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Optimisation of shape and process parameters in metal forging using genetic algorithms
Journal of Materials Processing Technology 146 (2004) 356–364
Optimisation of shape and process parameters in metal forging using genetic algorithms C.F. Castro∗ , C.A.C. António, L.C. Sousa Faculty of Engineering, University of Porto (FEUP), Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal Received 1 May 2003; accepted 28 November 2003
Abstract An approach to optimal design in forging is presented in this paper. The design problem is formulated as an inverse problem incorporating a finite element thermal analysis model and an optimisation technique conducted on the basis of an evolutionary strategy. A rigid viscoplastic flow-type formulation was adopted, valid for both hot and cold processes. In industrial forming processes most of the deformation energy is transformed into thermal energy. The generated heat causes the increase in temperature. External friction losses raise the temperature at the die–work-piece interface. Optimal solutions are obtained using a developed numerical algorithm based on a genetic search supported by an elitist strategy. The chosen design variables are work-piece preform shape and work-piece temperature. In order to demonstrate the efficiency of the inverse evolutionary search, specific forging cases are presented, considering the optimisation of the process parameters aiming the reduction of the difference between the realised and the prescribed final forged shape under minimal energy consumption and restricting the maximum temperature. © 2003 Elsevier B.V. All rights reserved. Keywords: Finite element method; Hot forging; Optimal design; Genetic algorithms
1. Introduction The computational modelling of forging is now well established. Within the last 20 years the finite element method (FEM) has become a powerful technique to simulate metal-forming processes. Forging of materials involves non-steady-state material flow, non-uniform distribution of strain and strain rate, variation of temperature distribution and heat transfer between work-material and forging dies. The objective is to produce a forged piece free of defects with minimal material loss and cost. In order to achieve this, a clear understanding of the influencing factors including the preform shape, the thermal behaviour and temperature variations of work-material and forging dies is required. FEM has been used to evaluate the material deformation, strain/strain rate distribution, to estimate the energy process requirement and the likelihood of defects. These include 2D or 3D simulations in forging under both non-isothermal and isothermal forging conditions. Non-isothermal forging ∗ Corresponding author. Tel.: +351-225081984; fax: +351-225081445. E-mail addresses: [email protected] (C.F. Castro), [email protected] (C.A.C. Ant´onio), [email protected] (L.C. Sousa). URL: http://www.fe.up.pt/smat.
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2003.11.027
is usually carried out at a much higher temperature of the work-piece material than that of the forging dies. Thus the forging process should be treated as a transient heat transfer problem, which has significant effect on the forged parts and consequently it needs to be dealt with so as to meet forging design specifications [1–4]. Optimisation techniques can play a crucial role in improving a given metal-forming process and its final product. The so-called inverse techniques developed for optimisation problems allow the calculation of the optimal solution using as entry parameters the specifications of the final product. Then it determines the design parameters that produce the required final product. Pioneer research work developed for inverse problems was successfully considered for rolling [5], extrusion [6] and forging processes [7]. Reported research work in this area considers gradient-based algorithms in order to find optimal solutions. Later other authors followed the same design problems improving from academic examples towards more industrial ones (e.g. Refs. [4,8]). The development of computer systems for automatic optimisation of preform shapes, tool geometries, tribological conditions and prestressing have also been considered [9]. In this paper, an evolutionary genetic algorithm is proposed to calculate the optimal work-piece shape geometry and work-piece temperature. Then the optimal solutions
C.F. Castro et al. / Journal of Materials Processing Technology 146 (2004) 356–364
are not gradient-dependent and consequently do not present numerical errors resulting from non-accurate sensitivity calculations. Discrete and continuous variables, including shape and process parameters, can be handled simultaneously which is an advantage of using genetic algorithms. These evolutionary searches, although computer is time-consuming when compared with developments of analytical methods, are accurate and provide free time for the researcher to develop more industrial applications.
2. Mechanical model The numerical model is composed of a solver for the direct problem and an optimisation algorithm for the inverse calculations. The computer program for forging simulation is based on a rigid viscoplastic thermal–mechanical problem. The constitutive model assumes a rigid, isotropic, strain hardening viscoplastic incompressive deformation; at any instant of time the viscoplastic strain rates for given directions and shear planes are proportional to the instantaneous deviatoric stresses and to the shear stresses. A flow rule based on the Von Mises criterion and the Perzyna model is considered. The deviatoric stress tensor sij is given by sij =
2 σ¯ ε˙ ij 3 ε˙¯
or
s = D˙
(1)
where ˙ is the strain rate tensor with tr(˙) = 0, ε˙¯ the effective strain rate and σ¯ a function of strain, strain rate and temperature. Friction has been characterised by an adhesive-type law, where a friction factor m is defined varying from 0 (no friction) to 1 (sticking friction). The shear stress is assumed constant and equal to that of the work-piece material and the mean interfacial or frictional shear stress τ f is then expressed as Y0 τf = m √ 3
(2)
where Y0 is the static yield stress. Friction and contact are modelled by interface elements of zero thickness, formulated on the basis of local normal and tangential relative displacements. The tangential stress τ is obtained from Eq. (2) and from the relative tangential velocity between die and work-piece ur : τ=
τf ur | ur |
(3)
The equilibrium equations, natural boundary conditions, incompressible constraint and constitutive equations for the viscoplastic deformation problem may be evaluated from a variational principle derived from the minimisation of the potential energy Φ(u, p) equated in terms of velocities u and
357
pressure p, and neglecting inertia forces, as 1 ˙ s : dΩ − p tr(˙ ) dΩ − bV · u dΩ Φ(u, p) = 2 Ω Ω Ω 1 t · u dΩ + − (4) 2 τ · ur ds δΩ
δ dΩ
where bV and t are the body forces and surface tractions vectors, respectively, Ω the work-piece part, δΩ the work-piece boundary and δ dΩ is the work-piece/tool contact zone. Seeking stationarity of the functional Φ(u, p) and approximating the independent variables by the usual manner in the finite element method a non-linear equation system is obtained. At any instant of time the system is solved iteratively by the following scheme: (Kd + Kf + Kα )u(i) = f (i−1) − Qp(i−1) , p(i) = p(i−1) + αQT u(i)
(5)
where α is a large penalty parameter, Kd = Ω BT DB dΩ, K = BT D B dΩ, Kα = αQT Q, f = Ω NuT bV dΩ + f T δ dΩ f f f T δΩ Nu t dΩ and Q = − Ω B mNp dΩ; B is the strain rate matrix, ˙ = Bu and D is the diagonal strain rate tensor-dependent matrix given by Eq. (1); Bf is the matrix that defines the relative velocity vector from the nodal velocities at the interface and Df relates the shear stress with the relative velocity between die and work-piece at the interface; Nu and Np are the considered form functions and mT = [ 1 1 1 0 0 0 ].
