Response Surface Methods for High Dimensional Structural Design Problems

xv
applicability of response surface techniques to problems that deal with more than a few
design variables.
The first approach is to alleviate computational costs by taking advantage of the
simple models that were in use a generation ago when computers were much less
powerful. These models, termed here as low-fidelity models, are less accurate. However,
they can be combined with more accurate but expensive high-fidelity models, through the
creation of correction response surfaces, to provide a good combination of high accuracy
and low computational cost. Correction response surface approximations are used to
solve two structural weight optimization problems efficiently: the design of the top cover
panel of a blended-wing-body airplane and the design of a wing composite blade-
stiffened panel subject to crack-propagation constraints.
The second approach uses response surface approximations that incorporate both
function and derivative data. In computer simulations, derivatives often are available,
frequently at a low computational cost. In theory the inclusion of derivatives data would
allow response surface approximations to be applied to a problem with many design
variables. However, derivatives are usually less accurate than function evaluations. The
present study attempts to identify the trade-offs of using derivatives in constructing
response surfaces through simple example.
1CHAPTER 1
INTRODUCTION
Designing an engineering system is often a challenging task. Deriving a suitable
mathematical formulation that accurately describes the behavior of the engineering
system can be even more challenging. When this is at all possible, finding a solution to
the mathematical formulation of the system might be even more difficult.
In many instances, however, it is possible to modify the original problem into a
slightly different approximate one and then to solve exactly the approximate problem.
The solution obtained may be viewed as an approximate solution to the original design
problem. In many other instances it may be possible to solve the mathematical
formulation of the original design problem using an approximation technique. The
solution is once again viewed as an approximate solution to the original design problem.
Of course, for a large variety of problems it is possible to solve the approximate problem
in an approximate way and still obtain a satisfactory solution to the original design
problem.
In this study, particular attention is paid to one approximation method called
response surface (RS) technique. Response surface methodology approximates the
response of a system as a function of some variables. The approximation is obtained by
fitting the system response for a number of selected combinations of the control variables
(design points). Response surface approximations have been shown to be accurate for a
small number of parameters. However as the number of parameters increases, traditional
2response surface techniques require an increasing number of evaluations to achieve
acceptable accuracy. This makes them computationally intractable.
The study focuses on methods that can be applied to response surface techniques
to design optimization problems with more than a few design variables. In particular the
two approaches that were studied are correction response surfaces (CRS) and derivative
based RS. The objective is to demonstrate the potential of these two techniques.
Chapter 2 provides a literature review on the state of the art in approximation
methods applied to optimization problems. It briefly introduces the most popular
approximation methods, dividing them into local, mid-range, and global approximations.
Particular attention is paid to global methods in the form of RS. The section on response
surface methodology, being the main focus of this work, is discussed in greater detail.
The mathematical and statistical bases of response surface methodology are summarized
briefly in Chapter 3. Chapter 4 applies of correction response surface to the optimization
of a hat-stiffened panel made of composite materials. Chapter 5 applies correction
response surface methodology to design optimization against crack propagation of a
bladed stiffened panel, also made of composite materials. Chapter 6 deals with response
surface based on both function and gradient data. The approach is applied to numerical
problem where control on the noise and modeling errors could be enforced. Concluding
remarks and proposed future research are provided in Chapter 7.
3CHAPTER 2
LITERATURE REVIEW
Approximation concepts in structural optimization were introduced in the
mid−seventies by Schmit and Miura, (1976) and by Schmit and Farshi, (1977).They
showed that applying non linear programming methods to large structural design
problems were cost effective, provided that suitable approximations concepts were
introduced. Several other publications on approximation concepts soon followed as the
approach attracted the interest of many other researchers.
Function Approximation Concepts
The use of approximation techniques to solve an optimization problem allows a
designer to minimize the computational effort needed to find a satisfactory solution and
to integrate different simulation codes if more then one code is needed during the design
process. The selection of appropriate approximating functions to replace the real
functions that describe an optimization problem is essential to ensure that the
optimization procedure is solved correctly and efficiently.
Barthelemy and Haftka (1993) distinguished approximation methods based on the
size of the domain of their validity, as local, global, and mid-range approximations.
In general, an optimization problem can be formulated as the minimization of an
objective function )(xF
Iixxx
KkxH
JjxG
xF
u
ii
l
i
k
j
,...,1
,...,10)(
,...,10)(
min)(
=≤≤
==
=≤
→
(2.1)
4where the vector
I
x ℜ∈ in Eq. (2.1) defines a point in the design space Ω whose
elements are the design variables. )(xG
j
and )(xH
k
are, respectively, the inequality and
equality constraints that, together with the lower and upper bounds of the design variables
l
i
x and
u
i
x , define the feasible design domain. These lower and upper bounds of the
design variables also are called side constraints.
