Optimal design of powder compaction processes via genetic algorithm technique

Finite Elements in Analysis and Design 46 (2010) 843–861

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Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel

Optimal design of powder compaction processes via genetic algorithm technique A.R. Khoei n, Sh. Keshavarz, S.O.R. Biabanaki Center of Excellence in Structures and Earthquake Engineering, Department of Civil Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran

a r t i c l e in fo

abstract

Article history: Received 24 November 2009 Received in revised form 10 May 2010 Accepted 23 May 2010 Available online 8 June 2010

In this paper, an optimal design is performed for powder die-pressing process based on the genetic algorithm approach. It includes the shape optimization of powder component, the optimal design of punch movements, and the friction optimization of powder–tool interface. The genetic algorithm is employed to perform an optimal design based on a fixed-length vector of design variables. The technique is used to obtain the desired optimal compacted component by verifying the prescribed constraints. The numerical modeling of powder compaction simulation is applied based on a large deformation formulation, powder plasticity behavior, and frictional contact algorithm. A Lagrangian finite element formulation is employed for large powder deformations. A cap plasticity model is used in numerical simulation of nonlinear powder behavior. The influence of powder–tool friction is simulated by a plasticity theory of friction to model sliding resistance at the powder–tool interface. Finally, numerical examples are analyzed to demonstrate the feasibility of the proposed optimization algorithm for designing powder components in the forming process of powder compaction. & 2010 Elsevier B.V. All rights reserved.

Keywords: Powder forming Optimal design Genetic algorithm Large deformation Cap plasticity Contact friction

1. Introduction Powder metallurgy considers the methods of producing commercial products from metallic powders by pressure. In powder compaction, the optimal design process is the main objective to obtain the desired compacted component based on the forming quality control, material saving, and reduction of manufacturing cost. One of the main difficulties that exist in the compaction forming process for powders includes a non-homogenous density distribution which has wide ranging effects on the final performance of the compacted part. The variation of density results in cracks and localized deformation in the compact, producing regions of high density surrounded by lower density material, leading to compact failure. Thus, the success of compaction forming depends on the ability of the process in imparting a uniform density distribution in the engineered part. In order to perform such analysis, the complex mechanisms of compaction process must be drawn into a mathematical formulation with the knowledge of material behavior. In powder metallurgy, numerical modeling of powder die-pressing has been extensively developed and presented by the senior author to control the properties of final product, including: the large deformations of powder compaction [1–3], powder plasticity behavior [4–8], contact friction between the powder and tools [9–11], and advanced computational algorithms in powder forming

n

Corresponding author. Tel.: + 98 21 6600 5818; fax: +98 21 6601 4828. E-mail address: [email protected] (A.R. Khoei).

0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2010.05.004

simulation [12–15]. The aim of this study is to develop an optimal design through the finite element simulation of powder compaction process. In the present paper, an optimization algorithm is developed for powder forming processes based on the genetic algorithm approach [16]. The goal of the optimization is to eliminate the work-piece defects that may arise during the powder compaction process. The genetic algorithm is used since it is suitable for discrete or noncontiguous variables, the derivative information is not required, a wide sampling of domain can be searched simultaneously, a list of optimal parameters can be provided, and all kinds of data, such as numerical data, experimental data, or analytical functions, can be employed. The genetic algorithm operator is used to increase the efficiency of the search algorithm and to design the optimal preform design of compacted component. The objective function of the optimization algorithm is associated with the quality of the final product. In order to demonstrate the efficiency of the proposed technique, several forming processes of powder compaction are simulated numerically. The plan of the paper is as follows; in Section 2, an overview of optimization methods are presented for forming processes. The genetic algorithm (GA) is described in this section for the desired optimal design by changing the design variables and verifying the prescribed constraints. In Section 3, the numerical modeling of powder compaction simulation is presented in the framework of a large deformation finite element (FE) formulation, the powder plasticity model based on single cap plasticity, and a frictional contact algorithm based on the penalty approach to model the

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powder–tool interface. The verification of powder die-pressing simulation is demonstrated in Section 4. In Section 5, numerical modeling of several complicated die geometries is presented to evaluate the capability of the GA in design optimization of powder compaction process. Finally, some concluding remarks are given in Section 6.

2. Optimal design technique Optimization technique based on the finite element method has become one of the international research interests in the field of metal forming [17]. The optimization methods include mathematical optimization, backward tracing, artificial intelligence, experiment optimization, and automatic control algorithm [18–21]. There are several optimization algorithms for metal forming processes, such as the simplex optimization algorithm [22], the genetic optimization algorithm [23], an inverse revolutionary search algorithm [24], and the gradient-based optimization algorithm [25]. The optimization process can be divided into three main tasks. The first step is to define the geometric and analytical model. In geometric model, the design variables are easily imposed and it allows an explicit integration with other design tools. The analytical model is used to obtain the structural response of the system subjected to external actions. The next step includes a sensitivity analysis to obtain a solution of the problem. Finally, an appropriate optimization algorithm needs to be performed to solve the problem in an effective and reliable way. The search of a robust optimization algorithm is necessary to survive in different environments. In this study, the genetic algorithm is employed due to the fact that it is theoretically and empirically proven to provide a robust search in complex space [26–27]. The optimal design method consists of the essential ingredients; including: the shape generation and control, mesh generation, nonlinear finite element analysis, sensitivity analysis, and design optimization. In order to make an optimal design, we first need to define the basic characteristics of the final product that need to be optimized. These features may be the shape or topological configuration, punch forces, and powder–tool friction. Consider an optimization process where we must optimize a set of variables either to maximize some target such as profit, or to minimize cost or some measure of error. The problem may be viewed as a black box with a series of control dials representing different parameters; the only output of the black box is a value returned by an evaluation function indicating how well a particular combination of parameter settings solves the optimization problem. There are interactions such that the combined effects of the parameters must be considered in order to optimize the output of the black box. In the genetic algorithm community, the interaction between variables is referred to as epistasis. The first assumption is that the variables representing parameters can be represented by bit strings. This means that the variables are discretized in a priori fashion, and that the range of the discretization corresponds to some power of 2. This assumes that the discretization provides enough resolution to make it possible to adjust the output with the desired level of precision. 2.1. Genetic algorithm technique A genetic algorithm comprises four main operations: fitness selection, crossover, elimination/substitution, and mutation. It starts from an initial population representing possible solutions of the problem. From these operators, crossover and mutation are applied to the population in improving the objective function value to form the new generation, in which the members have

