Struct Multidisc Optim (2011) 44:529–545 DOI 10.1007/s00158-011-0635-x
INDUSTRIAL APPLICATION
Die shape optimisation for net-shape accuracy in metal forming using direct search and localised response surface methods Bin Lu · Hengan Ou · Hui Long
Received: 6 January 2010 / Revised: 9 January 2011 / Accepted: 10 February 2011 / Published online: 13 April 2011 c Springer-Verlag 2011
Abstract In this paper, three direct search algorithms, i.e. a modified simplex, random direction search and enhanced Powell’s methods together with a new localised response surface method are presented and applied to solve die shape optimisation problems for achieving net-shape accuracy in metal forming processes. The main motivation is to develop efficient and easy to implement optimisation algorithms in metal forming simulations which often involve complex tool and workpiece interaction and coupled thermal and mechanical analysis. Three case studies are presented including a simple upsetting, a 2D blade forging and a forward extrusion problem. In all cases, the objective was to achieve net-shape accuracy of the formed parts, one important criterion for precision forming. C++ programs were developed to implement these algorithms and to automatically integrate optimisation computation and forging simulation. The optimisation results from the three case problems show that direct search based methods especially the modified simplex and the localised response surface methods are computationally efficient and robust for net-shape forging and extrusion optimisation problems. It is also suggested that these methods can be used in more complex forging
B. Lu Department of Plasticity forming Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China H. Ou (B) Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham, NG7 2RD, UK e-mail:
[email protected] H. Long Department of Engineering and Computing Sciences, Durham University, Durham, DH1 3LE, UK
problems where die shape design and optimisation are essential for achieving net-shape accuracy. Keywords Metal forming · Optimisation · Finite element · Direct search · Localised response surface
1 Introduction Metal forming processes such as forging and extrusion are competitive routes for manufacturing structural products due to the advantages of enhanced mechanical properties and microstructures as well as substantial benefits in reduced production cost. For the purpose of forming design and process verification, finite element (FE) based methods and software tools have been commonly used to evaluate material deformation, interfacial and thermal behaviour during and after forging operations (Chenot and Fourment 2004; Hartley and Pillinger 2006). To meet mechanical property and microstructure specifications and to achieve net-shape accuracy with minimum cost, in recent years significant efforts have been made to use various optimisation methods and computational tools as an integral part of whole metal forming process design and validation. As metal forming related process modelling often requires significant computing time to solve large material deformation problems involving complex tool/workpiece interactions, coupled thermal and mechanical behaviour, it is inevitable that some specific optimisation methods are more suitable than others for different types of metal forming optimisation problems such as optimised preform shape, homogeneity of material deformation and microstructure evolution as well as minimisation of forming force and energy. Gradient based methods, evolutionary algorithms and approximation approaches are among most commonly used
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methods for different metal forming optimisation problems (Bonte et al. 2008). Gradient based approach such as the BFGS (Broydon-Fletcher-Goldfarb-Shanno) approach is among one of the earlier methods used in forging optimisation in conjunction with FE simulation. As derivatives of the objective function with respect to design variables are required, calculation of the derivatives called sensitivity analysis may be obtained by direct differentiation or the adjoint state method (Fourment and Chenot 1996; Chung et al. 2003; Zhao et al. 1997; Gao and Grandhi 2000; Srikanth and Zabaras 2000). Many earlier gradient based optimisation methods were applied to preform design problems in metal forming. Depending upon the problems under consideration, the sensitivity analysis can be difficult due to either the complexity for differentiation or the discontinuity of the objective function within the problem domain especially when a large number of design variables and/or a large scale of computation have to be taken into account. To overcome this difficulty, a recent trend has been to incorporate gradient based method with other methods into so called hybrid optimisation methods for metal forming processes where the objective function and other parameters may be constructed using approximate representations so the gradients may be derived more easily (Fourment 2007a; Li et al. 2007; Di Lorenzo et al. 2010). Evolutionary methods including genetic algorithm (GA) have also been used to solve metal forming optimisation problems (Chung and Hwang 2002; Antonio and Dourado 2002; Castro et al. 2004; Naceur et al. 2004). GA based methods use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover in each generation of computations. In each generation of computation, a number of FE simulations are required to give a representative solution of the problem in the design space. Although no derivatives are required in metal forming optimisation computations, GA based optimisation methods can be computationally costly since a large number of metal forming simulations are often necessary for representative solutions in each generation. Due to the difficulty for the derivative results or sensitivity data in gradient methods and the demand for time consuming computations in genetic algorithms, different approximation methods using such as response surface method (RSM) have been developed for more efficient solutions of metal forming optimisation problems (Wei et al. 2008). In particular, response surface method in combination of moving least square approximation has recently been used in metal forming optimisations (Breitkopf et al. 2005; Oudjene et al. 2009). As this approach allows response surfaces to be constructed within a moving region of interests, it has been successfully used in such as sheet metal forming and clinching forming optimisation problems. Other approximation based methods including robust
