Multi-Level Design Process for 3-D Preform Shapececs.wright.edu/cepro/docs/thesis/Multi_Level... ·...

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 1999

2004 Wright State University

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

Richard J. Bethke, Ph.D. Chairman of Department

Committee on Final Examination 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

ABSTRACT

Thiyagarajan, Nagarajan M. S. Egr., Department of Mechanical and Materials Engineering, Wright State University, 2004. 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.

TABLE OF CONTENTS

1. Introduction …………………………………………………………….. 1

2. Background …………………………………………………………….. 5

2.1 Backward optimization …………………………………………….. 5

2.2 Discrete approach …………………………………………………... 7

2.3 Continuum approach ……………………………………………….. 8

3. Preform shape optimization methodology ……………………………… 10

3.1 Reduced basis method ……………………………………………… 10

3.2 Basis vector definition …………………………………………….... 11

3.3 Geometric scaling …………………………………………………... 13

3.4 Approximation model ……………………………………………..... 13

3.5 Optimization problem definition ………………………………….... 15

3.6 Multi-level optimization ……………………………………………. 16

3.7 Orthogonalization check ……………………………………………. 19

4. Case studies ……………………………………………………………... 21

4.1 Preform design for plane strain rail section ……………………….... 23

4.1.1 Single-level optimization ………………………………….... 24

4.1.2 Multi-level optimization of plane strain rail section ……….... 30

4.2 Preform design for 3-D metal hub ………………………………….. 39

4.3 Preform design for 3-D metal hub with

higher height to breadth ratio ………………………………………. 44

4.4 Preform design of 3-D spring seat …………………………………. 52

4.5 Preform design of 3-D steering link ………………………………... 56

5. Discussion and conclusions …………………………………………..… 70

Appendix …………………………………………………………….…. 72

References …………………………………………………………….... 81

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 4.7 3-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 4.10 3-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

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

LIST OF TABLES

Table 4.1 Performance characteristics of basis shapes and preform for rail section (single-stage optimization) 29

Table 4.2 Performance characteristics of basis shapes and preform for rail section (Level 1) 32



Table 4.5 Performance characteristics of basis shapes and preform for 3-D metal hub 43

Table 4.6 Performance characteristics of basis shapes and Level 1 optimum shape for 3-D metal hub (h/b =2) 47

Table 4.7 Performance characteristics of basis shapes and Level 2 optimum shape for 3-D metal hub (h/b =2) 51

Table 4.8 Performance characteristics of basis shapes and preform for spring seat (single-stage optimization) 55

Table 4.9 Performance characteristics of basis shapes and Level 1 optimum shape for steering link 63

Table 4.10 Performance characteristics of basis shapes and preform (Level 2) for steering link 69

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

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 co-

ordinates 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

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

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.

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.

2.1. 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

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

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: strain-

rate 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 plane-

strain 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.

2.2. 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

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.

2.3. 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

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 knowledge-

based 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-the-

fittest 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 re-

training), then the method loses its merit. The use of genetic algorithms is a powerful

technique that handles discrete design data with ease (e.g., number of stages in multi-

stage 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.

3.1. Reduced Basis Method

The main idea in this method is to construct basis functions or vectors, Y1, Y2,

Y3,…, Yn, with the large information content of each basis shape and to combine them

in

ii

c YaYY ∑+=1

(1)

linearly with the weighing factors a1, a2, a3,…,an that correspond to each basis vector.

And, Yc 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.

3.2. 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

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 Inaccurate boundary

(b) (a)

Desired boundary

Figure 3.1: Basis Vectors Definition

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

basis vector definition is useful for generating various possible shapes for optimization,

but scaling of the resultant shapes is essential to maintain volume constancy.

3.3. Geometric Scaling.

In order to understand the necessity of scaling, a simple problem of combining

two basis shapes is considered. The unknown lengths (l1, l2) and unknown breadths (b1,

b2), which are defined by their respective basis shapes (Y1 and Y2) having the same area,

form the initial shapes. The optimum shape Y is defined as

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡++

=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

bl

babalala

bl

abl

aY2211

2211

2

22

1

11

(2)

where a1 and a2 are the weights. The area of the resulting shape will be , 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.

bl ×

3.4. 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

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, xi are the design

variables, and y is the response. The design variables xi are the weights ai 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].

)0,2(

)2,0()0,0(

)1,1( −−

)1,1(

)0,0(

2=α

ts

r

Figure 3.2: Central Composite Design for Two Factors

In the case of CCD generation (α = 1.4142) 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, 0.1464), (0.5, 0.5), and (0.8536, 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.

3.5. 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, , 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):

2ws

∑

∑ −= i

wii ew2

)(

=

=

⎟⎠⎞

⎜⎝⎛

′−′ N

ii

N

w

wN

N

es

1

1

2

1 (4)

where ei is the observation, wi is the weight of ith observation, N is the number of weights

(elements), is the number of non-zero weights (in this case, N ′ N ′ = N), and we 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:

(5) actualdesire VVUnderfill −=

where Vdesire and Vactual 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, ai (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(ai)

Subject to: (6)

Underfill: g(ai) ≤ 0

Side bounds:

0 ≤ ai ≤ 1

3.6. 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

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

Reduced Basis Technique

Level ‘N’

Design of Experiments

Response Surface Method

OPTIMIZATION

Die fill ?

