Multi-Level Design Process for 3-D Preform Shape Optimization in Metal Forming Using the Reduced Basis Technique

Transcription

1 Multi-Level Design Process for 3-D Preform Shape Optimization in Metal Forming Using the Reduced Basis Technique A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering By Nagarajan Thiyagarajan B.E., Unviersity of Mysore, India Wright State University

2 WRIGHT STATE UNVERSITY SCHOOL OF GRADUATE STUDIES December 9, 2004 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Nagarajan Thiyagarajan ENTITLED Multi-Level Design Process for 3-D Preform Shape Optimization in Metal Forming Using the Reduced Basis Technique BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Masters of Science in Engineering Ramana V. Grandhi, Ph.D. Thesis Director Committee on Final Examination Richard J. Bethke, Ph.D. Chairman of Department Ramana V. Grandhi, Ph.D. Ravi C. Penmetsa, Ph.D. Henry D. Young, Ph.D. Joseph F. Thomas, Jr., Ph.D. Dean, School of Graduate studies

3 ABSTRACT Thiyagarajan, Nagarajan M. S. Egr., Department of Mechanical and Materials Engineering, Wright State University, Multi-Level Design Process for 3-D Preform Shape Optimization in Metal Forming Using the Reduced Basis Technique In this thesis, a 3-D preform shape optimization method for the forging process using the reduced basis technique is developed. Several critical techniques and new advances that enable the use of the reduced basis technique are presented. The primary objective is to reduce the enormous number of design variables required to define the 3-D preform shape. The reduced basis technique is a weighted combination of several trial shapes to find the best combination using the weights for each billet shape as the design variables. A multi-level design process is developed to find suitable basis shapes or trial shapes at each level that can be used in the reduced basis technique. Each level is treated as a separate optimization problem until the required objective--minimum strain variance and complete die fill--is achieved. Excess material, or the flash, is predetermined as per industry requirements and the process is started with geometrically simple basis shapes that are defined by their shape co-ordinates. This method is demonstrated on the preform shape optimization of a geometrically complex 3-D steering link.

4 TABLE OF CONTENTS 1. Introduction Background Backward optimization Discrete approach Continuum approach Preform shape optimization methodology Reduced basis method Basis vector definition Geometric scaling Approximation model Optimization problem definition Multi-level optimization Orthogonalization check Case studies Preform design for plane strain rail section Single-level optimization Multi-level optimization of plane strain rail section Preform design for 3-D metal hub.. 39

5 4.3 Preform design for 3-D metal hub with higher height to breadth ratio Preform design of 3-D spring seat Preform design of 3-D steering link Discussion and conclusions.. 70 Appendix.. 72 References... 81

6 LIST OF FIGURES Figure 3.1 Basis vectors definition 12 Figure 3.2 Central composite design for two factors 14 Figure 3.3 Multi-level design process 18 Figure 4.1 Rail section 23 Figure 4.2 Basis shapes and the corresponding forged billets with underfill 25 Figure 4.3 Optimized billet for rail section (Flash: 3%) 27 Figure 4.4 Stage 1 of plane strain rail section 31 Figure 4.5 Stage 2 of plane strain rail section 34 Figure 4.6 Stage 3 of plane strain rail section (Flash 3 %) 37 Figure D Metal hub (3/4 model) with section view (h/b = 1) 40 Figure 4.8 Basis shapes (1/4 model) assumed for 3-D Metal hub 40 Figure 4.9 Optimum preform shape and the forged part (Flash 1.5%) 42 Figure D Metal hub (3/4 model) with section view (h/b = 2) 44 Figure 4.11 Basis shapes (1/4 model) assumed for 3-D Metal hub_2 (Level 1) 45 Figure 4.12 Basis shapes (1/4 model) assumed for 3-D Metal hub_2 (Level 2) 48 Figure 4.13 Optimum preform shape and the forged part (Flash 2.0%) 50

7 Figure 4.14 Three dimensional spring seat 52 Figure 4.15 Basis shapes (1/4 model) assumed for spring seat 53 Figure 4.16 Optimized billet (Flash: 4.5%) 54 Figure 4.17 Steering link 56 Figure 4.18 Level 1 basis shapes for steering link 58 Figure 4.19 Constraint and objective function iteration history (Level 1) 59 Figure 4.20 Level 1 optimum billet 60 Figure 4.21 Cross-sections showing underfill 61 Figure 4.22 Level 2 basis shapes 64 Figure 4.23 Constraint and objective function iteration history (Level 2) 66 Figure 4.24 Final preform shape and forged part 67 Figure A.1 Basis shape with equidistant boundary points 72 Figure A.2 Basis shape with radial boundary points 73 Figure A.3 Sections lofted to 3-D shape 74

8 LIST OF TABLES Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9 Table 4.10 Performance characteristics of basis shapes and preform for rail section (single-stage optimization) 29 Performance characteristics of basis shapes and preform for rail section (Level 1) 32 Performance characteristics of basis shapes and preform for rail section (Level 2) 35 Performance characteristics of basis shapes and preform for rail section (Level 3) 38 Performance characteristics of basis shapes and preform for 3-D metal hub 43 Performance characteristics of basis shapes and Level 1 optimum shape for 3-D metal hub (h/b =2) 47 Performance characteristics of basis shapes and Level 2 optimum shape for 3-D metal hub (h/b =2) 51 Performance characteristics of basis shapes and preform for spring seat (single-stage optimization) 55 Performance characteristics of basis shapes and Level 1 optimum shape for steering link 63 Performance characteristics of basis shapes and preform (Level 2) for steering link 69

9 ACKNOWLEDGEMENT The author wishes to express his gratitude and appreciation to Dr. Ramana V. Grandhi for constant guidance throughout the graduate studies. This research is based upon work supported, in part, by the U.S. Department of Commerce, National Institute of Standards and Technology, Advanced Technology Program, and Cooperative Agreement Number 70NANB0H3014 (the Smartsmith project). Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the sponsors. Introduction In a forging process, an initial block of metal (billet) is compressed between two or more dies to produce a complex part. The shape of the initial billet is crucial in achieving the desired characteristics in the final forged part. Traditionally, an experienced designer uses his or her expertise and design data handbooks for optimizing the billet shape. With the advent of better computers, more robust and efficient shape optimization techniques are developed and are put to use in increasingly more industries. There are

10 many well-established 2-D preform shape optimization methodologies for various objectives, such as eliminating underfill and fold, minimizing energy consumption, achieving more uniform deformation, and optimizing the microstructure [1-3]. Most industrial components cannot be assumed as a 2-D cross-section therefore, the problem needs 3-D description. The sheer number of design variables required to define the 3-D preform shape coupled with the huge computational time required to simulate the 3-D forging process make the application of most preform shape optimization algorithms impractical. But, there have been some developments for 3-D preform shape optimization. The most notable techniques are presented in Reference 4, in which the optimization algorithm regards the shape of the initial billet as axisymmetric and finds the preform shape for a 3-D gear. Both deterministic and stochastic optimization algorithms are tested for a 3-D forging application with several objective functions. There has also been successful development of the sensitivity method for blank design in sheet metal stampings to find the optimal preform design in free forging applications to eliminate barreling [5]. The main challenge that is encountered while applying most of the optimization methods to more complex 3-D parts (with 3-D preform shapes) is the description of the preform shapes in terms of design variables. Generally, the finite element nodal coordinates are considered, which result in an exceedingly large number of design variables. Also, the resulting preform shapes may not have a smooth surface, which makes the preform shape impractical. Another approach is the use of B-splines and other blended functions. This may be practical in the case of 2-D problems or if the preform shape is relatively simple. However for 3-D parts, surfaces instead of edges (curves) have