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design, meta-modelling and kriging have also been reported (Tang and Chen 2009; Fourment 2007b; Jakumeit et al. 2005; Wang et al. 2010). In general the approximation methods offer much improved computational efficiency as less number of simulation samplings are required as compared to such as GA based evolutionary methods and at the same time without using derivative results, which is the case for gradient based algorithms. On the other hand, the robustness and effectiveness of approximation based methods for metal forming optimisation problems are depended upon how approximation is achieved and what search algorithms are adopted in searching for optimum. Having been developed for more than 40 years, direct search methods are a class of algorithms that sequentially examine and compare a few trial or sampling solutions and then determine the best possible search directions in the next iteration (Hooke and Jeeves 1961; Lewis et al. 2000; Kolda et al. 2003). Although direct search methods are best known for solving large scale structural optimisation problems, no attempt has been made to apply this type of methods to solve various metal forming optimisation problems. Though different in sampling methods and search strategies, direct search based methods have a distinctive derivative- and approximation-free feature with relatively small number of samplings required. This makes it especially attractive to metal forming optimisation. Other advantages include the availability of a wide range of methods for use and the flexibility for variation and adaptation. Therefore it is worth investigating and exploring direct search methods in solving metal forming optimisation problems and this is one motivation of this research. In metal forming optimisation, material formability, sheet thickness variation and minimisation of springback are usually used as objective functions for sheet metal forming processes (Naceur et al. 2004; Wei et al. 2008; Breitkopf et al. 2005; Tang and Chen 2009; Jakumeit et al. 2005; Wang et al. 2010), whilst in bulk metal forming optimisation, optimised preform design, microstructural evolution and uniform material deformation are common goals for optimisation (Fourment and Chenot 1996; Chung et al. 2003; Zhao et al. 1997; Gao and Grandhi 2000; Srikanth and Zabaras 2000; Fourment 2007a, b; Chung and Hwang 2002; Antonio and Dourado 2002; Castro et al. 2004). In preform design optimisation, the objective function is normally defined as the difference between the actual and desired shapes after preforming operation and this is achieved by varying the geometry of the billet. This is similar to die shape optimisation for net-shape specifications where the objective is to achieve the desired shape of the formed component at the end of forming operation. But the difference is that the optimisation computations are iterated by changing the die shapes rather than that of the billet in preform design. As the die shapes are normally defined as the
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desirable shapes of the formed component and the dimensional and shape errors of the formed component due to, for example, die elastic deformation and thermal distortions may be compensated for by perturbation of the original die shape, the search for optimum is normally within the vicinity of the initial design instead of a large design space. This is probably the reason that a number of empirical compensation approaches have been developed for net-shape forging and springback minimisation in sheet and bulk metal forming processes (Ou et al. 2006; Lu et al. 2009; Karafillis and Boyce 1996; Cheng et al. 2007). However little research has been reported in using rigorous optimisation methods and algorithms to solve die shape optimisation for net-shape forming operations and this is another motivation of this research. The aim of this paper is to develop direct search methods especially for die shape optimisation problems and to evaluate the efficiency and robustness of these methods for net-shape forming problems. In doing so, we evaluated three different types of direct search methods, i.e. a modified simplex method, a random direction search and an enhanced Powell’s method. To explore the simplicity and flexibility of direct search methods and the robustness of the response surface method, we also developed a new localised response surface method combining linear response surface approximation and direct search for minimisation of the objective function. In examining three case problems including a simple upsetting problem, a 2D blade forging problem and a forward extrusion problem, we were able to evaluate and demonstrate the accuracy, efficiency and suitability of these methods for die shape optimisation problems. The remaining part of the paper is organised into a number of sections. In Section 2, the concepts and optimisation procedures of each direct search method are reviewed and presented. Sections 3, 4, 5 summarise the optimisation definitions and results of the three case study problems. This is followed by a discussion and conclusions section in Section 6.
to search for the minimum from an initial design X0 = {x1 , x1 , . . . , xn }T Xk+1 = Xk + δk Dk k = 0, 1, . . .
(2)
where Xk+1 is the k + 1 iteration of design, Dk is the search direction and δk is the step length. The search direction and the step length are method dependent. Although they differ in detailed procedures and methods in implementation, what is essentially the same is that all direct search methods are only based on the ranks of a set of objective function values without requiring derivative results by either mathematical differentiation or constructing approximate gradients of the objective function (Lewis et al. 2000). This derivative- and approximation-free feature is attractive especially for simulation intensive problems such as various metal forming processes for a number of reasons. Firstly, as metal forming optimisation problems are largely based on FE simulations, in which gradient results of the objective function is often difficult to obtain (Fourment and Chenot 1996; Chung et al. 2003; Zhao et al. 1997; Gao and Grandhi 2000; Srikanth and Zabaras 2000). On the other hand, the use of response surface methods by approximating the objective function and computing approximated derivatives in a local or global region of consideration inevitably requires additional formulation and computation (Fourment 2007a; Li et al. 2007; Di Lorenzo et al. 2010; Wei et al. 2008; Breitkopf et al. 2005; Oudjene et al. 2009). Secondly, most direct search methods only require a small number of sampling of objective function values in each iteration, this is obviously advantageous as compared to such as GA based evolutionary optimisation methods (Chung and Hwang 2002; Antonio and Dourado 2002; Castro et al. 2004; Naceur et al. 2004). Finally, as only objective function values are required for evaluation and selection of new search directions, direct search methods are easy to implement and therefore simple to solve a variety of metal forming and other material processing problems. The developed direct search methods are summarised in the rest of this section.