3. Heat transfer model Hot forging involves heat generation due to the material plastic deformation in the work-piece material and frictional work at the material and die interface. The generated heat causes an increase in temperature. The heat is transferred by conduction into the work-piece and to the die by conduction, convection and radiation. Due to the much higher temperature of the work-piece material as well as the heat generated by the plastic deformation and friction work, the heat transfer during forging is mainly characterised by the conduction and friction heat flux at the work-piece material and die interface. The overall change in temperature results in a variation of the thermomechanical behaviour of the material and influences the load and energy necessary for the deformation to take place. This effect is taken into account in the mathematical simulation by including an isotropic Fourier law for heat flux. The thermal model is described by the Fourier equation: ∇ T K∇T + Q = ρc
∂T ∂t
(6)
where K is the conductivity, T the temperature, Q the rate of heat generation, ρ the density, c the specific heat and t the time.
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The heat generated by friction between the work-piece and die can be obtained from qf = τf s|ur |
(7)
where s is the contact area. If a lubricant is present its contribution to the heat conduction between the work-piece and the tools is considered by attributing its thermal properties to the same interface element used for friction. As a Lagrangian approach is adopted the heat is permanently transferred to the new mesh position and the set of differential equations obtained for the heat balance are of the form: ˙ =Q KT + CT
(8)
˙ the nodal temperature and nodal temperature with T and T time derivatives, respectively, and C the capacity matrix: cij = ρcNi Nj dΩ (9) Ω
where ρ is the density, c the specific heat per unit of volume and K the conductivity matrix: k∇Ni ∇Nj dΓ + hNi Nj dΓ Kij = Ω Γh ± hlub Ni Nj ds (10) δ dΩ
with k the conductivity, h the surface heat transfer coefficient, hlub the lubricant heat transfer coefficient (if existent) and Q the heat generation vector: T Qi = fs ε˙ Ni dΩ − q¯ Ni dΓ Ω
Γq
+
Γh
hT∞ Ni dΓ +
δ dΩ
qf ds
(11)
where q¯ is the heat flux at the boundary, qf the heat generated by friction, T∞ the temperature of the external environment and f the fraction of plastic work that turns into heat (85–95%).
4. Optimisation method With a reliable numerical simulation of the problem, optimisation can be addressed. The optimisation goal is the minimisation of a functional that quantifies several parameters of the process. We can select two of them that are taken into account in most forging sequences: the total energy of the process and the distance between the shape at the end of the process and the prescribed shape. The total energy is a measure of the actual cost of the process and it is given by t ϕe (b) = t · u dΩ dt (12) 0
δΩ
with b the design variable, t the applied traction vector and u the die velocity. The distance between the current shape
at the end of the process and the prescribed shape can be calculated as ||π(X) − X(b)||2 ds (13) ϕd (b) = δΩend
where π(X) is the projection of a material point X of the work-piece boundary δΩend onto the surface of the prescribed shape at the end of the simulation process. The optimisation problem to be solve is stated mathematically as follows: find the vector of design variables b = {b1 , . . . , bD } ∈ RD that minimises the objective functional Π(b) = β1 ϕe (b) + β2 ϕd (b) subject to T¯ end (b) ≤1 Ta bd − ≤ bd ≤ bd + ,
(14)
(15) d = 1, . . . , D
(16)
and to the state equations of the thermal–mechanical problem (Eqs. (5) and (8)). The number of design variables is D and bd − , bd + are the side constraints for each variable. The parameters βi are to be considered as regularisation parameters, Ta is the maximum allowed temperature and T¯ end is the maximum temperature registered in the work-piece along the forging process. The set of design variables b need not have the same units. Two sets of variables will be considered: shape design variables and process variables. In the numerical examples presented in this paper the shape variables are the variations of the active control points of a cubic B-spline defining the free boundary of the initial work-piece. The process variable considered is the initial temperature of the work-piece, T0 . The temperature of the work-piece will change along the forging process due to shear stress, die velocity and friction, among others, and its maximum value, Ta , strongly influences the quality of the final product.
5. Evolutionary search model The Genetic Algorithm (GA) method is a stochastic search method based on evolution and genetics, and exploits the concept of survival of the fittest [10]. For a given problem or design domain of significant complexity there exists a multitude of possible solutions that form a solution space. In a GA, a highly effective search of the solution space is performed, allowing a population of strings representing possible solutions to evolve through the basic random operators of selection, crossover and mutation. In a GA implementation the data codification is very important for further manipulation. The design vector b is codified using the binary code format with a different and independent space design for each variable. Clearly the dimension of the space design, even for a comparatively small chromosome structure, can be very large. During the operation of the algorithm the objective of the genetic operators is
C.F. Castro et al. / Journal of Materials Processing Technology 146 (2004) 356–364
to progressively reduce the space design driving the process into more promising regions. One important step for the evolutionary search is to define the fitness, which is related to the objective function and the constraints of the problem. In this work a hybrid method is adopted based on a graded penalisation of the solutions according to its constraint violation. The genetic algorithm will seek to increase the fitness as it operates. Then, the fitness function for the optimisation problem established from Eqs. (14)–(16) is defined as F(b) = F¯ − Π(b) − Ψ1 (b) with Ψ1 (b) =
0,
η T¯ end (b) ξ −1 , Ta
(17)
if T¯ end (b) ≤ Ta if T¯ end (b) > Ta
(18)
where ξ and η are calculated constants considering two degrees of violation of the constraints and F is a predefined constant to ensure a positive fitness function. The developed GA is based on four operators supported by an elitist strategy that always preserves a core of best individuals of the population whose genetic material is transferred into the next generations. A schematic representation of the GA is given in Fig. 1. A new population of solutions Pt+1 is generated from the previous Pt using the following genetic operators: selection, crossover, elimination/substitution and mutation aiming the improvement of the fitness. The operators are applied in the following sequence. Step1: Initialisation. Random generation of the initial population. Step 2: Selection. Population ranking according to solution fitness. Definition of the elite group that includes individuals highly fitted. Selection of the progenitors: one from the best-fitted group (elite) and another from
359
the least-fitted one. This selection is done randomly with an equal probability distribution for each solution. Transfer of the whole population Pt to an intermediate step where they join the offspring determined by the crossover operator. Step 3: Crossover. The crossover operator transforms two chromosomes (progenitors) into a new chromosome (offspring) having genes from both progenitors. The offspring genetic material is obtained using a modification of the parametrised uniform crossover technique [11]. This is a multipoint combination technique applied to the binary string of selected chromosomes. This crossover is applied with a predefined probability P(c) to the genetic material of the highest fitted chromosome. The new individuals created by crossover are also joined to the original population. Step 4: Elimination/substitution. The enlarged population solutions are ranked according to their fitness. Then, the elimination/substitution operator performs eliminating solutions with similar genetic properties and replacing them by new randomly generated individuals. After substitution and new ranking updating, another elimination is done corresponding to the deletion of the worst solutions. The exclusion of individuals with low fitness and also the natural death of old individuals are simulated by this operator. Now, the dimension of the population is smaller than the original one. The original size of the population will be recovered after including a group of new solutions obtained with the mutation operator. Step 5: Mutation. The mutation genetic operator is used to overcome the problem induced by selection and crossover operators where some generated solutions have a large percentage of equal genetic material. The implemented mutation is characterised by changing a set of bits of the binary string corresponding to one variable on a chromosome selected at random from the elite
Fig. 1. The developed GA for the forging optimisation problem.