11 to be defined, and this makes the use of blended functions very difficult for shape optimization. It is advantageous to use these functions when the designer has an idea of which regions to modify in the preform. Therefore, this research develops a unified algorithm applicable to a larger class of problems that can find a practical preform shape without the need for the engineer to have any industrial expertise in preform shape design. In this thesis, an innovative way of using an efficient design variable linking method, the reduced basis technique [6], is demonstrated to develop a preform shape optimization algorithm. This design process can be used for both 2-D and 3-D components. In the reduced basis technique, many initial billet shapes, called basis shapes, are combined linearly by assigning weights to each of the assumed basis shapes. Different resultant shapes can be generated by changing these weights. Therefore, the number of design variables (which may be huge) required to define the shape is reduced to equal the number of basis shapes. So, the weights assigned for each basis shape are the design variables and the optimization goal is to find the best possible combination of these weights to minimize the cost function. Reduced basis techniques have been widely used in shape optimization of structural problems [7, 8]. In order to develop suitable starting basis shapes, auxiliary loads are often applied to the structure, which will cause the structure to deflect without necessarily causing a large reduction in weight or a squeezing effect, and this technique will generate a smoother shape. Various basis shapes are generated using different boundary conditions for multiple sets of auxiliary loads. Another issue that must be

12 addressed is the adequacy of the basis shapes generated from the initial structure to actually define the optimal structure [9]. The availability of gradient information for objective functions and constraints is another important issue that has to be considered in optimization. Commercial 3-D forging analysis packages do not provide this information. The finite difference method may be used to build surrogate models on which optimization can be performed. Still, there are issues with the accuracy of finite difference gradients and the computational cost of simulations. Hence, this thesis focuses on the nongradient-based shape optimization technique: Response Surface Method (RSM). RSM is the combination of mathematical and statistical techniques used in the empirical study of relationships and optimization, in which several independent variables influence a dependent variable or response. This thesis starts with a brief description of the preform shape optimization techniques that are already developed, followed by their disadvantages and proceeds to the 3-D preform shape optimization method that is developed in this work. This method is demonstrated on many case studies, both 2-D and 3-D, in the following chapters. Two dimensional case studies are explained for the reader to have a better understanding of the methodology. The thesis concludes with a chapter on discussions and conclusions.

13 Background Several gradient and nongradient-based optimization techniques have been developed to optimize preform and die shapes. Some of the gradient methods for preform shape optimization are backward tracing, discrete and continuum approaches which are explained below Backward Optimization Park (1983) [10] et al developed the FEM based backward method for preform design. Since then several variations of this method have been studied for

14 solving specific problems. In the design of the forging process, the only information known beforehand is the final product shape and the material to be used. The backward tracing technique provides an avenue for preform design which starts with the final forging shape at a given stage and conducts the forging simulation in reverse, resulting in a preform shape at the end of the simulation. Because the deformation is dependent on the boundary conditions that are not known a priori, specific rules must be applied to determine how the material separates from the dies during backward tracing, which is not robust and requires expertise. Lanka (1991) et al. [11] implemented conformal mapping techniques to design intermediate shapes while mapping the initial shape to the final shape of closed die forgings. Hwang (1987) et al. [12] developed a backward tracing method for shell nose preform design. This method starts from the final product shape and a completely filled die, and the movement of the die is reversed in an attempt to reverse plastic deformation. During backward tracing, the workpiece boundary nodes are initially in contact with the die, and as the die is pulled back, nodes gradually separate from the die. The method iteratively checks whether the new workpiece geometry, obtained after each node separation during backward simulation, results in the desired final shape upon repeating the forward simulation. The starting shape, or preform, is obtained when all the boundary nodes have separated from the die. In the problem solved by Hwang et al., the die shape was simple and the sequence in which the nodes separate from the die is quite straightforward. This may not be true in general forging problems. Han (1993) et al. [13] introduced mathematical optimization techniques in a backward tracing method called Backward Deformation Optimization Method (BDOM). The objective of this method is

15 to obtain more uniform deformation by minimizing the strain-rate variation during deformation. This method combines the backward tracing method with numerical optimization techniques for determining a strategy for releasing nodes from an arbitrary die during reversed deformation. Two nodal detachment criteria are developed: strainrate based detachment and force-based nodal detachment. Kang (1990) et al. [14] established systematic approaches for preform design in blade forging in which each airfoil section was considered as a two-dimensional planestrain problem using the back-tracing scheme. This method, which is further extended by Zhao et al. [15], is called inverse die contact tracking method. This procedure starts with the forward simulation of a candidate preform into the final forging shape. A record of the boundary condition changes is documented by identifying when a particular segment of the die makes contact with the workpiece surfaces in forward simulation. This recorded time sequence is then optimized according to the material flow characteristics and the state of die fill to satisfy the requirement of material utilization and forging quality. Finally, the modified boundary conditions are used as the boundary conditions control criterion for the inverse deformation simulation. The method is used in preform design of complex plane strain forging. Zhao also established a node detachment criterion based on minimizing the shape complexity factor Discrete Approach Zhao (1997) et al. [16] derived the analytical sensitivities of the flow formulation after the domain discretization. An optimization approach for designing the first die shape in a two-stage operation is presented using sensitivity analysis. The control points

16 on the B-splines are used as the design variables. The optimization objective is to reduce the difference between the realized and desired final forging shapes. The sensitivities of the objective function with respect to the design variables are developed. Gao and Grandhi (1999) et al. [17] presented thermo-mechanical sensitivity calculations and shape optimization. Chung (2003) et al. [18] presented an adjoint variable method of sensitivity analysis for non-steady forming problems. This adjoint state method calculates the design sensitivities by introducing adjoint variables. The calculation of adjoint variables and design sensitivity of each incremental step is carried out backwardly from the last incremental step. Some special treatments are introduced for the contact algorithm, for remeshing, and for memory space problems. The developed methodology is applied to a simple upsetting problem and a single-stage forging process. In this approach the finite element constitutive equations have to be differentiated in order to obtain the sensitivities of path dependent variables such as velocities, strains and strain rates. For most problems the access to the finite element equations from the commercial packages is not available. Therefore there is a natural bias towards the methods which do not require the finite elements equations to calculate the sensitivities such as the continuum methods Continuum Approach Unlike the discrete approach the continuum approach differentiates the original continuum formulation first and discretizes it afterwards. While the discrete approach is easier to understand, requiring less knowledge of mathematics, the implementation needs much more effort and requires knowledge of the elemental stiffness matrix of the analysis

17 code, which is not possible if commercial software is used. Moreover, it has difficulty in treating the shape parameters in the finite element matrices. On the other hand, the continuum approach can be implemented independent of the analysis code without knowledge of it, because it just makes use of the output variables of the analysis. The above gradient methods deal with multi-disciplinary phenomena of deformation process mechanics which require large amount of mathematics and process constitutive laws. This makes the sensitivity computation and shape optimization for 3-D problems very difficult and expensive. Therefore this research focuses on non-gradient based design methods. Some of the non-gradient based methods include knowledgebased systems, genetic algorithms, neural networks, fuzzy logic techniques, and response surface methods. Chung, (1998) et al. [19] and Coulter (1993) et al. [20] have done research in the application of neural networks and genetic algorithms for the design of material processes. Neural networks are an artificial intelligence technique in which the network is trained using input-output data of various simulations of a process. Once trained, the neural network can be used for process design, obviating the need for a simulation. A genetic algorithm is a design technique that is based on the survival-of-thefittest design in a population of designs. The design variable is represented as a binary string. The optimal designs achieved after the optimization of generations of population are useful when one is concerned with the design of a single process for which different objectives may be required by the process engineer at various times. However, if the network has to deal with the design of different processes (new situations require retraining), then the method loses its merit. The use of genetic algorithms is a powerful

18 technique that handles discrete design data with ease (e.g., number of stages in multistage design). For large scale problems this method becomes inefficient due to the requirement of large number of FEM simulations. Hence an efficient preform shape methodology is developed by coupling a design variable linking technique with response surface method. Preform Shape Optimization Methodology The main emphasis in developing this algorithm is the design variable linking method: the reduced basis technique. Though this technique is widely used in structural shape optimization, it has to be adopted for metal forming applications. One of the main reasons is that in the former there is no mesh degradation or remeshing stages, unlike in forging applications. In aerospace structural design, the structure does not change its topology or configuration with time. In metal forming applications, changes to the billet shape, the number of elements, the element connectivities, element shapes, etc take place Reduced Basis Method The main idea in this method is to construct basis functions or vectors, Y 1, Y 2, Y 3,, Y n, with the large information content of each basis shape and to combine them Y n c i = Y + aiy (1)