higher quality. Each member in the population corresponds to a solution in the solution space. The quality of a member is represented by its fitness associated with the objective function value. The principle of survival of the fittest is taken as a rule in the search process. A genetic algorithm is called simple, if all operations stay constant over the course of the algorithm. Mutation models random change in the genetic information of creatures, and is inspired by random change of genetic information in living organisms. Crossover models the exchange of genetic information of creatures, and is inspired by exchange of genetic information in living organisms. Fitness selection models reproductive success of adapted organisms in their environment [16]. The first step in the implementation of genetic algorithm is to generate an initial population. In genetic algorithm, each member of population has a binary string, referred as a genotype, or a chromosome. In most cases the initial population is generated randomly. After creating an initial population, each string is evaluated and assigned a fitness value. The notion of evaluation and fitness are sometimes used interchangeably. However, it is useful to distinguish between the evaluation function and the fitness function used by a genetic algorithm. Here the evaluation function, or objective function provides a measure of performance with respect to a particular set of parameters. The fitness function transforms that measure of performance into an allocation of reproductive opportunities. The fitness can be assigned based on a string rank in the population, or by sampling method, such as tournament selection [16]. It is helpful to view the execution of the genetic algorithm as a two stage process. It starts with the current population. Selection is applied to the current population to create an intermediate population. The recombination and mutation are then applied to the intermediate population to create the next population. The process of going from the current population to the next population constitutes one generation in the execution of a genetic algorithm. In genetic algorithm, the probability that strings in the current population is duplicated and placed in the intermediate generation is proportion to their fitness. Fig. 1 describes the genetic algorithm employed here in the inserted area corresponding to the genetic operators. The operators are applied in the following sequence [21]: Initialization—A random generation of initial population is generated. Selection—This operator ranks the population according to the solution fitness. The progenitors are selected from two groups; one from the best-fitted group (elite) and another from the least fitted one. The selection operator is randomly made with an equal probability distribution for each solution. The two selection processes are independent and the based-fitness probability is considered to select each one of the parents. The selection operator chooses only the best solutions that will pass into the next population. Crossover—This operator generates the offspring population using couples of parents chosen with the selection operator. The crossover operator transforms two chromosomes (progenitors) into two new chromosomes (offspring) having genes from both progenitors. The offspring genetic material is obtained using a modification of the parameterized uniform crossover technique [21]. The technique is a multi-point combination approach applied to the binary string of the selected chromosomes. The crossover is applied with a predefined probability to the genetic material of the highest fitted chromosome. Elimination/substitution—This operator is applied to control the genetic similarity between individuals of the population. It is very important to avoid the incest resulting from the combination between parental solutions in the next generations. In this operator, the new individuals created by crossover are added to the original population. The enlarged population is newly ranked

A.R. Khoei et al. / Finite Elements in Analysis and Design 46 (2010) 843–861

Assign fitness and reproduction For every member numerical solution runs and fitness calculate then probability of living sized in proportion to its fitness.

Binary coding 11001 ………

101010 111000 …… …………. …… …… …………

Crossover 11 1100 00 0111

offspring1:110111 offspring1:001100

Mutation 110111

110101

110000

110001

Decoding and new generation

Convergence No

All strings converge to optimum solution

End of optimization Fig. 1. The employed genetic algorithm for shape optimization of powder compaction process.

according to their fitness and elimination of solutions with similar genetic properties and consequent substitution by new randomly generated individuals. The substitution and ranking updating are then followed by elimination corresponding to the deletion of the worst solutions. Mutation—The mutation 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. It is characterized by changing a bit of the binary string of a chromosome selected at random from the elite group. The mutation makes possible the exploration of previously unmapped space design regions and guarantees the diversity of the generated population. There are two different schemes for mutation; implicit mutation and adaptive mutation. In the first technique, new individuals are generated in a random way and inserted into the population. The new genetic material will be recombined with old individuals on further generations to produce the same effect as mutation. The second technique is

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based on an adaptive probability of mutation related with the chromosome fitness. 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. If the constraints of the problem are not satisfied then the evolutionary process continues. Consider that there is a feasible solution for the optimization 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 the second step.

2.2. Implementation of GA scheme In powder die-pressing, we optimize the process in different ways depending on various criteria, such as precise shape, die filling, friction coefficient, punch forces, or uniform distribution of mechanical properties. An optimal design criterion frequently encountered in powder die-pressing is the shape optimization to achieve the precise forming of finished component shape. Consider that we need to determine the optimal shape of preform work-piece at final compaction process, the geometry of the preform work-piece is considered as unknown design variables. The optimization algorithm is implemented to optimize the difference between the maximum and minimum values of stress on the final compacted component with the objective function defined as   Minimize f ðxÞ ¼ smax smin  ð1Þ where x denotes the design variable vector and the constraints ci(x) Z0, i¼1,2, ..., n are used to control the geometry of component. After identifying the design variables and search domain that represents the population phenotype, different solutions are represented by an appropriate code format called the genotype. The establishment of a code format is the main step of GA formulation. Here a binary code is developed with different number of bits for each design variable. The technique is performed by an initial population generated randomly using strings based on the design variable vector x. The numerical simulation is carried out for each string to calculate the fitness and reproduction. The process of reproduction is applied according to the value of objective function obtained at the end of compaction to copy the individual string. The reproduction operator is implemented in the algorithmic form based on a roulette wheel where each individual is represented by a space that proportionally corresponds to its fitness. The genetic algorithm described above generates a sequence of parameters to be tested using the system model, objective function, and the constraints. The genetic algorithm technique is employed to search the unconstrained objective functions. However, there are one or more constraints in powder compaction problems that need to be satisfied. Constraints are generally classified into equality, or inequality relations. Since equality constraints are subsumed into a system model, we deal with inequality constraints. We must evaluate the objective function, and check if any constraints are violated. If not, the parameter set is assigned the fitness value corresponding to the objective function evaluation. If constraints are violated, the solution is infeasible and thus has no fitness. As a result, we obtain information out of infeasible solutions by degrading their fitness ranking in relation to the degree of constraint violation, which is called as the penalty method. In this procedure, a constrained problem in optimization is transformed to an unconstrained problem by associating a cost, or penalty with all constraint

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violations. This cost is included in the objective function evaluation (1), and can be therefore transformed to the unconstrained form as Minimize

f ðxÞ þr

n X

f½ci ðxÞ

ð2Þ

i¼1

where f and r are the penalty function and penalty coefficient, in which a number of alternatives exists for the penalty function f. It must be noted that the implementation of an integrated genetic software into a highly nonlinear simulation, such as powder diepressing, plays an important role, since the nonlinear analysis cannot be restarted from the initial boundary condition due to large deformations and highly nonlinear material behavior. Here, an integrated genetic software is incorporated into a nonlinear FE code together with an automatic mesh generator to make the process of optimal design more efficient.