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group. The mutation makes possible the exploitation of previously unmapped space design regions and guarantees the diversity of the generated population. After mutation, the new population Pt+1 is obtained and the evolutionary process will continue until the stopping criteria are reached. Step 6: Stopping criterion. The stopping criterion used in the convergence analysis is based on the relative variation of the mean fitness of a reference group during a fixed number of generations and the feasibility of the corresponding solutions. The size for the reference group is predefined. If the constraints of the problem are not satisfied then the evolutionary process continues. Supposing that there is a feasible solution for the optimisation problem, the search is stopped if the mean fitness of the reference group does not evolve after a finite number of generations. Otherwise, the population evolves to the next generation returning to Step 2.
6. Numerical examples Two examples are presented here to show the efficiency of the genetic algorithm developed for hot forging optimisation. The objective is to design the preform shape and temperature of the work-piece such that, after forging, the required final part is obtained without defects. In one-step forging operations, the die shape is a prescribed datum. Two examples will be presented both considering the compression of a cylinder (20 mm in height by 25 mm in diameter) of aluminium alloy AL6016T6. The work-piece will be heated and the die will be considered at room temperature. The material behaviour is modelled considering the constitutive relations σ¯ = 600(¯ε + 0.003)0.056 MPa σ¯ = 100(¯ε + 0.003)0.25 MPa
(room temperature) (T = 660 K)
(19) (20)
experimentally carried out by Santos et al. [12]. A 50% height reduction is intended. Bilateral contact conditions are considered, the constant shear friction factor m is taken as 0.4 and the fraction of plastic work transformed
into heat is equal to 90%. The adopted material’s conductivity is k = 155 N/s K. The die matrix is considered rigid with no internal heat generation and an initial die temperature Tdie = 285 K. Since forging is symmetric about the vertical and the horizontal axis, the material deformation is also symmetric. To reduce the computation time, only one-quarter of the cross-section is considered for the finite element analysis. 6.1. Minimising barrelling in the upsetting of a cylinder It is well known that a disk, free of barrelling, cannot be produced from a cylindrical billet in a single-stage forging process using a pair of flat dies. Fig. 2 presents the 4-node linear element discretisation of the tool and work-piece as well as the deformed work-piece geometry after height reduction. As shown in Fig. 2, after forging the straight cylinder, the barrelling effect is quite remarkable. Our goal is to design the preform work-piece shape and to determine the initial work-piece temperature such that, after forging with a flat die, a disk with a minimum barrelling effect is produced, with minimum energy and where the temperature does not surpass an allowed maximum. Let us consider the shape of the free surface of the initial work-piece defined by a cubic B-spline curve. N control points determine the shape of the B-spline curve and the B-spline curve is bounded by its control polygon. For 2D problems each control point Pi , i = 1, . . . , N, has two co-ordinates (Pi (r), Pi (z)). The r-displacements of selected active control points of the B-spline function become the geometric design parameters. In this example, six control points were used to define the B-spline curve representing the preform die shape (Pi , i = 1, . . . , 6) corresponding to the first six components of the design vector b. The seventh component of the design vector b is the initial temperature of the work-piece, T0 . The optimisation problem has been solved using the developed genetic algorithm. The considered constraints were −4 ≤ bd ≤ 4 mm,
d = 1, . . . , 6
Fig. 2. Geometry and finite element mesh for the height reduction problem.
(21)
C.F. Castro et al. / Journal of Materials Processing Technology 146 (2004) 356–364
361
Fig. 3. Evolution of the fitness function components for the optimisation process: upset of a cylinder.
473 ≤ T0 = b7 ≤ 623 K
(22)
and the maximum allowed temperature during forging was Ta = 723 K. The parameters considered in the genetic algorithm are Npop = 12 and Ne = 5 for the population and elite group size, respectively. The number of bits in binary codifying for the geometric and temperature design variables is Nbit = 5. The evolutionary process will stop when convergence is achieved, that is when the mean fitness of the six best individuals does not change during five consecutive generations. As for the objective function defined in Eq. (14), equal weight was given to the energy and shape fitness components. After 46 generations the evolutionary process had converged. The evolution of the distance and energy components of the objective function given by Eq. (14) is presented in Fig. 3. The energy component decreases not only with the decrease of the initial volume of the work-piece but also with the increase of the initial temperature of the work-piece. The compromise between distance and energy components is given by the fitness evolution of the best solution along each generation presented in Fig. 3. The developed algorithm is very efficient in determining the optimum. The undeformed and deformed shapes for the optimal solution is shown in Fig. 4. The optimal initial solution corresponds to bT = [−0.33, 3.04, 0.26, −0.20, −1.74, −1.59, 521] where the first component corresponds to the r-displacements in mm of the B-spline control point at the highest value z = 10 mm and the sixth value at the equatorial
plane z = 0 mm. The seventh component is the initial temperature of the work-piece, Toptimal = 521 K. After forging, the barrelling effect is hardly seen. The cross-section of the resulting disk is almost rectangular so that the optimised result is very close to the desired shape. If all material outside the minimum radius were to be trimmed away, this would account for material savings by using the optimised design. The temperature distribution of the deformed optimal solution and die is presented in Fig. 5. No violation of the temperature constraint was detected in the optimal forging. The temperature transfer between work-piece and die is observable. Before forging the die temperature is considered constant and at room temperature Tdie = 285 K and according to the optimal solution the initial work-piece temperature is Toptimal = 521 K. After forging the highest achieved temperature in the work-piece is Tmax = 530 K at the equatorial plane and lowest Tmin = 450 K at the die–work-piece contact area. The decrease in temperature is due to the heat transfer between die and work-piece. 6.2. Preform design for efficient near-net shape manufacturing in upset forging In the second example, the design of a preform shape that encompasses a prescribed shape defined by a polynomial function is considered. After forging the prescribed shape rp is taken as a quadratic function of the height z from the equatorial plane of the final product:
z 2 (mm) (23) rp (z) = 15 + 2 5.5
Fig. 4. Optimal solution for the example of minimising the barrelling effect.