19 linearly with the weighing factors a 1, a 2, a 3,,a n that correspond to each basis vector. And, Y c is a vector added for generality. The basis vectors, which represent each basis shape or initial guess shape, will have the co-ordinates or shape parameters that define the respective basis shape. If the number of shape variables required to define a basis shape is m, then by applying the reduced basis method, the number of design variables is decreased from m to n (equal to the number of weighing factors). Generally, the value of m is more than 50 even for a simple shape and the number of basis shapes n required to define the optimum is typically about 5. It is a common practice to define the basis vector by the node data of the basis shapes. The most utilized is the auxiliary load method, in which the resulting nodal displacements of the fixed configuration structure are added to the original nodal locations to create the basis vectors. This approach cannot be considered in preform shape optimization because the forging analysis is nonlinear and time dependent and the designer will have less control on the resulting shape. To avoid these problems in the preform shape design, the basis vectors are defined by shape co-ordinates that define the basis shapes Basis vector definition. The geometrical features of the basis shapes can be defined by the x, y, and z co-ordinates of their boundary points. These co-ordinates define the basis vectors. All of the basis shapes have to be defined in the same fashion, and therefore all the resulting basis vectors will have the same dimension (Appendix A). This will help to add them

20 linearly with weights to each vector. The resulting vector will have a different shape than any of the basis shapes if at least two of the weights are non-zero. If the optimum billet is any of the basis shapes, then the corresponding weight will be one and the others will be simply zero. Boundary points Desired boundary Inaccurate boundary (a) Figure 3.1: Basis Vectors Definition (b) The important factor that must be heeded is that the number of shape parameters should be as plentiful as possible. That means that the locations at which the co-ordinates are extracted should be as close to each other as possible in order to facilitate splining and to get the detailed surface. Figure 3.1 (a) shows an edge defined by a set of points. There are 22 points that define the edge and the co-ordinates of these points form the basis vector. If a lesser number of points are used and the edge is defined by eight points, then the resultant edge would look much different than the original (Fig. 3.1(b)). To avoid this type of error, it is always safe to define the basis shapes with a large number of boundary points. Since this will not increase the number of design variables, there is no extra computational cost incurred by increasing the dimension of the basis vectors. This type of

21 basis vector definition is useful for generating various possible shapes for optimization, but scaling of the resultant shapes is essential to maintain volume constancy Geometric Scaling. In order to understand the necessity of scaling, a simple problem of combining two basis shapes is considered. The unknown lengths (l 1, l 2 ) and unknown breadths (b 1, b 2 ), which are defined by their respective basis shapes (Y 1 and Y 2 ) having the same area, form the initial shapes. The optimum shape Y is defined as Y = a 1 l1 + a b1 2 l b 2 2 a1l = a1b a + a 2 2 l2 l = b 2 b (2) where a 1 and a 2 are the weights. The area of the resulting shape will be l b, which is not equal to the area of Basis 1 or Basis 2. The resulting billet is scaled to a preset area or volume, which may be some percentage more than the actual volume of the part. By doing this, the amount of flash is predetermined for the part as per industry requirements, even before the actual optimization problem is started. This also negates the need of a constraint on the flash in the optimization routine Approximation model. Response surface methodology is used to build the approximation model and to perform optimization. Response surface methodology (RSM), in which several independent variables influence a dependent variable or response, is the combination of

22 mathematical and statistical techniques used in the empirical study of relationships and optimization. The goal of RSM is to secure an optimal response. In this design method, a quadratic RSM model (Eq. 3) with all interaction terms is built for the required responses: where β are the RSM parameters (coefficients), ε is the error, x i are the design variables, and y is the response. The design variables x i are the weights a i of equations (1) and (2). In order to determine the response surface parameters, several experimental designs are available. They attempt to approximate the equation using the least number of experiments possible. The most widely preferred class of response surface design is the Central Composite Design (CCD). CCD contains an imbedded factorial or fractional factorial design with center points that are augmented with a group of star points that allow estimation of curvature. If the distance from the center of the design space to a factorial point is ±1 unit for each factor, then the distance from the center of the design space to a star point is ±α with α > 1 [21]. (1,1) ( 2,0) α = 2 (0,0) (0,0) (0, 2) ( 1, 1) s t r Figure 3.2: Central Composite Design for Two Factors

23 In the case of CCD generation (α = ) for two factors (Fig. 3.2), the corresponding design points for (-1,-1), (0,0), and (1,1) when transformed to a new design space of (0,1) will be (0.1464, ), (0.5, 0.5), and (0.8536, ). Each of these three points, after scaling to a preset area or volume, will give the same billet shape, and this kind of numerical anomaly will be even more damaging for three or four variable design of experiment (DOE) points. One way to avoid this drawback is to employ the Latin Hypercube Sampling (LHS) technique, which is a stratified sampling technique with random variable distributions in which the selection of sample values is highly controlled, yet still able to vary. The basis of LHS is a full stratification of the sampled distribution with a random selection inside each stratum Optimization problem definition. In preform shape design the main emphasis is on the complete die fill criteria. Furthermore, quality forgings require a more uniform strain distribution throughout the forged part. The weighted strain variance, 2 s w, is a good measure of the strain distribution, for which the weighting coefficients are the area or the volume of each element for 2-D and 3-D forging simulations (Eq. 4): s 2 w N = i = 1 w i ( e e N N N i 1 i = 1 w w ) i 2 (4)

24 where e i is the observation, w i is the weight of i th observation, N is the number of weights (elements), N is the number of non-zero weights (in this case, N = N), and e w is the weighted mean of the observations. Underfill (Eq. 5) is measured as the volume of the die cavity in which the desired material flow or die fill was not achieved: Underfill = V desire V actual (5) where V desire and V actual are the desired volume (volume of the final part) and the actually realized volume of the final forgings, respectively. RSM models are fit for these responses as a function of the coefficients, a i (Eq.1). The optimization problem (Eq. 6) is formulated to minimize weighted strain variance in order to have a more uniform material deformation throughout the forged part while assigning constraints on the underfill. Minimize: Strain variance: f(a i ) Subject to: Underfill: g(a i ) 0 (6) Side bounds: 0 a i Multi-Level Optimization. The methodology demonstrated so far works well whenever appropriate starting basis shapes are provided for preform design. But in some instances in which the product

25 is completely new, complex, or a different material, it may not be possible to begin with a reasonable set of starting basis shapes. Our goal is to address the needs of designers when there is little or no information about forming a new product. In those cases, we may not obtain the optimum preform by solving the shape optimization problem just once. The problem can be solved in multiple levels (Fig. 3.3) in which the optimization guides the designer progressively in selecting viable basis shapes. In Level 1, the basis shapes may not be anywhere near to what they are supposed to be, but the optimizer takes the first set of basis shapes and determines a best combination from these uninformed first trial shapes. From this Level 1 resulting shape, 3 or 4 variants to this shape are constructed for starting the Level 2 design. This process may be repeated typically for 3 or 4 times before suitable basis shapes are developed for a complex 3-D problem. Irrespective of how impractical the starting shapes in any level are, the optimum shape (best possible combination) in that level will give a better die fill than the starting basis shapes; or, the optimizer will select one of them as the best basis shape by giving the weights of the other basis vectors as zero. After the completion of each level, the next level is started as a new problem and the best shape of the previous level becomes one of the basis vectors (Basis 1). A few additional basis shapes are chosen, which will be variants of Basis (shape) 1. Thus the designer is guided into the right path to reach the optimum shape because the basis shapes selected will be modifications of the best shape of the previous level. The algorithm does not take into consideration whether basis shapes from the subsequent level give more underfill because the reduced basis technique is used to generate only the shapes and does not consider the history of the basis. Even if all the

26 Assume simple basis shapes (N = 1) Level N Reduced Basis Technique Design of Experiments Update basis shapes (N = N + 1) Response Surface Method OPTIMIZATION Die fill? No Yes END Figure 3.3: Multi-Level Design Process

27 additional shapes are inappropriate in Level 2, the optimizer will give Basis 1 as the optimum shape, as it was the best shape in the previous level. It must be noted here that though the modified algorithm will take the designer to the optimum preform shape, the computational time increases because a new surrogate model has to be built in every level. An experienced designer can start from an intermediate level with practical basis shapes and reach the optimum in a single level. There is room for making use of a designer s experience or information from similar products as the trial shapes Basis vector independency check. A basis shape selected at any level should not itself be some combination of other basis shapes in the same level. This will unnecessarily increase the computational cost incurred in building the RSM. For this purpose, an dependency check is performed on the basis vectors to ascertain if all the basis shapes/vectors are linearly independent. The Gram-Schmidt orthogonalization method (Eq. 7.) generates orthogonal vectors if the original input vectors are linearly independent. Otherwise it produces zero vectors. This concept is utilized to check if the given or selected basis shapes are dependent or independent. u 1 = v 1 u 2 = v 2 - [(v 2. u 1 )/(u 1. u 1 )]u 1 u 3 = v 3 - [(v 3. u 1 )/(u 1. u 1 )]u 1 - [(v 3. u 2 )/(u 2. u 2 )]u 2 (7)...