Update basis shapes (N = N + 1)

No

Yes

END

Assume simple basis shapes(N = 1)

Figure 3.3: Multi-Level Design Process

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.

3.7. 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.

(7)

u1 = v1

u2 = v2 - [(v2 . u1)/(u1 . u1)]u1

u3 = v3 - [(v3 . u1)/(u1 . u1)]u1 - [(v3 . u2)/(u2 . u2)]u2

...

uk = vk - [(vk . u1)/(u1 . u1)]u1 - [(vk . u2)/(u2 . u2)]u2 - ... - [(vk . uk-1)/(uk-1 . uk-1)]uk-1

The Gram-Schmidt procedure takes an arbitrary basis (vk) and generates an

orthonormal one (uk). 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 (vk). A simple MatLab code is written to find the

orthonormal vectors (uk) 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.

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

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 1200o 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 Yc 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.

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 cm2. For

this example, the basis shapes are selected and scaled to a constant area of 153.5 cm2,

thereby specifying the scrap as 3%.

h2

h1

b2

b1

Symmetry axis

h1 = 1.25 x b1h2 = 1.50 x b2

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

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.

4.1.1. 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

Basis 1 [Y1]

Basis 4 [Y4] Basis 3 [Y3] Basis 2 [Y2]

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

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

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 a40 1 0.76 0

Optimized weights

Basis 2 carries an optimum weight of one, which means that 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

Figure 4.3: Optimized Billet for Rail Section (Flash: 3%)

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

Basis 1 Basis 2 Basis 3 Basis 4 Preform

Strain variance 0.24711 0.07602 0.07165 0.27377 0.0648

Flash (cm2) 11.38 (7.67%)

7.01 (4.72%)

6.41 (4.32%)

21.33 (14.37%)

4.51 (3.0%)

Underfill (cm2) 6.87 2.47 1.90 16.82 0.00

Load (KN) 292.39 276.08 274.78 407.01 278.67

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.

4.1.2. 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)

Basis 2 Basis 3Basis 1

BEST SHAPE (Level 1)

0.400.7a3a2a1

Optimum weights (Level 1) Figure 4.4: Level 1 of Plane Strain Rail Section

Basis 1 Basis 2 Basis 3 Preform

Strain variance 0.24711 0.3080 0.1002 0.1649

Flash volume (cm2) 11.38 (7.67%)

22.37 (15.01%)

12.15 (8.15%)

8.38 (5.62%)

Underfill volume (cm2) 6.87 17.87 7.65 3.88

Load (KN) 292.39 438.75 265.94 287.99

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 cm2, all three basis shapes give huge underfill of 6.87 cm2, 17.87 cm2, and

7.65 cm2, respectively. Three weights for each basis vector become the design variables

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 cm2 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

Basis shapes and their results (LEVEL 2)

Basis 2 Basis 3 Basis 1

110a3a2a1

Optimum weights (Level 2)

BEST SHAPE (Level 2)

Figure 4.5: Level 2 of Plane Strain Rail Section


Strain variance 0.1649 0.0985 0.1169 0.1301


6.22 (4.17%)

6.58 (4.42%)

5.14 (3.45%)


Load (KN) 287.99 297.97 261.46 324.44


up the respective basis vectors and FEM forging simulation of these basis shapes are

performed. All three basis shapes give underfill of 3.88 cm2, 1.72 cm2, and 2.08 cm2,

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 cm2 (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 cm2, 0.73 cm2, and 1.57 cm2,

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.

Basis 1

Figure 4.6:

Basis shapes and their results (LEVEL 3)

Basis 2 Basis 3

OPTIMUM SHAPE (Level 3)

0.7210

a3a2a1

Optimum weights (Level 3)

Level 3 of Plane Strain Rail Section (Flash 3 %)


Strain variance 0.1301 0.1143 0.0817 0.1059


5.23 (3.51%)

6.07 (4.07%)

4.50 (3.0%)


Load (KN) 324.44 337.41 260.22 368.72


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.

4.2. 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

b

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.


FEM simulations are performed in DEFORM 3D and a quarter model is

considered for the simulations, due to the quarter symmetry of the part. Basis 1 gives an

underfill at the bottom rectangular corner portion of the die cavity while filling the top

Figure 4.8: Basis Shapes (1/4 Model) Assumed for 3-D Metal Hub

die cavity, and has a strain variance of 0.0301. 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 0.0278. 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 0.0494, 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 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, 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

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%)


Strain variance 0.0301 0.0278 0.0494 0.0240

Flash volume (cm3) 298.06 (3.02%)

275.40 (2.79%)

635.26 (6.43%)

150.54 (1.5%)


Load (MN) 2.97 2.84 0.51 2.82

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

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.

4.3. 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.