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Fig. 5. Temperature distribution for the optimal solution of minimising the barrelling effect.
Fig. 6. Evolution of the fitness components for the optimisation process: near-net shape example.
Again the free surface of the initial work-piece is parametrised using a cubic B-spline as in the previous example. Same genetic parameters are adopted for this second example. Fig. 6 represents the variation of energy, distance and fitness of the best solution along generations. The two components, energy given by Eq. (12) and distance as measured by Eq. (13), were equally weighted to calculate the objective function. The fitness is shown in Fig. 6 and as defined by Eq. (17), it corresponds to the fitness of the best solution
in each generation. Fig. 7 presents the undeformed and deformed shapes for the optimal solution corresponding to bT = [4.13, −2.43, − 4.80, −6.96, −10.56, −10.83, 593], where the first component corresponds to the r-displacements in mm of the B-spline control points and the seventh component to the initial temperature of the work-piece, Toptimal = 593 K. After forging, the intended shape is obtained. The developed
Fig. 7. Undeformed and deformed shapes for the optimal solution for the near-net shape example.
C.F. Castro et al. / Journal of Materials Processing Technology 146 (2004) 356–364
363
Fig. 8. Temperature distribution for the optimal solution of the near-net shape example.
genetic algorithm found very efficiently an optimal solution for this complicated shape problem. The temperature distribution of the deformed optimal solution and die is presented in Fig. 8. Again, no violation of the temperature constraint was detected in the optimal forging. The temperature transfer between work-piece and die is quite significant. After forging the highest achieved temperature in the work-piece is Tmax = 590 K at the equatorial plane and lowest Tmin = 520 K at the die–work-piece contact area. The decrease in temperature is due to the temperature transfer between die and work-piece and to such an extensive contact area at the beginning of the forging.
7. Concluding remarks This paper describes the development of a new evolutionary search model for optimal design of hot metal-forming processes. The developed scheme is based on a genetic algorithm supported by an elitist search strategy and applied to a population with fixed dimension. The principal aspects of the approach are high fitness improvement and diversity based on small populations. The method was applied to design the optimal work-piece shape and temperature in hot forging processes. The two numerical examples of an inverse problem were efficiently solved considering the developed method: minimising the barrelling effect when upsetting a cylinder and fitting a prescribed polynomial free boundary shape. The final shapes are very close to the intended ones under energy-minimising constraints. To simulate a 50% height reduction with a straight die, using our developed thermal–mechanical code and a Pentium III takes only a few seconds. Each implemented example took 2–4 h to give an optimal solution. The developed genetic algorithm presents some advantages over sensitivity-based methods: the algorithm is not
sensitivity-dependent, runs well even for discontinuous derivative fields, discrete design variables and does not introduce iteration-dependent numerical errors. It can be concluded that the proposed genetic approach is efficient and gives promising results. At the moment, our future efforts will be dedicated to the optimisation of multistage forging processes. The computer running time will be higher due to the contact iteration process of geometrically complicated preform dies. But the approach is aimed towards the design of industrial pieces where material properties and shape design play important roles and where it is much more expensive to perform trial and error experiments than numerical optimisation simulation. References [1] R.W. Wagoner, J.L. Chenot, Metal Forming Analysis, Cambridge University Press, Cambridge, UK, 2001. [2] J.M. César de Sá, L. Costa Sousa, M.L. Madureira, Simulation model for hot and cold forging by mixed methods including adaptive mesh refinement, Eng. Comput. 13 (2–4) (1996) 339–360. [3] N. Rebelo, S. Kobayashi, A coupled analysis of viscoplastic deformation and heat transfer—applications, Int. J. Mech. Sci. 22 (1989) 707–718. [4] L.C. Sousa, C.F. Castro, C.A.C. António, A.D. Santos, Inverse methods in design of industrial forging processes, J. Mater. Process. Technol. 128 (2002) 266–273. [5] A.M. Maniatty, M.-F. Chen, Shape sensitivity analysis for steady metal-forming processes, Int. J. Num. Meth. Eng. 39 (1996) 1199– 1217. [6] M.S. Joun, S.M. Hwang, Die shape optimal design in three-dimensional shape metal extrusion by the finite element method, Int. J. Num. Meth. Eng. 41 (1998) 311–335. [7] J.-L. Chenot, E. Massoni, L. Fourment, Inverse problems in finite element simulation of metal forming processes, Eng. Comput. 13 (2–4) (1996) 190–225. [8] C.F. Castro, L.C. Sousa, C.A.C. António, J.M.A. César de Sá, An efficient algorithm to estimate optimal preform die shape parameters in forging, Eng. Comput. 18 (7–8) (2001) 1057–1077.
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[9] T. Rodic, I. Gresovnik, D. Jolovsek, J. Korelc, Optimisation and prestressing of cold forging die by using symbolic templates, in: Proceedings of the European Conference on Computational Mechanics (ECCM99), Munchen, Germany, 1999. [10] K. DeJong, Evolutionary computation: recent developments and open issues, in: K. Miettinen, M. Mäkelä, P. Neittaanmaki, J. Périaux (Eds.), Proceedings of the Evolutionary Algorithms in Engineering Computer Science (EUROGEN99), University of Jyväskylä, Finland, Wiley, Chichester, 1999, pp. 43–54.
[11] W.M. Spears, S.K. DeJong, On the virtues of parametrised uniform crossover, in: R.K. Belew, L.B. Booker (Eds.), Proceedings of Fourth International Conference on Genetic Algorithms, Morgan Kaufmann, San Mateo, CA, 1991, pp. 230–236. [12] A.D. Santos, J. Ferreira Duarte, A. Reis, A. Barata da Rocha, R. Neto, R. Paiva, Finite element simulation of closed die forging for prediction of material behaviour and optimisation of process, in: Proceedings of the Second ESAFORM Conference on Material Forming, Guimarães, Portugal, 13–17 April 1999, pp. 295–298.
