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Optimization of metal forming process parameters with damage minimization
Journal of Materials Processing Technology 80 – 81 (1998) 597 – 601
Optimization of metal forming process parameters with damage minimization P. Picart *, O. Ghouati, J.C. Gelin Laboratoire Me´canique Applique´e, R. Chale´at, UMR CNRS 6604, 24 rue de l’Epitaphe, 25030 Besanc¸on Cedex, France
Abstract A primary objective in metal forming is designing the geometry of the workpiece and dies in order to achieve a part with a given shape and microstructure. The problem is usually handled by extensive trial and error, using simulations or test forgings, or both, until an acceptable result is obtained. The purpose of this work is to present and apply an optimization technique to metal forming design problems with damage occurrence. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Optimization; Finite element; Simulation; Metal forming; Damage; Sensitivity
1. Introduction Due to its high productivity, and therefore its suitability for mass production, metal forming plays a decisive role in the manufacturing of mechanical components. Actually, the design and the manufacture of new components is still based on know-how, empirical and analytical methods confirmed by past and actual experiments and on trial and error, leading to the choice of the best adapted process, the associated adjustments of the tools and the forming parameters for a final expected product. Efficient development and optimization of metal forming technology need more and more use of simulation and finite element analysis. Many improvements have been accomplished in the field of numerical simulations of metal forming processes and various codes based on the finite element method (FEM) have been developed. These codes allow the prediction of the deformed shape of the workpiece as well as the distribution of strain, stress and internal variables. These codes can also account for all the specifications concerning the forming process under consideration. These specifications that can be represented by process parameters are for instance, constitutive equations describing the material behavior, operating conditions such as temperature, strain rate, friction condition, etc., * Corresponding author. 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00167-8
or the geometry of the process, including the shape of both the tools and the workpiece. The achieved improvements have made the FEM an accurate tool for forming process simulation. The up-to-date modeling of large strain in metal forming also requires taking into account the constitution of the material microstructure. Notably, for engineering materials such as steel and aluminum alloys, the initial microstructures are often less or more microvoided. Forging modeling of void-free material has been achieved already. Global results are often accurate enough for the daily management of forging plants. Considering refined analyses of workpieces to be forged, the microstructure is influential and so far, adapted porous material constitutive relations have been proposed. Up to now it seems that the most adapted to the problem is the constitutive potential for porous material, as defined in [1,2]. A realistic prediction of damage evolution can be obtained during the forming sequence by a constitutive model that first includes the nucleation of new microvoids by decohesion at inclusion–matrix interfaces, or by inclusion or matrix fractures, and secondly, as the strenghtening of the material continues, the macroscopic effective strain increases as does the growth of the existing microvoids. The analysis of microvoid effect on the plastic behavior has been implemented using a global model that involves an elasto-plastic potential proposed by Gurson. An elastic prediction with radial return on the yield
598
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
surface algorithm is used to compute stress and strain increments. A sensitivity analysis of the material parameters involved in the modelization has been achieved, and the influence of material parameters has been measured by a relative variation of the local and global mechanical variables for reference and industrial cases [3,4]. To avoid defect initiation during forming sequences, and preserve the mechanical and physical properties of the final product, a damage minimization must be performed. It was possible to consider the design of metal forming processes by formulating this problem as an inverse one [5]. This is performed by coupling the FEM with an optimization technique allowing the adjustment of process parameters in order to fulfill specified criteria. In this way, the control of process parameters is possible and allows us to obtain products with the desired specifications, such as the shape and/or material characteristics. In this paper an inverse method for the design of forming processes is proposed. The optimization uses an augmented Lagrangian technique that allows for constraint functionals as well as for constraints on process parameters. The direct problem is solved using a FEM providing the solution for the deformation, the stress and the internal variable fields for different forming processes under various mechanical conditions. The inverse method makes use of a gradient-base optimization technique. The sensitivity analysis required by the optimization is developed on a direct differentiation method that provides accurate sensitivity information with a low computational cost [6]. Application of this inverse process design method to the optimization of initial workpiece geometry in order to minimize damage evolution is proposed.
