Optimum Design of Forging Dies Using Fuzzy Logic in Conjunction with the Backward Deformation Method

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1 Optimum Design of Forging Dies Using Fuzzy Logic in Conjunction with the Backward Deformation Method F. R. Biglari, N. P. O Dowd and R. T. Fenner Mechanical Engineering Department, Imperial College of Science, Technology and Medicine, London SW7 2BX, UK. Abstract-A novel shape optimisation method is presented for the design of preform die shapes in multistage forging processes using a combination of the backward deformation method and a fuzzy decision making algorithm. In the backward deformation method, the final component shape is taken as the starting point, and the die is moved in the reverse direction with boundary nodes being released as the die is raised. The optimum die shape is thereby determined by taking the optimum reverse path. A fuzzy decision making approach is developed to specify new boundary condition for each backward time increment based on geometrical features and the plastic deformation of the workpiece. In order to demonstrate this approach, a design analysis for an axisymmetric disk forging is presented in this paper. 1. Introduction Forging is currently one of the most economical processes for the manufacture of engineering components. Material costs are an important fraction of the total cost of forging and any reduction of material waste during the operation has a direct effect on the price of the finished product. The need to reduce material waste and lead time has encouraged the trend towards the use of near net shape forgings and precision forging operations [1]. In precision forging, material flow patterns are a significant consideration and useful information can be obtained through numerical simulations of the forging operation. In this paper, forging die design is examined using the finite element method; in particular the design of preform dies is dealt with. Preform dies are used when the final shape of the product is complicated and a near net shape forging is required. If the final shape can not be achieved in a single stage forming, the workpiece is first deformed to an intermediate shape, known as the preform. For very complicated component geometries a number of preform dies may be required. The preforming sequence must ensure the appropriate control of material flow in order to fill the die cavity and to obtain an acceptable surface finish. Since there is currently no standard preform die design procedure, most preform die design is a trial and error process which can be costly and time consuming and indeed may not give the optimum preform shape. An optimum preform shape should minimise material waste in the workpiece through the use of flashless forging, reduce die wear to minimise manufacturing cost and also minimise residual stress and plastic strain gradients in the finished workpiece. Since the final component shape and the material specification of the product are the only technical information given, reversing the forming process can be a suitable method to predict the preform shapes. The preform die shape is then given by the shape of the Corresponding author

2 workpiece boundary obtained from the reversed deformation. This method is generally known as the backward tracing method, (Hwang and Kobayashi [2, 3, 4]). 2. The Backward Deformation Method In a multistage forging process, the final shape of each stage is in fact the preform shape of the next stage. The backward deformation method is used to predict the shape of the workpiece at any stage in the deformation process, when the geometry and process conditions are known at the final stage. In forward simulation of the forging process, the finite element method is used to obtain the position of the workpiece and other field variables when the material flows into the die cavity. Hence, the forward simulation starts from the initial shape and ends with the final shape. The backward tracing method was developed by Hwang and Kobayashi [2] to reverse the forming process in a rolling operation. Later, this method was used to design preform die shape in forging of axisymmetric shapes [3, 4]. Lanka, et al. [5] proposed a technique for the design of die shapes for plane strain forgings. A design procedure was introduced to obtain the number of stages and the shape of each preform die for manufacturing a desired product. Staging criteria are developed from the results of the forging simulation and the number of stages is based on the stress ratio and strain rate gradient information. Grandhi, et al. [6] introduced the optimum control design algorithm into process parameter design of the forging process. The optimum ram velocity for maintaining the specified strain-rate in the billet, for an isothermal disk forging, was generated through this approach. These analyses were all carried out using the finite element method with a rigid viscoplastic material formulation. Design constraints on strain-rates and temperature variation are imposed to achieve the desired forging conditions. Han, et al. [7] developed an optimisation method for the design of intermediate die shapes for axisymmetric forging operations. The approach was also based on backward deformation simulation and assumed a rigid viscoplastic material response. The advantage of this optimisation approach is that it can determine the intermediate die shapes from the final product shape by applying constraints on the plastic deformation of the material. A preform design method was presented by Osman and Bramley [8-9] for forging of the rotationally symmetric parts using upper bound modelling based technique. The preform design methodology used in this technique is dependent on the geometrical features of the product and was shown for preform design of a gear forging. The gear geometry was divided into five regions and the velocity fields were calculated for each region based on the related flow functions. The reverse computer simulation was then carried out to achieve the preform shape which reduces the forging force. In this work, the finite element method is used to simulate the forging process. The deformation of a two-dimensional element during forging, is illustrated schematically in Fig. 1. This element is a four noded element which consists of nodes A, B, C and D. The initial configuration of the nodes are shown by A 0, B 0, C 0 and D 0 and after a small time increment t the nodal coordinates are updated to A 1, B 1, C 1 and D 1. Finally, in the nth time increment, the elemental nodes are located as A n, B n, C n and D n. Similar to the