Optimisation of shape and process parameters in metal forging using genetic algorithms C.F. Castro∗ , C.A.C. António, L.C. Sousa Faculty of Engineering, University of Porto (FEUP), Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal Received 1 May 2003; accepted 28 November 2003
Abstract An approach to optimal design in forging is presented in this paper. The design problem is formulated as an inverse problem incorporating a finite element thermal analysis model and an optimisation technique conducted on the basis of an evolutionary strategy. A rigid viscoplastic flow-type formulation was adopted, valid for both hot and cold processes. In industrial forming processes most of the deformation energy is transformed into thermal energy. The generated heat causes the increase in temperature. External friction losses raise the temperature at the die–work-piece interface. Optimal solutions are obtained using a developed numerical algorithm based on a genetic search supported by an elitist strategy. The chosen design variables are work-piece preform shape and work-piece temperature. In order to demonstrate the efficiency of the inverse evolutionary search, specific forging cases are presented, considering the optimisation of the process parameters aiming the reduction of the difference between the realised and the prescribed final forged shape under minimal energy consumption and restricting the maximum temperature. © 2003 Elsevier B.V. All rights reserved. Keywords: Finite element method; Hot forging; Optimal design; Genetic algorithms
1. Introduction The computational modelling of forging is now well established. Within the last 20 years the finite element method (FEM) has become a powerful technique to simulate metal-forming processes. Forging of materials involves non-steady-state material flow, non-uniform distribution of strain and strain rate, variation of temperature distribution and heat transfer between work-material and forging dies. The objective is to produce a forged piece free of defects with minimal material loss and cost. In order to achieve this, a clear understanding of the influencing factors including the preform shape, the thermal behaviour and temperature variations of work-material and forging dies is required. FEM has been used to evaluate the material deformation, strain/strain rate distribution, to estimate the energy process requirement and the likelihood of defects. These include 2D or 3D simulations in forging under both non-isothermal and isothermal forging conditions. Non-isothermal forging ∗ Corresponding author. Tel.: +351-225081984; fax: +351-225081445. E-mail addresses: [email protected] (C.F. Castro), [email protected] (C.A.C. Ant´onio), [email protected] (L.C. Sousa). URL: http://www.fe.up.pt/smat.
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2003.11.027
is usually carried out at a much higher temperature of the work-piece material than that of the forging dies. Thus the forging process should be treated as a transient heat transfer problem, which has significant effect on the forged parts and consequently it needs to be dealt with so as to meet forging design specifications [1–4]. Optimisation techniques can play a crucial role in improving a given metal-forming process and its final product. The so-called inverse techniques developed for optimisation problems allow the calculation of the optimal solution using as entry parameters the specifications of the final product. Then it determines the design parameters that produce the required final product. Pioneer research work developed for inverse problems was successfully considered for rolling [5], extrusion [6] and forging processes [7]. Reported research work in this area considers gradient-based algorithms in order to find optimal solutions. Later other authors followed the same design problems improving from academic examples towards more industrial ones (e.g. Refs. [4,8]). The development of computer systems for automatic optimisation of preform shapes, tool geometries, tribological conditions and prestressing have also been considered [9]. In this paper, an evolutionary genetic algorithm is proposed to calculate the optimal work-piece shape geometry and work-piece temperature. Then the optimal solutions
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are not gradient-dependent and consequently do not present numerical errors resulting from non-accurate sensitivity calculations. Discrete and continuous variables, including shape and process parameters, can be handled simultaneously which is an advantage of using genetic algorithms. These evolutionary searches, although computer is time-consuming when compared with developments of analytical methods, are accurate and provide free time for the researcher to develop more industrial applications.
2. Mechanical model The numerical model is composed of a solver for the direct problem and an optimisation algorithm for the inverse calculations. The computer program for forging simulation is based on a rigid viscoplastic thermal–mechanical problem. The constitutive model assumes a rigid, isotropic, strain hardening viscoplastic incompressive deformation; at any instant of time the viscoplastic strain rates for given directions and shear planes are proportional to the instantaneous deviatoric stresses and to the shear stresses. A flow rule based on the Von Mises criterion and the Perzyna model is considered. The deviatoric stress tensor sij is given by sij =
2 σ¯ ε˙ ij 3 ε˙¯
or
s = D˙
(1)
where ˙ is the strain rate tensor with tr(˙) = 0, ε˙¯ the effective strain rate and σ¯ a function of strain, strain rate and temperature. Friction has been characterised by an adhesive-type law, where a friction factor m is defined varying from 0 (no friction) to 1 (sticking friction). The shear stress is assumed constant and equal to that of the work-piece material and the mean interfacial or frictional shear stress τ f is then expressed as Y0 τf = m √ 3
(2)
where Y0 is the static yield stress. Friction and contact are modelled by interface elements of zero thickness, formulated on the basis of local normal and tangential relative displacements. The tangential stress τ is obtained from Eq. (2) and from the relative tangential velocity between die and work-piece ur : τ=
τf ur | ur |
(3)
The equilibrium equations, natural boundary conditions, incompressible constraint and constitutive equations for the viscoplastic deformation problem may be evaluated from a variational principle derived from the minimisation of the potential energy Φ(u, p) equated in terms of velocities u and
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pressure p, and neglecting inertia forces, as 1 ˙ s : dΩ − p tr(˙ ) dΩ − bV · u dΩ Φ(u, p) = 2 Ω Ω Ω 1 t · u dΩ + − (4) 2 τ · ur ds δΩ
δ dΩ
where bV and t are the body forces and surface tractions vectors, respectively, Ω the work-piece part, δΩ the work-piece boundary and δ dΩ is the work-piece/tool contact zone. Seeking stationarity of the functional Φ(u, p) and approximating the independent variables by the usual manner in the finite element method a non-linear equation system is obtained. At any instant of time the system is solved iteratively by the following scheme: (Kd + Kf + Kα )u(i) = f (i−1) − Qp(i−1) , p(i) = p(i−1) + αQT u(i)
(5)
where α is a large penalty parameter, Kd = Ω BT DB dΩ, K = BT D B dΩ, Kα = αQT Q, f = Ω NuT bV dΩ + f T δ dΩ f f f T δΩ Nu t dΩ and Q = − Ω B mNp dΩ; B is the strain rate matrix, ˙ = Bu and D is the diagonal strain rate tensor-dependent matrix given by Eq. (1); Bf is the matrix that defines the relative velocity vector from the nodal velocities at the interface and Df relates the shear stress with the relative velocity between die and work-piece at the interface; Nu and Np are the considered form functions and mT = [ 1 1 1 0 0 0 ].