The bottom rectangular portion destroys the axisymmetric nature of the top

portion, making the problem 3-D. But unlike the previous case study, die 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

Figure 4.10: 3-D Metal Hub (3/4 Model) with Section View (H/B = 2)

b

h

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.


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

underfill value of 406.12 cm3 and has a strain variance of 0.0254. Basis 2 gives a huge

underfill (1464.56 cm3) and also has a higher strain variance of 0.0644. 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 139.60 cm3, 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.


Strain variance 0.0254 0.0644 0.0328 0.0328

Flash volume (cm3) 42128.12(2.02%)

43186.56(2.07%)

41861.60(2.01%)

41861.60 (2.01%)


Load (MN) 10.60 11.55 9.16 9.16

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 a1 = 0, a2 = 0, a3 = 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

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.


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.

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 139.60

cm3 and 137.20 cm3, respectively. Basis 3 gives slightly more underfill (165.52 cm3)

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 (a1 = 0.603, a2 = 1.0, a3 = 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

has a strain variance of 0.0404 (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.


Strain variance 0.0328 0.0353 0.0454 0.0404

Flash volume (cm3) 41861.60(2.01%)

41859.20(2.01%)

41887.52(2.01%)

41722.00 (2.00%)


Load (MN) 9.16 7.92 10.67 8.39

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.

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 0.0353, 0.0534, and 0.0939, 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).


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

appropriate basis shapes and save significant computation time to design a preform

shape.

Figure 4.16: Optimized Billet (Flash: 4.5%)


Strain variance 0.0353 0.0534 0.0939 0.0414


51.72 (6.84%)

84.27 (11.14%)

34.01 (4.5%)


Load (MN) 0.60 0.52 3.28 0.54

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 viewFront end

Rear end

Isometric view

Top view

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.

(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 mm3, which is 5% more than

the volume of the part.


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

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.

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

1 3 5 7 9 11 13

O

Iteration

Scal

ed O

bj./C

onst

. Vio

latio

ns

bjective

Constraint

Objective

Constraint

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 mm3. 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

The optimum weights in Level 1 are 0, 0.7649, and 1.0, respectively. The

underfill is reduced to 4.57% and the strain variance of the resultant shape (Fig. 4.20) is

0.0485. 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.

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.

(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.


Strain variance 0.037 0.072 0.045 0.0485

Flash volume (mm3) 15475.89(12.60%)

14702.03(11.97%)

14579.2 (11.87%)

11778.56 (9.59%)

Underfill volume (mm3) 9310.89 8537.03 8414.19 5613.56

Load (MN) 1.45 1.62 2.20 1.50

Table 4.9: Performance Characteristics of Basis Shapes and Level 1 Best Shape for Steering Link

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 mm3.

Basis 4 Basis 3

Basis 2 Basis 1

Figure 4.22: Level 2 Basis Shapes

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 0.0415 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 0.088 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 0.0981. 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

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. 6.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 2 4 6 8 10 12

Objective

Constraint

Objective Constraint

Scal

ed O

bj./C

onst

. Vio

latio

ns

Iteration

Figure 4.23: Constraint and Objective Function Iteration History (Level 2)

Optimum weights that give complete die fill are achieved in eleven iterations

(Fig. 4.23). Optimum weights are 1.0, 0.6002, 0.7608, 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

The resulting preform shape has a strain variance of 0.0753. This may be more

than some of the basis shapes, but this increase in strain variance is due to material flow

at the deep die cavities where material flow was achieved. Since most of the material

flow aids in filling the die cavities, unlike the basis shapes, flash was reduced to 5%, a

realization of one of the goals (Table 4.10).

A preform shape that gives complete die fill was achieved in two levels in this

example and further increases in the design levels will give a preform shape with less

strain variance. Also, an increase in the number of basis shapes in Level 1 and Level 2

would have produced a better prefom shape. The increase in strain variance of the

preform shape is the penalty that has to be paid for satisfying the underfill constraint.

Basis 1 Basis 2 Basis 3 Basis 4 Preform

Strain variance 0.0485 0.0981 0.0415 0.0888 0.0753

Flash (mm3) 11778.56(9.59%)

11168.07(9.09%)

8882.11 (7.23%)

9832.85 (8.00%)

6165.00 (5.01%)

Underfill (mm3) 5613.56 5003.07 2717.11 3667.85 0.00

Load (MN) 1.50 2.03 1.70 1.41 1.89

Table 4.10: Performance Characteristics of Basis Shapes and Preform (Level 2) for Steering Link

Discussion and Conclusions

A three-dimensional preform shape optimization technique is demonstrated in this

research by using the reduced basis technique. This design process can be used for both

2-D and 3-D preform shape optimization. The reduced basis technique is a design

variable linking technique that has originally been used extensively in large scale

structures for shape optimization. In metal forming, this concept has to be used in

conjunction with scaling so that only the resulting shapes can be used for optimization,

while keeping the volume constant. This technique also makes the use of RSM methods

practical when gradient information is not available. RSM models are also useful for

predicting the observations within the design space.