2. Optimization
2.1. Formulation of the mechanical problem Consider a workpiece S occupying a domain V0 with boundary G0 in the initial configuration, and the domain V with boundary G in the current configuration. On boundaries Gu and Gs, the displacements u and the traction forces fs are prescribed. The workpiece undergoes a deformation process specified by the displacement field ut (p, t), where t is time or a loading parameter for the quasi-static process and p is the vector of process parameters. In the finite deformation process, the initial configuration V0 is deformed into V with: xt = x0 + ut The weak form associated with the equilibrium equations for the workpiece S is used in the FEM. The solution of this nonlinear equation is performed using a
Newton iterative scheme. A tangent operator consistent with the algorithm is evaluated to determine stress and internal variables. In the displacement-based FEM, the discretized form of the problem is used to calculate an estimated incremental displacement dun + 1 between tn and tn + 1 following an implicit iterative scheme; (i) s (i) [KT ](i) n + 1du n + 1 = [Fext]− [Fs (9n + 1ut (p))]n + 1
(1)
where [Fext] represents the vector of external loads, [Fs (9sn + 1ut (p))] the vector of internal loads, 9sn + 1ut (p) the symmetrical gradient of ut (p) and [KT ] the tangent stiffness matrix. A hyperelastic formulation of the constitutive equations is used in the elasto-plastic case to determine the stress tensor and internal variables following a prediction–correction scheme and a multiplicative split of the deformation gradient.
2.2. Damage modelization The measurement of the microvoid evolution is made by a single scalar parameter f, which represents the current void volume fraction. Based on many microstructural observations of strained steel specimens, porosity changes mainly with strain gradient. So far, strain-controlled equations for void nucleation rate are therefore considered in the typical form: (2)
f: n = Ao; m
with o; m being the effective plastic strain rate in the material matrix and A depends, in some complicated fashion, on the statistics of the particle spacing and is chosen so that the initiation of the cavities follows a normal distribution about some mean value on of the effective plastic strain as A=
fn Sn 2p
exp −
1 om − on 2 Sn
n 2
(3)
In this relationship, om is the current effective plastic strain in the matrix, Sn is the S.D. of the second phase particles Gauss distribution and fn is the nucleating void volume fraction in relation to the inclusions’ volume fraction. Current reference values for on, fn and Sn are, respectively, 0.2, 0.04 and 0.1 [1–4]. Considering now the strained porous material, the voids are assumed to grow as spherical cavities once they nucleate, and the material of the matrix is considered incompressible, so the rate of void volume fraction due to microvoids growth is well represented by: f: g = (1− f)D p : I
(4)
where D p is the material plastic strain rate tensor and I the second-order identity tensor. The total rate of increase of the void volume fraction in a mechanical structure until incipient coalescence is given by f: = f: n + f: g
(5)
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
599
The damage elasto-plastic flow is described from the Gurson model by f(s, sm, f)=
3 sm s 2eq +2q1 cosh q2 −(1 +q3 f 2) 2 sM s 2m
in which sM is the effective isotropic yield stress of the matrix, sm the macroscopic hydrostatic stress and seq the von Mises effective stress. q1, q2 and q3 are three specific parameters of the material introduced by Tvergaard for which reference values are, respectively, 1.5, 1 and 2.25. The macroscopic dissipated power is equal to that dissipated in the matrix; s : D p = (1− f)sMo; m
(6)
The material plastic strain rate tensor is given by the flow rule D p =l:
(f (s
(7)
with s the Cauchy stress tensor and l: the plastic multiplier determined from the consistency relationship f =0.
Fig. 2. Final void volume fraction distribution in the notched part of the specimen.
2.3. Formulation of the design problem The solution of the design problem can be expressed in the form of a nonlinear mathematical minimization problem as [7,8] : (8)
S(p, u) min p subject to constraints g i (p, u)= 0 (15 i5 n c) h j (p,u)5 0 (15j 5n ic)
where S is the cost function to optimize, gi and hj are the constraint functions which usually concern the nodal displacements, internal variables as strains and stresses, technological limitations etc., nc is the number of equality constraints and nic is the number of inequality constraints. To solve this constrained optimization problem, an augmented Lagrangian method is used by the definition of the Lagrangian associated with the problem as follows: Lr(p, u, l)= S(p, u)+A1(p, l, r)+ A2(p, l, r) with nic
{A 1(p, l, r)= % [lici + rc 2i ]
i=1
c i = max hi (p),
− li 2r
n
(9)
nc
{A 2(p, l, r)= % [li gi + rg 2i ] i=1
Fig. 1. Geometry and dimensions for the notched tensile test.