3 forward simulation, in backward simulation, in order to find the previous shape of the workpiece, the whole process is divide into small time increments. In this method, we start from the final shape of the element (A n, B n, C n and D n ) and try to predict the shape of the element in the previous time increment (A n-1, B n-1, C n-1 and D n-1 ) until reaching the state (A 0, B 0, C 0 and D 0 ). In preform shape design, the displacement of each node must be predicted from the previous time increment. A typical backward deformation process for a time increment of t is shown in Fig. 2. The horizontal axis represents the time and the vertical axis represents the design variable which is the x coordinate of the node A in this figure. At time t n the node A is located at x n and the coordinate x n-1 is required to be predicted when there is a time interval of t between t n and t n-1. The backward deformation prediction is deemed to be successful when forward loading from x n-1 gives the coordinate x n after t (to within a specified tolerance). In forward loading, the relationship between x n and x n-1 may be expressed by (j) [F(j)] xn = xn 1+ vn 1 t, (1) where v [F(j)] is the forward velocity and j is the number of forward iterations for any time increment. In backward deformation, the corresponding relationship cx is (j) [B(j)] x 1 = x v t, n n n (2) where v [B(j)] is the backward velocity and j is the number of backward iterations for any time increment. [ B( 1) ] In the backward tracing method, shown schematically in Fig. 2, an initial guess, v n, is made to predict the velocity of node A in the reverse direction. Using this velocity, the ( 1) ( 1) point A n- 1 is evaluated by Equation (2). During forward loading, the point A n and the ( 1) ( 1) value of x n are obtained based on Equation (1). Next, the coordinate x n is compared [ B( 2 )] ( 2 ) with x n and the second guess for the velocity v n is made. The point A n- 1 is then ( 2 ) obtained and again forward simulation is carried out and coordinate x n is compared with x n. If they are close enough, based on the tolerance that has been set for this design ( 2 ) variable, the point A n 1 is accepted as the previous position of the node A at the time increment (n-1). In Fig. 2, it is shown that the jth guess for the reverse velocity v [B(j)] n gives (j) point A n- 1 and the forward velocity obtained is almost equal to the reverse velocity. Thus (j) (j) the points A n and A are very close to the points A n- 1 n and A n-1 respectively and the point (j) A is accepted as the previous location of node A n- 1 n. Similarly the location of the node A for all previous time increments can be predicted. Since the number of iterations for each time increment can be high, an optimisation method is used to determine the location of (j) the node A at the previous time increment A. n- 1