2 Direct search algorithms 2.1 A modified simplex method A general unconstrained optimisation problem may be defined as (Fletcher 1987; Powell 1998): minimise
f (X)
(1)
where f (X) is the objective function and X (x1 , x2 , . . . , xn ) is a set of design variables. In cases of constraints in presence, the optimisation problem becomes a constraint optimisation problem g j (X) ≤ 0 ( j = 1, 2, . . . , m). In the direct search method, iterative computations are carried out
In the modified simplex method, a n + 1 set of objective function values in a R n space is utilised to find the descent direction of the objective function. As shown in Fig. 1 for a 2D optimisation problem, three random points X p = {X1 , X2 , X3 } are initially selected as the three vertices of a triangle or a simplex in general case. The values of the objective function are calculated at the three vertices. Assuming that X1 is the point with the highest objective function value (denoted by Xh ), the movement of the simplex is achieved by three operations, i.e. reflection,
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Xh with Xe , otherwise the reflection process is restarted by replacing Xh with Xr . 3. Contraction: Two contraction operations may be made in the modified simplex method. Firstly, if the reflected point Xr produces second highest value of the objective function only with the exception of Xh , a contraction from the reflection may be made to calculate the first contraction point Xc1 using the following equation
x2 Xe
X2 Xr Xc1 X0 Xc2
X1=Xh
Xc1 = βXr + (1 − β) X0 Xc3
X3
x1
where β(0 ≤ β < 1) is the contraction coefficient. If the reflected point Xr produces the highest value of the objective function of all points, the second contraction operation may be made to calculate the second contraction point Xc2 by
Fig. 1 Modified simplex method
contraction and expansion (Nelder and Mead 1965; Rao 2009): 1. Reflection: as the X1 is the point of highest value of the objective function of the simplex, reduction of the objective function may be obtained by reflecting point X1 over the other two points, i.e. X2 and X3 . Hence the reflected point Xr may be given by Xr = (1 + α) X0 − αX h
(3)
where X0 is the centroid of the points on the reflecting line between X2 and X3 in the 2D case and X0 = n+1 1 Xi in general; α > 0 is the reflection coefficient, n i =1 i = h
which is the ratio of the distance between X0 Xr and X1 X0 and normally defined to be α = 1. If the objective function of Xr lies between the lowest and highest values of the objective function, a new simplex is started by replacing point Xh (=X1 ) with Xr . 2. Expansion: If the objective function of Xr produces the lowest value from the simplex, an expansion operation is used so as to further reduce the value of the objective function by expanding the reflected point Xr along the direction X0 Xr to a new expansion point Xe using the following equation Xe = γ Xr + (1 − γ ) X0
(5a)
(4)
where γ > 1 is the expansion coefficient, the ratio of the distance between X0 Xe and X0 Xr . If the expansion point Xe produces the lowest value of the objective function, the reflection process is restarted by replacing
Xc2 = βX h + (1 − β) X0
(5b)
Equation (5a) is a new contraction operation used in the modified simplex method in this research. It is recognised that this operation may be achieved within the reflection operation as given in (3). However, the use of (5a) as a separate operation in contraction operation gives additional flexibility for simplex search iterations. If none of the two contraction points, i.e. Xc1 or Xc2 produces a smaller value of the objective function than that of Xh , a further contraction is to replace the current simplex (X1 X2 X3 ) with a new simplex (Xc3 X3 X0 ) by reducing its size toward point X3 assuming that X3 produces the lowest value of the objective function and Xc3 is the centroid of X1 (=Xh ) and X3 , as shown in Fig. 1. The iteration of the above operations continues until the standard deviation of the objective function at n + 1 points over the centroid X0 of the current simplex reaches a specified tolerance. To further improve the efficiency and robustness of the simplex method, a few measures may be used to prevent situations where there is a collapse of the simplex into a hyperplane or a simplex straddles the valley of the objective function (Lewis et al. 2000; Kolda et al. 2003; Nelder and Mead 1965). 2.2 Random direction search The random direction search is to randomly generate several directions around an initial design point with a unit length. This is followed by determining the steepest descent direction among the randomly generated directions and exploring the minimum in the chosen direction. Then the obtained minimum is considered as the initial design point in the next iteration and this procedure is repeated until
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the optimum is obtained (Rao 2009; Solis and Wets 1981; Anderson 1953). The procedure for the random search method may be summarised in the following steps: 1. Initialisation: a feasible design is chosen as X0 . 2. Generation of random unit directions e j : ⎧ ⎫ j ⎪ ⎪ r ⎪ ⎪ 1 ⎨ ⎬ 1 . j . e = ( j = 1, 2, . . . , k) . 1 ⎪ n 2 2 ⎪ ⎪ ⎩ rj ⎪ ⎭ j ri n
(6)
i=1 j
where ri (i = 1, 2, . . . , n; j = 1, 2, . . . , k) are a set of random numbers each lying in the interval [−1, 1] used to formulate the unit random vector ej . 3. Computation of the objective function of random points X j around initial design X0 : X j = X0 + a j e j
( j = 1, 2, . . . , k)
(7)
where aj is the initial step length. By comparing all the values of the objective function f (Xj ), the random design point Xl that produces lowest value of the objective function f (X l ) = min{ f (X j )| j=1,2,...,k } is obtained. If f (Xl ) ≥ f (X0 ), repeat step 3 and reduce the initial step length by half; otherwise start a new search along the descent direction d1 = Xl − X0 . 4. Descent direction search: the descent direction search continues until the minimum is obtained, in which an expansion coefficient τ = 1.3 may be used so as to accelerate the descent direction search. This whole process is illustrated in Fig. 2.
