INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 39, 51-65 (1996)

OPTIMAL DESIGN FOR NON-STEADY-STATE METAL FORMING PROCESSES-11. APPLICATION OF SHAPE

OPTIMIZATION IN FORGING

L. FOURMENT, T. BALAN AND J. L. CHENOT

CEMEF, Ecole des Mines de Paris, BP 207, U R A CNRS 1374,06904 Sophia Antipolis Cedex, France

SUMMARY This paper is the second part of a two-part article about shape optimization of metal forming processes. This part is focused on numerical applications of the optimization method which has been described in the first paper. The main feature of this work is the analytical calculations of the derivatives of the objective function for a non-linear, non-steady-state problem with large deformations. The calculations are based on the differentiation of the discrete objective function and on the differentiation of the discrete equations of the forging problem. Our aim here is to show the feasibility and the efficiency of such a method with numerical examples. We recall the formulation and the resolution of the direct problem of hot axisymmetrical forging. Then, a first type of shape optimization problem is considered: the optimization of the shape of the initial part for a one-step forging operation. Two academic problems allow for checking the accuracy of the analytical derivatives, and for studying the convergence rate of the optimization procedure. Both con- strained and unconstrained problems are considered. Afterwards, a second type of inverse problem of design is considered: the shape optimization of the preforming tool, for a two-step forging process. A satisfactory shape is obtained after few iterations of the optimization procedure.

KEY WORDS finite element method; shape optimization; sensitivity analysis; forging preform design; optimal design

1. INTRODUCTION

In the field of metal forming processes, just like in most other fields, engineers and technicians are not so much concerned by the direct analysis of the problem, which can usually be solved by a finite element commercialized software, but by the inverse analysis of the forming process, such as design problems, or such as finding the optimal process data. Of course, from a numerical standpoint, inverse problems actually arise when direct problems have already been solved in quite a satisfactory manner, and this is now the case in forging. In fact, several works show the efficiency of numerical simulation, and more particularly for axisymmetrical problems.' Simula- tion is used for the design of process parameters, such as the shape of the preforming tool as described in Reference 2. It makes it possible to substitute the repeated numerical trials and errors procedure for expensive and time-consuming experiments. Meanwhile, the basic approach is still the same, based on trials and errors, even though it can be made more systematic, for instance as has been done for e x t r u ~ i o n . ~ - ~

We are then concerned by the following inverse problem: finding the optimal process condi- tions and the optimal shape of the preforming tools which make it possible to obtain the required part for a minimal industrial cost with good metallurgical properties. This problem can be solved in a relatively simple manner: the forging code is considered a black box and is inserted into an

CCC 0029-598 1 /96/0 1005 1 - 15 0 1996 by John Wiley & Sons, Ltd.

Received 20 January 199.5 Revised 23 January 1995

52 L. FOURMENT. T. BALAN AND J. L. CHENOT

optimization algorithm.' An objective function is defined, the gradients of which are calculated numerically, If a large number of iterations (of forging simulations) are required to reach a satisfactory solution, then this approach is quite computational time consuming. Moreover, as mentioned in Reference 6, numerical gradients could be quite inaccurate if the actual sensitivities are of the order of the numerical errors. Therefore, more complex methods have been investi- gated. For preforms designs, Kobayashi and colleague^^*^ introduced the specific so-called 'backward tracing technique' and 'traced back the loading path in the actual forming process from a given final configuration'. Another approach is to analytically compute the gradients of the objective function.

We are more inclined toward the second approach, as has been used for instance by Kusiak and Thompsong for the steady-state extrusion process. Moreover, this problem can be handled using different approaches. Either the equations of the continuous problem are considered, differentiated and then discretized, or the equations of the discretized problem are considered and then differentiated. The first approach has been followed by Zabaras and colleagues' while the second approach has been chosen by Kusiak'' and ourselves." Both show good agreement between numerical and analytical gradients.'.'

In this paper, we are concerned with the shape optimization of the preforming tools for a two-step axisymmetrical forging sequence: the initial shape of the part and the shape of the finishing tools are given data of the process, the final shape of the part is prescribed. This work shows the first results which have been obtained using the method described in Reference 11. By way of introduction, another problem has also been studied: the shape optimization of the initial part for a one-step forging sequence and a given shape of the tooling.