3. Powder compaction simulation The numerical simulation of powder compaction process is central to an understanding of the mechanics of powder behavior. Since powder compaction forming is a process involving large deformations, nonlinear material behavior, and friction, the numerical analysis of such a highly nonlinear process is a formidable computational problem. Hence, the large deformation analysis of metal powder during the cold compaction process is simulated by the finite element method based on an updated Lagrangian formulation. As the compaction process involves a very large reduction in volume, the formulation adopted is capable of representing this physical process. A generalized cap plasticity model is employed together with an efficient contact friction algorithm within the framework of large FE deformation in order to predict the non-uniform relative density distribution during large deformation of powder die-pressing.

been proposed and applied within finite element algorithms—in particular the perturbed technique by introducing a certain amount of penetration as a function of the normal contact load [39], and augmented Lagrangian approach by using the combination of the two basic constraint methods [40,41]. In this study, the finite element approach adopted is characterized by the use of penalty approach in which a Coulomb friction law [10,11] is incorporated to simulate sliding resistance at the powder–tool interface. 3.1.1. Modeling of contact constraints There are various approaches established for resolving the contact problem. One of these techniques applied for imposing contact conditions in the normal direction is the formulation of non-penetration condition, as a purely geometrical constraint. For the tangential direction, the sticking and sliding states can be distinguished by the development of elastic–plastic constitutive laws. In sticking interfaces, either a geometrical constraint equation, or a constitutive law for the tangential relative micro displacements between the contacting bodies can be applied. For tangential sliding between bodies, a special constitutive equation for friction must be employed. In this study, a simple and efficient algorithm is employed to model the frictional contact in powder–die interface. The contact constraints is implemented based on the penalty approach by imposing the normal and tangential springs at the contact interface, in which the stiffness of tangential spring is modified according to the Coulomb friction law for the frictional slip. The node-to-segment (NTS) contact element is one of the most commonly used discretizations in large deformation finite element simulation of contact problems [10,11]. Consider that the discrete slave point s with coordinate xs comes into contact with the master segment (1)–(2) defined by the nodal coordinates m xm 1 and x2 . By introducing the surface coordinate x along the master surface, we have m m x^ m ðxÞ ¼ xm 1 þ ðx2 x1 Þx

3.1. Modeling of frictional contact In powder compaction process, the friction between the powder and tools limits the performances of the process and the mechanical characteristics of the parts. Friction can result in poor density distributions, which leads to differential retreat during compaction and sintering, and the heterogeneity of the final properties of the part [28]. After compaction, when the tooling retreats on a multi-levels press, or during ejection, the force due to friction may result in fatal cracks of different size. A number of experimental and numerical investigations into frictional effects and its impact on the compaction process of powder have been reported in the literature; including: the effects of stress and density on friction [29,30], the micromechanical friction mechanisms between the powder and tool [31], the evolution of voids in the die forging process under different frictional conditions [32], the influence of powder–die and powder–punch friction [33], and the large frictional contact modeling in powder die-pressing [9–11,34]. Basically, there are two main constraint methods of solution employed in the finite element solution of contact problems; the method of Lagrangian multipliers and the penalty approach. In Lagrangian multipliers approach, the contact forces are taken as primary unknowns and the non-penetration condition is strictly enforced [35,36]. In penalty method, the penetration between two contacting boundaries is introduced approximately, or allows infinitesimal penetration, and the normal contact force is related to the penetration by a penalty parameter [37,38]. Based on the two basic constraint methods, other constraint techniques have

ð3Þ

The normalized tangent vector tm to the master segment can be computed as tm ¼

1 m 1 x^ ðxÞ, x ¼ ðxm xm 1Þ ‘ ‘ 2

ð4Þ

m where ‘ ¼ :xm 2 x1 :. The unit normal nm to the segment (1)–(2) can be then computed by nm ¼ e3  tm , with e3 denoting the unit vector perpendicular to the plane. The minimal distance gN between the slave point s and the master segment (1)–(2) can be obtained as m gN ¼ ðxs ð1xÞxm 1 xx2 Þ:nm

ð5Þ

where 1 s m ðx x1 Þ:tm ð6Þ ‘ In order to perform the contribution of NTS element into the weak form of equilibrium equation, a new approach is applied by introducing the contact constraints based on the potential energy of springs imposed at the normal and tangential directions. In this technique, two springs are defined in the normal and tangential directions of contact interface between the slave node and master segment. The shape functions of the slave-master at the contact interface are defined as " # 1 0 ð1xÞ 0 x 0 N¼ ð7Þ 0 1 0 ð1xÞ 0 x

x¼

The relative displacement between the slave and master is defined by Ndu, with du denoting the nodal displacements. The normal and tangential relative displacements are derived using

A.R. Khoei et al. / Finite Elements in Analysis and Design 46 (2010) 843–861

the normal and tangential shape functions as

If dfti 4ðdfti Þmax , the values of dfti and at are corrected as

Nn ¼ ðnm  nm ÞN Nt ¼ ðInm  nm ÞN

ait ¼ ð8Þ

in which the normal and tangential relative displacements are defined by dun ¼ nTm Nn du and dut ¼ tTm Nt du, respectively. In order to incorporate the contact constraints into the equilibrium equation, the potential energy of contact interface is decomposed into the normal and tangential directions as 1 1 an ðdun Þ2 þ at ðdut Þ2 2 2 1 1 T T ¼ ðduÞ Nn an Nn ðduÞ þ ðduÞT NTt at Nt ðduÞ 2 2

P¼

ð9Þ

where an and at are the normal and tangential penalty parameters, as shown in Fig. 2. Taking the derivative from relation (9), the normal and tangential stiffness matrices at the contact interface are defined as Kcn ¼ NTn an Nn Kct ¼ NTt at Nt

ð10Þ

3.1.2. Contact friction algorithm In order to implement the contact constraint of frictional slip, the Coulomb friction law is incorporated in the tangential spring. According to the numerical procedure described in preceding sections, a computational algorithm is applied based on the Newton–Raphson technique. For iteration i within the time step n, the active nodes and segments are firstly determined. The normal and tangential stiffness matrices Kcn and Kct are then evaluated by relation (10) using the parameters an and at, and assembled into a global stiffness matrix. At the first iteration of the first time step, at is taken as a practical value and for subsequent iterations, if required, it is modified with the Coulomb friction law. The global i system of equilibrium equation Kitotal d ui ¼ d f is solved to i compute the incremental nodal displacement d u at iteration i. The total displacement at iteration i becomes ui ¼ ui1 þ d ui . The tangential and normal forces are determined at each slave–master point by dfti ¼ at Nt dui and dfni ¼ an Nn dui . The maximum frictional force is then computed based on the Coulomb friction law by ðdfti Þmax ¼ Cf þ mf dfni , with Cf and mf denoting the cohesion and friction coefficient of contact interface.

