R. Venketesan / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3758-3761
EXTRUSION DIE PROFILE DESIGN
USING SIMULATED ANNEALING
ALGORITHM AND PARTICLE
SWARM OPTIMIZATION
R.VENKETESAN*
*
Department of Mechanical Engineering, Sona College of Technology,
Salem - 636005.
Abstract:
In this paper a new method has been proposed for optimum shape design of extrusion die. The Design problem is formulated as an unconstrained optimization problem. Here nontraditional optimization techniques like Simulated Annealing Algorithm and Particle Swarm Optimization are used to minimize the extrusion force by optimizing the extrusion ratio and die cone angle. Internal power of deformation is also calculated and results are compared.
Keywords: Extrusion, Optimization, Particle Swarm Optimization (PSO), Simulated Annealing Algorithm (SA).
1. Introduction
A simple approach for determining die pressure is to use slab analysis to account for friction on extruding through conical die. Hill [1] has performed this analysis for coulomb sliding friction. While this analysis considers die friction, it does not allow for redundant deformation. However, extrusion probably is most often studied using slip line field theory. Upper bound analysis [2] has used a different velocity field, which gives good agreement with experiments on hydrostatic extrusion
Various factors affecting the extrusion process are the material properties, blank temperature, friction at the die work interface, extrusion ratio, die cone angle and die length. Richmond and Devenpeck [3] has done a pioneer work on theoretical consideration of an optimal die shape for metal extrusion. Slip line field method was applied to find the ideal die profile associated with minimum forming energy. Upper Bound [4] method has been used for optimal die shape design. An incremental slab method has been developed by Wifi [5] to calculate the extrusion pressure for an arbitrary curved die and to generate the optimum curved profile for the hot forward rod extrusion process.
Nontraditional optimization techniques like simulated annealing algorithm [6] and Particle Swarm optimization [7] are used for minimizing the Die force by optimizing die cone angle and extrusion ratio. The specification of a practical design can be formulated into an objective function and constraints for optimization problems. Here the objective function is minimization of extrusion force. The parameters to be varied are the die cone angle and extrusion ratio. In this problem, the coefficient of friction is kept as constant.
2. Design Model
The objective function is minimization of extrusion force by optimizing the die cone angle and the extrusion ratio. [8]
Die force, d O
R
B
B
B
P
1
1
(1)Where, B = cot
Boundary condition R = 1 to 5
R. Venketesan / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3758-3761 = 0 to / 2 deg or 0 to 1.571 rad
Extrusion force=
P
d+ 4*Ti*L (2)D
Internal power of deformation Wi=2**
O*Vf*Rf*lnR (3)3. Simulated Annealing Algorithm
Simulated Annealing method resembles the cooling process of molten metals through annealing. At high temperature, the atoms in the molten metal can move freely with respect to each other but as the temperature is reduced, the movement of atoms gets restricted. The formation of crystal mainly depends on the cooling rate. If the temperature is reduced at very fast rate, the crystalline state may not be achieved at all, instead the system may end up in a polycrystalline state, which has high-energy state than the crystalline state.
This search technique tries to avoid local minima by jumping out of them early in the computation. Towards the end of the computation, when the temperature, or probability of accepting a worse solution is nearly zero, this simply seeks the bottom of the local minimum. Thus effective use of this technique depends on finding a cooling schedule, which gets good enough solutions without taking excessive time for the problem.
Simulated annealing algorithm is treated as a computational methodology, not a fixed algorithm, or program. Two different cooling schedules are generally used in practice. The first reduces the temperature by a constant amount in each phase, while the second reduces the temperature by a constant factor (e.g. 10%) The first method allows the simulation to precede for the same number of Monte Carlo steps in the high, intermediate and low temperature regimes, while the latter causes the simulation to spend more time in the low temperature regime than the high one. When attempting to solve an optimization problem using Simulated Annealing Algorithm, the most obvious representation of the control variables is usually appropriate. The solution generator should introduce only small random changes, and thus allow possible solutions to be reached. But in the later stages the time spent searching around the optimal will be more. Thus the optimal possible solution is obtained.
3.1 General Procedure
1) Simulated Annealing procedure simulates the process of slow cooling of molten metal to achieve the minimum function value in a minimization problem.
2) The cooling phenomenon is simulated by controlling temperature like parameter introduced the concept of Boltzmann probability distribution. According to the boltzmann probability distribution, a system in thermal equilibrium at temperature ‘T’ has its energy distributed probabilistically according to
P (E)=e(E/KT)
Where K is Boltzmann constant
3)
At any instant current point at the X (t), the functional value at that point E (t)=f (x (t))4)
Using the metropolis algorithm we can say that probability of the next point being at X(t1), depend on the value at two points E=E (t+1)-E (t) and is calculated using the boltzmann probability distributionP (E (t+1)=Min ([1,eE/KT]
5)
If the function value at X) 1 (t
is better than X(t), then the point X
) 1 (t
must be accepted.
