of the Bézier curve is equal to 3 or 4 in two dimensions.

10. Method according to one of the claims

2-7, characterized in that the degree of the Bezier surface is between 2x3 and 4x3 in three dimensions.