28 u k = v k - [(v k. u 1 )/(u 1. u 1 )]u 1 - [(v k. u 2 )/(u 2. u 2 )]u [(v k. u k-1 )/(u k-1. u k-1 )]u k-1 The Gram-Schmidt procedure takes an arbitrary basis (v k ) and generates an orthonormal one (u k ). It does this by sequentially processing the list of vectors and then generating a vector perpendicular to the previous vectors in the list. For the process to succeed in producing an orthonormal set, the given vectors must be linearly independent. If the given vectors are not linearly independent, indeterminate or zero vectors may be produced. By doing the orthogonality check, the designer can eliminate or change one or more basis shapes that are not linearly independent. Once the basis shapes are generated, the coordinates of the surface points are extracted to build the basis vectors. These basis vectors are the arbitrary basis (v k ). A simple MatLab code is written to find the orthonormal vectors (u k ) according to the equations 7. If any of the basis vectors are linearly dependent, then indeterminate orthonormal vectors will be generated prompting the designer to eliminate or change the dependent basis shapes.

29 Case Studies The feasibility of the methodology is demonstrated through 2-D as well as 3-D preform shape design of forged mechanical components. The methodology starts with intuitive or practical guess shapes to obtain the optimum preform shape. However, expert knowledge is often not available for complex 3-D products. In these situations, it is wise to start with geometrically simple and readily available billets as basis shapes. To achieve the optimum shape from these simple starting shapes, the developed methodology is modified to accommodate these basis shapes by using the multi-level optimization algorithm. The developed algorithm aids in achieving preform shapes from simple basis shapes in a minimum number of levels. Preform shape optimization of a 2-D plane strain rail section and a 3-D metal hub are considered for the case studies. Finite element packages DEFORM 2D and 3D are used to analyze the metal forming process and to conduct DOE. In a DEFORM model of a forging sequence the workpiece is represented by a deformable mesh of 2-D (quadrilateral) or 3-D (tetrahedral) elements and the dies are represented by lines or surfaces that define the rigid die surfaces. Mechanical and thermal properties are ascribed to the mesh. Once these values are prescribed, the dies are moved in small incremental steps by incorporating automatic remeshing, and a solution is calculated for each step. These forging simulations aid in

30 predicting the responses, such as the underfill and loads, and also localized responses, such as elemental strains and strain rates that can be used to build the RSM. AISI-1045 steel, listed in the material library of DEFORM software, is assigned as the workpiece material, and a mechanical press of constant die velocity is used for the hot forging simulations. Isothermal conditions are considered, and the billet temperature is 1200 o C with no heat transfer between the billet and the ambience. Generally H-13 is used as the die material in industries, which is very hard compared to the billet material at high temperature; therefore, the dies are considered as rigid since there is no die deformation. Elastic effects, such as residual stress and spring-back of the deformed billet, can become insignificant in hot forging. Therefore, a rigid-viscoplastic material property is applied to the analysis when elastic effects are overshadowed by thermal effects and by the large plastic deformations involved. Frequently, deformation is brought about during contact between a tool and a workpiece. This inevitably results in friction if there is any tangential force at the contacting surfaces. The coefficient of friction is reasonably constant and a friction value of 0.3 is assigned for the simulations. In this preform shape optimization method, various billet shapes for the DOE are generated for different combinations of weights and are scaled to a constant area or volume. The modeling package I-DEAS is used for this purpose. A MatLab file, which is used to generate an I-DEAS programming file, is shown in the appendix. This programming file aids in modeling various billet shapes even if the basis vectors are very large. Normally, in the reduced basis method a constant vector Y c is used for generality, but in this research it is assumed as zero, since it does not have any affect on the optimum weights. Different flash percentages are assumed for each example.

31 4.1. Preform design for plane strain rail section A two-dimensional plane strain rail section (Fig. 4.1), in which there is no material flow in the Z direction, is used to evaluate the optimization methodology before applying it on the 3-D part. The rail section is symmetric about the Y axis, as shown in the figure; thus, a half model is used for the forging simulations. The upper and the lower cavities towards the outer end have different height-to-breadth ratios of 1.25 and 1.50, respectively. Complete die fill at these cavities is difficult to achieve when the allowable flash percentage is less than 5% of the total cross-sectional area, which is 149 cm 2. For this example, the basis shapes are selected and scaled to a constant area of cm 2, thereby specifying the scrap as 3%. b 1 h 1 b 2 h 2 h 1 = 1.25 x b 1 h 2 = 1.50 x b 2 Symmetry axis Figure 4.1: Rail Section An optimum preform shape that gives complete die fill can be achieved by the proposed methodology in two ways: 1. Starting shapes are guessed intuitively by a

32 relatively experienced designer, or 2. A less-experienced user starts with geometrically simple guess shapes and reaches the optimum shape in multiple-levels. Both cases are investigated individually for the same part Single-level optimization We may have many preform shapes available that are used in industry and are designed using the metal forming design data hand books or other optimization schemes. Furthermore, there may be many practical guess shapes, that may or may not give a complete die fill, or may be close to the optimum, but can be further improved in terms of performance characteristics of the forged part while satisfying the complete die fill constraint. Thus, it is reasonable to use previous designs or practical guess shapes as basis shapes. In this case, four basis shapes are considered with an area of 3% more than the cross-section of the final part. A typical forging process starts with a simple rectangular or cylindrical billet that is deformed to the preform shape in the buster stage. Therefore, Basis 1 is just a rectangular block selected for universality, whereas Basis 2 and Basis 3 are practical guess shapes (Fig. 4.2). Since the rail section has deeper cavities at the outer end, it is a common practice to provide more material at that location of the preform in order to fill the die cavities. Therefore, Basis 2 is guessed with this knowledge, making it a

33 Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3] Basis 4 [Y4] Figure 4.2: Basis Shapes and the Corresponding Forged Billets with Underfill practical guess shape. Among the two die cavities, the bottom cavity is deeper than the top cavity; thus, it is reasonable to provide more material depth at the bottom than

34 at the top edge, as seen in Basis 3. Basis 4, with more material in the middle and less at the ends, which is contrary to the physics of the problem, has an impractical starting shape, but is selected to check the efficiency of the method. The contribution of Basis 4 should come out as zero. Finite element forging simulations of the basis shapes are performed in DEFORM 2D. Underfill and strain variance responses are obtained from these simulations for preliminary analysis (Fig. 4.2). As expected, Basis 1 gives more underfill at the bottom compared to the top cavity, because of the higher h/b ratio at the bottom. Since there is more material flowing outside of the die cavities instead of filling them, Basis 1 gives a flash of 7.67%. Basis 2 provides more uniform material distribution and also considerably less underfill at the top cavity than Basis 1, but the underfill at the bottom cavity is almost the same and the flash percentage for Basis 2 is 4.72%. Basis 3 gives underfill at the top cavity and complete die fill at the bottom cavity and has a flash of 4.32%. Basis 4 gives huge underfill at both of the cavities because more material is flowing outside the dies, which increases the flash percentage to 14.37%. The strain variance and the load required to forge this basis shape is also high because of the higher material deformation. Hence, it is supposed to have less of a, or no, contribution towards the optimum shape; however, the tail end can play a small role in reducing the material depth at the outer end. Each basis shape is defined by 64 shape variables, i.e., (x, y) co-ordinates at 32 locations along the edges of the shape. These shape variables form the basis vectors for the corresponding basis shape. Weights are assigned to each basis vectors and are combined linearly. By changing these weights, it is possible to obtain various resultant