3. Heat transfer model Hot forging involves heat generation due to the material plastic deformation in the work-piece material and frictional work at the material and die interface. The generated heat causes an increase in temperature. The heat is transferred by conduction into the work-piece and to the die by conduction, convection and radiation. Due to the much higher temperature of the work-piece material as well as the heat generated by the plastic deformation and friction work, the heat transfer during forging is mainly characterised by the conduction and friction heat flux at the work-piece material and die interface. The overall change in temperature results in a variation of the thermomechanical behaviour of the material and influences the load and energy necessary for the deformation to take place. This effect is taken into account in the mathematical simulation by including an isotropic Fourier law for heat flux. The thermal model is described by the Fourier equation: ∇ T K∇T + Q = ρc
∂T ∂t
(6)
where K is the conductivity, T the temperature, Q the rate of heat generation, ρ the density, c the specific heat and t the time.
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The heat generated by friction between the work-piece and die can be obtained from qf = τf s|ur |
(7)
where s is the contact area. If a lubricant is present its contribution to the heat conduction between the work-piece and the tools is considered by attributing its thermal properties to the same interface element used for friction. As a Lagrangian approach is adopted the heat is permanently transferred to the new mesh position and the set of differential equations obtained for the heat balance are of the form: ˙ =Q KT + CT
(8)
˙ the nodal temperature and nodal temperature with T and T time derivatives, respectively, and C the capacity matrix: cij = ρcNi Nj dΩ (9) Ω
where ρ is the density, c the specific heat per unit of volume and K the conductivity matrix: k∇Ni ∇Nj dΓ + hNi Nj dΓ Kij = Ω Γh ± hlub Ni Nj ds (10) δ dΩ
with k the conductivity, h the surface heat transfer coefficient, hlub the lubricant heat transfer coefficient (if existent) and Q the heat generation vector: T Qi = fs ε˙ Ni dΩ − q¯ Ni dΓ Ω
Γq
+
Γh
hT∞ Ni dΓ +
δ dΩ
qf ds
(11)
where q¯ is the heat flux at the boundary, qf the heat generated by friction, T∞ the temperature of the external environment and f the fraction of plastic work that turns into heat (85–95%).
4. Optimisation method With a reliable numerical simulation of the problem, optimisation can be addressed. The optimisation goal is the minimisation of a functional that quantifies several parameters of the process. We can select two of them that are taken into account in most forging sequences: the total energy of the process and the distance between the shape at the end of the process and the prescribed shape. The total energy is a measure of the actual cost of the process and it is given by t ϕe (b) = t · u dΩ dt (12) 0
δΩ
with b the design variable, t the applied traction vector and u the die velocity. The distance between the current shape
at the end of the process and the prescribed shape can be calculated as ||π(X) − X(b)||2 ds (13) ϕd (b) = δΩend
where π(X) is the projection of a material point X of the work-piece boundary δΩend onto the surface of the prescribed shape at the end of the simulation process. The optimisation problem to be solve is stated mathematically as follows: find the vector of design variables b = {b1 , . . . , bD } ∈ RD that minimises the objective functional Π(b) = β1 ϕe (b) + β2 ϕd (b) subject to T¯ end (b) ≤1 Ta bd − ≤ bd ≤ bd + ,
(14)
(15) d = 1, . . . , D
(16)
and to the state equations of the thermal–mechanical problem (Eqs. (5) and (8)). The number of design variables is D and bd − , bd + are the side constraints for each variable. The parameters βi are to be considered as regularisation parameters, Ta is the maximum allowed temperature and T¯ end is the maximum temperature registered in the work-piece along the forging process. The set of design variables b need not have the same units. Two sets of variables will be considered: shape design variables and process variables. In the numerical examples presented in this paper the shape variables are the variations of the active control points of a cubic B-spline defining the free boundary of the initial work-piece. The process variable considered is the initial temperature of the work-piece, T0 . The temperature of the work-piece will change along the forging process due to shear stress, die velocity and friction, among others, and its maximum value, Ta , strongly influences the quality of the final product.
5. Evolutionary search model The Genetic Algorithm (GA) method is a stochastic search method based on evolution and genetics, and exploits the concept of survival of the fittest [10]. For a given problem or design domain of significant complexity there exists a multitude of possible solutions that form a solution space. In a GA, a highly effective search of the solution space is performed, allowing a population of strings representing possible solutions to evolve through the basic random operators of selection, crossover and mutation. In a GA implementation the data codification is very important for further manipulation. The design vector b is codified using the binary code format with a different and independent space design for each variable. Clearly the dimension of the space design, even for a comparatively small chromosome structure, can be very large. During the operation of the algorithm the objective of the genetic operators is
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to progressively reduce the space design driving the process into more promising regions. One important step for the evolutionary search is to define the fitness, which is related to the objective function and the constraints of the problem. In this work a hybrid method is adopted based on a graded penalisation of the solutions according to its constraint violation. The genetic algorithm will seek to increase the fitness as it operates. Then, the fitness function for the optimisation problem established from Eqs. (14)–(16) is defined as F(b) = F¯ − Π(b) − Ψ1 (b) with Ψ1 (b) =
0,
η T¯ end (b) ξ −1 , Ta
(17)
if T¯ end (b) ≤ Ta if T¯ end (b) > Ta
(18)
where ξ and η are calculated constants considering two degrees of violation of the constraints and F is a predefined constant to ensure a positive fitness function. The developed GA is based on four operators supported by an elitist strategy that always preserves a core of best individuals of the population whose genetic material is transferred into the next generations. A schematic representation of the GA is given in Fig. 1. A new population of solutions Pt+1 is generated from the previous Pt using the following genetic operators: selection, crossover, elimination/substitution and mutation aiming the improvement of the fitness. The operators are applied in the following sequence. Step1: Initialisation. Random generation of the initial population. Step 2: Selection. Population ranking according to solution fitness. Definition of the elite group that includes individuals highly fitted. Selection of the progenitors: one from the best-fitted group (elite) and another from
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the least-fitted one. This selection is done randomly with an equal probability distribution for each solution. Transfer of the whole population Pt to an intermediate step where they join the offspring determined by the crossover operator. Step 3: Crossover. The crossover operator transforms two chromosomes (progenitors) into a new chromosome (offspring) having genes from both progenitors. The offspring genetic material is obtained using a modification of the parametrised uniform crossover technique [11]. This is a multipoint combination technique applied to the binary string of selected chromosomes. This crossover is applied with a predefined probability P(c) to the genetic material of the highest fitted chromosome. The new individuals created by crossover are also joined to the original population. Step 4: Elimination/substitution. The enlarged population solutions are ranked according to their fitness. Then, the elimination/substitution operator performs eliminating solutions with similar genetic properties and replacing them by new randomly generated individuals. After substitution and new ranking updating, another elimination is done corresponding to the deletion of the worst solutions. The exclusion of individuals with low fitness and also the natural death of old individuals are simulated by this operator. Now, the dimension of the population is smaller than the original one. The original size of the population will be recovered after including a group of new solutions obtained with the mutation operator. Step 5: Mutation. The mutation genetic operator is used to overcome the problem induced by selection and crossover operators where some generated solutions have a large percentage of equal genetic material. The implemented mutation is characterised by changing a set of bits of the binary string corresponding to one variable on a chromosome selected at random from the elite
Fig. 1. The developed GA for the forging optimisation problem.