ðdfti Þmax dut

and

dut  dfti ¼ ðdfti Þmax  dut 

M2 ( = 1) t n nm

M1 ( = 0)

tm



Fig. 2. Modeling of contact constraints in normal and tangential directions.

ð11Þ

Finally, the out of balance, or residual forces of contact constraints are evaluated, and the computational algorithm is repeated until the norm of residual forces and the maximum residual are both less than the prescribed tolerance. 3.2. Powder plasticity model The mechanical behavior of powders involves several interacting micromechanical processes. First, at low pressure, particle sliding occurs leading to particle re-arrangement. The second stage involves both elastic and plastic deformation of the particles via their contact areas leading to geometric hardening (i.e. plastic deformation and void closure). Lastly, at very high pressure, the flow resistance of the material increases rapidly due to material strain hardening. Therefore, it is necessary that the constitutive model of powder captures various behaviors of compaction process. It has been experimentally observed that the constitutive modeling of geological and frictional materials can be utilized to construct the suitable phenomenological constitutive models [42,43], which capture the major features of the response of initially loose powders to the complex deformation processing histories encountered in the manufacture of engineering components by powder metallurgy techniques. A number of constitutive models have been proposed for metal powders, including: micromechanical models [44,45] and macromechanical models [4–8,46–53]. Based on above requirements, the following plasticity model is developed for powder materials using the invariants of stress states, J1 and J2D, as [7]  2 f Fðr, ZÞ ¼ J2D þ d J12 fd2 ¼ 0 ð12Þ fh where J1 is the first invariant of stress tensor and J2D the second invariant of deviatoric stress tensor. fh is a positive increasing function of relative density, which controls the intersection of J1 axis with yield surface function at the maximum value of compression. fd is a function of the first invariant of stress tensor and relative density, which controls the shear strength of material. This parameter indicates the size of yield surface in the perpendicular direction to J1 axis. The polynomial functions are proposed for the dependence of fh and fd on relative density as   1 fh ¼ ð13Þ ða0 þ a1 Z þa2 Z2 þ a3 Z3 Þ 1Zb fd ¼ ðabJ1 Þ þ ðc1 Z þc2 Z2 þ c3 Z3 þ c4 Z4 Þ

S

847

ð14Þ

where Z is the relative density, i.e. Z ¼ r=rs , with r denoting the current total density, rs the density of solid particles, in which Z0 indicates the position of first yield surface according to the initial value of relative density, and b, a0, a1, a2, a3, a, b, c1, c2, c3, c4 are parameters of the powder. Fig. 3 presents the 2D and 3D representation of yield surface (12) for the isotropic hardening behavior of material. It is worth mentioning that the yield surface (12) is very similar to the double-surface cap models, i.e. a combination of Mohr–Coulomb or Drucker–Prager and elliptical surfaces, which has been extensively used by researchers to demonstrate the behavior of powder and granular materials. However, the double-surface plasticity consists of two different yield functions, and special treatment should be made to avoid numerical difficulties in the intersection of these two surfaces [51,52].

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J1/2 2D

Increasing η

-J1 Fig. 3. Trace of single-cap plasticity on the meridian plan for different values of relative density; (a) the 2D cone-cap yield function and (b) the 3D cap plasticity surface.

3.2.1. Computation of material property matrix In order to derive the powder elasto-plastic constitutive matrix, we need to calculate the plastic hardening/softening modulus and flow direction vector in the matrix Dep, defined as Dep ¼ De 

ð@F=@rÞT DTe De ð@Q =@rÞ H þ ð@F=@rÞT De ð@Q =@rÞ

ð15Þ

where n ¼ @F=@r and ng ¼ @Q =@r are the normal vector to the yield and potential plastic surfaces, respectively, and H is the hardening plastic modulus. The plastic hardening modulus and flow direction vector are defined as [28] H¼

n¼

@F @Z @e ng @Z @e @ep

@F @F @J1 @F @J2D @F @J3D ¼ þ þ @r @J1 @r @J2D @r @J3D @r

ð16Þ

ð17Þ

where

In order to evaluate and verify the accuracy of proposed computational algorithm for large plastic deformation of powder die-pressing, two examples are presented numerically. The first example is the extrusion of an aluminum billet chosen to demonstrate the efficiency and accuracy of computational algorithm by comparison with available numerical results reported in literature. Due to significant changes in geometry of components, the capability of proposed NTS algorithm for handling the large deformation under frictional contact behavior is verified. The second example is chosen to demonstrate the efficiency and accuracy of computational algorithm in modeling of die-powder pressing of a shaped-charge liner. Both numerical examples have been solved under displacement control condition by increasing the punch movement and predicting the die-pressing forces at different displacements.

4.1. Extrusion of an aluminum billet

f2 J2 @F ¼ 2J1 d2 2bfd 12 2bfd @J1 fh fh @F ¼ 1:0 @J2D

ð18Þ

@F ¼0 @J3D Substituting the yield surface (12) into Eq. (16), the hardening plastic modulus can be computed as   @F @fh @F @fd Z2 H¼ þ ng ð19Þ @fh @Z @fd @Z Z0 ii where ! @F 2 2 2 ¼ fd J1 @fh fh3 @F 2 ¼ fd J12 2 @fd fh

4. Verification of computational algorithm

! ð20Þ

@fh bZb1 1 ¼ ða0 þa1 Z þ a2 Z2 þa3 Z3 Þ þ ða1 þ 2a2 Z þ 3a3 Z2 Þ @Z 1Zb ð1Zb Þb @fd ¼ c1 þ 2c2 Z þ 3c3 Z2 þ 4c4 Z3 @Z

The first example is of the extrusion of an aluminum billet, which is applicable typically in metallurgy. This example illustrates the capability and accuracy of proposed node-tosurface technique, which can be particularly useful in modeling of contact constraints. Two bodies with frictional surfaces are in contact, large deformations are expected, and considerable geometrical nonlinearity behavior is included in the mechanical description. The smoothed and non-smoothed contact friction were proposed by Padmanabhan and Laursen [54] for this example, and used here for comparison. Some geometrical complications are included in this problem by inclining the master surfaces. An efficient search algorithm is used for each slave node, which can be easily determined by the relative master segment at different angle of master segments. The bottom nodes of the billet are subjected to a uniform upward displacement and the outer boundaries of rigid die are considered fixed. The material properties of the billet are chosen as; K ¼63.8 GPa and G¼ 26.1 GPa. The billet is forced to move upward with the total movement of 30 cm by using an incremental displacement control approach. The initial geometry and boundary conditions are shown in Fig. 4(a) [54]. The finite element model at the initial configuration together with the deformed meshes at half and final configurations of process are depicted in this figure. In Fig. 5, the variations of vertical force with displacement are shown at different friction coefficients. As can be observed, the force– displacement curve of friction coefficient m ¼0.1 is identical with that obtained by Padmanabhan and Laursen [54].