The methodology is also extended to accommodate even relatively simpler billet

shapes as basis shapes and can still reach the optimum shape. The concept of a multi-

level design process is introduced, which aids the designer in the selection of practical

basis shapes that will give complete die fill. However, this will also increase the number

of FEM simulations to build the surrogate model. It is important to mention that if expert

knowledge is available, then practical basis shapes can be selected and the optimum

preform shape can be obtained in a single level. Increasing the number of basis shapes

also enables the designer to obtain a better preform shape, but the computation time also

increases to build an approximation model.

Most preforms obtained by this method are practical and can be forged in a single

stage. However, if the basis shapes are complicated, which may be the case for some

parts, the obtained preform shape will also be complicated. Therefore, it is prudent to

start from very simple starting shapes. This design method does not take into account the

strain variance that may already be present in the starting shapes while designing the

preform, which may be significant for some complicated preform shapes. Continuing the

design levels even after obtaining a complete die fill (constraint) will further minimize

the strain variance (objective), but at the expense of computational time, which has to be

decided by the designer. The forging simulations conducted for the DOE can be

computationally expensive depending on the complexity of the part geometry. For a very

complex part such as the crankshaft, each forging simulation may take about one day and

building a surrogate model at each level as in the multi-level design process is

impractical.

Important points that can be considered for the efficient use of this method are:

1. The selection of basis shapes should depend on the shape complexity of the part to be

forged. It is advantageous to use practical basis shapes, which need not necessarily

give a die fill.

2. Increasing the number of basis shapes gives more flexibility to the optimizer and may

result in a better preform shape; however, DOE points for the RSM will also increase.

3. Shape co-ordinates used to define the basis shapes should be as plentiful as possible

(>50), depending on their shape complexity.

Appendix

A. Basis vector generation There are two main methods for creating basis vectors by defining the boundary

points:

1. Equidistant boundary points: As the name suggests, all the boundary points are

equally spaced from each other as shown in the figure below.

Figure A.1: Basis shape with equidistant boundary points

Boundary

5 6 7 4 8 3 2

Boundary point

9 1

10 20

11 19 16 15 12 17 14 13 18

It is important to define all of the basis shapes in the same way with same number

of boundary points. The numbering should be consistent (either clock-wise or

counter clock-wise) for all of the basis shapes. Also, the origin (0, 0) for all the

basis shapes should be same.

2. Radial boundary points: In this method a point called the center point is defined

within the basis shape and many radial lines that have the same angle from each

other are generated. Boundary points are defined at the junction where the radial

lines meet the boundary of the basis shape. Here the center point co-ordinates and

the origin for all the basis shapes should be same.

Center point

Figure A.2: Basis shape with radial boundary points

The above two methods define the generation of basis vectors for 2-D basis

shapes. In the case of 3-D basis shapes, many sections are selected at regular intervals

along the length or height of the basis shapes, and boundary points are generated for each

of these 2-D sections. These boundary points in the 3-D space will have x, y, and z

coordinates. All of these boundary points are numbered and the boundary point co-

ordinates form the respective basis vector. While recreating the basis shapes from the

basis vectors or building a resultant shape from the reduced basis technique, the basis

vector or the resultant vector will produce the various 2-D sections in 3-D space. These

sections can be lofted to produce the 3-D shape.

Sections along the length

Z

Y X

Z

Y X

Figure A.3: Sections lofted to 3-D shape

These are the two techniques used in this work. Other techniques can be used to define

the basis shapes to form the basis vectors as long as all the basis shapes are defined in

that same way and also the boundary is accurately captured.

B. The MatLab code to generate programming file for Steering Link (Level 1) resultant DOE billets clc clear all %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % basis 1 - Basis vector 1 [Y1] % basis 2 - Basis vector 2 [Y2] % basis 3 - Basis vector 3 [Y3] [basis1] = basis_1; [basis2] = basis_2; [basis3] = basis_3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DOE points a = [ 0.23296518617966 0.07497660112866 0.07852550654911 0.38871294272705 0.94431604008723 0.24548328803734 0.03168363398385 0.24446032883861 0.62748688186693 0.89058648911109 0.67461964454401 0.72465322190072 0.86558683440171 0.76802087962077 0.04970971619867 0.14420989627890 0.82927888481497 0.94646460678270 0.47976918748262 0.35893942296648 0.15547694041743 0.08681434465281 0.42254006567154 0.53242490981470 0.56920748122749 0.49222445100137 0.36256137535225 0.32782582547357 0.62099066665161 0.30302927079686 0.61195219941856 0.16598979819713 0.87300338533074 0.78713199880936 0.01906927703100 0.44785607659208 0.43697635145480 0.89898614003513 0.86233104307448 0.99913614469690 0.55960894705597 0.76822720764405 0.71275054689182 0.26888649135641 0.53441920545365]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%DOE Number doe_n = 2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Reduced Basis Technique DOE = a(doe_n,1)*basis1 + a(doe_n,2)*basis2 + a(doe_n,3)*basis3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Generating .prg (I-DEAS programming file) delete Resultant_shape diary('Resultant_shape') diary on fprintf('AP: 1 8 Change View\n') fprintf('AP: 1 0 0 0 0\n') fprintf('AP: 0.0 0.0 0.0\n') fprintf('AP: 1.000000 0.0 0.0\n') fprintf('AP: 0.0 1.000000 0.0\n') fprintf('AP: 0.0 0.0 1.000000\n') fprintf('AP: 0.1050000 1.000000 1.000000 15.00000\n') fprintf('AP: -1.000000 -1.000000 -1.000000\n') fprintf('AP: 1.000000 1.000000 1.000000\n') fprintf('AP: 2 11 Program File User Preferences\n') fprintf('AP: 0 1 0 0 0 0 0 1 0\n') fprintf('AP: 1 -0.1000000 -0.1000000 0.1000000 0.1000000\n') fprintf('AP: 2 0.0 0.0 0.0 0.0\n') fprintf('AP: 3 1 0 0\n') fprintf('AP: 4 0 1 0\n') fprintf('AP: 5 0 0 1\n') fprintf('AP: 6 0 0 0\n') fprintf('AP: 7 0 0 0 10 10 0.05000000 0.05000000 0.05000000 30.00000\n') fprintf('AP: 8 0\n') fprintf('AP: 9 1 1\n') fprintf('AP: 10 1 0 0 20 2.000000\n') fprintf('K : $ return\n') for j = 1:5 fprintf('K : $ /cr m3 sp\n')