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group. The mutation makes possible the exploitation of previously unmapped space design regions and guarantees the diversity of the generated population. After mutation, the new population Pt+1 is obtained and the evolutionary process will continue until the stopping criteria are reached. Step 6: Stopping criterion. The stopping criterion used in the convergence analysis is based on the relative variation of the mean fitness of a reference group during a fixed number of generations and the feasibility of the corresponding solutions. The size for the reference group is predefined. If the constraints of the problem are not satisfied then the evolutionary process continues. Supposing that there is a feasible solution for the optimisation problem, the search is stopped if the mean fitness of the reference group does not evolve after a finite number of generations. Otherwise, the population evolves to the next generation returning to Step 2.
6. Numerical examples Two examples are presented here to show the efficiency of the genetic algorithm developed for hot forging optimisation. The objective is to design the preform shape and temperature of the work-piece such that, after forging, the required final part is obtained without defects. In one-step forging operations, the die shape is a prescribed datum. Two examples will be presented both considering the compression of a cylinder (20 mm in height by 25 mm in diameter) of aluminium alloy AL6016T6. The work-piece will be heated and the die will be considered at room temperature. The material behaviour is modelled considering the constitutive relations σ¯ = 600(¯ε + 0.003)0.056 MPa σ¯ = 100(¯ε + 0.003)0.25 MPa
(room temperature) (T = 660 K)
(19) (20)
experimentally carried out by Santos et al. [12]. A 50% height reduction is intended. Bilateral contact conditions are considered, the constant shear friction factor m is taken as 0.4 and the fraction of plastic work transformed
into heat is equal to 90%. The adopted material’s conductivity is k = 155 N/s K. The die matrix is considered rigid with no internal heat generation and an initial die temperature Tdie = 285 K. Since forging is symmetric about the vertical and the horizontal axis, the material deformation is also symmetric. To reduce the computation time, only one-quarter of the cross-section is considered for the finite element analysis. 6.1. Minimising barrelling in the upsetting of a cylinder It is well known that a disk, free of barrelling, cannot be produced from a cylindrical billet in a single-stage forging process using a pair of flat dies. Fig. 2 presents the 4-node linear element discretisation of the tool and work-piece as well as the deformed work-piece geometry after height reduction. As shown in Fig. 2, after forging the straight cylinder, the barrelling effect is quite remarkable. Our goal is to design the preform work-piece shape and to determine the initial work-piece temperature such that, after forging with a flat die, a disk with a minimum barrelling effect is produced, with minimum energy and where the temperature does not surpass an allowed maximum. Let us consider the shape of the free surface of the initial work-piece defined by a cubic B-spline curve. N control points determine the shape of the B-spline curve and the B-spline curve is bounded by its control polygon. For 2D problems each control point Pi , i = 1, . . . , N, has two co-ordinates (Pi (r), Pi (z)). The r-displacements of selected active control points of the B-spline function become the geometric design parameters. In this example, six control points were used to define the B-spline curve representing the preform die shape (Pi , i = 1, . . . , 6) corresponding to the first six components of the design vector b. The seventh component of the design vector b is the initial temperature of the work-piece, T0 . The optimisation problem has been solved using the developed genetic algorithm. The considered constraints were −4 ≤ bd ≤ 4 mm,
d = 1, . . . , 6
Fig. 2. Geometry and finite element mesh for the height reduction problem.
(21)
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361
Fig. 3. Evolution of the fitness function components for the optimisation process: upset of a cylinder.
473 ≤ T0 = b7 ≤ 623 K
(22)
and the maximum allowed temperature during forging was Ta = 723 K. The parameters considered in the genetic algorithm are Npop = 12 and Ne = 5 for the population and elite group size, respectively. The number of bits in binary codifying for the geometric and temperature design variables is Nbit = 5. The evolutionary process will stop when convergence is achieved, that is when the mean fitness of the six best individuals does not change during five consecutive generations. As for the objective function defined in Eq. (14), equal weight was given to the energy and shape fitness components. After 46 generations the evolutionary process had converged. The evolution of the distance and energy components of the objective function given by Eq. (14) is presented in Fig. 3. The energy component decreases not only with the decrease of the initial volume of the work-piece but also with the increase of the initial temperature of the work-piece. The compromise between distance and energy components is given by the fitness evolution of the best solution along each generation presented in Fig. 3. The developed algorithm is very efficient in determining the optimum. The undeformed and deformed shapes for the optimal solution is shown in Fig. 4. The optimal initial solution corresponds to bT = [−0.33, 3.04, 0.26, −0.20, −1.74, −1.59, 521] where the first component corresponds to the r-displacements in mm of the B-spline control point at the highest value z = 10 mm and the sixth value at the equatorial
plane z = 0 mm. The seventh component is the initial temperature of the work-piece, Toptimal = 521 K. After forging, the barrelling effect is hardly seen. The cross-section of the resulting disk is almost rectangular so that the optimised result is very close to the desired shape. If all material outside the minimum radius were to be trimmed away, this would account for material savings by using the optimised design. The temperature distribution of the deformed optimal solution and die is presented in Fig. 5. No violation of the temperature constraint was detected in the optimal forging. The temperature transfer between work-piece and die is observable. Before forging the die temperature is considered constant and at room temperature Tdie = 285 K and according to the optimal solution the initial work-piece temperature is Toptimal = 521 K. After forging the highest achieved temperature in the work-piece is Tmax = 530 K at the equatorial plane and lowest Tmin = 450 K at the die–work-piece contact area. The decrease in temperature is due to the heat transfer between die and work-piece. 6.2. Preform design for efficient near-net shape manufacturing in upset forging In the second example, the design of a preform shape that encompasses a prescribed shape defined by a polynomial function is considered. After forging the prescribed shape rp is taken as a quadratic function of the height z from the equatorial plane of the final product:
z 2 (mm) (23) rp (z) = 15 + 2 5.5
Fig. 4. Optimal solution for the example of minimising the barrelling effect.