2. STATEMENT OF THE DIRECT FORGING PROBLEM

For hot forging of metals, the material is assumed incompressible and to follow the viscoplastic Norton-Hoff law:

div u = 0 (1)

(2)

where u is the velocity field, B is the strain rate tensor, i is the equivalent strain rate (i = fi), a is the Cauchy stress tensor, s is the deviatoric part of a, K is the consistency of the material and m is the strain rate sensitivity coefficient.

s = 2 K ( $ ; ) " ' - ' B

At the interface between the tools and the part, a Norton friction law is assumed:

where T is the shear stress (7 = a n - (an -n)n), n is the normal to the tool surface, Au, is the relative tangential velocity (Au, = (u - udie) - [ (u - Udie).n]n), Udie is the prescribed velocity of the tool forging die, and u and q are the friction law coefficients.

At any time in the process, the forged part i2 is in equilibrium, being in contact with a tool on the ddi2 part of the surface, while the &i2 part is free (see Figure 1):

div a = 0 (4)

METAL FORMING PROCESSES. PART I I 53

Figure 1 . Workpiece and tool

node coming into contact sliding contact

Figure 2. Projection of a node on the tool surface at time t + Ar

Using the finite element method and the Galerkin formulation,' 3*14 these equations are dis- cretized into:

Vk, & ( X , v) = 0 (7)

V k E d d Q ( V k - V d i , ) ' n = O

&(X, v) = [Q2K(\/3&,)m-1 dh:Bk dw + ctK l / A V h , 1 / 4 - 1 AU~,,'N'dS 1., + p ( jQe dw)- jne div Oh dw jne tr(Bk) dw (8)

where the subscript h is used for the discrete variables (& = zk X k N ~ , uh = zk v k Nk), X and V are the arrays of the nodal coordinates and velocities of the finite element mesh, Bk is a linear operator which is defined according to d h = xk v k Bk and p is a large numerical coefficient.

Isoparametric finite elements are used. The incompressibility constraint (1) is imposed using a penalty method. Equation (8) is written for a piecewise interpolation of the pressure field, which is compatible with either linear quadrangular elements or quadratic triangular elements. Equation (7) is solved at each time step of the process by a Newton-Raphson algorithm. After calculation of V , the normal stress value of the boundary node belonging to ad12 is checked: if it is not compressive, the node is released from ads and a new V field is calculated, until V verifies the unilateral contact condition (5).

At time t + At, the domain a' is updated according to

Vk, XL+A' = X i + 1'; At (9) A node which penetrates the tool during the time interval A?, or a node which belongs to the tool surface at time t and which must remain on the tool surface during the time interval At, has to be projected onto the tool surface at time t + At as shown in Figure 2.

54 L. FOURMENT, T. BALAN AND J. L. CHENOT

3. SHAPE OPTIMIZATION OF THE INITIAL PART

3.1. Problem statement

For a one-step forging operation and for a given design of the forging tools, we look for the optimal initial part which allows for forming the prescribed part at the end of forging (see Figure 3). In fact, this problem is almost equivalent to the problem of design which is tackled in Section 4. The difference here is that the shape of the preforming tools which produce the preform is not a straight result of the calculation. The calculations only provide the shape of the preform part. Meanwhile, as this problem is slightly simpler, it is used to carry out some preliminary numerical tests.

Therefore, the optimization objective is minimizing the function @A which measures the distance between the forged part geometry and the prescribed part geometry R, (see Figure 3):

where n ( x ) is the projection of the current point x onto the surface of the prescribed shape. From a practical standpoint, the formed part is usually machined after forging. Therefore, it is

not required that the prescribed part be perfectly formed. On the other hand, at the end of forging the workpiece should perfectly overlap the domain Qp (see Figure 3). Thus, the following object- ive function is used:

where 1 is the characteristic function of R,:

if x ER,, I,(x) = 1 if x 6 a,, l,(x) = 0

If we use @i as an objective function, the problem is ill-posed because the solution is obviously not unique. Among the set of solutions which verify 02 = 0, we decide to select those which minimize Qe, the total energy of the process, me can be considered as a regularization function:

Figure 3. Shape optimization of the initial part

METAL FORMING PROCESSES. PART 11 55

- initial curve ,d -Q- podified curve - - - t normal to the Fume

0 active and passive points

Figure 4. Cubic spline and design parameters

3.2. Shape optimization method

The shape of the initial part is discretized by a cubic spline function. The displacements of some selected points of the spline are considered as design parameters, p , of the optimization problem. In order to reduce the number of parameters and in order to avoid the problem being ill-posed, only displacements in the normal direction of the initial curve (see Figure 4) are considered. After discretization, the objective and regularization functions are written:

where L(xk) is a segment length associated with the boundary node Xk, and Ninc is the number of time increments.