4 2.1 Optimisation Problem The optimisation problem can be defined as follows: The design variable u [ B( j)] is determined which minimises the following objective function: F j L ( u) = u u j [ ( )] [ B( j)] (3) where u [ B( j)] is the backward displacement and u [ F( j)] is the solution of the forward simulation for a time increment. The matrix representation of the governing equation for forward simulation is ( j ) K( x, εε, ) [F(j)] ( j) u - f( x )= 0 n 1 n 1 (4) where K is the finite element stiffness matrix, u the nodal displacements and f the nodal forces. Substituting Equation (4) into Equation (3), the objective function can be expressed in matrix form as B j B j L( ) K(,,, B j u = x u εε) 1 f( x, u ) u [ ( )] [ ( )] [ ( )] [ B( j)] n n (5) Based on the general iterative optimisation procedure [10], the minimum of the objective function L( u) can be found numerically by iterating on the following equation: [ B( j)] [ B( j 1)] [ B( j)] u = u α L( u ) (6) [ where L( u B ( j )] ) is the gradient of the objective function with respect to the design variable u [ B( j)] and the scalar quantity α defines the distance that we wish to move in [ direction Lu ( B ( j )] ) Sensitivity analysis The gradient of the objective function with respect to the design variable is calculated by the design sensitivity analysis procedure [11]. The design variable is the nodal displacement in the reversed direction ( u [ B( j)] ). Using Equation (3), the derivative of the objective function can be expressed as [ B( j)] Lu ( ) [ B( j)] u [ F( j)] u = [ B( j)] 1 u (7) substituting Equation (4) into Equation (7), we get [ B( j)] L( u ) 2 f K = K K f B j B j B j u u u [ ( )] [ ( )] [ ( )] 1 where K -2 =(K -1 ) 2. Alternatively, we can obtain the derivative of forward displacement with respect to the backward displacement, required in Equation (7) directly with following equation: (8)

5 u u [ F( j)] [ B( j)] = u u 2β [ F( j)] [ F( j)] 2 1 (9) where u F j u B j [ ( )] 2 and u F [ ( j )] 1 are the numerical solutions of the forward simulation for [ ( )] + β and u [ B( j)] β respectively and β is the distance from the sampling points to the last search point. 3. Release of Boundary Nodes During the backward deformation process, as the die moves in the reverse direction, nodes which were in contact with the die are released. Therefore, it is necessary to predict which nodes were not in contact with the die in the previous time increment. This then sets the boundary conditions for the forward simulation and a different strategy in releasing nodes can give rise to different preform shapes. When the number of boundary nodes in contact with the die is large, the number of combinations for releasing the nodes become enormously high. Since releasing a node has an important effect on the deformation for the next time increments, an adequate node releasing strategy should be used. The decision making process carried out in this work is based on the fuzzy logic concept and used for releasing the boundary nodes in each backward time increment. Rao [12, 14] described a fuzzy optimisation approach to structural design using the multiobjective optimisation method. The fuzzy set concept is adopted in the shape optimisation of structures based on stress analysis of the structural components. An application of fuzzy optimisation techniques to multi-objective, multiple-attribute decision making problems has also been introduced by Yager [15]. More recently fuzzy optimisation was used in the control of the feed rate of CNC machines in small-diameter drilling processes [16]. In the present work, the fuzzy optimum decision making concept is used to specify optimum boundary conditions during the backward deformation process. 3.1 Node Releasing Strategy Using Fuzzy logic The fuzzy decision making approach described in this paper is developed based on optimum fuzzy controllers. In most fuzzy systems, the known information about the process is accumulated in the knowledge repository. This information can be classified through fuzzy rules and gathered by the simulation of the process and studying the process constraints. In the forging process there are a number of factors that should be carefully studied during the forward deformation of the workpiece. These may then be used in the decision making process for node release during backward deformation. The most important factors are discussed below for an axisymmetric disk and related forging die and illustrated in Fig. 3 and Fig. 4. This geometry will be analysed in detail later in the paper. Since in axisymmetric forging, the material is spread out by the die, the areas of the die surface which will come into contact with the boundary nodes of the component near the end of the process are :