Approximation techniques replace the problem formulated in Eq. (2.1) by a series
of approximate problems. The solution of each approximate problem is called a cycle. At
each cycle the original problem in Eq. (2.1) is replaced by
Iixxx
KkxH
JjxG
xF
pu
ii
pl
i
p
k
p
j
p
,...,1
,...,10)(
~
,...,10)(
~
min)(
~
,,
=≤≤
==
=≤
→
(2.2)
where p is the cycle number, and )(
~
xF
p
, )(
~
xG
p
j
,and )(
~
xH
p
k
are approximate
expressions for the objective function, inequality constraints, and equality constraints,
respectively, and the side constraints satisfy the condition
l
i
pl
i
xx ≥
,
and
u
i
pu
i
xx ≤
,
(2.3)
The lower and upper bounds of the design variables change from cycle to cycle in
order to limit the search for the solution to a region where the approximations are valid.
The number of cycles needed to reach an acceptable design may vary from one, in the
case of very accurate global approximations; to thousands, in the case of local
approximations that are valid only in small regions of the design domain.
5Problems in Engineering Optimization based on Computer Simulations
Numerical simulations enable engineers to perform detailed analyses of complex
systems. However, the use of such simulation tools in design optimization is associated
with several problems. A numerical simulation provides a designer with only a single
point in the design space rather than an overall view of the system behavior within the
design space. This is in contrast with the pre-computer era, when algebraic expressions
and design charts furnished a global view of the system behavior in the design space. The
algebraic expressions and design charts allowed the designer to spot promising design
regions and structural configurations. However analysis methods available in the pre-
computer era were not accurate enough for the design of complex structures. It would be
beneficial if it were possible to generate a global view of the design space by producing
algebraic expressions based on accurate numerical simulations.
A second problem with computer simulations is that human errors and numerical
noise usually are present when dealing with large, complex, and cumbersome simulation
software such as finite element codes. Numerical noise can be caused by mesh generator
procedures that update the element mesh as the design changes in order to maintain
required accuracy. Other sources of noise include less than fully converged iterative
procedures and round-off errors. Because the number of elements and grid points in a
finite element (FE) model is discrete by nature, finite jumps in the evaluated response can
be introduced when an infinitesimal change in the FE model leads to adaptive meshing
that changes the number of nodes and elements or the topology of the mesh. Human and
numerical errors of considerable magnitude are easily located and corrected, so normally
the remaining errors have little influence on the accuracy of the results. However, the
remaining numerical noise may cause non-smooth behavior of the response, trapping the
6optimizer in spurious local optima and making it difficult to find the best design (Giunta
et al., 1994).
When analyzing a complex structure using a finite element code, a source of noise
is introduced implicitly in the discretization procedure.VanKeulen et al. (1997)
performed an extensive study of the influence of numerical noise on the optimal solution
of several structural design problems for different discretization densities. They showed
that high levels of noise can be tolerated at the early stages of optimization, but as the
optimization progresses and more information about the behavior of the response is
gathered, the level of noise that can be tolerated is reduced.
Optimization is a procedure that searches through the design domain for the
optimum design. During this search, the optimization algorithm continuously calls the
analysis code and, based on the data obtained, it decides on the next design point to
evaluate. This requires an interface between the analysis and optimization programs.
Some general-purpose finite element programs, like NASTRAN, come equipped with an
optimization algorithm and an interface between optimization and analysis. However,
often the optimization code and the analysis code are separate, or one might want to use
an optimization algorithm different from the one furnished with the analysis code. In
these situations the user must write an interface between the two codes, possibly a
cumbersome and frustrating task.
Finally, some optimization problems such as global optimizations or stochastic
optimization require a large number of analyses and, except for simple problems; such a
large number of analyses are computationally impractical.
As we show, the use of approximations, especially global approximations, helps
alleviate these four problems. Approximations can provide the designer with a global
7view of the design space, filter out numerical noise, help in interfacing analysis and
optimization software, and permit us to perform global or stochastic optimization.
Local Approximations
Local approximations are probably the most commonly applied approximations in
structural optimization. Generally, they are based on the function value and sensitivities
at one point of the design domain. Much literature has been written over the years on
these types of approximations (e.g., Barthelemy and Haftka, 1993). As their name
suggests, local approximations are only accurate in the vicinity of the point at which they
are generated.