35 shapes. Therefore, the optimization goal is to find the best combination of these weights, which are the design variables. It can be seen that the number of design variables (weights of each basis) is equal the number of basis shapes, thereby reducing the optimization design variables from 64 (shape variables) to 4 (weights). LHS techniques are used to generate 25 DOE points for these four design variables to conduct forging simulations. The strain variance and underfill are calculated from these simulations to construct the RSM. It is interesting to note that none of the 25 DOE points give a complete die fill. These response surface models are used for optimization. Optimization as per the problem formulation (Eq. 5) is performed in MatLab, and takes six iterations to reach the optimum weights in order to satisfy the underfill constraint and to minimize the strain variance. The resulting preform shape with optimum weights is shown in Figure 4.3. Preform Forged part a1 a2 a3 a Optimized weights Basis 2 Figure carries 4.3: an Optimized optimum weight Billet for of Rail one, Section which means (Flash: that 3%) the contribution of Basis 2 to the preform shape is maximum compared to the other basis shapes. The contribution of Basis 3 is 0.76, which is also relatively higher, and this adds more

36 material near the bottom cavity where the h/b ratio is higher and is more difficult to achieve die fill. These two contributions ensure complete die fill in both die cavities. As previously predicted, the contribution of Basis 4 is zero because more material should be in the outer end of the preform shape to fill the cavities and to minimize the strain variance. The optimum weight for Basis 1 is also zero for the same reasons, since any contribution of Basis 1 will reduce the material depth at the outer end and will increase the strain variance even if the underfill constraint is satisfied. It can also be verified from Table 4.1 where it can be seen that the strain variance for Basis 1 and Basis 4 is significantly higher than Basis 2 and Basis 3, thereby validating the result. Another interesting fact is that even though the strain variance and the underfill are slightly higher for Basis 2 than Basis 3, the contribution of Basis 2 towards the optimum shape is higher. The methodology is good for predicting the best combination of these two weights to reduce the strain variance of the preform, which is lower than all four basis shapes in this example. Furthermore, the underfill constraint is still satisfied. Also, the flash percentage is reduced (3%) compared to the basis shapes because all of the material flow was into the die cavities to achieve complete die fill, rather than out of them. FEM forging simulation of the preform shape is performed to verify these results (Fig. 4.3); and, by achieving complete die fill and more uniform strain variance, they are in accordance with the approximation models. It can be seen that

37 Basis 1 Basis 2 Basis 3 Basis 4 Preform Strain variance Flash (cm 2 ) (7.67%) 7.01 (4.72%) 6.41 (4.32%) (14.37%) 4.51 (3.0%) Underfill (cm 2 ) Load (KN) Table 4.1: Performance Characteristics of Basis Shapes and Preform for Rail Section (Single-Stage Optimization) the geometrically simple Basis 1, which may be guessed by an inexperienced designer, does not aid in reaching the optimum and only the practical or intuitive basis shapes play the most important role. Therefore, it is important to modify the method to accommodate even simple basis shapes to reach the optimum. For this purpose, the same rail section is considered and the multi-level optimization is demonstrated.

38 Multi-level optimization of plane strain rail section The above-described optimization scheme works well when the designer considers practical billet shapes as basis shapes and applies the reduced basis technique. This design process is further enhanced to accommodate even geometrically simple starting shapes as basis shapes for reaching the preform shape in more than one optimization stage or level using the multi-level design process. (a) Level 1: Three simple basis shapes (Fig. 4.4) are assumed. Basis 1 is rectangular in shape, which is same as the Basis 1 in the single level optimization; Basis 2 and Basis 3 are trapezoidal with tapers on opposite sides for each basis. Basis 2 has more material at the center (left end), which is contrary to the physics of the problem, and Basis 3 has more material at the outer end and less at the center. All three basis shapes are defined by straight lines in this level; therefore, any combination of these basis will also have only straight lines. Each basis is defined by 64 shape variables, which make up the respective basis vectors, as in the single-level optimization. FEM Basis shapes and their results (LEVEL 1)

39 Basis 1 Basis 2 Basis 3 BEST SHAPE (Level 1) a1 a2 a Optimum weights (Level 1) Figure 4.4: Level 1 of Plane Strain Rail Section

40 Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (cm 2 ) (7.67%) (15.01%) (8.15%) 8.38 (5.62%) Underfill volume (cm 2 ) Load (KN) Table 4.2: Performance Characteristics of Basis Shapes and Preform for Rail Section (Level 1) forging simulation of these basis shapes are performed and, compared to the total part area of 149 cm 2, all three basis shapes give huge underfill of 6.87 cm 2, cm 2, and 7.65 cm 2, respectively. Three weights for each basis vector become the design variables

41 and 15 DOE points are generated by the LHS technique. Forging simulations are conducted at these points to extract the underfill and strain variance response. Optimization is performed on these RSM models, and the optimum shape (Fig. 4.4), which is a weighted (a1 = 0.7, a2 = 0, a3 = 0.4) combination of Basis 1 and Basis 3 in Level 1, gives 3.88 cm 2 as underfill, which is significantly less than that of the basis shapes (Table 4.2). The weight for Basis 2 is zero since the material depth is at the wrong location and it has no contribution towards the optimum. Unlike in the single-level optimization, the rectangular Basis 1 has the maximum contribution of 0.7 because it is relatively better than the other two simple basis shapes, which are not intuitively guessed in this level. The small contribution of Basis 3 makes the preform shape in this level slightly tapered with more material at the outer edge than at the center, as can be seen in Figure 4.4. Level 2 is performed in order to further reduce the underfill. (b) Level 2: The optimum shape of Level 1 is considered as Basis 1 in Level 2 and two more basis shapes (Fig. 4.5) are selected that are variations of Basis 1. The top and bottom edges of Basis 2 and Basis 3 in this level are made of curves, and if these basis shapes are unsuitable, the optimizer will give weights of zero for them and one for Basis 1. Each basis is again defined by 64 shape variables, which make

42 Basis shapes and their results (LEVEL 2) Basis 1 Basis 2 Basis 3 BEST SHAPE (Level 2) a1 a2 a Optimum weights (Level 2) Figure 4.5: Level 2 of Plane Strain Rail Section

43 Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (cm 2 ) 8.38 (5.62%) 6.22 (4.17%) 6.58 (4.42%) 5.14 (3.45%) Underfill volume (cm 2 ) Load (KN) Table 4.3: Performance Characteristics of Basis Shapes and Preform for Rail Section (Level 2)

44 up the respective basis vectors and FEM forging simulation of these basis shapes are performed. All three basis shapes give underfill of 3.88 cm 2, 1.72 cm 2, and 2.08 cm 2, respectively. Fifteen DOE points are generated, and forging simulations are conducted to build the RSM and to perform optimization. The resulting billet (a1 = 0, a2 = 1, a3 = 1) gives a very small underfill of 0.64 cm 2 (Fig. 4.5). Basis 2 and Basis 3 have equal contributions, but there is no contribution from Basis 1, which is built of just straight lines. The underfill in this level is also significantly reduced (Table 4.3), and optimization in Level 3 is performed to eliminate the underfill completely. (c) Level 3: Again, the optimum shape of the previous level (Level 2) is assumed as Basis 1 in this level and two more basis shapes are obtained. All three basis shapes in this level (Fig. 4.6) are practical shapes and it can be seen that, even though the design process is started with simple guess shapes (Level 1), eventually a stage is reached in which all the guess shapes are viable. FEM forging simulations of these basis shapes are performed and all three basis shapes give underfill of 0.64 cm 2, 0.73 cm 2, and 1.57 cm 2, respectively (Table 4.4). Basis vectors with 64 shape variables are generated and weights are assigned to conduct DOE and forging simulations to build the RSM. Optimization of these models results in a preform shape (a1 = 0.0, a2 = 1.0, a3 = 0.7) that gives complete die fill (Fig. 4.6). Basis 2 and Basis 3 play the most important role towards the preform shape, even though they have more underfill compared to Basis 1. It is seen that the history (response) of the basis shapes from each level is not carried to the next level and only the shapes play a role in reaching the optimum, by considering each level as a separate problem.