A gradient-based method is used to minimize Lr(p, u, l) with respect to p. The iterative solution process is as follows. At the kth iteration, for r (k) and l (k) fixed, an estimation p* for the design parameters is evaluated in a way to minimize Lr(p, u, l). The Lagrange multipliers are updated using the equations:
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
600
Fig. 3. Evolution of x-displacement for boundary nodes in the notched part of the specimen versus function evaluations. (k) + 1) l (k =l (k) gi (p*) for i= 1, n c i i +2r (k + 1) (k) (k) (k) (k) li =l i +2r max[hi (p*), (−l i )/(2r )] for i=1, n ic
(10) The penalty coefficients are evaluated as follows: r (k + 1) =r (k)b
with b \1
The solution process needs the evaluation of the gradient of the Lagrangian function and thus of both the cost function S and the constraint functions gi and hj. This is performed by means of a sensitivity analysis.
2.4. Sensiti6ity analysis A sensitivity analysis for metal forming problems has been developed on the basis of direct differentiation of the solution process for the mechanical problem [9]. The increment of displacement dun + 1 is obtained when the equilibrium is satisfied, which from Eq. (1) can be written as: Rn + 1 =[Fext]−[Fs (9sn + 1ut (p))](i) n + 1 =0
(11)
Sensivity of displacements is obtained, in the implicit case, by solving the following equation
d(dun + 1) ( (K = − [KT ] − 1 [Fs ]n + 1 − T dun + 1 dp (p (p
n
(12)
The inverse of the stiffness matrix can be obtained from the solution of the direct problem. It remains therefore, to evaluate the sensitivity of the internal load vector as follows:
dFs = dp
&
dBT sn dV + V(p) dp
&
&
BT V(p)
dsn dV dp
du BTsn n n dS + dp G(p) with n the external unit vector to the boundary G of the workpiece and B the strain interpolation matrix. Then, sensitivity of the external load vector is evaluated as d[Fext]n dNT df NT s dS = fs dS + dp dp dp Gs(p) Gs(p)
&
&
&
dun n dL dp (Gs(p) with N the displacement interpolation matrix. Sensitivity of the Cauchy stress tensor s is needed and evaluated in a consistent manner with the algorithm of evaluation of stresses and internal variables [10]. +
NTfs
3. Application To illustrate the use of the optimization algorithm described above, a reference numerical application addresses the optimization of the initial geometry in the tensile test of a notched specimen in order to minimize the void volume fraction. In this study, two-dimensionnal simulations in plane stress conditions are used with POLYFORM code developed by the authors. Fig. 1 represents a schematic diagram of the notched tensile test for a plate specimen. The material is mild steel with an elasto-plastic behavior and an isotropic hardening law represented by a
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
Swift-type flow rule as follows: sM =650(1+o 0.18 m )
N mm − 2
The finite element simulation uses plane stress triangular shell elements. For evident symmetry reasons, only one quater of the test is considered. The tensile test of a notched plate specimen is analyzed when porous material is involved. Porosity is introduced by the nucleation of new voids with the reference parameter values. The initial void volume fraction is 0.001. A first computation is achieved for a 5 mm axial prescribed displacement on each side of the notched plate. Fig. 2 shows the final void volume fraction map in the notched part of the specimen. To illustrate the optimization procedure, the process parameter considered here is the initial geometry of the specimen, especially the displacement along x-axis of the nodes located in the notched part of the specimen. The design parameter is the void volume fraction f. The cost function S introduced in Eq. (8) represents the S.D. of the void volume fraction distribution in all specimen. The constraint 05f is introduced in the solution process to ensure that the void volume fraction stays in a realistic range. After optimization, the notched part of specimen have vanished. Fig. 3 represents the evolution of x-displacement for the boundary nodes in the notched part of the specimen versus function evaluations during the optimization process.