6 1- Those areas of the die which are at a greater distance from the rotational symmetry axis line 2- Those areas of the die which are a greater distance from the plane of symmetry. 3- The top and bottom die separation line (in flashless forging). These 3 may be considered to be geometric constraints on the forging process. We can also specify a process constraint viz. 4- To have a near uniform residual stress and to reduce the likelihood of forging defects in the workpiece, the plastic strain deviation from the mean should be minimised. As can be seen in Fig. 4, region a is close to the rotational symmetry axis, therefore it will come into contact with the workpiece at the very beginning of the forging process, while regions b and c will not come into contact with the die until much later. In addition, region b has the largest distance from the plane of symmetry while it is not too far away from the separation line. Therefore it will come into contact with the die close to the end of the process. Thus for this component there are three geometrical features that must be considered to obtain an appropriate decision. This can be done by defining a number of fuzzy sets. In this work, the distances of boundary nodes from the rotational symmetry axis, the plane of symmetry and separation line are represented by five fuzzy sets which are Very Close (VC), Close (C), Average (A), Far (F) and Very Far (VF). The input variables which are the distance of a boundary node from the plane of symmetry (D PS ), rotational symmetry axis (D RS ) or separation line (D SL ) can be fuzzified by the linear membership functions shown in Fig. 5. The effective plastic strain deviation (E d ) is fuzzified by the fuzzy membership functions also shown in Fig. 5. The effective strain deviation is characterised by five fuzzy sets which are Very Small (VS), Small (S), Medium (M), Large (L) and Very Large (VL). Similarly, the output variable which is the priority of node releasing (P NR ) is divided into five fuzzy sets which are Very Low (VL), Low (L), Average (A), High (H) and Very High (VH). The degree of membership is calculated by Equation (10) based on the fuzzy sets and type of input and output feature. µ( X) = bx + g if k<x<e (10) Where µ( X ) is the degree of membership, X is the input variable and b, g, k, and e are the constants that are declared in Tables 1, 2 and 3. Note that the distance of a boundary node from the plane of symmetry is scaled by considering the die displacement in each time increment. Similarly, the effective strain deviation is also scaled based on the amount of plastic deformation done in the workpiece. The following tables are given based on the first time increment and a linear change of maximum range of the effective strain deviation through all time increments.

7 3.2 Fuzzy Decision Making Rules The relationship between inputs and output is characterised by a set of linguistic statements (fuzzy rules). The fuzzy rules are defined based on expert knowledge and observations from experimental work. In this work a set of computer simulations have been carried out and used to define the fuzzy rules. Generally, fuzzy decision making and reasoning process are based on a compositional rule which is in form of antecedent-consequent pairs. The antecedent is normally referred to the relationship between input variables and known information about the process. In this work, the input variables are grouped into two sets. The first set consists of the first three input variables. These are the distance of a boundary node to the plane of symmetry, die separation line and rotational symmetry axis. The second set consists of the last input variable which is the effective strain deviation. The maximum number of fuzzy rules can be obtained based on the number of input and output sets. In this case, there are four input variables and one output variable each of which is classified into five fuzzy regions. In this work, a total number of 75 rules are defined and classified into 15 main groups which are shown in Table 4. In Table 4, each fuzzy rule will be identified for convenience as R i, j in which i is row and j is column number. For example, the fuzzy rule R 42, in Table 4 can be rewritten as R42, : IF ( DSL is VeryClose AND Ed is Large) THEN ( PNR is VeryHigh) Using these fuzzy rules, the logical decision is carried out by the fuzzy intersection (AND) and fuzzy union (OR) operators. Mathematically, the fuzzy (AND) and the fuzzy (OR) operators are symbolically used as and respectively. Hence, the following equation can be generated to calculate the fuzzy membership values for the output variable. m { } µ ( P ) = µ (D ) µ (E ), µ (D ) µ (E ), µ (D ) µ (E ) NR w PS d SL d RS d i= 1 (11) where µ( P NR ) w is the fuzzy membership value for node releasing priority, (w=1,2,..,5) is the number of fuzzy sets and m is the number of fuzzy rules which are fired for each node in each time increment. The numerical procedure of fuzzy decision making for a typical time increment and boundary node is shown in Fig. 5. Assuming that the input variables such as D PS, D SL, D SL and E d for a typical time increment t and a boundary node n, are given. These input variables are fuzzified by the membership function of each individual fuzzy set. The degree of membership values of 0.67 and 0.33 are calculated using the fuzzy membership functions of the first input feature D PS, for the fuzzy sets Very Far and Far respectively. [ ] µ( ) = 067., 033. D PS VeryFar Far In addition, the degree of membership values of 0.86 Close and 0.14 VeryClose for D SL, 0.56 Far and 0.44 VaryFar for D RS and 0.76 Medium and 0.24 Large for the input feature E d are determined.