Therefore, two objective functions are considered for our shape optimization problem:

where w is a regularization parameter. @ ( p ) can also be interpreted as the objective function of a penalty method: the minimization of

@,(p) under the inequality constraint @ i ( p ) = 0 (because of the characteristic function Q,, this is actually an inequality), the constraint being enforced by an external penalty method. In this case, ljw is a penalty coefficient.

@h(p) or @ $ ( p ) are minimized using a BFGS alg~ri thm: '~

(0) initial guess of p ( O ) ,

(i) p(") being known, calculation of @&'")) and (d@,,/dp) (p'")), (ii) calculation of H t i o s , the BFGS approximate of (dzQO/dp2) (p(")),15

(iii) calculation of the parameter correction, Ap'") = - (d@,/dp) (p'")), (iv) calculation of the line search coefficient 1 which minimizes @o(p'") + AAp(")), (q p ( n + l ) = p ( n ) + LAP("),

(vi) return to (i) until convergence is reached.

The analytical calculation of (d@,jdp) (p'")) has been described in the previous paper. Although this is a tough coding task, numerical differentiation of @o(p(")) would be too time consuming in

56 L. FOURMENT. T. BALAN AND J. L. CHENOT

Figure 5. Academic problem of upsetting between flat dies

f ~ r g i n g . ~ Moreover, more accurate results are expected from analytical calculations, especially when remeshing or adapted remeshing is necessary.

3.3. Numerical applications

The following academic problem is considered (see Figure 5). A cylinder is upset between flat dies. We assume sticking contact conditions between workpiece and tools. The material is viscoplastic with rn = 0.15. We want to optimize the shape of the free surface of the initial part, in order to obtain, at the end of forging, a prescribed free surface.

3.3. I . Accwaty of the derivntiv~s. The accuracy of the analytical derivatives is estimated by comparing these values with the numerical values which are obtained by a finite difference scheme:'

where E is a small numerical perturbation of the parameter value. The cubic spline which represents the shape is defined with five parameters. The simulation of

the forging process requires 24 time increments. At the end of the first increment, the discrepancy between the numerical (d@Jdp)(p)Inu,,, (p"') and analytical (d@,,/dp) (p")) values is around 001 per cent. At the end of forging, it is around 0.7 per cent. Although the agreement is quite satisfactory, we cannot conclude whether the increasing of the discrepancy is due to the accumulation of numerical errors, to the manner in which we compute the analytical derivatives, or to the poor accuracy of the numer id derivatives themselves. Meanwhile, we have noticed that the agreement increases with decreasing time step size. Regarding (d@,/dp) (p")), which is only computed at the end of forging, the discrepancy fluctuates between 0.5 and 10 per cent, depending on the parameter which is studied. This mediocre agreement can be explained by some oversim- plifications which have been introduced in the calculation of the discrete function a,,, and which have made it locally non-differentiable. Meanwhile, globally the discrepancy between the deriva- tives is less than 10 per cent, which is considered satisfactory and sufficient for the optimization algorithm.

3.3.2. Unconstrained aradenzic problem. After forging of a straight cylinder, the free surface bulges (see Figures h and 7). We look for the initial design of the cylinder which produces a straight side at the end of forging (see Figure 6). This consists in a simple minimization problem of @ ( p ) . Seven parameters are used to describe the shape (see Figure 7) and the simulation of the forging process still requires 24 time increments.

METAL FORMING PROCESSES. PART 11 57

Figure 6. Unconstrained academic problem

I

Figure 7. Rulging of an uphet cylinder between flat dies with sticking contact conditions; prescribed final geometry of the workpiece; location of the Seven design parameters

Figure 8 shows the intermediate shapes which have been tested by the optimization algorithm. Figure 9 shows the convergence of the method in terms of @A. Using only four optimization parameters (instead of seven). 1 1 iterations of the BFGS algorithm-- including one line search computation--are enough to reach a relative accuracy of with respect to 0; (top of Figure 10). Using seven optimization parameters, the number of iterations is slightly increased: 14 iterations of the BFGS algorithm-including five line search computations-are required to reach the same accuracy (bottom of Figure 10). The optimal shape and the result of forging are shown in Figure 10.