4. Conclusions An algorithm for process optimization is presented.
.
601
This algorithm consists of an optimization technique based on a gradient method and a FEM. The FEM used allows a simulation of the process studied with sufficient accuracy making the adjustement of the design parameters possible. Associated to this algorithm, a sensitivity analysis is developed on the basis of a direct differentiation of the finite elment formulation. A reference application is presented to demonstrate efficiency of the proposed method for metal forming optimization relative to damage occurrence.
References [1] P. Picart, J. Oudin, B. Bennani, J. Mater. Proc. Technol. 32 (1992) 179 – 188. [2] B. Bennani, P. Picart, J. Oudin, Int. J. Damage Mech. 2 (2) (1993) 118 – 136. [3] L. Lazzarotto, P. Picart, J. Oudin, Int. J. Damage Mech. 5 (3) (1996) 259 – 277. [4] P. Picart, G. Piechel, J. Oudin, Damage influence in the finite element computations for large strains elasto-plastic mechanical structures, Proc. 3rd Int. Conf. on Material Processing Defects, Cachan, France, 1997. [5] A. Tarantola, Inverse Problem Theory, Elsevier, Amsterdam, 1987. [6] D.A. Tortorelli, P. Michaleris, Inverse Prob. Eng. 1 (1994) 71 – 105. [7] M. Kegl, B.J. Butinar, M.M. Oblak, Int. J. Numer. Methods Eng. 38 (1995) 3227 – 3242. [8] O. Ghouati, J.C. Gelin, Sensitivity analysis and optimization of shape and process parameters in metal forming, Engineering Systems Design and Analysis Conference ‘96, Vol. 3, Montpellier, France, 1996, pp. 221 – 226. [9] J.C. Gelin, O. Ghouati, Comput. Mech. 16 (3) (1995) 143–151. [10] L.J. Leu, S. Mukherjee, Int. J. Numer. Methods Eng. 37 (1994) 3843 – 3868.
Optimization of metal forming process parameters with damage minimization P. Picart *, O. Ghouati, J.C. Gelin Laboratoire Me´canique Applique´e, R. Chale´at, UMR CNRS 6604, 24 rue de l’Epitaphe, 25030 Besanc¸on Cedex, France
Abstract A primary objective in metal forming is designing the geometry of the workpiece and dies in order to achieve a part with a given shape and microstructure. The problem is usually handled by extensive trial and error, using simulations or test forgings, or both, until an acceptable result is obtained. The purpose of this work is to present and apply an optimization technique to metal forming design problems with damage occurrence. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Optimization; Finite element; Simulation; Metal forming; Damage; Sensitivity
1. Introduction Due to its high productivity, and therefore its suitability for mass production, metal forming plays a decisive role in the manufacturing of mechanical components. Actually, the design and the manufacture of new components is still based on know-how, empirical and analytical methods confirmed by past and actual experiments and on trial and error, leading to the choice of the best adapted process, the associated adjustments of the tools and the forming parameters for a final expected product. Efficient development and optimization of metal forming technology need more and more use of simulation and finite element analysis. Many improvements have been accomplished in the field of numerical simulations of metal forming processes and various codes based on the finite element method (FEM) have been developed. These codes allow the prediction of the deformed shape of the workpiece as well as the distribution of strain, stress and internal variables. These codes can also account for all the specifications concerning the forming process under consideration. These specifications that can be represented by process parameters are for instance, constitutive equations describing the material behavior, operating conditions such as temperature, strain rate, friction condition, etc., * Corresponding author. 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00167-8