45 Basis shapes and their results (LEVEL 3) Basis 1 Basis 2 Basis 3 OPTIMUM SHAPE (Level 3) a1 0 a2 1 a Optimum weights (Level 3) Figure 4.6: Level 3 of Plane Strain Rail Section (Flash 3 %)

46 Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (cm 2 ) 5.14 (3.45%) 5.23 (3.51%) 6.07 (4.07%) 4.50 (3.0%) Underfill volume (cm 2 ) Load (KN) Table 4.4: Performance Characteristics of Basis Shapes and Preform for Rail Section (Level 3)

47 It is also observed from the above multi-level optimization example that the optimum preform shape is reached in three levels, and in each level the underfill is reduced until complete die fill is achieved. This is made possible because the knowledge gained in each level is utilized to select better basis shapes in the subsequent levels, thereby guiding the user into the right path. Another important point to be noticed is that four basis shapes were selected in single-level optimization and three basis shapes were selected in each level of the multi-level optimization scheme. If four or more basis shapes were selected in each level of the multi-level optimization scheme, the optimum preform shape could have been reached faster. Also the perform shape will be different depending on the selection of the basis shapes with a different value for the objective function Preform design for 3-D metal hub Preform design for a plane strain part is demonstrated above and in this example, a 3-D metal hub is considered. The top portion of the part is axisymmetric, whereas the bottom rectangular portion destroys the 2-D assumption, thereby making the part 3-D. Height-to-breadth ratio of the hub is one (Fig. 4.7), making it difficult to achieve die fill at the cavity while attaining complete die fill at the bottom corner of the rectangular portion of the metal hub. The optimization goal is to design a preform shape that gives 1.5% flash with complete die fill and has a more uniform strain variance. h

48 Estimating the starting shapes is tricky for this part because there are two distinct zones of underfill, as explained above. The intention is to reach the optimum shape in multi-levels, and for this purpose the first level is started with geometrically simple guess shapes. Three basis shapes (Fig. 4.8) are selected as starting shapes: cylindrical for Basis 1, tapered cylindrical for Basis 2, and rectangular block for Basis 3. All of the three basis shapes have a material volume of 1.5% more than the final part. Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3] FEM simulations are performed in DEFORM 3D and a quarter model is considered Figure for the 4.8: simulations, Basis Shapes due (1/4 to Model) the quarter Assumed symmetry for 3-D of the Metal part. Hub Basis 1 gives an underfill at the bottom rectangular corner portion of the die cavity while filling the top

49 die cavity, and has a strain variance of Basis 2 also does not fill the bottom die cavity but has slightly less underfill than Basis 1 because of the tapered basis shape. The tapered shape facilitates relatively more material flow towards the bottom die cavity than the top die cavity, thereby attaining a top die cavity fill at the end of the die stroke. This makes the material deformation more uniform with a strain variance of Basis 3 gives a complete die fill at the bottom corner because of the rectangular shape of the basis; however, there is an underfill at the top die cavity because the material flows outside the die cavity faster and flash is formed at a very early stage compared to Basis 1 and Basis 2. The strain variance of Basis 3 is , which signifies that material deformation is not uniform as in the other basis shapes. All three basis shapes give underfill, but at different locations, and have a flash more than 1.5% because material flows out of the die cavitites instead of filling them. Each basis vector that defines the corresponding basis shape has 171 shape coordinates (x, y, and z). Weights are assigned to these vectors and combined linearly, thereby making the weights the design variables. Fifteen DOE points are generated, and 3-D forging simulations are conducted at these points, yet none of the 15 forging simulations give a complete die fill. RSM models are developed on which optimization is performed. The optimum shape that gives complete die fill is reached in the first level (Fig. 4.9) and the optimum weights are 0.10, 0.71, and 0.62, respectively. It can be observed that most of the contribution is from the tapered cylindrical Basis 2 and the rectangular Basis 3; Basis 2 gives die fill at the stub of the metal hub and Basis 3 gives die fill at the corner of the rectangular portion. The resultant preform is tapered and its profile is a

50 combination of cylindrical and rectangular shapes (Fig. 4.9). The cylindrical nature of the preform reduces the material flow at the sides of the rectangular bottom die, but the rectangular nature, coupled with the taper of the preform, enhances the material flow at the bottom die corner. Therefore, the material flow toward the corner is faster than at the sides of the bottom die and uniformly fill the entire bottom die before flash formation. At the same time, the height of the preform shape, which is more than Basis 3 and less than both Basis 1 and Basis 2, is adequate to fill the top die cavity. The slight contribution of Basis 1, which is the tallest of the basis, aids in the selection of the appropriate height. Also, the strain variance of the preform shape is minimized to 0.024, which is less than that of the basis shapes. The reasons for this are the more uniform material flow and die fill at cavities occurring at more or less the same time. Since all of the material flow aids in filling the cavities, the scrap percentage is reduced to 1.5%, a realization of one of the goals. Comparison of the performance characteristics are tabulated in Table 4.5. Preform shape Forged part with complete die fill Figure 4.9: Optimum Preform Shape and the Forged Part (Flash 1.5%)

51 Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (cm 3 ) (3.02%) (2.79%) (6.43%) (1.5%) Underfill volume (cm 3 ) Load (MN) Table 4.5: Performance Characteristics of Basis Shapes and Preform for 3-D Metal Hub The preform shape in this example is reached in the first level, which may not always happen. The number of levels depend on the basis shapes guessed; if the basis shapes give underfill at different locations and if these are the only locations (cavities) where the die fill is difficult to achieve, the chances of finding a preform shape which

52 will be some combination of these basis is high. It is also important to mention that, though the preform shape gives a complete die fill, the strain variance need not necessarily be better than the basis shapes; in most cases the increased strain variance is the penalty that has to be paid to satisfy the underfill constraint Preform design for 3-D metal hub with higher height-to-breadth ratio. In the previous example, the preform shape that gives complete die fill is reached in a single level, which may not always happen. Here, a more complex metal hub (Fig. 4.10) is selected for which the height-to-breadth ratio is 2.0. The optimization goal is to design a preform shape that gives 2.0 % flash with complete die fill and has a more uniform strain variance. h b The bottom rectangular portion destroys the axisymmetric nature of the top portion, Figure making 4.10: the problem 3-D Metal 3-D. Hub But (3/4 unlike Model) the with previous Section case View study, (H/B = die 2) fill at the bottom rectangular corner portion is not difficult to obtain because material flow at the rectangular portion occurs much sooner than at the top die cavity. The preform design

53 process is started with geometrically simple basis shapes, as in the multi-level design process. (a) Level 1: Three basis shapes (Fig. 4.11) are selected as starting shapes: cylindrical for Basis 1, tapered cylindrical for Basis 2, and rectangular block for Basis 3. All three basis shapes have a material volume of 2.0% more than the final part. All three basis shapes are the same as in the previous case study. It can be seen how the influence of the basis shapes changes when the problem changes. Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3] Figure 4.11: Basis Shapes (1/4 Model) Assumed for 3-D Metal Hub 2 (Level 1) Three dimensional forging simulations of the basis shapes are performed to find the underfill and the strain variance for preliminary analysis (Table 4.6). A quarter model is considered for the simulations, due to the quarter symmetry of the part. As mentioned above, all three basis shapes give underfill only in the top die cavity. Basis 1 gives an

54 underfill value of cm 3 and has a strain variance of Basis 2 gives a huge underfill ( cm 3 ) and also has a higher strain variance of This is because most of the material flows out of the die cavities as flash instead of filling the deeper die cavities. The material flow as flash is facilitated by the rectangular nature of the basis shape. Basis 3, which is a tapered cylinder, performs better than both of the other basis and gives an underfill value of cm 3, which is significantly less than other basis shapes. The strain variance (0.0328) for this basis is less than Basis 2 because the material deforms more uniformly and less material flows outside the die cavities as flash. The strain variance for this part is slightly higher than Basis 1 because for Basis 3 the die fill at the bottom rectangular portion occurs sooner than the former, yet there is still material flow into the top die cavity. Each basis vector that defines the corresponding basis shape is defined by 171 shape co-ordinates (x, y, and z). Weights are assigned to these vectors and combined linearly, thereby making the weights the design variables. Fifteen DOE points are generated, and 3-D forging simulations are conducted at these points. It is interesting to note that all 15 DOE points give more underfill than Basis 3. RSM models are developed on which optimization is performed.