A distinction should be made between local function approximation and local
problem approximation. Local function approximations are variations on Taylor series
expansions, while local problem approximations try to reduce the size of the active
constraint set or the design variable set.
One of the first robust optimization algorithms was the simplex algorithm
(Dantzig, 1963) for the solution of linear optimization (known as linear programming)
problems. It was only natural that the first approximation methods developed used linear
programming. Nonlinear optimization problems were linearized about a point at each
optimizaton cycle and the simplex method was used to find the optimum in the small
neighborhood. The cycle was repeated until convergence leading to the so-called
sequential linear programming (SLP) method.
For a linear approximation, the functions )(
~
xF
p
, )(
~
xG
p
j
,and )(
~
xH
p
k
in Eq. (2.2)
are linear Taylor expansions. For example
∑
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−+=≈
I
i
i
ii
p
x
x
xF
xxxFxFxF
1
0
00
)(
)()()(
~
)(
8where
0
x is the design point about which linearization is performed.
It was realized soon that for some problems more accurate approximations could
be obtained for the same computational cost. The first to come was probably the
reciprocal approximation, where the linear approximation of )(xF is replaced by a linear
approximation in )(yF where
i
i
x
y
1
= (2.4)
This substitution usually helps because in many structural design problems the
design variables are truss cross-sectional areas, moment of inertia of beams and
thicknesses of plates. Displacements and stresses of statically determinate structures are
linear in the reciprocals of these design variables, and for statically indeterminate
structures approximate linearity often holds. One problem with the reciprocal
approximation is that it becomes unbounded when the design variables approach zero.
One way to avoid the problem was to make use of the modified reciprocal
approximations (Haftka and Shore, 1979). Another first-order approximation method,
called the conservative approximation, was introduced by (Starnes and Haftka, 1979).
This method is a hybrid form of the linear and reciprocal approximations methods that is
more conservative than either. The original or reciprocal approximations are used
according to the sign of the product.
o
i
io
x
x
xF
x
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂ )(
(2.5)
One advantage of the conservative approximation is that it renders the problem
convex and therefore is guaranteed to have only one global minimum. Several
researchers proposed different approximation methods that had the desirable property of
9convexity; the most successful one may be the method of moving of asymptotes
introduced by Svanberg (1987). However the reciprocal and the conservative methods as
well as other convex approximation destroy the linearity of the problem, thereby
rendering impossible the use of SLP.
Higher derivatives can be used in local approximations, but the increase in
accuracy often is not worth the computational cost of the additional derivatives.
Quadratic approximations primarily have been used in solving eigenvalue problems.
However, Murthy and Haftka (1988) showed that for almost the same computational cost
it is possible to obtain a third order approximation by using a linear approximation to the
eigenvectors in the Rayleight quotient. Often the high cost of obtaining second
derivatives dictates a compromise leading to the use of only the diagonal elements of the
second derivatives matrix (Fleury, 1989).
Using approximations, it is not necessary to approximate all response quantities
considered in the optimization process. Using the process of local problem approximation
we can ignore constraints that are not critical or near critical for the current cycle in the
optimization process. This approach, referred to as constraint deletion or constraint
screening, was first introduced by Haug and Aurora (1979). It should be noted that
constraint deletion is used only to reduce the computational effort but that it does not
affect the final optimum achieved. For a more complete overview of the process see
Vanderplaats (1999). Moreover when dealing with complex FE models, many elements
in a small region of the structure have approximately the same stress. It is possible to use
the most critical element to represent the response in that region. This approach, known
as regionalization, was introduced by Schmit and Miura (1976). The result of these two
simple techniques is that while the structure may have several thousand constraints, the
10
sensitivity of only a small fraction of them has to be evaluated, leading to considerable
savings in computational cost.
Local approximations have been successfully applied to numerous problems but
they have some drawbacks. First, they do not provide the designer with a global view of
the design space. Second, local approximations provide only limited protection against
numerical noise. Finally, local approximations are not well suited for parallel computing
without non-trivial software changes.
Mid-Range Approximations
Mid-range approximations are valid in a region of the design space that is
considerably larger than local approximations but smaller than the entire design space.
Unlike local approximations, data from more than one design point are used to construct
mid-range approximations. In local methods, a new approximation is constructed at a
new design point, discarding previous analyses. Some mid-range approximation methods
work in a similar way, but now at each cycle more than one point is used to obtain an
approximation valid in a region of the design domain. Mid-range methods use data from
previously generated design points to enhance the accuracy of the approximations. In the
literature two types of mid-range approximation methods that use the previously
generated data can be distinguished. The first type uses data from individual points along
the optimization path in the design space. Such approximations are called single-point
path (SSP) approximations. Approximations derived from data computed in clusters of
design points along the optimization path are called multiple point path (MPP)
approximations.