fprintf('K : KEY\n') fprintf('K : KEY\n') pt = DOE(:,:,j); for i = 1:(length(pt)) fprintf('K : %6.4f,%6.4f,%6.4f\n',pt(i,1),pt(i,2),pt(i,3)) end fprintf('K : %6.4f,%6.4f,%6.4f\n',pt(1,1),pt(1,2),pt(1,3)) fprintf('K : DON\n') fprintf('K : OKAY\n') fprintf('K : DON\n') end fprintf('K : $ REDI\n') fprintf('AP: 1 8 Change View\n') fprintf('AP: 1 0 0 0 0\n') fprintf('AP: 0.0 0.0 0.0\n') fprintf('AP: 1.000000 0.0 0.0\n') fprintf('AP: 0.0 1.000000 0.0\n') fprintf('AP: 0.0 0.0 1.000000\n') fprintf('AP: 0.1050000 0.2620000 0.2620000 15.00000\n') fprintf('AP: -1.000000 -1.000000 -1.000000\n') fprintf('AP: 1.000000 1.000000 1.000000\n') fprintf('K :\n') fprintf('K : $ REDI\n') fprintf('AP: 1 8 Change View\n') fprintf('AP: 1 0 0 0 0\n') fprintf('AP: 0.0 0.0 0.0\n') fprintf('AP: 1.000000 0.0 0.0\n') fprintf('AP: 0.0 1.000000 0.0\n') fprintf('AP: 0.0 0.0 1.000000\n') fprintf('AP: 0.1050000 0.2820000 0.2820000 15.00000\n') fprintf('AP: -1.000000 -1.000000 -1.000000\n') fprintf('AP: 1.000000 1.000000 1.000000\n') fprintf('K : $ mpos :; /F PR E\n') fprintf('E : **** END OF SESSION ****\n') diary off C. I-DEAS Programming file (Resultant_shape.prg) that should be executed in I-DEAS to generate the DOE billet

AP: 1 8 Change View AP: 1 0 0 0 0 AP: 0.0 0.0 0.0 AP: 1.000000 0.0 0.0 AP: 0.0 1.000000 0.0 AP: 0.0 0.0 1.000000 AP: 0.1050000 1.000000 1.000000 15.00000 AP: -1.000000 -1.000000 -1.000000 AP: 1.000000 1.000000 1.000000 AP: 2 11 Program File User Preferences AP: 0 1 0 0 0 0 0 1 0 AP: 1 -0.1000000 -0.1000000 0.1000000 0.1000000 AP: 2 0.0 0.0 0.0 0.0 AP: 3 1 0 0 AP: 4 0 1 0 AP: 5 0 0 1 AP: 6 0 0 0 AP: 7 0 0 0 10 10 0.05000000 0.05000000 0.05000000 30.00000 AP: 8 0 AP: 9 1 1 AP: 10 1 0 0 20 2.000000 K : $ return K : $ /cr m3 sp K : KEY K : KEY K : 0.0000,28.8338,0.0000 K : 0.0000,28.7727,2.5173 K : 0.0000,28.5897,5.0411 K : 0.0000,28.2864,7.5793 K : 0.0000,27.8649,10.1420 K : 0.0000,27.3285,12.7435 K : 0.0000,26.6814,15.4045 K : 0.0000,25.9283,18.1552 K : 0.0000,25.0751,21.0405 K : 0.0000,24.1282,24.1282 K : 0.0000,23.0949,27.5234 K : 0.0000,20.4884,29.2604 K : 0.0000,17.3283,30.0135 K : 0.0000,14.2973,30.6606 K : 0.0000,11.3548,31.1970 K : 0.0000,8.4721,31.6185 K : 0.0000,5.6287,31.9218 K : 0.0000,2.8088,32.1048 K : 0.0000,0.0000,32.1659 K : 0.0000,-2.8088,32.1048 K : 0.0000,-5.6287,31.9218