A.R. Khoei et al. / Finite Elements in Analysis and Design 46 (2010) 843–861

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Fig. 4. The extrusion of an aluminum billet; (a) geometry and boundary conditions, (b) initial FE configuration, (c) deformed mesh at 50% of pressing, and (d) deformed mesh at final stage of pressing.

Table 1 The material model parameters for the single cap plasticity. No friction Friction coefficient µ = 0.1 Friction coefficient µ = 0.2 Padmanabhan & Laursen (2001)

3E+07

Force (N)

2.5E+07

fd

fh b a0 a1 a2 a3

2E+07

(MPa) (MPa) (MPa) (MPa)

1.2908 1087.135  2875.383 4406.547  2568.542

a (MPa) b c1 c2 c3 c4

(MPa) (MPa) (MPa) (MPa)

248.620 1.6990 e 2 0.0 0.0 0.0 0.0

1.5E+07

1E+07

5E+06

0

0

5

10

15

20

25

30

Displacement (cm) Fig. 5. The variations of vertical force with displacement for an aluminum billet extrusion.

form a shaped-charge liner from iron powder along with the geometry and initial FE mesh of powder before compaction are presented in Fig. 6. The loading characteristics are achieved by the use of prescribed nodal displacements for the top punch movement. The die and upper punch are modeled as rigid surfaces. The simulation is performed using the remaining pressing distance of 20 mm from above. In Fig. 6, the deformed finite element meshes of component are presented at half and final stages of compaction. In Fig. 7, the variations of top punch force with displacement are plotted for different values of friction coefficient. The results are in good agreement with those reported by Keshavarz et al. [34].

4.2. A shaped-charge liner

5. Numerical simulation of optimal design

The next example is of a shaped-charge liner, which is extensively used for civilian oil and steel sectors in geophysical prospecting, mining, and quarrying. Most liners used in the civilian sector are often made from a mixture of different metallic powders. This component has been simulated by Keshavarz et al. [34] using the node-to-node contact friction algorithm. In the present simulation, the NTS algorithm is applied to model the shaped-liner pressed from the iron powder with material parameters given in Table 1 [7]. The schematic of process to

In order to illustrate the efficiency and applicability of the genetic algorithm developed for powder compaction process; the optimal design of a set of powder components is presented. The large FE deformation formulation together with the frictional contact algorithm and the powder elasto-plasticity constitutive matrix, presented in Section 3, have been implemented in the framework of genetic algorithm into a nonlinear finite element code to evaluate the capability of the model in design optimization of powder compaction process. The simulation is presented

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Fig. 6. A shaped-charge liner; (a) geometry of punch and powder, (b) initial FE mesh, (c) deformed mesh at 50% of compaction, and (d) deformed mesh at final stage of compaction.

5000 Top punch reaction (kN)

µ = 0.1 Keshavarz et al. (2008) µ = 0.1 NTS algorithm (present model)

4000

µ = 0.3 NTS algorithm (present model)

3000

2000

1000

0 0

0.5

2.0

1.5

2.0

Displacement (cm) Fig. 7. The variations of vertical force with displacement for a shaped-charge liner. Fig. 8. An industrial component; the geometry and boundary conditions.

for efficiency and accuracy in shape optimization of powder component, the optimal design of punch movements, and the friction optimization of powder–tool interface. The design optimization is performed on an industrial component and an automotive component. The problems have been solved with displacement control by increasing the punch movement and predicting the optimal design variables at the final stage of compaction. The distribution of stress and relative density contours are presented at different generations of optimization process. The initial relative density is r0 ¼ 0:4. The variation of the Youngs modulus with relative density for iron powder is assumed as E ¼ 3640r3:9 , with r denoting the relative density [5]. The powder parameters in the single plasticity model are presented in Table 1 [7]. In the FE simulations, the tools are modeled as rigid bodies, since the elastic deformation of the tools has only an

insignificant influence on the density distribution in the green component. The powder–tool interface is modeled by using the friction coefficient m ¼0.1 and cohesion c¼5.0 MPa.

5.1. An industrial component The first example is of an engineering industrial component in hard metal powder that includes several complications caused by discontinuities in geometry, such as flow around corners due to complex geometry of upper punch. The numerical modeling of this component was performed by Keshavarz et al. [34] to show the applicability of their frictional plasticity model in large deformations. The compacted specimen has an initial height of

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22 mm and the inner and outer radiuses of 6 and 12 mm, respectively. In the production of this component the compaction starts from loose powder by moving the upper punch downward to its final position 15 mm, while the bottom punch is fixed. The schematic of the process to form an industrial component along with the geometry of powder and punch before compaction are presented in Fig. 8. In order to obtain the optimal design of preform die shape at final compaction, the geometry of top surface of the preform work-piece is considered as the unknown design variables with two control points, as shown in Fig. 8. The optimization algorithm is used to optimize the difference between the maximum and minimum values of stress on the final compacted component with the objective function defined as   Minimize f ðx1 ,x2 Þ ¼ smax smin  ð21Þ and the constraints of 6.0 rx1 r 9.0 and 9.0 rx2 r 12.0 to control the geometry of z-shape component at the top punch

3rd Generation

7th Generation

851

surface. The optimization process is performed using the genetic algorithm technique with an initial population generated randomly using 50-string based on two control points x1 and x2. The numerical simulation is performed using 2D axisymmetric FE mesh of 3-noded elements for each string to calculate the fitness and reproduction. The process of reproduction is applied according to the value of objective function obtained at the end of compaction to copy the individual string. The reproduction operator is implemented in the algorithmic form based on a roulette wheel where each individual is represented by a space that proportionally corresponds to its fitness. The evolutionary process is converged after 15 generations with the optimal design variables for two control points as X T ¼ ½6:7,11:6. In Fig. 9, the deformed FE meshes of compacted component are presented at four generations of optimization process. The optimal preform shape of final component is shown in Fig. 9(d). In Figs. 10 and 11, the distribution of normal stress sy contours and relative density distributions are presented at four

10th Generation

18th Generation

Fig. 9. An industrial component; the deformed FE meshes of compacted component at four generations of optimization process.