or the geometry of the process, including the shape of both the tools and the workpiece. The achieved improvements have made the FEM an accurate tool for forming process simulation. The up-to-date modeling of large strain in metal forming also requires taking into account the constitution of the material microstructure. Notably, for engineering materials such as steel and aluminum alloys, the initial microstructures are often less or more microvoided. Forging modeling of void-free material has been achieved already. Global results are often accurate enough for the daily management of forging plants. Considering refined analyses of workpieces to be forged, the microstructure is influential and so far, adapted porous material constitutive relations have been proposed. Up to now it seems that the most adapted to the problem is the constitutive potential for porous material, as defined in [1,2]. A realistic prediction of damage evolution can be obtained during the forming sequence by a constitutive model that first includes the nucleation of new microvoids by decohesion at inclusion–matrix interfaces, or by inclusion or matrix fractures, and secondly, as the strenghtening of the material continues, the macroscopic effective strain increases as does the growth of the existing microvoids. The analysis of microvoid effect on the plastic behavior has been implemented using a global model that involves an elasto-plastic potential proposed by Gurson. An elastic prediction with radial return on the yield
598
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
surface algorithm is used to compute stress and strain increments. A sensitivity analysis of the material parameters involved in the modelization has been achieved, and the influence of material parameters has been measured by a relative variation of the local and global mechanical variables for reference and industrial cases [3,4]. To avoid defect initiation during forming sequences, and preserve the mechanical and physical properties of the final product, a damage minimization must be performed. It was possible to consider the design of metal forming processes by formulating this problem as an inverse one [5]. This is performed by coupling the FEM with an optimization technique allowing the adjustment of process parameters in order to fulfill specified criteria. In this way, the control of process parameters is possible and allows us to obtain products with the desired specifications, such as the shape and/or material characteristics. In this paper an inverse method for the design of forming processes is proposed. The optimization uses an augmented Lagrangian technique that allows for constraint functionals as well as for constraints on process parameters. The direct problem is solved using a FEM providing the solution for the deformation, the stress and the internal variable fields for different forming processes under various mechanical conditions. The inverse method makes use of a gradient-base optimization technique. The sensitivity analysis required by the optimization is developed on a direct differentiation method that provides accurate sensitivity information with a low computational cost [6]. Application of this inverse process design method to the optimization of initial workpiece geometry in order to minimize damage evolution is proposed.
2. Optimization
2.1. Formulation of the mechanical problem Consider a workpiece S occupying a domain V0 with boundary G0 in the initial configuration, and the domain V with boundary G in the current configuration. On boundaries Gu and Gs, the displacements u and the traction forces fs are prescribed. The workpiece undergoes a deformation process specified by the displacement field ut (p, t), where t is time or a loading parameter for the quasi-static process and p is the vector of process parameters. In the finite deformation process, the initial configuration V0 is deformed into V with: xt = x0 + ut The weak form associated with the equilibrium equations for the workpiece S is used in the FEM. The solution of this nonlinear equation is performed using a
Newton iterative scheme. A tangent operator consistent with the algorithm is evaluated to determine stress and internal variables. In the displacement-based FEM, the discretized form of the problem is used to calculate an estimated incremental displacement dun + 1 between tn and tn + 1 following an implicit iterative scheme; (i) s (i) [KT ](i) n + 1du n + 1 = [Fext]− [Fs (9n + 1ut (p))]n + 1
(1)
where [Fext] represents the vector of external loads, [Fs (9sn + 1ut (p))] the vector of internal loads, 9sn + 1ut (p) the symmetrical gradient of ut (p) and [KT ] the tangent stiffness matrix. A hyperelastic formulation of the constitutive equations is used in the elasto-plastic case to determine the stress tensor and internal variables following a prediction–correction scheme and a multiplicative split of the deformation gradient.