4 Particle Swarm Optimization
PSO is an evolutionary computation technique inspired by social behavior of birds flocking or fish schooling. In PSO, the potential solutions called particles are flown through the problem space by following the current optimum particles. Each particle keeps track of its coordinates in the problem space, which are associated with the best solution obtained so far. This value is called Pbest.Another best value that is tracked by the particle swarm optimizer, obtained so far by any particle in the neighbors of the particle. This location is called gbest. Acceleration is weighted by random term, with separate random numbers being generated for acceleration towards pbest and lbest locations.
4.1 PSO Algorithm:
1) Initialize each particle.
R. Venketesan / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3758-3761 2) Calculate fitness value for each particle. If fitness value is better than the best fitness value (pbest) in
history. Set current value as the new best.
3) Choose particle with best fitness value of all particles as the gbest. 4) For each particle, calculate velocity according to the equation
V []=V []+C1*rand ()*(pbest []-present [])+c2*rand( )*(gbest[]-present []) Present []=present []+V []
V [], is the particle velocity.
Present [] is the current particle solution. Rand( ) is a random number between 0 and 1. C1, C2 are learning factors. Usually C1=C2=2.
5)
Particle velocities on each dimension are clamped to a minimum velocity Vmax. If the sum of acceleration would cause the velocity on that dimension to exceed Vmax, which is a parameter specified by the user. Then, the velocity on the dimension is limited to Vmax.5.0 Results and Discussion
In this paper the optimal value of Die cone angle () and Extrusion ratio(R) are found out by using Simulated Annealing Algorithm and particle swarm optimization Algorithm.
Simulated Annealing Results
0 50 100 150 200 250 300 350 400 450 500
1 6 11 16 21 26 31 36 41 46
Numbe r of ite ration
Die pressure(Mpa)
min Die pressure=123.14 mpa
Particle Swarm Optimization Results
0 30 60 90 120 150 180 210 240 270 300
1 6 11 16 21 26 31 36 41 46
Numbe r of ite ration
Die pressure(Mpa)
min Die pressure=118.77 mpa
Figure 1 & 2 Variation of Die Pressure with Number of iterations
The results obtained by using Simulated Annealing Algorithm are: Minimum Die Pressure =123.141Mpa
Optimal die cone angle =21.4681 deg Optimal Extrusion ratio =1.45399 Internal power of deformation=7.822 KW
R. Venketesan / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3758-3761 The results obtained by using Particle swarm optimization Algorithm are:
Minimum Die Pressure =118.77Mpa Optimal die cone angle =22.9 deg Optimal Extrusion ratio =1.444 Internal power of deformation=7.784 KW
6 Conclusion
Die profile is designed only for a straightly converging die. PSO and SA can also be applied for designing a streamlined die, where the objective function will become a cubic polynomial equation. Better Results can be obtained by combining SA and PSO. This is termed as MEMETIC ALGORITHM.
The present investigation demonstrates that the effectiveness of an objective function selected for the die shape optimal design and the resulting die profile depends upon the given process condition and geometric parameters that are to be varied. A designer in spite of this has to consider some other factors like material properties, temperature, etc
7 Nomenclature
d
P
Die forceO
Flow stress D Diameter of billet (mm) d diameter of extruded rod vf Final VelocityR Extrusion Ratio
Die cone angle
Friction coefficient L Length of billet in container liner Mean strain rate
Ti uniform interface shear stress between billet and container liner.
References
[1] R.Hill,“Ideal forming for perfectly plastic solids”, Journal of Mechanical Physical Solids, P 223,1967. [2] Avitzr.B, “Metal Forming Processes and Analyses”, McGraw Hill, New York, 1968.
[3] O.Richmond and M.L.Devenberg,“A die profile for maximum efficiency in strip drawing”, ASME, 4th US congress on Applied Mechanics, p1053, 1962.
[4] Z.Zimerman, B.Avitzur “Analysis of the effect of strain hardening on central bursting defects in drawing and extrusion”, Journal of Engineering for Industry, 1970.
[5] A.S.Wifi,“An Optimum curved die profile for the hot forward rod extrusion process”, Journal of Material Processing Technology 97-108,1997.
[6] M.C.Chen, “A Simulated Annealing Approach for optimization of multi-pass turning operation”, International Journal of Production Research, 1996.
[7] Hirotaka Yoshida Yoshikazu Fukuyama, Kenichi Kawata, “A Particle Swarm Optimization for reactive power and voltage control considering voltage security assessment” IEEE Trans. on Power Systems, Vol.15, No.4, pp.1232-1239, November 2001.
[8] George E. Dieter, “Mechanical Metallurgy”, McGraw Hill, New York, P 626-628, 1988.