264.2 250.3 236.4 222.5 208.6 194.7 180.8 166.9 153.0 139.1 125.1 111.2 97.37 83.46 69.55

258.7 244.3 229.9 215.5 201.2 186.8 172.4 158.1 143.7 129.3 114.9 100.6 86.23 71.86 57.49

258.0 243.6 229.3 215.0 200.6 186.3 172.0 157.6 143.3 129.0 114.6 100.3 86.00 71.66 57.33

236.9 223.8 210.6 197.4 184.3 171.1 157.9 144.8 131.6 118.4 105.3 92.15 78.98 65.82 52.65

Fig. 10. An industrial component; the distribution of normal stress sy contours at four generations of optimization process.

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0.846 0.819 0.793 0.766 0.740 0.714 0.687 0.661 0.634 0.608 0.581 0.555 0.528 0.502 0.476

0.841 0.820 0.798 0.776 0.755 0.733 0.712 0.690 0.669 0.647 0.625 0.604 0.582 0.561 0.539

0.857 0.835 0.813 0.791 0.769 0.747 0.725 0.703 0.681 0.659 0.637 0.615 0.593 0.571 0.549

0.825 0.804 0.783 0.762 0.741 0.719 0.698 0.677 0.656 0.635 0.614 0.592 0.571 0.550 0.529

Fig. 11. An industrial component; the distribution of relative density contours at four generations of optimization process.

280

Generation 3 Generation 7 Generation 10 Generation 18

1200

1000

260

240 Objective value

Load (kN)

800

600

400

220

200

180

200

160

0 0

3

6 9 Top punch movement (mm)

12

15 0

5

10

15

Generation number Fig. 12. The variations of the top punch force with displacement at four generations of optimization process.

generations of optimization process. At the end of the optimization process, the contours display the highest values of stress and density at the bottom corner of upper punch, which decreases gradually to the bottom of component. In Fig. 12, the variations of top punch force with displacement are shown at four generations. This figure clearly shows the performance of genetic algorithm on the top punch reaction. Finally, the variation of the objective function is plotted with the number of generation in Fig. 13.

Fig. 13. The variations of the objective value with generation number.

5.2. An automotive component The next example is of an axisymmetric automotive part that is compacted from iron powder with a mechanical press and a multi-platen die set. This practical example is chosen to present the capability of proposed computational algorithm in shape optimization of automotive component, the optimal design of punch movements, and the friction optimization of powder–tool interface. The measured density distribution is obtained for this

A.R. Khoei et al. / Finite Elements in Analysis and Design 46 (2010) 843–861

853

Fig. 14. An automotive component; the geometry and boundary conditions.

Fig. 15. The deformed FE meshes of compacted component at four generations of optimization process; (a)–(d) 3-noded FE meshes and (e)–(h) 6-noded FE meshes.

854

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component by Shen et al. [55]. The numerical simulation result is performed for this example by Khoei et al. [14], where the effects of friction were neglected. The shape of powder together with the die and punches in their position before compaction are presented in Fig. 14(a). On the virtue of symmetry, the automotive part is analyzed employing an axisymmetric FE mesh.

379.9 360.9 341.9 322.9 303.9 284.9 265.9 246.9 227.9 208.9 189.9 170.9 151.9 132.9 113.9

5.2.1. Shape optimization of automotive component The first objective of proposed die-pressing is to design the preform die shape so that, after the final compaction, the required final product is obtained without defects. The shape of the geometry of free surface of the preform work-piece is considered as the unknown design variables with three control points, as

235.6 223.8 212.0 200.3 188.5 176.7 164.9 153.1 141.3 129.6 117.8 106.0 94.26 82.48 70.69

180.7 171.7 162.6 153.6 144.6 135.5 126.5 117.4 108.4 99.41 90.37 81.33 72.30 63.26 54.22

S tre s s 162.6 155.8 149.0 142.3 135.5 128.7 121.9 115.2 108.4 101.6 94.87 88.09 81.32 74.54 67.76

Fig. 16. The normal stress sy contours using 3-noded FE meshes at four generations of optimization process.

206.5 196.2 185.9 175.5 165.2 154.9 144.5 134.2 123.9 113.6 103.2 92.95 82.62 72.29 61.97

159.4 151.5 143.5 135.5 127.5 119.6 111.6 103.6 95.68 87.71 79.73 71.76 63.78 55.81 47.84

165.1 156.9 148.6 140.4 132.1 123.8 115.6 107.3 99.11 90.85 82.59 74.33 66.07 57.81 49.55

152.7 146.6 140.5 134.4 128.3 122.2 116.1 109.9 103.8 97.76 91.65 85.54 79.43 73.32 67.21

Fig. 17. The normal stress sy contours using 6-noded FE meshes at four generations of optimization process.

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shown in Fig. 14(b). The compaction is employed by means of the action of two punches; a top punch and a lower outer punch. The simulation has been performed using the remaining pressing distance of top punch of 7.5 mm and lower outer punch of 20 mm. The optimization scheme is used to optimize the difference

0.999 0.972 0.945 0.918 0.891 0.864 0.837 0.810 0.783 0.756 0.729 0.702 0.675 0.648 0.621

855

between the maximum and minimum values of stress on the final compacted component. The objective function for this example is as follows: Minimize

0.984 0.959 0.934 0.908 0.883 0.858 0.833 0.807 0.782 0.757 0.732 0.706 0.681 0.656 0.631

  f ðx1 ,x2 ,x3 Þ ¼ smax smin 

ð22Þ

0.997 0.931 0.864 0.798 0.731 0.665 0.598 0.532 0.465 0.399 0.332 0.266 0.199 0.133 0.066

0.860 0.843 0.826 0.808 0.791 0.774 0.757 0.740 0.722 0.705 0.688 0.671 0.654 0.636 0.619

Fig. 18. The relative density distributions using 3-noded FE meshes at four generations of optimization process.