55 Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (cm 3 ) (2.02%) (2.07%) (2.01%) (2.01%) Underfill volume (cm 3 ) Load (MN) Table 4.6: Performance Characteristics of Basis Shapes and Level 1 Best Shape for 3-D Metal Hub (H/B =2) The optimum billet in this Level is Basis 3, because the optimum weights obtained were a 1 = 0, a 2 = 0, a 3 = 1. This shows that Basis 1 and Basis 2 selected in this level were unsuitable and should be discarded, as explained in the multi-level design process. This also clearly shows that if the designer starts with impractical basis shapes, the design method rejects those shapes by selecting only the appropriate shapes that may

56 be basis shapes or some combination of suitable basis shapes. With the knowledge of what suitable basis shapes should be, we can proceed to the next level. (b) Level 2: In this level, three starting shapes are selected based on the knowledge obtained from Level 1. Basis 1 in this level is the optimum shape from Level 1. This shape is assumed as a basis shape because if the other basis shapes in this level are more unsuitable than Basis 1, then the optimizer will select Basis 1 by default as the optimum shape. Basis 1 will be given a weight of one and the others will be given zero. The design process can proceed to the next level without any further decrease in the underfill value. Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3] Figure 4.12: Basis Shapes (1/4 Model) Assumed for 3-D Metal Hub 2 (Level 2) The other two basis shapes (Fig. 4.12) are modifications of Basis 1. The top portions of Basis 2 and Basis 3 have opposing profiles, as can be seen in the figures. Any combination of these basis shapes will reduce the material volume near the central axis or at the periphery of the resultant shape. So some combination of these shapes may provide enough material to aid in filling the top die cavity. All three basis shapes in this level are axisymmetric, even though the part is 3-D. This is because it is difficult to achieve complete die fill only at the top die cavity and not at the rectangular corner at the bottom.

57 The taper angle of the basis shapes is different from each other and some taper, which will reduce the strain variance, will be selected by the optimizer. A preliminary forging analysis shows that all three basis shapes give some underfill. The underfill for Basis 1 and Basis 2 are almost the same, which are cm 3 and cm 3, respectively. Basis 3 gives slightly more underfill ( cm 3 ) because the material volume at the periphery is less than Basis 2. Some contribution from this basis shape will change the profile of the resultant shape, which may prove to be useful in achieving the desired goal. Since all of the basis shapes are axisymmetric, fewer boundary points can be used to define the respective shapes even though they are more complex than those in Level 1. Thirty-six shape co-ordinates are used to define each basis shape, which form the respective basis vector. Weights are assigned to each vector and are combined linearly to obtain various shapes. Fifteen DOE points are selected for the forging simulations to build a RSM for optimization. Optimum weights (a 1 = 0.603, a 2 = 1.0, a 3 = 0.304) that give complete die fill are achieved in this level (Fig. 4.13). Basis 2 makes the most contribution towards the preform shape and this contribution increases the material volume towards the periphery of the preform. There is also a significant contribution from Basis 1, and this has increased the taper of the preform. While this contribution is useful to achieve a die fill, the strain variance to the part is increased because the die fill at the rectangular bottom corner is achieved at an earlier stage. A slight contribution from Basis 3 increased the material volume at the center of the preform and changed the profile, which has also played a critical role in satisfying the underfill constraint. The resulting preform shape

58 has a strain variance of (Table 4.7) which is higher than Basis 1 and Basis 2. This increase in strain variance is because of the increase of material deformation at the top die cavity, which aided in achieving a complete die fill. Basis 1 Basis 2 Basis 3 Preform Strain variance

59 Flash volume (cm 3 ) (2.01%) (2.01%) (2.01%) (2.00%) Underfill volume (cm 3 ) Load (MN) Table 4.7: Performance Characteristics of Basis Shapes and Level 2 Optimum Shape for 3-D Metal Hub (H/B =2) 4.4. Preform design of 3-D spring seat Spring seats (Fig. 4.14) are used in heavy drilling machines to provide cushioning effects to the tool, which prevents metal chips from breaking away. This part cannot be assumed as plane strain or axisymmetric. The optimization goal is to design a preform shape with 4.5% flash that gives more uniform strain variance with zero underfill.

60 In this example, intuitive basis shapes are selected to check if the optimum shape is reached in a single level. Three basis shapes (Fig. 4.15) are assumed, and all of these shapes give underfill at different locations. Basis 3, which is cylindrical in shape, is assumed for generality. Basis 1 and Basis 2 have material depth at different locations so that the optimizer can find the best shape, which will be some combination of these shapes. The strain variance for each basis is , , and , respectively. Basis 3 has a higher strain variance because the cylindrical shape does not aid in filling the die cavities, but flows out as flash, producing huge underfill (Table 4.8). Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3]

61 Each basis shape is defined by 58 shape co-ordinates (x, y, and z). These basis shapes are all of the same volume, 4.5 % more than the volume of the part. Fifteen DOE points are generated, and 3-D forging simulations are conducted at these points. A quarter model is considered for forging simulation. The RSM is fit at all these points for strain variance and the underfill. Optimization is done on these RSM models, and the resulting optimum weights are 0.87, 0.18, and 0, respectively. A preform shape that gives complete die fill is reached in a single stage. Basis 1 makes a significant contribution towards the optimum shape, but the minor contribution of Basis 2 is crucial to achieve a complete die fill. Basis 3 has no contribution towards the optimum shape, which would have increased both the underfill value and the resulting strain variance. A 3-D forging simulation is conducted on the optimized shape (Fig. 4.16) to verify this result, which gives zero underfill in accordance with the approximation models. This case study proves that if the design process is started with practical or intuitive guess shapes, then the optimum preform that gives complete die fill can be reached in a single level. If expert knowledge is available, it can be exploited to select

62 appropriate basis shapes and save significant computation time to design a preform shape. Figure 4.16: Optimized Billet (Flash: 4.5%) Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (cm 3 ) (5.84%) (6.84%) (11.14%) (4.5%) Underfill volume (cm 3 ) Load (MN)

63 Table 4.8: Performance Characteristics of Basis Shapes and Preform for Spring Seat (Single-Stage Optimization) 4.5. Preform design of 3-D steering link The preform shape optimization of a 3-D steering link is demonstrated in this case study. The steering link translates the rotation of the steering wheel into the linear action of pushing the tires in the desired direction of travel. The steering link is a complex 3-D part with varying cross-sections (Fig. 4.17) along all three axes (X, Y, and Z). This part is almost always produced by hot forging, and obtaining a die fill at the big end is particularly difficult. The part geometry cannot be assumed as plane strain or axisymmetric about any axis; therefore, none of the 2-D assumptions can be used for preform shape optimization. Side view Front end Rear end Top view Isometric view

64 A steering link is a high-volume forged component and any saving in the material loss is highly desirable. There is usually about 30% flash (material waste) in manufacturing this product. In this research work, the goal is to design a preform with 5% flash that gives complete die fill and has more uniform strain variance. The steering link forging process consists of three main stages: the buster, blocker, and finisher stages. The buster stage is where the initial billet, which may be cylindrical or rectangular, is forged to a preform shape. Following this is the blocker stage, where most of the material deformation takes place to produce a near-finished part. The finisher stage aids in fine-tuning the sharp corners and fillets of the forged part. Since most of the material deformation takes place during the blocker stage, the optimization methodology is applied to this stage of preform shape design. A steering link is a complicated part to forge and finding useful basis shapes that in some combination would give a die fill is difficult to obtain. Therefore, the design process is started with geometrically simple basis shapes, as the process is multi-level design.

