Application of Deterministic Operations Research
for Structural Optimization
ARCHVEvs
INSTITIJTE
MASSACHUSET
OF I ECHNOLOWY
by
JUL U2 2015
Yue Chen
LIBRARIES
B.Eng. in Civil and Structural Engineering
The Hong Kong University of Science and Technology, 2014
SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVRONMENTAL ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING IN CIVIL AND ENVIRONMENTAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2015
02015 Yue Chen. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly
paper and electronic copies of this thesis document in whole or in part in any medium
now known or hereafter created
Signature of Author:
Signature redacted
Department of Civil and Environmental Engineering
May 15, 2015
n
Certified by:
Signature redacted
Jerome J. Connor
Professor of Civil and Environmental Engineering
The)4s upervisor
Accepted by:
Signature redacted
V
I
f
HeidN
f
Donald and Martha Harleman Professor of Civil and Environmental Enginee ng
Chair, Departmental Committee for Graduate Students
Application of Deterministic Operations Research
for Structural Optimization
by
Yue Chen
Submitted to the Department of Civil and Environmental Engineering
on May 15 h, 2015 in Partial Fulfillment of the Requirements
for the Degree of Master of Engineering in Civil and Environmental Engineering
Abstract
This thesis discusses the application of operations research theories for structural
optimization problems, while the discussion is restricted to deterministic system. The
methodology employed follows the general methodology of operations research:
mathematical models are utilized to represent real-world problems and making decisions
based on the solutions generated through mathematical models.
The discussion is focused on three general categories of structural optimization problems:
sizing optimization, shape optimization and topology optimization. Simple structures are
included as examples to illustrate the application of operations research theories. Abstract
operations research models are formulated to represent the general case for each category of
structural optimization.
The thesis shows that operations research models provide mathematical insights for structural
optimization problems and the theories are of significant value for solving high-dimensional
structural optimization problems. Operations research model formulation and solving
techniques are also discussed for a more efficient computation of the optimal answers.
Thesis Supervisor: Jerome J. Connor
Title: Professor of Civil and Environmental Engineering
Acknowledgments
To Prof. Jerome Connor, for the careful guidance and help he offered me that made this thesis
possible, and the kindest advice as an amiable grandpa.
To Prof. John Ochsendorf, Dr. Pierre Ghisbain, Dr. Eric Adams, Prof. Caitlin Mueller and all
the other professors and scholars that I have met during my journey at MIT this year, whose
contribution in providing structural engineering knowledge and general wisdom helped me to
be a better engineer and person.
To my mom and dad, to whom I owe whatever I have achieved in my life.
To my professors in HKUST, who helped me to be well prepared for the challenges I
encountered in MIT.
To my classmates in the M.Eng. Class of 2015, who are the most inspiring and hardworking
group that I am forever proud to be part of.
Table of Contents
Key Sym bols and Abbreviations ..........................................................................................
9
List of Figures..........................................................................................................................10
Chapteri. Introduction.......................................................................................................
12
1.1.
B ackgrou nd ..........................................................................................................................
13
1.2 .
T hesis O bjective ...................................................................................................................
17
1.3 .
T h esis Ou tline ......................................................................................................................
18
Chapter2. Problem Formulation for Structural Optimization. ...................
19
2.1.
M ethodology for Deterministic Operations Research......................................................
20
2.2.
General M athematical Form for Structural Optimization ...............................................
22
2.3.
Classifications of Structural Optimization Problem ........................................................
23
2.4.
Problem Formulation Procedures of Structural Optimization.........................................
25
Chapter3. Structural Sizing Optimization Problem .........................................................
3.1.
W eight M inimization of a Truss System..........................................................................
3.1.1. Two-Bar Truss System Subject to Stress Constraints ................
26
27
27
3.1.2. Two-Bar Truss System Subject to Stress and Displacement Constraints ... 31
3.1.3. Two-Bar Truss System Subject to Stress and Instability Constraints ......... 35
3.2.
Abstract OR Model for Sizing Optimization Problems ...................................................
37
3.3.
Linear Programming and Simplex M ethod .....................................................................