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Fig. 5. Temperature distribution for the optimal solution of minimising the barrelling effect.
Fig. 6. Evolution of the fitness components for the optimisation process: near-net shape example.
Again the free surface of the initial work-piece is parametrised using a cubic B-spline as in the previous example. Same genetic parameters are adopted for this second example. Fig. 6 represents the variation of energy, distance and fitness of the best solution along generations. The two components, energy given by Eq. (12) and distance as measured by Eq. (13), were equally weighted to calculate the objective function. The fitness is shown in Fig. 6 and as defined by Eq. (17), it corresponds to the fitness of the best solution
in each generation. Fig. 7 presents the undeformed and deformed shapes for the optimal solution corresponding to bT = [4.13, −2.43, − 4.80, −6.96, −10.56, −10.83, 593], where the first component corresponds to the r-displacements in mm of the B-spline control points and the seventh component to the initial temperature of the work-piece, Toptimal = 593 K. After forging, the intended shape is obtained. The developed
Fig. 7. Undeformed and deformed shapes for the optimal solution for the near-net shape example.
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Fig. 8. Temperature distribution for the optimal solution of the near-net shape example.
genetic algorithm found very efficiently an optimal solution for this complicated shape problem. The temperature distribution of the deformed optimal solution and die is presented in Fig. 8. Again, no violation of the temperature constraint was detected in the optimal forging. The temperature transfer between work-piece and die is quite significant. After forging the highest achieved temperature in the work-piece is Tmax = 590 K at the equatorial plane and lowest Tmin = 520 K at the die–work-piece contact area. The decrease in temperature is due to the temperature transfer between die and work-piece and to such an extensive contact area at the beginning of the forging.
7. Concluding remarks This paper describes the development of a new evolutionary search model for optimal design of hot metal-forming processes. The developed scheme is based on a genetic algorithm supported by an elitist search strategy and applied to a population with fixed dimension. The principal aspects of the approach are high fitness improvement and diversity based on small populations. The method was applied to design the optimal work-piece shape and temperature in hot forging processes. The two numerical examples of an inverse problem were efficiently solved considering the developed method: minimising the barrelling effect when upsetting a cylinder and fitting a prescribed polynomial free boundary shape. The final shapes are very close to the intended ones under energy-minimising constraints. To simulate a 50% height reduction with a straight die, using our developed thermal–mechanical code and a Pentium III takes only a few seconds. Each implemented example took 2–4 h to give an optimal solution. The developed genetic algorithm presents some advantages over sensitivity-based methods: the algorithm is not
sensitivity-dependent, runs well even for discontinuous derivative fields, discrete design variables and does not introduce iteration-dependent numerical errors. It can be concluded that the proposed genetic approach is efficient and gives promising results. At the moment, our future efforts will be dedicated to the optimisation of multistage forging processes. The computer running time will be higher due to the contact iteration process of geometrically complicated preform dies. But the approach is aimed towards the design of industrial pieces where material properties and shape design play important roles and where it is much more expensive to perform trial and error experiments than numerical optimisation simulation. References [1] R.W. Wagoner, J.L. Chenot, Metal Forming Analysis, Cambridge University Press, Cambridge, UK, 2001. [2] J.M. César de Sá, L. Costa Sousa, M.L. Madureira, Simulation model for hot and cold forging by mixed methods including adaptive mesh refinement, Eng. Comput. 13 (2–4) (1996) 339–360. [3] N. Rebelo, S. Kobayashi, A coupled analysis of viscoplastic deformation and heat transfer—applications, Int. J. Mech. Sci. 22 (1989) 707–718. [4] L.C. Sousa, C.F. Castro, C.A.C. António, A.D. Santos, Inverse methods in design of industrial forging processes, J. Mater. Process. Technol. 128 (2002) 266–273. [5] A.M. Maniatty, M.-F. Chen, Shape sensitivity analysis for steady metal-forming processes, Int. J. Num. Meth. Eng. 39 (1996) 1199– 1217. [6] M.S. Joun, S.M. Hwang, Die shape optimal design in three-dimensional shape metal extrusion by the finite element method, Int. J. Num. Meth. Eng. 41 (1998) 311–335. [7] J.-L. Chenot, E. Massoni, L. Fourment, Inverse problems in finite element simulation of metal forming processes, Eng. Comput. 13 (2–4) (1996) 190–225. [8] C.F. Castro, L.C. Sousa, C.A.C. António, J.M.A. César de Sá, An efficient algorithm to estimate optimal preform die shape parameters in forging, Eng. Comput. 18 (7–8) (2001) 1057–1077.
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[9] T. Rodic, I. Gresovnik, D. Jolovsek, J. Korelc, Optimisation and prestressing of cold forging die by using symbolic templates, in: Proceedings of the European Conference on Computational Mechanics (ECCM99), Munchen, Germany, 1999. [10] K. DeJong, Evolutionary computation: recent developments and open issues, in: K. Miettinen, M. Mäkelä, P. Neittaanmaki, J. Périaux (Eds.), Proceedings of the Evolutionary Algorithms in Engineering Computer Science (EUROGEN99), University of Jyväskylä, Finland, Wiley, Chichester, 1999, pp. 43–54.
[11] W.M. Spears, S.K. DeJong, On the virtues of parametrised uniform crossover, in: R.K. Belew, L.B. Booker (Eds.), Proceedings of Fourth International Conference on Genetic Algorithms, Morgan Kaufmann, San Mateo, CA, 1991, pp. 230–236. [12] A.D. Santos, J. Ferreira Duarte, A. Reis, A. Barata da Rocha, R. Neto, R. Paiva, Finite element simulation of closed die forging for prediction of material behaviour and optimisation of process, in: Proceedings of the Second ESAFORM Conference on Material Forming, Guimarães, Portugal, 13–17 April 1999, pp. 295–298.