2.2. Damage modelization The measurement of the microvoid evolution is made by a single scalar parameter f, which represents the current void volume fraction. Based on many microstructural observations of strained steel specimens, porosity changes mainly with strain gradient. So far, strain-controlled equations for void nucleation rate are therefore considered in the typical form: (2)
f: n = Ao; m
with o; m being the effective plastic strain rate in the material matrix and A depends, in some complicated fashion, on the statistics of the particle spacing and is chosen so that the initiation of the cavities follows a normal distribution about some mean value on of the effective plastic strain as A=
fn Sn 2p
exp −
1 om − on 2 Sn
n 2
(3)
In this relationship, om is the current effective plastic strain in the matrix, Sn is the S.D. of the second phase particles Gauss distribution and fn is the nucleating void volume fraction in relation to the inclusions’ volume fraction. Current reference values for on, fn and Sn are, respectively, 0.2, 0.04 and 0.1 [1–4]. Considering now the strained porous material, the voids are assumed to grow as spherical cavities once they nucleate, and the material of the matrix is considered incompressible, so the rate of void volume fraction due to microvoids growth is well represented by: f: g = (1− f)D p : I
(4)
where D p is the material plastic strain rate tensor and I the second-order identity tensor. The total rate of increase of the void volume fraction in a mechanical structure until incipient coalescence is given by f: = f: n + f: g
(5)
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
599
The damage elasto-plastic flow is described from the Gurson model by f(s, sm, f)=
3 sm s 2eq +2q1 cosh q2 −(1 +q3 f 2) 2 sM s 2m
in which sM is the effective isotropic yield stress of the matrix, sm the macroscopic hydrostatic stress and seq the von Mises effective stress. q1, q2 and q3 are three specific parameters of the material introduced by Tvergaard for which reference values are, respectively, 1.5, 1 and 2.25. The macroscopic dissipated power is equal to that dissipated in the matrix; s : D p = (1− f)sMo; m
(6)
The material plastic strain rate tensor is given by the flow rule D p =l:
(f (s
(7)
with s the Cauchy stress tensor and l: the plastic multiplier determined from the consistency relationship f =0.
Fig. 2. Final void volume fraction distribution in the notched part of the specimen.
2.3. Formulation of the design problem The solution of the design problem can be expressed in the form of a nonlinear mathematical minimization problem as [7,8] : (8)
S(p, u) min p subject to constraints g i (p, u)= 0 (15 i5 n c) h j (p,u)5 0 (15j 5n ic)
where S is the cost function to optimize, gi and hj are the constraint functions which usually concern the nodal displacements, internal variables as strains and stresses, technological limitations etc., nc is the number of equality constraints and nic is the number of inequality constraints. To solve this constrained optimization problem, an augmented Lagrangian method is used by the definition of the Lagrangian associated with the problem as follows: Lr(p, u, l)= S(p, u)+A1(p, l, r)+ A2(p, l, r) with nic
{A 1(p, l, r)= % [lici + rc 2i ]
i=1
c i = max hi (p),
− li 2r
n
(9)
nc
{A 2(p, l, r)= % [li gi + rg 2i ] i=1
Fig. 1. Geometry and dimensions for the notched tensile test.
A gradient-based method is used to minimize Lr(p, u, l) with respect to p. The iterative solution process is as follows. At the kth iteration, for r (k) and l (k) fixed, an estimation p* for the design parameters is evaluated in a way to minimize Lr(p, u, l). The Lagrange multipliers are updated using the equations:
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
600
Fig. 3. Evolution of x-displacement for boundary nodes in the notched part of the specimen versus function evaluations. (k) + 1) l (k =l (k) gi (p*) for i= 1, n c i i +2r (k + 1) (k) (k) (k) (k) li =l i +2r max[hi (p*), (−l i )/(2r )] for i=1, n ic
(10) The penalty coefficients are evaluated as follows: r (k + 1) =r (k)b
with b \1
The solution process needs the evaluation of the gradient of the Lagrangian function and thus of both the cost function S and the constraint functions gi and hj. This is performed by means of a sensitivity analysis.