2.3 Enhanced Powell’s method Enhanced Powell’s method employs a set of linearindependent directions to search for the minimum (Rao 2009; Powell 1964). For a 2D optimisation problem as shown in Fig. 3, the search is started from an initial design point X0 along two linearly independent directions normally the coordinate directions, s1 and s2 , respectively. By obtaining the minimum along each of the coordinate directions, i.e. X01 and X02 , a further search converges to X 10 along direction dr = X02 − X0 . In the next cycle of iterations, X10 is used as the new initial design point. The first coordinate direction s1 is replaced by d1 so s2 and d1 are used as the search directions in this iteration. This procedure continues until the minimum of the objective function is found. As all constructed search directions such as d1 and d2 (= X12 − X10 ), etc are conjugate to the positive definite matrix A of a quadratic function, the Powell’s method proves to be quadratic convergence (Rao 2009). To avoid nearly dependent search directions in iteration, it is sometimes necessary to allow old set of linearly independent search directions to be used again in search iterations based on the following criteria (Powell 1964) f X10 < f (X0 )
(8)
2 f X0 − 2 f X02 + f X10 f X0 − f X02 − m 2 < 0.5m f X0 − f X10
(9)
where m is the maximum descent i.e. in thisiteration, m = max f (X 0 ) − f X 10 , f X 10 − f X 20 . The first inequality, (8), indicates that there may be further reduction for the objective function f (X) in the search direction d1 = X02 − X0 . The second inequality, (9),
x2 X01
x2 d1=X20-X0
X02
d1=Xl-X0 Xl ej
X21
X01 X20
d2=X21-X01 X11
X0 s2
x1 Fig. 2 Random direction search
X0
s1
X10
Fig. 3 Enhanced Powell’s method
x1
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indicates that the reduction in the objective function f (X) in the direction of greatest reduction m was a major part of the total decrease. If the conditions in (8) and (9) are both satisfied, new search directions are used in the next search iteration. Otherwise, old search directions are used. 2.4 A localised response surface method Response surface method (RSM) is a widely used computational tool for optimisation (Cornell 1984; Myers and Montgomery 1995). In metal forming optimisations, to ensure the approximation of the objective functions is sufficiently representative in the local area of consideration, a number of pattern search based moving least squares RSM have been developed in sheet forming optimisation for springback control (Breitkopf et al. 2005; Oudjene et al. 2009). Other researchers have used randomly generated set of experiments in sequential and successive approximations in optimisation computations (Rais-Rohani and Singh 2004; Kok and Stander 1999). Nevertheless, one area for further enhancement of localised RSM is to use exploratory measures in directional search in optimisation iterations, which is to be explored in this research. To take the advantage of the derivative-free direct search methods and to utilise the robustness of RSM based approximation approaches, a new localised RSM is developed in this research. The central concept of the localised RSM is to combine the computation of successive response surfaces to derive necessary gradient information of the objective function in the design space with least amount of approximation computation and the flexibility of direct search to determine a feasible steepest descent search direction. As shown in Fig. 4 for an optimisation problem of two design variables, the optimisation iteration of the localised RSM may be achieved by the following steps 1. Construction of linear response surface: a linear response surface from n + 1 random sample points
is firstly constructed. The approximated function in matrix form is given as Y = XA + B where ⎧ ⎪ ⎪ ⎪ ⎨ Y= ⎪ ⎪ ⎪ ⎩
(10)
f (X1 ) f (X2 ) .. .
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
f (Xn+1 ) ⎧ 1 x11 x12 ⎪ ⎪ ⎪ ⎨ 1 x21 x22 X= .. .. ⎪ . . ⎪ 1 ⎪ ⎩ 1 x(n+1)1 x(n+1)2 ⎧ ⎫ ⎧ ⎫ a0 ⎪ ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ a1 ⎬ ⎨ ε2 ⎪ ⎬ A= B= .. .. ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ an εn+1
Φ (X0)
X0 _
Φ (X01)
Φ (X0)
x2
X01
_
Φ(X01)
X02
x1
Fig. 4 Localised response surface method
···
x1n x2n .. . x(n+1)n
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
where B = {ε1 , ε2 · · · , εn }T is an error vector. The linear regression coefficients A = {a0 , a1 , · · · , ak }T may be obtained by minimising the sum of the squared errors (Myers and Montgomery 1995): n+1 i=1
(εi )2 =
n+1
⎛ ⎝ yi − a0 −
i=1
n
⎞2 a j xi j ⎠
(11)
j=1
of which the solution is given by −1 A = XT X XT Y
(12)
2. Determination of steepest descent direction: as no minima may be found from the constructed linear response n ai xi , the gradient of the consurface (X) = a0 + i=1
f(X)
··· ··· .. .