65 (a) Level 1: Three basis shapes (Fig. 4.18) are selected in this level. Since most of the forging processes start with a simple cylindrical or rectangular billet, Basis 1 is assumed as cylindrical and Basis 3 as a rectangular block. From the geometry of the part, it is evident that more material is needed at the front end, compared to the rear end of the steering link. Therefore, Basis 2 is selected as a tapered cylindrical block, and more material is provided at the end to correspond to the front end of the steering link. All three of the basis shapes have the same volume of 129,000 mm 3, which is 5% more than the volume of the part. Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3] Figure 4.18: Level 1 Basis Shapes for Steering Link Three dimensional forging simulations of the basis shapes are performed to find the underfill and the strain variance for preliminary analysis. All of the basis shapes give

66 underfill because more material flows outside the die cavities as flash instead of filling them. Underfill for Basis 1, Basis 2, and Basis 3 are 7.58%, 6.95%, and 6.85%, respectively. Basis 2 and Basis 3 have less underfill compared to Basis 1. In the case of Basis 3, the rectangular shape has potential to fill the die cavities and the extra material provided at one end of Basis 2 aided in filling the cavities. Also, all three of the basis shapes give more underfill at the front end of the steering link than at the rear end. The strain variance of the basis shapes are 0.037, 0.072, and 0.045, respectively. The higher strain variance for Basis 2 results from more material deformation at the front end of the steering link, which also aids in filling the die cavities. From this preliminary analysis, it can be said that the rectangular shape is more successful than the other two shapes in filling the cavities and also that the material deformation is more uniform for this shape. Therefore, the contribution of Basis 3 must be more than the other basis shapes, which must be recognized by the optimizer Scaled Obj./Const. Violations Objective Objective Constraint Constraint Iteration

67 Figure 4.19: Constraint and Objective Function Iteration History (Level 1) Each basis shape is defined by 648 shape variables, (x, y, and z co-ordinates) at 216 locations along the surface of the basis shapes. These shape variables form the respective basis vectors. The reduced basis technique is applied to these basis vectors and the number of design variables is decreased to three, which are the weights for each basis vector. By changing these weights it is possible to obtain various resultant billet shapes for the optimizer to find the best combination of these weights. Fifteen DOE points are generated by the LHS technique to conduct forging simulations. All of the resultant billet shapes are scaled to maintain a constant volume of 129,000 mm 3. Forging simulations are conducted at these DOE points to find the underfill and strain variance and to build the RSM models for optimization. Optimization is performed in MatLab to minimize the strain variance and to eliminate the underfill (Eq. 6). The underfill constraint is not satisfied, and it takes eleven iterations (Fig. 4.19) to reach the best possible combination of the three basis shapes. Figure 4.20: Level 1 Best Billet

68 The optimum weights in Level 1 are 0, , and 1.0, respectively. The underfill is reduced to 4.57% and the strain variance of the resultant shape (Fig. 4.20) is It can be clearly seen that most of the contribution is from Basis 3 because the rectangular nature of the billet is crucial in achieving more material flow into the die cavities. There is also a significant contribution from Basis 2 towards the Level 1 optimum shape. This contribution increases the billet material close to the front end of the steering link by reducing the material at the rear end. This is because most of the extra material provided at the rear end flows out as flash after filling the cavity at that location and does not aid in the material flow at the front end. Therefore, the optimizer shifted the material to the front end by accepting the significant contribution from Basis 2. This contribution has slightly increased the strain variance, but is significantly less than Basis 2. The weight for Basis 1 is zero, since the cylindrical shape is not a practical basis shape. It can be clearly seen that even if the designer starts with impractical starting shapes, the optimizer aids in discarding those shapes by giving them zero weights or reducing their contribution.

69 Also, it cannot be said that Basis 2 and Basis 3 are suitable basis shapes because they did not satisfy the underfill constraint (Fig. 4.21). If expert knowledge is available, it could have been possible to guess practical starting shapes that would have given complete die fill. But even without expert knowledge, the results of Level 1 have shown that the optimum shape that may give a die fill should have a rectangular nature coupled with a tapering profile. In this level, three basis shapes were selected, but a different resultant shape might have been achieved if the number of starting shapes were increased. But, since this work is to show the capability of the multi-level design process in aiding the designer to select good basis shapes and to reach the optimum shape that gives complete die fill, only three simple starting shapes were selected. The performance characteristics of the basis shapes and the Level 1 optimum shape are shown in Table 4.9. With the knowledge of what suitable basis shapes should be, we can proceed to the next level.

70 (b) Level 2: In this level, four starting shapes are selected based on the knowledge obtained from Level 1. Basis 1 in this level is the best shape from Level 1. The design process can proceed to the next level without any further decrease in the underfill value. Basis 1 Basis 2 Basis 3 Preform Strain variance Flash volume (mm 3 ) (12.60%) (11.97%) (11.87%) (9.59%) Underfill volume (mm 3 ) Load (MN) Table 4.9: Performance Characteristics of Basis Shapes and Level 1 Best Shape for Steering Link

71 The other basis shapes (Fig. 4.22) are variants of Basis 1. Basis 2 has the same cross-section, but has a slightly different profile along the length of the basis. Basis 3 has the same profile as Basis 1 along the length, but has a different cross-section to aid in filling the die cavities. In Basis 4, more material has been added at the front end and the taper has been slightly increased to check if it has potential to fill the die cavities. All of the basis shapes are of the same volume as in Level 1, which is 129,000 mm 3. Basis 1 Basis 2 Basis 3 Basis 4 Figure 4.22: Level 2 Basis Shapes

72 A preliminary forging analysis shows that all four basis shapes give some underfill, which is 4.574%, 4.073%, 2.212%, and 2.986% of the part volume, respectively. It can be seen that Basis 3 and Basis 4 have less underfill than the Level 1 best shape (Basis 1) owing to their shapes. Even though the underfill is reduced for Basis 3, the strain variance is at a lesser value of because of its cross-section, which aids in material flow into the die cavities (at front end) at the same time as the material flow at the other regions of the dies. Basis 4 has a higher strain variance value of because of more material deformation at the front end of the steering link. Higher strain value and less underfill mean that the die fill is due just to the more material volume present and not because of the shape. Basis 2 also gives a very high strain variance, which is The main reason to select Basis 2 and Basis 3 is because both basis shapes have different profiles along two different (perpendicular) planes, along the length for the former and about the cross-section for the latter. Some combination of these basis shapes may give a better shape. If these shapes are not viable, then the optimizer will give zero weights for them. The geometry of the Level 2 basis shapes is slightly more complicated than that of the Level 1 basis shapes. The number of shape co-ordinates for all basis vectors are increased to 1125 i.e., x, y, and z co-ordinates at 375 boundary points. These shape variables form the respective basis vectors and it can be seen that the number of design variables (weights) are reduced to four even though the number of shape variables are

73 huge. By just changing these weights, it is possible to obtain many different possible shapes and the optimizer has a better chance to find the optimum weights that may give a die fill than in Level 1. This is mainly because of two reasons: (a) A higher number of basis shapes and (b) Practical basis shapes selected based on the knowledge of Level 1. Twenty-five DOE points are generated by the LHS technique to build a good RSM and the resulting billets are scaled to a constant volume and 3-D forging simulations are performed. Underfill and strain variance results are extracted from the simulations to build the approximation models and optimization is performed on these models, as per Eq Scaled Obj./Const. Violations Objective Objective Constraint Constraint Iteration Figure 4.23: Constraint and Objective Function Iteration History (Level 2)

74 Optimum weights that give complete die fill are achieved in eleven iterations (Fig. 4.23). Optimum weights are 1.0, , , and 0.0, respectively for the four basis shapes. Basis 1 has the most contribution towards the preform shape (Fig. 4.24), even though the underfill for Basis 1 is at a maximum compared to the other basis shapes. This is because the contribution from Basis 1 has reduced the curvature of Basis 2 along the length and the cross-sectional profile of Basis 3. There is no way to increase the material depth of the Basis 3 cross-sectional profile because there is no other basis that contributes towards this. If this would have been permitted by selecting another basis, the optimizer would have given the maximum weight to Basis 3, but the preform shape will be more complicated to manufacture. The contributions from Basis 2 and Basis 3 are nearly the same and this gives the preform shape the characteristics of both Basis 2 and Basis 3. Basis 4 has zero contribution even though it aids in providing more material at the front end of the steering link, since any contribution of this basis will increase the strain variance. Figure 4.24: Final Preform Shape and Forged Part