K : 0.0000,-8.4721,31.6185 K : 0.0000,-11.3548,31.1970 K : 0.0000,-14.2973,30.6606 K : 0.0000,-17.3283,30.0135 K : 0.0000,-20.4884,29.2604 K : 0.0000,-23.0949,27.5234 K : 0.0000,-24.1282,24.1282 K : 0.0000,-25.0751,21.0405 K : 0.0000,-25.9283,18.1552 K : 0.0000,-26.6814,15.4045 K : 0.0000,-27.3285,12.7435 K : 0.0000,-27.8649,10.1420 K : 0.0000,-28.2864,7.5793 K : 0.0000,-28.5897,5.0411 K : 0.0000,-28.7727,2.5173 K : 0.0000,-28.8338,0.0000 K : 0.0000,-28.7727,-2.5173 K : 0.0000,-28.5897,-5.0411 K : 0.0000,-28.2864,-7.5793 K : 0.0000,-27.8649,-10.1420 K : 0.0000,-27.3285,-12.7435 K : 0.0000,-26.6814,-15.4045 K : 0.0000,-25.9283,-18.1552 K : 0.0000,-25.0751,-21.0405 K : 0.0000,-24.1282,-24.1282 K : 0.0000,-23.0949,-27.4072 K : 0.0000,-20.4884,-29.2604 K : 0.0000,-17.3283,-30.0135 K : 0.0000,-14.2973,-30.6606 K : 0.0000,-11.3548,-31.1970 K : 0.0000,-8.4721,-31.6185 K : 0.0000,-5.6287,-31.9218 K : 0.0000,-2.8088,-32.1048 K : 0.0000,0.0000,-32.1659 K : 0.0000,2.8088,-32.1048 K : 0.0000,5.6287,-31.9218 K : 0.0000,8.4721,-31.6185 K : 0.0000,11.3548,-31.1970 K : 0.0000,14.2973,-30.6606 K : 0.0000,17.3283,-30.0135 K : 0.0000,20.4884,-29.2604 K : 0.0000,23.0949,-27.4072 K : 0.0000,24.1282,-24.1282 K : 0.0000,25.0751,-21.0405 K : 0.0000,25.9283,-18.1552 K : 0.0000,26.6814,-15.4045

K : 0.0000,27.3285,-12.7435 K : 0.0000,27.8649,-10.1420 K : 0.0000,28.2864,-7.5793 K : 0.0000,28.5897,-5.0411 K : 0.0000,28.7727,-2.5173 K : 0.0000,28.8338,0.0000 K : DON K : OKAY K : DON K : $ /cr m3 sp K : KEY K : KEY K : 130.4694,26.1548,0.0000 K : 130.4694,26.1038,2.2838 K : 130.4694,25.9514,4.5759 K : 130.4694,25.6986,6.8859 K : 130.4694,25.3474,9.2257 K : 130.4694,24.9005,11.6113 K : 130.4694,24.3613,14.0650 K : 130.4694,23.7338,16.6186 K : 130.4694,23.0228,19.3185 K : 130.4694,22.2339,22.2339 K : 130.4694,21.3728,25.4711 K : 130.4694,18.9517,27.0659 K : 130.4694,15.9888,27.6934 K : 130.4694,13.1651,28.2326 K : 130.4694,10.4385,28.6795 K : 130.4694,7.7788,29.0307 K : 130.4694,5.1635,29.2835 K : 130.4694,2.5753,29.4359 K : 130.4694,0.0000,29.4869 K : 130.4694,-2.5753,29.4359 K : 130.4694,-5.1635,29.2835 K : 130.4694,-7.7788,29.0307 K : 130.4694,-10.4385,28.6795 K : 130.4694,-13.1651,28.2326 K : 130.4694,-15.9888,27.6934 K : 130.4694,-18.9517,27.0659 K : 130.4694,-21.3728,25.4711 K : 130.4694,-22.2339,22.2339 K : 130.4694,-23.0228,19.3185 K : 130.4694,-23.7338,16.6186 K : 130.4694,-24.3613,14.0650 K : 130.4694,-24.9005,11.6113 K : 130.4694,-25.3475,9.2257 K : 130.4694,-25.6986,6.8859