0.855 0.834 0.812 0.791 0.770 0.748 0.727 0.705 0.684 0.663 0.641 0.620 0.598 0.577 0.556

0.852 0.831 0.810 0.788 0.767 0.746 0.724 0.703 0.682 0.660 0.639 0.618 0.597 0.575 0.554

0.899 0.876 0.854 0.831 0.809 0.786 0.764 0.741 0.719 0.696 0.674 0.652 0.629 0.607 0.584

0.804 0.790 0.777 0.763 0.750 0.737 0.723 0.710 0.696 0.683 0.670 0.656 0.643 0.629 0.616

Fig. 19. The relative density distributions using 6-noded FE meshes at four generations of optimization process.

856

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with the constraints of 62.5r x1 r67.5, 45.0rx2 r50.0, 45.0rx3 r50.0, x2  x3 Z0.0, and x2 1.875x3 r  35.2. The first three constraints are applied to control the geometry of the top and bottom punches, and the last two constraints to control the slope of bottom punch surfaces. The optimization process for this example is performed using the genetic algorithm technique, as described in Section 2. The initial population is generated randomly using 50-string based on three control points x1, x2, and x3. To calculate the fitness and reproduction, the numerical simulation is performed for each string. The compaction process of each string is modeled by using 2D axisymmetric FE mesh of 3-noded and 6-noded elements, as shown in Fig. 15. For each string, the value of objective function is calculated at the end of compaction. The process of reproduction is applied according to the fitness function to copy the individual string. The reproduction operator is implemented in the algorithmic form based on a

roulette wheel where each individual is represented by a space that proportionally corresponds to its fitness. It has been observed that the evolutionary process is converged after 15 generations. The optimal design variables for three control points corresponding to the FE mesh of 3-noded elements is obtained as X T3 ¼ ½67:5,46,45 and the FE mesh of 6-noded elements as X T6 ¼ ½67:5,46,45. Obviously, both FE modeling converge to similar design variables. In Fig. 15, the deformed FE meshes of compacted component are presented at four generations of optimization process using 3-noded and 6-noded FE analyses. The optimal preform shapes of final component are shown in Figs. 15(d) and (h). A good agreement can be seen between the 3-noded and 6-noded FE meshes for the obtained optimal shape. The optimal shape is similar to that measured experimentally by Shen et al. [55]. In Figs. 16 and 17, the distribution of normal stress sy contours are presented at four generations of optimization process using the 3-noded and 6-noded FE meshes, respectively. Also plotted in

200

Generation 1 Generation 4 Generation 6 Generation 10

500

190

400

Objective value

Load (kN)

180

300

200

170 160 150 140

100

130 0 0

1

2

3

4

5

6

7

5

10 Generation number

15

Fig. 21. The variation of the objective function with the number of generation.

Generation 1 Generation 2 Generation 5 Generation 15

500

120 0

Top punch movement (mm)

Load (kN)

400

300

200

100

0 0

1

2

3

4

5

6

7

Top punch movement (mm) Fig. 20. The variations of the top punch force with displacement at four generations of optimization process; (a) 3-noded FE meshes and (b) 6-noded FE meshes.

Fig. 22. An automotive component; the optimal design of punch movements.

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Figs. 18 and 19 are the predicted relative density distributions at four generations of optimization process for 3-noded and 6-noded FE meshes. The result of final density distribution can be compared with that measured by Shen et al. [55] experimentally. At the end of compaction, the relative density contour shows the highest density values at the bottom punch surface and the topright corner for the optimal component, which reduces gradually to the left-hand side. Clearly, it can be seen from the contours of density distribution at the final generation of optimization process that the results are almost identical for the 3-noded and 6-noded FE meshes. In Fig. 20, the variations of the top punch force with displacement are shown at four generations using the 3-noded and 6-noded FE meshes. This figure clearly shows the accuracy of the genetic algorithm on the top punch reaction. Finally, the variation of the objective function with the number of generation is plotted in Fig. 21.

857

5.2.2. Optimal design of punch movements In the second stage, the optimal shape of an automotive component obtained from previous section is used to determine the optimal punch movements. The top and bottom punch movements of the preform work-piece is considered as the unknown design variables, as shown in Fig. 22. The optimization scheme is used to optimize the difference between the maximum and minimum values of stress on the final compacted component. The objective function for this example is as follows:   Minimize f ðdt ,db Þ ¼ smax smin  ð23Þ with the constraints of 15rdb r25 and dt + dt ¼27.5. The first constraint is applied to control the bottom punch motion and the second constraint to control the total movement of the top and bottom punches. The optimization process is performed using the genetic algorithm technique with an initial population generated

Fig. 23. The deformed FE meshes of compacted component at four generations of optimization process of punch movements; (a) dt ¼3.3 and db ¼ 24.2 mm, (b) dt ¼8.2 and db ¼19.3 mm, (c) dt ¼6.6 and db ¼20.9 mm, and (d) dt ¼ 7.4 and db ¼ 20.1 mm.

263.81 249.16 234.50 219.84 205.19 190.53 175.88 161.22 146.56 131.91 117.25 102.59 87.94 73.28 58.63

205.47 194.05 182.64 171.22 159.81 148.39 136.98 125.56 114.15 102.73 91.32 79.90 68.49 57.07 45.66

146.31 138.19 130.06 121.93 113.80 105.67 97.54 89.41 81.29 73.16 65.03 56.90 48.77 40.64 32.51

Fig. 24. The normal stress sy contours at four generations of optimization process.

160.36 151.45 142.54 133.63 124.72 115.81 106.90 98.00 89.09 80.18 71.27 62.36 53.45 44.54 35.63

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0.837 0.808 0.779 0.751 0.722 0.693 0.664 0.635 0.606 0.577 0.548 0.520 0.491 0.462 0.433

0.949 0.913 0.876 0.840 0.803 0.767 0.730 0.694 0.657 0.621 0.584 0.548 0.511 0.475 0.438

0.959 0.924 0.888 0.853 0.817 0.782 0.746 0.711 0.675 0.640 0.604 0.569 0.533 0.497 0.462

0.852 0.820 0.789 0.757 0.726 0.694 0.663 0.631 0.599 0.568 0.536 0.505 0.473 0.442 0.410

Fig. 25. The relative density distributions at four generations of optimization process; (a) dt ¼3.3 and db ¼24.2 mm, (b) dt ¼ 8.2 and db ¼ 19.3 mm, (c) dt ¼6.6 and db ¼ 20.9 mm, and (d) dt ¼7.4 and db ¼20.1 mm.