3.3.3. Constrained academic problem. In order to study a constrained optimization problem, an arbitrary final shape is prescribed (see Figures 11 and 12). We look for the initial shape of the cylinder which allows for filling the prescribed shape at the end of forging, and which minimizes the total energy of the process. Using a penalty method, this problem is equivalent to an unconstrained minimization problem of @;(p) . Four design parameters are used to describe the shape (their location can be seen in Figure 12) and 35 time increments are required for the simulation of the upsetting problem. At the first iteration of the optimization, O,(p'O') = 5.13 x lo6 and @dz(p(o)) = 2.32 x 10'. According to this, (0 is taken as 0.1, so that the regulariz- ation becomes effective when Od(p) has decreased by a factor of

Figure 13 shows the convergence of the BFGS algorithm during the first 25 iterations. After nine iterations-including one line search computation-a perfect filling of the prescribed part is obtained (see the top of Figure 14). Twenty-three additional iterations-including five

58 L. FOURMENT, T. BALAN AND J. L. CHENOT

100

80

60

40

20

0

OplimizatiOn Ot the iniiial part - Stmhht side cylinder - 7 parameters I B I I

initial shape linal shape --

80 100 120 140 180 180 200

Figure 8. Shape optimization procedure: tested shapes (full lines for the shape of the workpiece at the beginning of forging and dotted lines for the corresponding shape at the end of forging)

le+10

1 e+W

1 e+08

- 1 lea' a .8! le+o6 v F 3 10Ml00

lo000

lo00

Shape Optimization of the initial part (7 parameters)

\ ', \

\ \

\

I

30 100

0 5 10 15 20 25 iteration number

Figure 9. Convergence of the optimization procedure (log of the objective function with respect to the number)

iteration

METAL FORMING PROCESSES, PART I1 59

Figure 10. Optimal shapes for the unconstrained academical problem, using four design parameters (top) and Seven design parameters (bottom)

Figure 11. Constrained academic problem

I

--

Figure 12. Bulging of the cylinder after forging with sticking contact conditions; arbitrary prescribed geometry of the final workpiece; location of the four design parameters

60 L. FOURMENT, T. BALAN A N D J. L. CHENOT

le+ll

le+10

1 e+09

1 e+08

le+07

1 ec06 0 5 10 15 20 25

Figure 13. Convergence of the constrained optimization procedure (log of the objective function with respect to the first 25 iteration numbers)

Figure 14. Optimal shapes for the constrained academical problem: intermediate solution with perfect filling of the prescribed shape (top) and final solution which minimizes the forging energy (bottom)

additional line search computations-are required to minimize the total energy of the process (see the bottom of Figure 14). @,(p) is reduced by a factor of 0.4 per cent, which is quite satisfactory with respect to the accuracy of the finite element calculations.

Similar numerical applications can be found in references 12 and 16.

METAL FORMING PROCESSES, PART 11 61

Figure 15. Two feasible deformation paths for the forging of a rib from a cylinder

Figure 16. Inverse design problem of shape optimization of the preforming tool

4. SHAPE OPTIMIZATION OF THE TOOL O F THE PREFORMING OPERATION

4.1. Problem statemen1

Very often, a part cannot be forged in a single operation: either the required forging force is too great, some forming defects occur, or the microstructural properties of the final part are not satisfactory. Then, at least two forging operations are often required in which several deformation paths could be considered. For instance in Figure 15, several preforms might be used for the forging of a rib, but some of them produce folding defects. Usually, the preform is designed according to industrial know-how. Finding an adequate preform-the workpiece at the end of the first forging operation-is actually an inverse problem of design of the shape of the preforming tools. It can be stated as follows: for a given initial part and a given geometry of the tools of the second forging operation, find the optimal shape of the preforming tool which allows for forming a prescribed part (see Figure 16). From a mathematical standpoint, this problem is rather similar to the problem of Section 3, but here the evolution of the contact surface under a bilateral contact condition with friction is included. As a consequence, remeshing procedures have to be considered.