In SPP approximation, the design points used for the approximations are the
solutions of the previous optimization cycles. Vanderplaats (1979) developed Taylor
11
series expansions about the current design point in optimizing airfoils using function
values from all previous design points. Two and three-point approximations were
proposed by Haftka et al. (1987). The authors generated approximations based on the
projection of a design point on the line connecting two points, or onto the plane
connecting three points, approximating the functions by Hermite polynomials at the
projection point. While tests showed improvement over local linear approximations,
reciprocal methods still gave better results. In addition the results indicated that the
approximations were good for interpolations, while the improvements in accuracy were
marginal for extrapolations. Fadeletal.(1990) presented a two-point exponential
approximation in terms of the intervening variables
i
p
ii
xy = , i = 1, ,I. The exponent p
i
for each design variable was determined by matching the derivatives of the approximate
function with the previous data point gradients.
Grandhi and co-workers (e.g., Wang and Grandhi, 1994, Wang and Grandhi,
1995 and Xu and Grandhi, 1999) constructed a family of algorithms based on intervening
variables called TANA, where the intervening variables were defined as in Fadelelal.as
i
p
ii
xy = for i = 1, ,I. For example, the TANA3 algorithm of Xu and Grandhi (1999)
uses a second-order Taylor series expansion in terms of the intervening variables, with a
diagonal Hessian matrix. The exponent of each design variable and the diagonal second-
order terms are evaluated by matching the derivatives and the function value of the
approximate function with the previous function and gradient data. In other words, unlike
the approach followed by Fadeletal.,TANA3 matches the function value and the
gradient of a previous design point rather than only the gradient. Rodriguez et al. (1998)
implemented an algorithm that is based on an approximation of the Hessian matrix. The
authors approximated the Hessian matrix for their problem by matching the function and
12
gradient values of the response surface approximation with the values obtained from
numerical experiments in a region of the design space at additional design points around
the current one.
An improvement to local methods, known as the accumulated function
approximation, was developed by Rasmussen (1996). Starting from an arbitrary design
point, a local approximation is constructed and a possible optimum point is found in a
design space larger than the space of validity of the local approximation. Based on the
function value and possibly the gradients at the starting point and newly obtained point, a
new approximation is found for the design space. The process allows including
information from more points as they become available. A weighting procedure is used
so that points closer to the last point have more weight in defining the approximation.
Rasmussen (1996) showed that even a considerable level of noise in the sensitivities can
be tolerated by this method. No noise in the function values was considered.
The MPP methods generate one or more extra design points around each solution of an
optimization cycle, enabling the approximations to use these additional points. The
additional points can be chosen according to design of experiments discussed in Khuri
and Cornell (1996).Freeetal.(1987) used a central composite design (CCD) scheme to
select design points used to construct low-order polynomials in a region of the design
domain. They reported competitive efficiency with conventional optimization algorithms
when no noise is present. When noise is present, the method was more efficient than
conventional algorithms. A general MMP concept was formulated by Toropov et al.
(1999) in which any function )(
~
xF in Eq. (2.2) can be used as an approximation
function. These models can be fitted to a function )(xF by means of weighted least-
squares method as
13
()
∑∑
==
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
+−
I
i
n
j
i
ii
ijii
x
x
F
x
F
wxFxFw
11
2
,
2
~
)(
~
)((min (2.7)
where w
i
contains the weight factors expressing the relative contribution of each design
point. Note that in this approach the use of first-order derivatives is not mandatory. The
procedure to construct function approximation is comparable to global methods. The
difference is that the region where the approximation is used, bounded by move limits,
moves through design space as the optimization progresses. Van Keulen et al. (1995)
used the multi-point-path-approximation mid-range methods successfully to solve of a
shape optimization problem. A similar approximation concept was applied successfully
by Etman et al. (1996).
Global Approximations
Forsomeproblems,itcouldbeprofitabletoconstructaglobalapproximation
instead of a local or mid-range one. One of the most commonly used methods for global
approximations is the response surface (RS) technique, originally developed for the
planning and analysis of physical experiments (e.g., Box and Draper, 1987). Because of
similarities between physical experiments and computer simulations, response surface
techniques have been used to build explicit approximations to structural response from
numerical experiments in the form of finite element simulations.
Response Surface Techniques
When dealing with complex problems the true response or performance measures
as a function of the design variables usually is not known explicitly. In these situations
design optimization based on experiments may be the only tool a designer has.