8 Based on the fuzzy rules in Table 4, the fuzzy AND ( ) operator is used to calculate the fuzzy values 0.76 High, 0.33 High, 0.56 High, 0.67 High, and 0.44 High from the fuzzy rules R 3,5, R 3,10, R 3,12, R 3,13, and R 3,15 and the fuzzy values 0.14 VeryHigh and 0.14 High from the fuzzy rules R 4,2 and R 3,2 and the value 0.24 High from the fuzzy rules R 4,12, R 4,5, and R 4,10 and the value 0.24 VeryHigh from the two fuzzy rules R 4,13, and R 4,15. The fuzzy OR operation which determines the union of the logical propositions is used to take the effect of all input variables which were initially treated by the AND operator into account. Since all the fuzzy values 0.76 High, 0.33 High, 0.56 High, 0.67 High, 0.24 High, 0.14 High and 0.44 High belong to the fuzzy set High, the value 0.76 can be obtained using the fuzzy OR operation. 7 { } { } µ ( P ) = µ (D ) µ (E ), µ (D ) µ (E ), µ (D ) µ (E ) NR High PS d SL d RS d i= 1 = 0. 67, 0. 33, 056., 0. 24, 0. 44, 0. 76, 014. = Similarly, the fuzzy values 0.24 VeryHigh and 0.14 VeryHigh were the results of fuzzy AND operation of the rules R 4,2, R 4,13, and R 4,15, the fuzzy value 0.24 which belongs to the set VeryHigh is then obtained. 2 { } t h t { } µ ( P ) = µ (D ) µ (E ), µ (D ) µ (E ), µ (D ) µ (E ) NR VeryHigh PS d SL d RS d i= 1 = ,. = 024. The outputs of the inference process are 0.76 High and 0.24 VeryHigh which are still fuzzy values and they need to be defuzzified. The defuzzification is basically a mapping from a space of fuzzy values into that of nonfuzzy universe. The most commonly used strategy is the centroid defuzzification method which takes the effect of all inference outputs into account by calculating the centre of area of the releasing node priority fuzzy sets. h h P NR X = X µ ( X) XdX µ ( X) dx (12) where X is the numerical value of the node releasing priority and µ( X ) is the fuzzy membership function for P NR. In this example, the value µ( P NR ) High = 0.76 intersects the fuzzy set High at 68% and 79.5% while the value µ( P NR ) VeryHigh = 0.24 intersects the fuzzy sets High at 92% and Very High at 79.5% and 121.4%. Therefore, the defuzzified value for P NR can be calculated through the use of Equation (10), Table 3 and Equation (12). P NR = 92% µ ( X ) High X dx + µ ( X ) VeryHigh X dx 92% % =76.3% µ ( X) dx + µ ( X) dx 68% 92% 68% High % 92% VeryHigh

9 Hence, using centroid defuzzification method, nonfuzzy P NR, i.e. the releasing node priority is estimated. Then, the decision for the detaching a boundary node is made based on the highest priority. 4. Simulation of the Forging of an Axisymmetric Disk A three-dimensional illustration of the upper half of a H-section disk is given in Fig. 3. A two stage forging process of this geometry is considered and the optimum preform shape is obtained by the backward deformation method. The commercial finite element program ABAQUS [17] is used for the simulations. 4.1 Constitutive Model and Material Definition The deformation process is assumed to be isothermal and the workpiece material is taken to be a low carbon steel. The constitutive model used for the steel is a rate dependent model with von-mises yielding and isotropic hardening. The elastic response is assumed linear and the rate of plastic strain is given by the flow rule: D ε = 3 D ε 2 S pl pl ij ij σ (13) where S ij is the deviatoric part of the stress tensor, σ the effective stress and D ε pl the equivalent plastic strain rate. The strain rate dependence is given by a power law of the form Dε pl σ = D 1 σ 0 p (14) Where D and p are material parameters and σ 0 is the flow stress. For this formulation the only internal variable is the effective plastic strain ε pl. The material parameters chosen were those used for modelling of the forging of a steel billet in [17], i.e. D=40/sec; p=5; initial static yield stress=700 MPa and work hardening rate=300 MPa. The above material formulation is for small deformation theory. For a large deformation analysis as is used in this case the equivalent plastic strain, D ε pl, is replaced by the symmetric part of the rate of deformation tensor, L p,where p p p L = F ( F ) 1 (15) with F p is the plastic part of the deformation gradient, F. The stress measure used is the Cauchy stress. Further details of the material formulation for small and finite strain is given in [17]. 4.2 Numerical Results for Forward Simulation For comparison purposes a forward simulation is first carried out using a cylindrical billet. This is illustrated in Fig. 6. Contours of plastic strain are shown, the darker regions