K : 130.4694,-25.9514,4.5759 K : 130.4694,-26.1038,2.2838 K : 130.4694,-26.1548,0.0000 K : 130.4694,-26.1038,-2.2838 K : 130.4694,-25.9514,-4.5759 K : 130.4694,-25.6986,-6.8859 K : 130.4694,-25.3475,-9.2257 K : 130.4694,-24.9005,-11.6113 K : 130.4694,-24.3613,-14.0650 K : 130.4694,-23.7338,-16.6186 K : 130.4694,-23.0228,-19.3185 K : 130.4694,-22.2339,-22.2339 K : 130.4694,-21.3728,-25.3549 K : 130.4694,-18.9517,-27.0659 K : 130.4694,-15.9888,-27.6934 K : 130.4694,-13.1651,-28.2326 K : 130.4694,-10.4385,-28.6795 K : 130.4694,-7.7788,-29.0307 K : 130.4694,-5.1635,-29.2835 K : 130.4694,-2.5753,-29.4359 K : 130.4694,0.0000,-29.4869 K : 130.4694,2.5753,-29.4359 K : 130.4694,5.1635,-29.2835 K : 130.4694,7.7788,-29.0307 K : 130.4694,10.4385,-28.6795 K : 130.4694,13.1651,-28.2326 K : 130.4694,15.9888,-27.6934 K : 130.4694,18.9517,-27.0659 K : 130.4694,21.3728,-25.3549 K : 130.4694,22.2339,-22.2339 K : 130.4694,23.0228,-19.3185 K : 130.4694,23.7338,-16.6186 K : 130.4694,24.3613,-14.0650 K : 130.4694,24.9005,-11.6113 K : 130.4694,25.3474,-9.2257 K : 130.4694,25.6986,-6.8859 K : 130.4694,25.9514,-4.5759 K : 130.4694,26.1038,-2.2838 K : 130.4694,26.1548,0.0000 K : DON K : OKAY K : DON K : $ /cr m3 sp K : KEY K : KEY K : 260.9387,23.4758,0.0000

K : 260.9387,23.4350,2.0503 K : 260.9387,23.3131,4.1107 K : 260.9387,23.1109,6.1925 K : 260.9387,22.8300,8.3094 K : 260.9387,22.4725,10.4791 K : 260.9387,22.0412,12.7255 K : 260.9387,21.5393,15.0820 K : 260.9387,20.9706,17.5964 K : 260.9387,20.3395,20.3395 K : 260.9387,19.6508,23.4189 K : 260.9387,17.4151,24.8714 K : 260.9387,14.6493,25.3733 K : 260.9387,12.0329,25.8046 K : 260.9387,9.5222,26.1621 K : 260.9387,7.0854,26.4430 K : 260.9387,4.6983,26.6452 K : 260.9387,2.3418,26.7671 K : 260.9387,0.0000,26.8079 K : 260.9387,-2.3418,26.7671 K : 260.9387,-4.6983,26.6452 K : 260.9387,-7.0854,26.4430 K : 260.9387,-9.5222,26.1621 K : 260.9387,-12.0329,25.8046 K : 260.9387,-14.6493,25.3733 K : 260.9387,-17.4151,24.8714 K : 260.9387,-19.6508,23.4189 K : 260.9387,-20.3395,20.3395 K : 260.9387,-20.9706,17.5964 K : 260.9387,-21.5393,15.0820 K : 260.9387,-22.0412,12.7255 K : 260.9387,-22.4725,10.4791 K : 260.9387,-22.8300,8.3094 K : 260.9387,-23.1109,6.1925 K : 260.9387,-23.3131,4.1107 K : 260.9387,-23.4350,2.0503 K : 260.9387,-23.4758,0.0000 K : 260.9387,-23.4350,-2.0503 K : 260.9387,-23.3131,-4.1107 K : 260.9387,-23.1109,-6.1925 K : 260.9387,-22.8300,-8.3094 K : 260.9387,-22.4725,-10.4791 K : 260.9387,-22.0412,-12.7255 K : 260.9387,-21.5393,-15.0820 K : 260.9387,-20.9706,-17.5964 K : 260.9387,-20.3395,-20.3395 K : 260.9387,-19.6508,-23.3027

K : 260.9387,-17.4151,-24.8714 K : 260.9387,-14.6493,-25.3733 K : 260.9387,-12.0329,-25.8046 K : 260.9387,-9.5222,-26.1621 K : 260.9387,-7.0854,-26.4430 K : 260.9387,-4.6983,-26.6452 K : 260.9387,-2.3418,-26.7671 K : 260.9387,0.0000,-26.8079 K : 260.9387,2.3418,-26.7671 K : 260.9387,4.6983,-26.6452 K : 260.9387,7.0854,-26.4430 K : 260.9387,9.5222,-26.1621 K : 260.9387,12.0329,-25.8046 K : 260.9387,14.6493,-25.3733 K : 260.9387,17.4151,-24.8714 K : 260.9387,19.6508,-23.3027 K : 260.9387,20.3395,-20.3395 K : 260.9387,20.9706,-17.5964 K : 260.9387,21.5393,-15.0820 K : 260.9387,22.0412,-12.7255 K : 260.9387,22.4725,-10.4791 K : 260.9387,22.8300,-8.3094 K : 260.9387,23.1109,-6.1925 K : 260.9387,23.3131,-4.1107 K : 260.9387,23.4350,-2.0503 K : 260.9387,23.4758,0.0000 K : DON K : OKAY K : DON K : $ REDI AP: 1 8 Change View AP: 1 0 0 0 0 AP: 0.0 0.0 0.0 AP: 1.000000 0.0 0.0 AP: 0.0 1.000000 0.0 AP: 0.0 0.0 1.000000 AP: 0.1050000 0.2620000 0.2620000 15.00000 AP: -1.000000 -1.000000 -1.000000 AP: 1.000000 1.000000 1.000000 K : $ mpos :; /F PR E E : **** END OF SESSION ****

References

1. E. Wright and R. V. Grandhi, Integrated Process and Shape Design in Metal Forming

with Finite Element Sensitivity Analysis, Design Optimization: International Journal

for Product and Process Improvement, Vol. 1, No. 1, 1999, pp. 57-78.