structed local response surface, i.e. −∇ (X), which is the steepest descent direction of the approximated objective function, is calculated and then used as the search direction for the minimum before constructing another localised response surface around the updated minimum. 3. Specification of the sub-region for the linear response surface: to ensure that the linear response surface is sufficiently representative in the sub-region around the design point, i.e. X0 , X10 , X20 . . . for a new descent direction search, the sub-region for generating n + 1 random sample points is defined by the following equation: (i) (i) (13) l j ≤ 1 ± ω(i) η(i) x j ( j = 1, 2, . . . , n)
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(i)
where l j is the size or bandwidth of the sub-region of the linear response surface; ω(i) is the sub-region size coefficient; η(i) is a size or bandwidth reduction coefficient, respectively. For net-shape metal forming optimisation, it is found that the sub-region size coefficient may be defined to be ω(i) = 0.05 ∼ 0.1 so the scope of the design variables gives sufficient variation of the objective function. A size reduction coefficient η(i) = 1 is used if the steepest descent direction is found from the linear response surface approximation. Otherwise the size reduction coefficient is reduced to η(i) = 0.5 for a new linear response surface. This procedure continues until the optimisation convergence criteria are met or the size reduction coefficient becomes sufficiently small, e.g. η(i) = 0.001.
3 Case study 1: upsetting of a cylinder
3.1 Objective function and design variables In forging optimisation for net-shape accuracy, the objective is to minimise forging errors of the forged components. The forging errors may be defined as the deviation of the actual dimensions of the forged component from that of the desired ones. Due to the variability of forging load and process conditions, die/tool elastic deformations and thermal distortions, greater forging errors than specified tolerances often occur even if the die shapes are manufactured to a high level of precision. Therefore in real forging production, die shape modifications are often necessary to compensate for
top die
Δδ Tj
top die
y
P3 P2
P0
y2
P1
x
Fig. 6 Section profile and design variables of die shape
forging errors due to such as die elastic deformation and thermal distortions. Considering the upsetting of a cylinder as shown in Fig. 5(a), the objective is to ensure two flat surfaces of the cylinder to be formed at the end of deformation. This may be achieved by modifying the die surfaces so that any dimensional errors due to such as die-elasticity is minimised, as illustrated in Fig. 5(b) for the top die surface modification (Ou et al. 2006). Therefore the objective function may be defined as the average forging error of the upset cylinder, i.e. the difference between the actually deformed and desired top and bottom surfaces as given by f (X) =
N TD i=1
yiT D − y0T D
2
+
N BD j=1
y Bj D − y0B D
(N T D + N B D ) × (i = 1, . . . , N T D , j = 1, . . . , N B D )
modified die shapes
errors duo to die-elasticity
Δδ Bj
bottom die
bottom die
(a) initial die shape top die modified die shape TD
TD
top surface of the upset cylinder
(xi , yi ) Y X
(x0TD, y0TD)
desired top surface of the upset cylinder
(b) Fig. 5 Forging optimisation for net-shape accuracy a Die shape optimisation b Dimensional error of the top surface
y3
Fig. 7 Case study 1: FE mesh of cylinder upsetting
2
(14)
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Table 1 Case study 1: mechanical and material properties of the workpiece and dies Young’s modulus
Poisson’s
Density (kg/m3 )
Heat Capacity Cp
Conductivity k (N/s/◦ C)
Thermal expansion (1/◦ C)
(N/mm2 /◦ C)
E (GPa)
ratio ν
ρ
Workpiece
−0.078T + 206
0.3
8170
0.002T + 3.49
9.7 × 10−3 T + 12.9
6 × 10−9 T + 10−5
Dies
206
0.3
7900
4.04
48.11
1.2 × 10−5
T temperature (◦ C)
Fig. 8 Flow stresses of the workpiece
500
Flow Stress (MPa)
400
300
200
100
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Strain Strain Rate:0 Temp:982ºC
Strain Rate:2.6 Temp:982ºC
Strain Rate:27.3 Temp:982ºC
Strain Rate:0 Temp:1038ºC
Strain Rate:2.6 Temp:1038ºC
Strain Rate:27.3 Temp:1038ºC
Strain Rate:0 Temp:1121ºC
Strain Rate:2.6 Temp:1121ºC
Strain Rate:27.3 Temp:1121ºC
Fig. 9 Stress σ Z and equivalent plastic strain distributions a Stress σ Z b Effective strain
(a)
(b)
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Optimised Die Shape
direct search algorithms and the localised RSM and to integrate optimisation computation and FE simulation using DEFORM 3D software.