10 corresponding to regions of high plastic strain. It may be seen that this choice of billet gives rise to a fold-over defect in the final component due to the workpiece material separating from the die. This pattern of deformation was also observed in [7] where a similar axisymmetric forging was analysed. The fold-over defect also leads to a high stress concentration and thus large plastic strains as may be seen in Fig. 6h. Clearly this is an undesirable feature so an optimum die design would seek to avoid such large plastic strain gradients. We next attempt a backward simulation of the process and show that the resultant preform die gives rise to considerably reduced plastic strain levels in the simulated finished component. 4.3 Backward Deformation Analysis The simulation of the deformation in the reverse direction which was carried out using the fuzzy decision making method, is shown in Fig. 7. The backward deformation starts with all the boundary nodes in contact with the die surface and ends when all of the boundary nodes are released. In other words, it is assumed that the backward deformation process is started from the finished workpiece. The initial value for the internal variable (effective plastic strain) in taken from the last step of the forward simulation analysis. The boundary nodes are classified into four sets which are designated I, II, III and IV. These nodes are released sequentially as determined by the fuzzy node releasing algorithm. In Fig. 7b it is illustrated that two sets of nodes (III and IV) have been detached from the die surface. These two sets are released at the very beginning of the backward deformation process because they are expected to come into contact with the die at the very end of the forward deformation process. Set IV is the set of boundary nodes which are very close to the die separation line and set III the set of boundary nodes located in the depth of the die cavity very far from the plane of symmetry. The boundary nodes which are located in region II are detached sequentially in Fig. 7d and 7e. Finally, the boundary nodes in region I which are located very close to the rotational symmetry axis are detached from the die when the backward deformation process is finished. This sequence of node release is consistent with the discussion of Section 3.1. An additional design parameter must be incorporated into the simulation as discussed below. In a typical forward deformation process, the workpiece is pushed from above and below. Most of the deformation therefore takes place in the centre and plane of symmetry so a barrel shape is normally obtained. Similarly, in the backward deformation process, the middle and central region of the workpiece are deformed backward more than the other regions and as a result of that the workpiece may become narrow in these regions (lower area of the region IV). In order that the preform itself may be produced by a single stage forging, (i.e. two stages for the complete operation) this must be avoided as obtaining a workpiece with a narrow waist is very difficult for a closed die forging operation. Therefore, a constraint on the movement of the boundary nodes located in region IV is essential to prevent the preform becoming narrow at the workpiece waist. This is achieved by constraining the nodes in region IV to lie along a line of constant angle. Within the context of the backward deformation analysis, this is relatively simple to achieve. The final preform shape is shown in Fig. 7f. Once the backward deformation has been completed, to verify that the preform shape obtained is indeed appropriate, i.e. does not

