multi-level design process for 3-d preform shapececs.wright.edu/cepro/docs/thesis/multi_level... ·...
TRANSCRIPT
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 4.3 Performance characteristics of basis shapes and preform for rail section (Level 2) 35
Table 4.4 Performance characteristics of basis shapes and preform for rail section (Level 3) 38
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
Basis 1 Basis 2 Basis 3 Preform
Strain variance 0.1649 0.0985 0.1169 0.1301
Flash volume (cm2) 8.38 (5.62%)
6.22 (4.17%)
6.58 (4.42%)
5.14 (3.45%)
Underfill volume (cm2) 3.88 1.72 2.08 0.64
Load (KN) 287.99 297.97 261.46 324.44
Table 4.3: Performance Characteristics of Basis Shapes and Preform for Rail Section (Level 2)
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 %)
Basis 1 Basis 2 Basis 3 Preform
Strain variance 0.1301 0.1143 0.0817 0.1059
Flash volume (cm2) 5.14 (3.45%)
5.23 (3.51%)
6.07 (4.07%)
4.50 (3.0%)
Underfill volume (cm2) 0.64 0.73 1.57 0.00
Load (KN) 324.44 337.41 260.22 368.72
Table 4.4: Performance Characteristics of Basis Shapes and Preform for Rail Section (Level 3)
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.
Basis 3 [Y3] Basis 2 [Y2] Basis 1 [Y1]
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%)
Basis 1 Basis 2 Basis 3 Preform
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%)
Underfill volume (cm3) 147.52 124.86 484.72 0.00
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.
Basis 3 [Y3] Basis 2 [Y2] Basis 1 [Y1]
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.
Basis 1 Basis 2 Basis 3 Preform
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%)
Underfill volume (cm3) 406.12 1464.56 139.60 139.60
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.
Basis 3 [Y3] Basis 2 [Y2] Basis 1 [Y1]
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.
Basis 1 Basis 2 Basis 3 Preform
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%)
Underfill volume (cm3) 139.60 137.20 165.52 0.00
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).
Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3]
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%)
Basis 1 Basis 2 Basis 3 Preform
Strain variance 0.0353 0.0534 0.0939 0.0414
Flash volume (cm3) 44.22 (5.84%)
51.72 (6.84%)
84.27 (11.14%)
34.01 (4.5%)
Underfill volume (cm3) 10.21 17.71 50.26 0.00
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.
Basis 1 [Y1] Basis 2 [Y2] Basis 3 [Y3]
Figure 4.18: Level 1 Basis Shapes for Steering Link
Three dimensional forging simulations of the basis shapes are performed to find
the underfill and the strain variance for preliminary analysis. All of the basis shapes give
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.
Basis 1 Basis 2 Basis 3 Preform
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.