2.4. Sensiti6ity analysis A sensitivity analysis for metal forming problems has been developed on the basis of direct differentiation of the solution process for the mechanical problem [9]. The increment of displacement dun + 1 is obtained when the equilibrium is satisfied, which from Eq. (1) can be written as: Rn + 1 =[Fext]−[Fs (9sn + 1ut (p))](i) n + 1 =0
(11)
Sensivity of displacements is obtained, in the implicit case, by solving the following equation
d(dun + 1) ( (K = − [KT ] − 1 [Fs ]n + 1 − T dun + 1 dp (p (p
n
(12)
The inverse of the stiffness matrix can be obtained from the solution of the direct problem. It remains therefore, to evaluate the sensitivity of the internal load vector as follows:
dFs = dp
&
dBT sn dV + V(p) dp
&
&
BT V(p)
dsn dV dp
du BTsn n n dS + dp G(p) with n the external unit vector to the boundary G of the workpiece and B the strain interpolation matrix. Then, sensitivity of the external load vector is evaluated as d[Fext]n dNT df NT s dS = fs dS + dp dp dp Gs(p) Gs(p)
&
&
&
dun n dL dp (Gs(p) with N the displacement interpolation matrix. Sensitivity of the Cauchy stress tensor s is needed and evaluated in a consistent manner with the algorithm of evaluation of stresses and internal variables [10]. +
NTfs
3. Application To illustrate the use of the optimization algorithm described above, a reference numerical application addresses the optimization of the initial geometry in the tensile test of a notched specimen in order to minimize the void volume fraction. In this study, two-dimensionnal simulations in plane stress conditions are used with POLYFORM code developed by the authors. Fig. 1 represents a schematic diagram of the notched tensile test for a plate specimen. The material is mild steel with an elasto-plastic behavior and an isotropic hardening law represented by a
P. Picart et al. / Journal of Materials Processing Technology 80–81 (1998) 597–601
Swift-type flow rule as follows: sM =650(1+o 0.18 m )
N mm − 2
The finite element simulation uses plane stress triangular shell elements. For evident symmetry reasons, only one quater of the test is considered. The tensile test of a notched plate specimen is analyzed when porous material is involved. Porosity is introduced by the nucleation of new voids with the reference parameter values. The initial void volume fraction is 0.001. A first computation is achieved for a 5 mm axial prescribed displacement on each side of the notched plate. Fig. 2 shows the final void volume fraction map in the notched part of the specimen. To illustrate the optimization procedure, the process parameter considered here is the initial geometry of the specimen, especially the displacement along x-axis of the nodes located in the notched part of the specimen. The design parameter is the void volume fraction f. The cost function S introduced in Eq. (8) represents the S.D. of the void volume fraction distribution in all specimen. The constraint 05f is introduced in the solution process to ensure that the void volume fraction stays in a realistic range. After optimization, the notched part of specimen have vanished. Fig. 3 represents the evolution of x-displacement for the boundary nodes in the notched part of the specimen versus function evaluations during the optimization process.
4. Conclusions An algorithm for process optimization is presented.
.
601
This algorithm consists of an optimization technique based on a gradient method and a FEM. The FEM used allows a simulation of the process studied with sufficient accuracy making the adjustement of the design parameters possible. Associated to this algorithm, a sensitivity analysis is developed on the basis of a direct differentiation of the finite elment formulation. A reference application is presented to demonstrate efficiency of the proposed method for metal forming optimization relative to damage occurrence.
References [1] P. Picart, J. Oudin, B. Bennani, J. Mater. Proc. Technol. 32 (1992) 179 – 188. [2] B. Bennani, P. Picart, J. Oudin, Int. J. Damage Mech. 2 (2) (1993) 118 – 136. [3] L. Lazzarotto, P. Picart, J. Oudin, Int. J. Damage Mech. 5 (3) (1996) 259 – 277. [4] P. Picart, G. Piechel, J. Oudin, Damage influence in the finite element computations for large strains elasto-plastic mechanical structures, Proc. 3rd Int. Conf. on Material Processing Defects, Cachan, France, 1997. [5] A. Tarantola, Inverse Problem Theory, Elsevier, Amsterdam, 1987. [6] D.A. Tortorelli, P. Michaleris, Inverse Prob. Eng. 1 (1994) 71 – 105. [7] M. Kegl, B.J. Butinar, M.M. Oblak, Int. J. Numer. Methods Eng. 38 (1995) 3227 – 3242. [8] O. Ghouati, J.C. Gelin, Sensitivity analysis and optimization of shape and process parameters in metal forming, Engineering Systems Design and Analysis Conference ‘96, Vol. 3, Montpellier, France, 1996, pp. 221 – 226. [9] J.C. Gelin, O. Ghouati, Comput. Mech. 16 (3) (1995) 143–151. [10] L.J. Leu, S. Mukherjee, Int. J. Numer. Methods Eng. 37 (1994) 3843 – 3868.