2. N. Zabaras, S. Ganapathysubramanian, and Q. Li, A Continum Sensitivity Method for

Design of Multi-Stage Metal Forming Process, International Journal of Mechanical

Sciences, Vol. 45, No. 2, 2003, pp. 325-358.

3. S. H. Chung, L. Fourment, J. L. Chenot, and S. M. Hwang, Adjoint State Method for

Shape Sensitivity Analysis in Non-Steady Forming Applications, International

Journal for Numerical Methods in Engineering, Vol. 57, No. 10, 2003, pp.

1431-1444.

4. T. T. Do, L. Fourment, and M. Laroussi, Sensitivity Analysis and Optimization

Algorithms for 3D Forging Process Design, Materials Processing and Design:

Modeling, Simulation and Applications, Numiform 2004, Copy 712, pp.

2026-2031.

5. Hyundo Shim, Optimal Preform Design for the Free Forging of 3D Shapes by

Sensitivity Method, Journal of Materials Processing Technology, Vol. 134,

September 2003, pp. 99-107.

6. G. N. Vanderplaats, Numerical Optimization Techniques for Engineering Design,

Vanderplaats Research and Development, Inc., 1999.

7. S. Candan, J. Garcelon, V. Balabanov, and G. Venter, Shape Optimization Using

ABAQUS and VisualDOC, 8th AIAA/USAF/NASA/ISSMO Symposium on

Multidisciplinary Analysis and Optimization, AIAA-2000-4769, Sept. 2000.

8. R. M. Hicks, and P. A. Henne, Wing Design by Numerical Optimization, AIAA

Aircraft Systems and Technology Conference, Seattle, Washington, 1977, pp.

22-24.

9. C. E. White, Shape Optimal Design of a Vented Fuselage Panel Subject To Internal

Blast, M. S. Thesis, Wright State University, 1995.

10. J. J. Park, N. Rebelo, and S. Kobayashi, A New Approach to Preform Design in Metal

Forming with Finite Element Method, Int. J. Mach. Tool. Design. Res., Vol. 23, 1983,

pp. 71 – 99.

11. S. S. Lanka, R. Srinivasan, and R. V. Grandhi, A Design Approach for Intermediate

Die Shapes in Plane Strain Forgings, ASME Journal of Material Shaping Tech., Vol.

9, 1991, pp. 193 – 205.

12. S. M. Hwang, and S. Kobayashi, Preform Design in Shell Nosing at Elevated

Temperatures, Intl. J. of Machine Tool Design, Vol. 27, No. 1, 1987, pp. 1 – 14.

13. C. S. Han, R. V. Grandhi, and R. Srinivasan, Optimum Design of Forging Die Shapes

Using Non-Linear Finite Element Analysis, AIAA J, Vol. 31, 1993, pp. 17 – 24.

14. B. K. Kang, N. Kim, and S. Kobayashi, Computer-Aided Preform Design in Forging

of an Airfoil Section Blade, Int. J. Mach. Tools Manufact., Vol. 30, 1990, pp. 43 – 52.

15. G. Zhao, E. Wright, and R. V. Grandhi, Computer Aided Preform Design in Forging

using the Inverse Die Contact Tracking Method, Int. J. Mach. Tools. Manufact., Vol,

36, No. 7, 1996, pp. 755 – 769.

16. G. Zhao, R. Huff, A. Hutter, and R. V. Grandhi, Sensitivity Analysis Based Preform

Die Shape Design using the Finite Element Method, Journal of Materials Engineering

and Performance, Vol. 6, No. 3, June 1997, pp. 303-310.

17. R. V. Gradhi, Computer-Aided Optimization for Improved Process Engineering

Productivity of Complex Forgings, Chapter 25, Multidisciplinary Process and

Optimization, July 2003, pp. 368 – 376.

18. S. H. Chung, L. Fourment, J. L. Chenot, and S. M. Hwang, Adjoint State Method for

Shape Sensitivity Analysis in Non-Steady Forming Applications, International

Journal for Numerical Methods in Engineering, Vol. 57, No. 10, 2003, pp.

1431 – 1444.

19. K, Chung and S. M. Hwang, Application of a Genetic Algorithm to the Optimal

Design of the Die Shape in Extrusion, Journal of Materials Processing Technology,

Vol. 72, 1998, pp. 69 – 77.

20. K, Chung, and S. M. Hwang, Application of a Genetic Algorithm to Process Optimal

Design in Non-Isothermal Metal Forming, Journal of Materials Processing

Technology, Vol. 80 – 81, 1998, pp. 136 – 143.

21. E. P. Box, and J. S. Hunter, Statistics for Experiments, John Wiley & Sons, Inc.,

1978.

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