11 lead to any forging defects or large plastic strain gradients a standard forward analysis may be carried. 4.5 Forward Simulation of Two Stage Forging Process Fig. 8 and Fig. 9 illustrate the complete two stage forging operation of the H-section disk. First, a solid cylindrical billet is deformed to produce the preform for the second forging stage shown in Fig. 8. The die shape in the first forging stage is the negative of the final shape of the backward deformation analysis (Fig. 7f). The second forging stage starts with the preformed workpiece and the die cavity is the negative shape of the finished workpiece. As shown in Fig. 9, the deformation begins when the upper die comes to contact with the highest region of the workpiece. The top-central region of the preform is the first area that comes to contact with the die. During the deformation, the boundary nodes come into contact with the die in a gradual manner. This helps to push the workpiece material smoothly into the die cavity on the top-right hand side. In order to assess the performance of the preform design procedure, the results of the preceding analysis are compared with those from the analysis without the use of an optimum preform. In Fig. 10, the comparison of the accumulated plastic strain for the two cases is shown. In Fig. 10a, the accumulated plastic strain contours for the forging simulation was carried out without using the optimum preform. In Fig. 10b, the forging simulation is performed using the optimum preform which was explained before. In Fig. 10a, the maximum and minimum plastic strains are 2.5 and 0.11 respectively. Moreover, the average plastic strain is 1.27 and the plastic strain deviation of maximum strain from average strain is In Fig. 10b, the maximum and minimum plastic strains are 1.83 and 0.11 respectively. In addition, the average plastic strain is 0.96 and the plastic strain deviation of maximum strain from average strain is Thus the deviation of the plastic strain from the mean value is decreased by about 20% by using the optimum preform. It may also be observed that the contours of plastic strain for the optimum preform, Fig. 10a, are more uniform than for the single stage forging, Fig. 10b. Finally, the use of the optimum preform does not lead to fold-over development and the formation of a forging defect as was observed for the simulated single stage forging. 5. Conclusions A new method has been presented for the design of preform shapes for multi-stage forging. The backward deformation method in combination with a fuzzy decision making approach is used to predict the shape of the workpiece at each backward time increment. The resultant boundary shape can then be used to design the preform die. An example of the forging of a H-section disk was presented and it was shown that the backward deformation method in conjunction with the fuzzy logic algorithm leads to a more uniform plastic strain in the simulated final component. The method presented can easily be generalised to more complex geometries and is independent of the finite element method and material model employed. References

12 [1] Dean, T. A., Concepts and Practice in the Precision Forging, 7 th International Cold Forging Congress, pp , [2] Hwang, S. M., and Kobayashi, S., Preform Design in Plain-Strain Rolling by the Finite Element Method, International Journal of Machine Tool Design Research, Vol. 24, No. 4, pp , [3] Hwang, S. M., and Kobayashi, S., Preform Design in Disk Forging, International Journal of Machine Tool Design Research, Vol. 26, No. 3, pp , [4] Hwang, S. M., and Kobayashi, S., Preform Design in Shell Nosing at Elevated Temperatures, International Journal of Machine Tool Manufacturing, Vol. 27, No. 1, pp. 1-14, [5] Lanka, S., Srinivasan, R., and Grandhi, R. V., Design Approach for Intermediate Die Shapes in Plane Strain Forgings, Journal of Material Shaping Technolog, Vol. 9, No. 4, pp , [6] Grandhi R. V., Cheng, H, and Kumar S. S, Optimum Design of Forging Process With Deformation and Temperature Constraints, ASME Advances in Design Automation, Vol. 2, pp , 1993 [7] Han C. S., Grandhi R. V., and Srinvasan, R., Optimum Design of Forging Die Shapes Using Nonlinear Finite Element Analysis, AIAA Journal, Vol. 31, No. 4, [8] Osman, F. H, Bramley, A. N., and Ghbrial, M. I., " Preform Design for Forging Rotationally Symmetric Parts", Annals of the CIRP, Vol. 44/1/1995. [9] Osman, F. H, and Bramley, A. N., Preform Design for Forging Rotationally Symmetric Parts, Annals of the CIRP, Vol. 44/1/1995. [10] Vanderplaats, G. N., Numerical Optimization Techniques for Engineering Design, Mc Graw-Hill Series in Mechanical Engineering, [11] Haug, E. J., Choi, K. K. and Komkov, V., Design Sensitivity Analyses of Structural Systems, Academic press inc., [12] Rao, S. S., Description and Optimum Design of Fuzzy Structural Systems, Journal of Mechanisms, Transmission, and Automation in Design, Vol. 109, No. 1, pp , [13] Rao, S. S., Multi-Objective Optimisation of Fuzzy Structural Systems, International Journal for Numerical Methods in Engineering, Vol. 24, pp , [14] Rao, S. S., Optimisation Using Fuzzy Set Theory, Structural Optimisation Status and Promise, Volume 150 Progress in Astronautics and Aeronautics, Published by American Institue of Aeronautics and Astronautics [15] Yager, R. R., Multiple Objective Decision Making Using Fazzy Sets, International Journal of Man-Machine Studies, Vol. 9, pp , [16] Biglari, F. R., and Fang, X. D., Real-time Fuzzy Logic Control for Maximising the Tool Life of Small-diameter Drills, International. Journal of Fuzzy Sets and Systems Vol. 72, pp , [17] ABAQUS, Version 5.4, (1995). HKS Inc, Pawtucket, RI 02860, USA.