5.2.3. Friction optimization of powder–tool interface In the final stage of optimization process, the optimal shape together with the optimal punch movements of an automotive component obtained from previous sections are used to evaluate the optimal value of friction at the powder–tool interface. In order to obtain the optimal value of friction coefficient, the friction at the powder–tool interface is considered as the only unknown

350

300 Objective value

randomly using 50-string based on two design variables dt and db. The computational modeling is performed for each string to calculate the fitness and reproduction. The process of reproduction is applied according to the value of objective function obtained at the end of compaction to copy the individual string. The reproduction operator is implemented in the algorithmic form based on a roulette wheel where each individual is represented by a space that proportionally corresponds to its fitness. The evolutionary process is converged after 20 generations with the optimal design variables for two design variables as dt ¼7.4 and db ¼20.1 mm. In Fig. 23, the deformed FE meshes of compacted component are presented at four generations of optimization process. The optimal top and bottom punch movements of final component is shown in Fig. 23(d). In Figs. 24 and 25, the distribution of normal stress sy contours and the relative density distributions are presented at four generations of optimization process. At the end of compaction, it has been observed once again the highest density values at the bottom punch surface and the top-right corner for the optimal component, which reduces gradually to the left-hand side. In Fig. 26, the variation of the objective function is plotted with the number of generation.

250

200

150

100 0

5

10 Generation number

15

20

Fig. 26. The variation of the objective function with the number of generation.

design variable. The optimization algorithm is used to optimize the difference between the maximum and minimum values of stress on the final compacted component with the objective function defined as   Minimize f ðmÞ ¼ smax smin  ð24Þ

A.R. Khoei et al. / Finite Elements in Analysis and Design 46 (2010) 843–861

312.75 295.37 278.00 260.62 243.25 225.87 208.50 191.12 173.75 156.37 139.00 121.62 104.25 86.87 69.50

256.77 242.51 228.24 213.98 199.71 185.45 171.18 156.92 142.65 128.39 114.12 99.86 85.59 71.33 57.06

859

152.30 143.84 135.38 126.92 118.46 110.00 101.54 93.07 84.61 76.15 67.69 59.23 50.77 42.31 33.85

151.02 142.63 134.24 125.85 117.46 109.07 100.68 92.29 83.90 75.51 67.12 58.73 50.34 41.95 33.56

Fig. 27. The normal stress sy contours at four generations of optimization process; (a) m ¼ 0.984, (b) m ¼ 0.73, (c) m ¼0.0, and (d) m ¼ 0.111.

0.926 0.892 0.858 0.824 0.789 0.755 0.721 0.686 0.652 0.618 0.583 0.549 0.515 0.480 0.446

0.907 0.874 0.840 0.806 0.773 0.739 0.706 0.672 0.638 0.605 0.571 0.538 0.504 0.470 0.437

0.852 0.821 0.789 0.757 0.726 0.694 0.663 0.631 0.600 0.568 0.537 0.505 0.473 0.442 0.410

0.860 0.828 0.797 0.765 0.733 0.701 0.669 0.637 0.605 0.574 0.542 0.510 0.478 0.446 0.414

Fig. 28. The relative density distributions at four generations of optimization process; (a) m ¼ 0.984, (b) m ¼ 0.73, (c) m ¼0.0, and (d) m ¼ 0.111.

and the constraint of 0 r m r1 applied to control the friction coefficient at the powder–tool and powder–punch interfaces. The optimization process is performed using the genetic algorithm

technique with an initial population generated randomly using 50-string. The numerical simulation is performed for each string to calculate the fitness and reproduction. The process of

860

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160

0.73, 0.0, and 0.111. This figure clearly shows the accuracy of the genetic algorithm on the top punch reaction. Finally, the variation of the objective function with the number of generation is plotted in Fig. 30.

Generation 3 Generation 7 Generation 10 Generation 18

140 120

6. Conclusion

Load (kN)

100 80 60 40 20 0 0

4

8 12 Top punch movement (mm)

16

20

Fig. 29. The variations of the top punch force with displacement at four generations of optimization process, i.e. m ¼ 0.984, m ¼ 0.73, m ¼ 0.0, and m ¼ 0.111.

350

Object value

300

250

200

150

In the present paper, an optimization algorithm was presented for powder forming processes based on the genetic algorithm approach. The goal of optimization is to eliminate the work-piece defects that may arise during the powder compaction process. The genetic algorithm operator was used to increase the efficiency of the search algorithm and to produce an optimal design. The genetic algorithm was used based on a fixed-length vector of design variables to obtain the desired optimal compacted component by verifying the prescribed constraints. The numerical modeling of powder compaction took into account a large deformation formulation, powder plasticity behavior, and frictional contact algorithm. An updated-Lagrangian FE formulation was employed for large powder deformations. A generalized cap plasticity model was used in numerical simulation of nonlinear powder behavior. The influence of powder–tool friction was simulated by using the penalty enforcement of node-to-surface contact algorithm, in which a plasticity theory of friction was incorporated to model sliding resistance at the powder–tool interface. Finally, several numerical examples were analyzed to demonstrate the feasibility of the proposed optimization algorithm for designing powder components in the forming process of powder compaction. In order to illustrate the applicability of the genetic algorithm in powder compaction process, the algorithm was implemented in a full nonlinear FEM code with the aforementioned capabilities. The optimization process was performed on an industrial and an automotive component to obtain the optimal shape of compacted component, the optimal movements of top and bottom punches, and the optimal value of friction coefficient at the powder–tool and powder–punch interfaces. The distribution of stress and relative density contours were presented at different generations of optimization process. It is shown how the proposed genetic algorithm can be efficiency used in the highly nonlinear modeling of powder compaction process to perform an optimal design of compacted component.

100 0

5

10

15

20

References

Generation number Fig. 30. The variation of the objective function with the number of generation.

reproduction is applied according to the value of objective function obtained at the end of compaction to copy the individual string. The evolutionary process is converged after 15 generations with the optimal friction coefficient of m ¼0.111. In Fig. 27, the distribution of normal stress sy contours are presented at four generations of optimization process. Also plotted in Fig. 28 are the predicted relative density distributions at four generations of optimization process, i.e. m ¼0.984, 0.73, 0.0, and 0.111. It must be noted that the optimization process is performed on an automotive component with the optimal shape of Fig. 15(d) and the optimal punch movements of Fig. 23(d) at different values of friction coefficient. Obviously, the relative density contour shows the highest density values at the bottom punch surface and the top-right corner for the optimal value of friction m ¼0.111. In Fig. 29, the variations of the top punch force with displacement are shown at four generations, i.e. m ¼0.984,

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