4.2. Shape optimization method

The shape optimization method itself is rather similar to the method of Section 3. Meanwhile, the initial shape of the workpiece is a prescribed datum, the preforming tools are discretized using

62 L. FOURMENT. T. BALAN AND J . L. CHENOT

mwRn*Mrn,,<"I

Figure 17. Forging of a rib in a single operation, initial and final configuration (at the end of forging the rib is not filled)

Figure 18. Two-step forging process, first iteration of the optimization procedure: flat preforming tool

200

180

160

140

120

initial

0 20 40 60 80 100 120

Figure 19. Tested shapes for the tool shape optimization procedure

METAL FORMING PROCESSES. PART I1 63

12

11

10

1: 7 -

6 -

5 -

4 -

3 -

2 -

1

cubic spline functions, and two forging operations are performed. The displacement of selected points of the spline curve become the design parameters of the problem (still in the normal direction of the initial spline curve). The prescribed shape of the part is defined at the end of the second operation. A flash being allowed, our objective function is @ i ( p ) rather than @ i ( p ) . The total energy of the two forging operations, @,(p) , is again introduced as a regularization function. Therefore, the optimization problem reduces to another minimization problem of @ i ( p ) . The same BFGS algorithm is used. As design parameters have changed, the calculation of the analytical derivatives, (d@,/dp) (p'")), is slightly modified as explained in Reference 1 1. Actually, the changes are caused by the treatment of boundary conditions.

Due to the large deformations of the workpiece during the forging process, the finite element mesh is highly distorted. Several remeshing steps are required. These are performed when an internal or boundary element is distorted, or when a certain fraction of the mesh penetrates the tool. A totally new mesh is generated" and state variables have to be interpolated from the old mesh to the new mesh. Moreover, as (d@,,/dp)(p) and @ i ( p ) depend on (dX/dp)(pFthe deriva- tives of the nodal co-ordinates of the mesh+dX/dp)(p) must also be transferred. A simple procedure is used. For any node nnew of the new mesh, the closest node nold of the old mesh is found. Some neighbouring nodes nnei of nold are found too. For any variable cp to be transferred, q(nnew) is equal to the weighted average of q((n,,ld) and q(nnei).

I 8 I

volume gap piQ] +

- \

'\ \

'\

0 2 ,'L, 4 6 8 , 10 , I 12

4.3. Numerical application

We study the forging of an axisymmetrical rib, starting from a straight cylinder. The shape of the forging dies and the initial workpiece are shown in Figure 17. The material is viscoplastic with m = 0.15 and the friction coefficients are q = 0.15 and c( = 0.6. As previously mentioned, bilateral contact conditions are considered. Initially, the forging simulation is carried out with a single

Figure 20. Percentage of unfilled volume with respect to the number of iterations of the optimization procedure

64 L. FOURMENT, T. BALAN AND J. L. CHENOT

operation. Eighty-two time increments and 11 remeshings are required. However, at the end of forging, the rib is not filled as can be seen in Figure 17. In order to overcome this problem, a preforming operation is added. The shape of the preforming tool becomes the unknown of our design problem.

The shape of the preforming die is described using a cubic spline function with three active parameters. Their locations are shown in Figure 18, for the initial guess of the shape. The prescribed shape of the part R, (at the end of forging) is very similar to the male shape of the finishing die. However it is cut where the flash starts (Figure 18) because we do not aim to prescribe the shape of the flash.

The first guess for the shape of the preforming die is a straight line (flat die). Figure 18 shows that it does not allow for filling the rib. Meanwhile, after five iterations of the optimization procedure the filling is achieved. Figure 19 shows the various shapes which have been tested. Figure 20 shows the convergence rate of the optimization procedure in terms of percentage of the unfilling of the rib (relative values of the square root of @). Finally, Figure 21 shows the simulation of the two forging operations, using the optimal shape of the preforming die. A perfect filling of the die can be noticed.

Figure 21. Two-step forging: optimal sequence with the optimal shape of the preforming tool

METAL FORMING PROCESSES, PART I I 65

5. CONCLUSION

This study shows the feasibility of the method which has been described in Reference 11. Although we have followed an approach different from that of Zabaras and colleagues,6 we find that the discrepancy between numerical and analytical derivatives of the objective function is rather similar. Based on this result, our optimization method has been first tested on academic problems, for optimizing the initial shape of the part. It is found that the convergence rate of the BFGS algorithm is quite satisfactory and that an acceptable solution can be reached in a reduced number of iterations, for both unconstrained and constrained problems. Regarding the actual problem of shape optimization of the preforming tool, although the direct forging problem itself is already complex because of the contact evolution and the numerous remeshing procedures (for the studied example), a satisfactory solution is obtained after only six iterations.

These results are quite encouraging and our future efforts will be dedicated toward more complex industrial forging problems, involving unilateral contact conditions and folding defects.

ACKNOWLEDGEMENTS

This work has been carried out in the framework of a cooperative study funded by the French Forging Profession and its Technical Center, CETIM.

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