0.055
Height (mm)
0.045 0.035
3.2 FE modelling
0.025 0.015 0.005
-0.005 0
2
4
6
8 10 12 Radius (mm)
Initial Random Direction Search Localised RSM
14
16
18
20
Modified Simplex Enhanced Powell's search
Fig. 10 Optimised die shapes (Case study 1: upsetting of a cylinder)
where yiT D and y Bj D are the vertical nodal positions on the top and bottom surfaces of the upset cylinder; y0T D and y0B D are the desired values of the nominal positions on the top and bottom surfaces of the cylinder, respectively. To define die shapes, the die surfaces may be parameterised by using a B-Spline or other methods. In this case study, a B-Spline curve with four control points is used to define the die surface profiles. As shown in Fig. 6, the die surface is defined by sweeping the B-Spline curve around the vertical axis which passes the first control point P0 . Thus the four control points: P0 , P1 , P2 , and P3 determine the die shape. The vertical coordinates of control points P2 and P3 are chosen as design variables (y2 and y3 ), while other coordinates are fixed in optimisation computations. The initial design of the top and bottom die surfaces is defined by the vertical coordinates of the two control points, i.e. y2 = 0.01 mm, y3 = 0.03 mm, respectively. The tolerance for the upsetting case problem is set to be ε = 0.0005 mm and the optimisation iterations stop when the average forging error calculated by (14) falls into the tolerance zone. C++ programs were developed to implement each of the
Table 2 Case study 1: upsetting optimisation results
The FE model of the cylinder upsetting case problem is shown in Fig. 7. Due to symmetry a quarter of the full upsetting model was used in FE simulation. The radius and height of the dies are 20 and 10 mm and that of the workpiece are 8 and 6 mm, respectively. The mechanical and material properties of the workpiece and dies are defined in Table 1. The flow stresses of the workpiece as a function of strain, strain-rate and temperature are illustrated in Fig. 8 (Ou et al. 2006). The initial workpiece temperature is 1010◦ C and the initial die temperature is 200◦ C. At the interface between the workpiece and dies, the heat transfer coefficient is defined to be h f = 11 kW/m2 ◦ C and the friction coefficient is assumed to be μ = 0.2. The forging simulation was implemented by DEFORM 3D. In the upsetting process, the stroke of the top die is 4 mm with a constant speed of 50 mm/s. To simplify the FE simulation, only the forging step was implemented without the consideration of unloading and cooling steps. Figure 9 show the results of the final upset component and the distributions of the stress component (σ Z ) and the effective strain.
3.3 Optimisation results Figure 10 shows the cross-section of the optimised die shapes from different methods. As shown in the figure, the profiles of the die surface using the direct search and localised response surface methods are close to each other in the area where the radius is less than 13 mm shown by the vertical dotted line, while there are obvious differences outside of the region. Since the radius of the final upset cylinder is less than 12 mm, these differences in the die profile has little effect on the optimised die shape and minimised forging errors.
y2 (mm)
y3 (mm)
Forging
No. of
No. of
error (mm)
iterations
simulations
Modified simplex
0.0514
0.0360
0.00038
10
16
Random direction
0.0504
0.0395
0.00043
9
39
Enhanced Powell’s method
0.0480
0.0405
0.00049
10
45
Localised RSM
0.0490
0.04259
0.00049
7
32
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Fig. 11 Optimisation simulation history (Case study 1: upsetting of a cylinder)
Comparision of Optimisatoin Algorithms 0.007
Average Errors (mm)
0.006 0.005 0.004 0.003 0.002
0.001 Tolerance 0 1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Simulation Number
Modified Simplex Enhanced Powell's Search
Table 2 gives the final average forging error results of the upset cylinder and Fig. 11 shows the forging error reduction history using different optimisation methods. Both the table and figure reveal that similar optimisation results are obtained using the developed methods. This suggests that all direct search algorithms and the localised response surface method are sufficiently robust in finding optimum of the cylinder upsetting problem with the modified simplex method producing maximum forging error reduction. Also shown in Table 2 are the total numbers of iteration and simulation required by each method. Due to the difference in
Fig. 12 Contour plot of the objective function and search paths of different methods (Case study 1: upsetting of a cylinder)
Randmon Direction Search Localised RSM
sampling approach and direction search, each optimisation method requires different numbers of iteration and simulation in optimisation computations. However, as simulation takes most computing time, the computational efficiency is largely determined by the number of simulations instead of search iterations. As given in Table 2, the modified simplex method is most efficient using only 16 simulations from 14 iterations. The second best method is the localised RSM, which requires 32 simulations from seven iterations. The least efficient method is the enhanced Powell’s methods using 45 simulations. It is also shown in Fig. 11 that the objective function results of both the modified simplex and localised RSM descend quickly in the first few simulations but they differ in that the modified simplex converges quickly after a few oscillations whilst the localised RSM takes more simulations to converge. The oscillations of the modified simplex method are due to the reflection and expansion operations, while the slower convergence of the localised RSM is because the sub-region for constructing the response surface is far larger than required when approaching the minimum and therefore it takes more simulations before the bandwidth is reduced to the specified level. On the other hand, although relatively stable, both the random direction search and enhanced Powell’s methods take more forging simulations to converge. Figure 12 shows the contour plots of the objective function in the design space and search paths of all samplings from each method, which shows how each method takes a different search path to the optimum. In particular, the search along the coordinate directions makes the enhanced Powell’s method very rigid hence requiring more simulations and computing time than other methods.