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14 Table captions Table 1: Constants of fuzzy membership functions for the boundary nodes location. Table 2: Constants of fuzzy membership functions for effective strain deviation. Table 3: Constants of fuzzy membership functions for node release priority. Table 4: Fuzzy decision making rules.

15 Figure captions Fig. 1: Typical nodal displacements during workpiece deformation. Fig. 2: Geometrical representation of the backward tracing method. Fig. 3: Upper half of axisymmetric H-section component. Fig. 4: Schematic illustration of an axisymmetric forging die cavity. Fig. 5: The fuzzy decision making process diagram. Fig. 6: Simulation of one stage forging operation. Fig. 7: Backward deformation using the fuzzy decision making method. Fig. 8: Simulation of the first stage of the forging of a H cross section. Fig. 9: Simulation of the second stage of the forging of a H cross section Fig. 10: Comparison of accumulated plastic strain for two cases; (a) without optimum preform; (b) with optimum preform.

16 Linguistic levels of boundary nodes Location Constants of linear fuzzy sets for the following geometrical features Distance of the boundary nodes from: Plane of symmetry Separation line Rotational symmetry axis (D PS) (D Sl) (D RS) b g k e b g k e b g k e Very Close Close Average Far Very Far

17 Linguistic levels of effective strain Constants for linear fuzzy sets Effective strain deviation (E d ) deviation b g k e Very Small Small Medium Large Very Large

18 Linguistic levels of node releasing priority Constants for linear fuzzy sets Releasing node priority (P NR ) b g k% e% Very Low Low Average High Very High

19 Linguistic levels of strain deviation Linguistic levels of distance of a boundary node form D PS, D SL and D RS Very Close (VC) Close (C) Average (A) Far (F) Very Far (VF) E d D PS D SL D RS D PS D SL D RS D PS D SL D RS D PS D SL D RS D PS D SL D RS Very Small (VS) VL L VL VL L VL VL VL VL L VL L L VL L Small (S) VL A VL L A L L L L A L A A VL A Medium (M) L H L A H A A A A H A H H L H Large (L) A VH A A H A H H H H A H VH A VH Very Large (VL) H VH H H VH H H H H VH H VH VH H VH

20 y D n-1 D n C n D 0 D 1 C n-1 C1 C 0 A n A n-1 B n A 0 A 1 B 0 B n-1 B 1 x

21 x(t) (1) x n-1 (1) x n () 1 A n-1 [F(1)] v n-1 () 1 A n x n (2) x n (2) x n-1 x n (j) ( 2) A n-1 [F(2)] v n-1 [B(1)] v n [B(2)] v n [B(j)] v n ( 2) A n A n (j) A n [F(j)] v n-1 = v n-1 x n-1 (j) x n-1 (j) A n-1 A n-1 t t n-1 Time t n t

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23 b Separation line Plane of symmetry a R otational sym m etry axes c

24 µ(d PS) 1 Input Feature1: D PS Very Close Close Average Far Very Far µ(d PS ) VF= 0.67 Fuzzy AND Operator 0.5 µ(d PS ) F= D PS (n,t) D PS µ(d SL) 1 Input Feature 2: D SL Very Close Close Average Far Very Far µ(d SL ) C= Fuzzy OR Operator 0 µ(d RS ) D SL (n,t) Input Feature 3: D RS Very Close Close Average Far Very Far D RS (n,t) D SL D RS µ(d SL ) VC= 0.14 µ(d RS ) F= 0.56 µ(d RS ) VF= µ(p NR ) Very Low Low Average High Very High 0% 50% 100% Releasing node priority P NR (n,t)=76% P NR µ(e d ) 1 Input Feature 4: E d Very Small Small Medium Large Very Large µ(e d) M= µ(e d ) L= E d (n,t) E d

25 Plastic strain (a) (b) Die Separation (c) (d) (e) Fold Over (f) (g) (h)

26 III I II IV (a) (b) (c) (d) (e) (f)

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29 n Plastic strain Fold-over (a) (b)