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4 Case study 2: forging of aerofoil sections 4.1 FE modelling and forging optimisation definition
P1
P2
P0 P4
P3 P5
P6
P7
Fig. 13 Case study 2: 2D forging of aerofoil section
Fig. 14 Comparison the of stress (σz ) distributions
In the second case study, a 2D blade forging problem is evaluated for achieving net-shape accuracy of forged aerofoil section. The developed C++ programs were used for optimisation computations while ABAQUS software was used as FE solver for forging simulations. The FE model of the 2D blade forging is illustrated in Fig. 13. The same workpiece and die material properties were used as in the upsetting problem. The initial workpiece temperature is defined to be 1010◦ C and that of the initial die temperature is 200◦ C. The heat transfer coefficient is defined to be h f = 11 kW/m2 ◦ C and the friction coefficient is μ = 0.2. In the forging process, three steps are defined including forging (0.2 s), unloading (0.1 s) and cooling (7200 s), respectively. As shown in Fig. 13, both the top and bottom die surfaces are defined by B-Spline curves with four control points each. For the top die, the vertical coordinates of control points P1 and P2 are selected as design variables. For the bottom die, the vertical coordinates of control points P5 and P6 are selected as design variables. Thus in this case
540
problem, four design variables are defined, i.e. P1y , P2y , P5y and P6y . The same objective function as the upsetting case problem given in (14) is used in this case study which quantifies the discrepancies between the nominal blade shape (i.e. the initial die shape) and the forged aerofoil shape. The tolerance of the forging error for the aerofoil sections is set to be 0.04 mm.
4.2 Optimisation results Figure 14 shows the contour plots of stress component σz at the end of forging obtained of the initial model and the optimised models using different optimisation methods. As shown in the figure, similar stress distributions can be observed for all cases in the workpiece. However, larger stress values are predicted from the optimisation results than that from the initial die shape without optimisation. This is due to the increased material deformation and hence stress level by the die shape modification to compensate for die elastic deformation induced forging errors. Similarly as shown in Fig. 15, smaller maximum equivalent plastic strain results are obtained from the initial die design, whilst larger identical maximum equivalent plastic strain
Fig. 15 Comparison of equivalent plastic strain distributions
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values are obtained at the end of forging using different optimisation methods. This suggests that a small change of die shapes causes an increase in material deformation and maximum strain values in the workpiece. Figure 16 shows the initial and optimised die shapes using different optimisation methods suggesting that similar but not exactly the same optimised die shapes are obtained at the end of forging optimisation. Nevertheless, all optimised die shapes using different optimisation methods have achieved the same level of forging error reduction from an average error of 0.13 mm from the initial design to an average of 0.039 mm to the end of forging optimisations as given in Table 3. Figure 17 shows forging error reduction history using different optimisation methods. These results are consistent with the results obtained from an empirical based compensation method and experimental measuring data (Lu et al. 2009; Ou et al. 2008), in which the average forging errors without and with optimisation are 0.16 ∼ 0.17 mm and 0.035 ∼ 0.04 mm, respectively. The localised RSM is the most efficient method requiring only three iterations and 22 simulations as compared to a similar result of 20 iterations and 27 simulations by using the modified simplex method. The enhanced Powell’s method is most expensive requiring 9 iterations and 39 simulations to finally converge.
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Top Die Surface
Comparison of Optimisation Algorithms 0.15 Average Error (mm)
45.8 45.3 Y (mm)
44.8 44.3 43.8
-16 -14 -12 -10 -8
-6
-4
0.13 0.11 0.09 0.07
0.05 Tolerance 0.03
43.3 -2 0
2
X (mm) Initial Random Direction Search Localised RSM
4
6
8
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
10 12 14 16
Simulation Number
Modified Simplex Enhanced Powell's Search
Random Direction Search Localised RSM
Modified Simplex Enhanced Powell's search
Fig. 17 Optimisation simulation history (Case study 2: 2D blade forging)
(a) Bottom Die Surface 41.1 40.6 Y (mm)
40.1 39.6 39.1
-16 -14 -12 -10 -8
-6
-4
38.6 -2 0
2
4
6
8
10 12 14 16
X (mm) Modified Simplex Initial Random Direction Search Enhanced Powell's Search Localised RSM
(b)
of extruded parts are main design considerations (Long and Balendra 1998). Various optimisation techniques have been recently developed for die shape optimisation (Kim et al. 2001; Ettinger et al. 2004; Lebaala et al. 2009). In this research, an extrusion die shape optimisation for netshape accuracy of the extruded component is also evaluated. As shown in Fig. 18 of the 2D axis-symmetric extrusion model, the objective is to achieve net-shape accuracy of the extruded part by optimising the extrusion die shape in the transition region, which is defined by a B-spline curve with seven control points (P1 , . . . , P7 ) and the horizontal coordinates of P2x , P3x , P4x , P5x , and P6x are selected as design variables. The objective function is given by
Fig. 16 Optimised die shapes (Case study 2: 2D blade forging) a Top die surface b Bottom die surface
f (X) =
N
i=1
xiD P − x0D P N
2 (i = 1, . . . , N )
(15)
5 Case study 3: forward extrusion of components 5.1 FE modelling and extrusion optimisation definition Extrusion is widely used in mass production for a wide range of metallic and non-metallic parts. Uniform material flow, minimised energy consumption and precision
Table 3 Case study 2: blade forging optimisation results
where xiD P is the horizontal nodal coordinate on the die profile and x0D P is the desired nominal positions of the extruded parts, also used as initial extrusion die shape. The objective is to minimise geometric errors of the extruded part due to die elastic deflection and thermal distortion. The tolerance of the geometric error is defined to be 0.01 mm.
P1y
P2y
P5y
P6y
Forging
No. of
No. of
(mm)
(mm)
(mm)
(mm)
error (mm)
iterations
simulations
Modified simplex
46.162
46.004
40.217
40.369
0.039
20
27
Random direction
46.038
46.110
40.463
40.126
0.039
4
32
Enhanced Powell’s method
46.015
46.160
40.387
40.124
0.039
9
39
Localised RSM
46.053
46.071
40.355
40.290
0.039
3
22
542
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as a function of strain and temperature at given strain rates (Long and Balendra 1998; Long 2004). The friction coefficient between the workpiece and the die is defined to be 0.05. The initial temperature of the workpiece and the die is assumed to be room temperature (20◦ C). The heat transfer coefficients during extrusion and cooling are defined to be 20 and 0.02 kW/m2 C, respectively.
Fig. 18 Case study 3: forward extrusion model
5.2 Optimisation results
P1
P2
P3 P5
P4 P6
P7
As shown in Fig. 18, the workpiece has a diameter of 40 mm and length of 60 mm.The diameters of the coneshaped die are 20 and 40 mm and the die angle is 22.5◦ . The punch stroke is defined to be 35 mm and the punch velocity is 150 mm/s. The material of the workpiece is pure aluminium (DIN: Al99.5) and the flow stresses are defined
Fig. 19 Comparison of stress (von Mises) distributions
Figure 19 shows the contour plots of von Mises stress distributions of the FE model at the end of extrusion from the initial and optimised die shapes using different optimisation methods. As shown in Fig. 19, similar von Mises stress distributions can be observed for all optimised models with some slight variations from that of the initial die shape without optimisation. Concerning the equivalent plastic strain results shown in Fig. 20, almost identical strain distributions are obtained, which indicate that the die shape modifications in optimisation are less sensitive to the strain distribution results. Figure 21 compares the die shapes at the transition region before and after optimisation. Figure 22 shows forging error reduction history using different optimisation methods. As shown in the figure, the average error of the extruded part is 0.050 mm from the initial die design without optimisation as compared to 0.05 ∼ 0.055 mm from published data
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Fig. 20 Comparison of equivalent plastic strain distributions
(Long and Balendra 1998; Long 2004). After die shape optimisation, the average errors are significantly reduced to the region of 0.007 ∼ 0.01 mm, as given in Table 4. From both Fig. 22 and Table 4, it is evident that the modified simplex and localised RSM perform significantly better
Die surface shape -1.00
-6.00
than the other two methods in computational efficiency. The modified simplex and localised RSM only require 19 and 23 simulations to converge, whilst the enhanced Powell’s and the random direction search take 87 and 143 simulations to reach the converged solutions. In comparing the results obtained from the other two cases, it is suggested that the convergence rate of the modified simplex and localised RSM is not affected by the number of design variables. But in the random direction search and enhanced Powell’s method, a modest increase of design variables results in a significantly increased number of simulations and thus reduced computational efficiency.
Y (mm)
-11.00
-16.00
-21.00
-26.00
-31.00
-36.00 5
10
15
20
25
X (mm) Initial Random Direction Search Localised RSM
Modified Simplex Enhanced Powell's Search
Fig. 21 Comparison of die shapes at transition area before and after optimisation (×50)
Fig. 22 Optimisation simulation history (Case study 3: forward extrusion)
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Table 4 Case study 3: extrusion optimisation results
P2x
P3x
P4x
P5x
P6x
Forging
No. of
No. of
(mm)
(mm)
(mm)
(mm)
(mm)
error (mm)
iterations
simulations
Modified simplex
19.981
17.747
14.511
11.429
9.892
0.008
13
19
Random direction
19.980
17.751
14.504
11.435
9.882
0.010
11
143
Enhanced Powell’s
19.979
17.747
14.496
11.434
9.874
0.008
7
87
19.979
17.760
14.504
11.443
9.878
0.007
3
23
method Localised RSM
6 Discussion and conclusions To evaluate and validate the developed direct search methods (modified simplex, random direct search and enhanced Powell’s) and a new localised response surface method, three case problems including upsetting of cylinder, 2D blade forging and forward extrusion are examined all involving die shape optimisation for net-shape accuracy. The results show that all methods are able to derive converged solutions to specified tolerances but at different computational cost, an important question for simulation intensive optimisation applications. The modified simplex and localised response surface methods are more computationally efficient and less dependent upon the type of problems and number of design variables than the enhanced Powell’s and random direction search methods. This is, to a large extent, due to the difference in the flexibility and versatility of each method used in sampling and constructing search directions. For example, the simplexs in the modified simplex method are flexible and adaptable and the localised response surface method derives robust descent directions from the line response surfaces. On the other hand, in the enhanced Powell’s method the search along the coordinate directions exhibits a particular slow pace to convergence. It is also noted that the correlation between the number of iteration and simulation is method depended but in the end the computational efficiency is mainly determined by the number of simulations required in optimisation computation. The results from the three case problems demonstrate that the derivative- and approximation-free (with the exception of the localised response method in strict terms) direct search methods are easy to implement and to integrate with metal forming simulations. There is a clear potential for applying this type of methods for solving more challenging and complex metal forming optimisation problems. To do so, there is a scope for further research so their overall capability and possible limitations for general metal forming and other material processing problems can be fully established.
Acknowledgement The authors would like to thank the UK Engineering and Physical Science Research Council (EPSRC) for funding of the present research (EP/C004140/1 and EP/C004140/2).
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