39
Chapter4. Structural Shape Optimization Problem .........................................................
4.1.
4.2.
42
Shape Representation .......................................................................................................
43
4.1.1. Bezier Spline............................................................................................
45
4.1.2. B-Spline ...................................................................................................
47
Geometric Design Constraints..........................................................................................
52
4.2. 1. Nodal Sensitivity and Spline Continuity .................................................
52
4.2.2. Continuity between the Splines ................................................................
53
7
4.2.3. Continuity on the Point of Symmetry......................................................54
4.3.
Abstract OR Model for Optimal Shape Design Problems...............................................
56
Chapter5. Structural Topology Optimization Problem..................................................
57
5.1.
Problem Formulation for Minimum Compliance Truss Design...................58
5.2.
Abstract OR Model for Minimum Compliance Design Problems ...................................
60
Chapter6. Conclusions............................................................................................................63
6 .1.
C onclu sion s ..........................................................................................................................
64
6.2.
Suggestions for Future Studies........................................................................................
65
BIBLIOGRAPHY ...................................................................................................................
8
66
Key Symbols and Abbreviations
x
f(x)
g (x)
S
x*
so
-i
Ai
6
j,k
PC
W
Vi
r(u)
Bi,n(u)
ti
T
J(n, u)
K
ki, ke
U
F
A
Ii
Vmax
F
Ft
t(x)
p (x)
C(u)
E(p, u, u)
e(u)
U
Ee
Ead
Decision variables
Objective function
Constraint function
Feasible set
Optimal solution
Maximum allowable stress
Maximum deflection value
Axial stress in the ith element
Slenderness ratio of the ith element
Displacements in the kth direction of the
jth joint
Euler buckling load
Overall weight of the structure
Control vertex
Shape function
Bernstein polynomial
Knot for B-Splines
Knot sequence
Cost function for shape optimization
problem
System stiffness matrix
Element stiffness matrix
Displacement vector
Force vector
Cross-sectional areas of the truss system
Element length
Maximum material volume
Reference domain
Domain occupied by the structure
Boundary of the structure
Free boundary of the structure
Boundary where connected to the support
Force per unit length
Density distribution
Compliance of the structure
Strain energy of the structure
Specific strain energy
Admissible displacement set
Stiffness for each finite element
Admissible stiffness tensors set
9
OR
SO
s.t.
LP
R
Rn
BFS
Operations research
Structural optimization
Subject to
Linear programming
Real number
n-tuples of real numbers
Basic feasible solution
List of Figures
Figure 1.1. Overall design process for engineering projects ------------------------------------
14
Figure1.2. Comparison between conventional design approach and design method with
structural optimization --------------------------------------------------------------------------------
15
Figure2.1. General methodology for operations research ----------------------------------------
20
Figure2.2 -Three categories of structural optimization (Sigmund, 2004) ----------------------- 24
Figure3.1. Two-bar truss system subject to stress constraints ----------------------------------
27
Figure3.2. Graphical representation of the optimization problem for the two-bar truss system
subject to stress constraints ----------------------------------------------------------------------------
29
Figure 3.3: Two-bar truss system subject to stress and displacement constraints -------------- 31
Figure 3.4: Graphical representation of the optimization problem for the two-bar truss system
subject to stress and displacement constraints ---------------------------------------------------
33
Figure 3.5: Two-bar truss system subject to stress and instability constraints ------------------ 35
Figure 3.6: Graphical representation of simplex algorithm as a 3D polyhedron -------------- 39
Figure 3.7: Flowchart for the simplex algorithm --------------------------------------------------
41
Figure 4.1: Control vertices of a spline curve ---------------------------------------------------
43
Figure 4.2: Moving space for the control vertex Vi -------------------
- - - - - - - - - - - - - - - - - - - - - - - - - - - - 44
Figure 4.3: 3rd degree B-splines constructed with different knot sequences ------------------- 51
Figure 4.4: Two linked 3rd degree Bezier splines ----------------------------------------------
53
Figure 4.5: Two symmetrical 3rd degree Bezier splines ---------------------------------------
55
Figure 5.1: Two-dimensional elastic domain l ---------------------------------------------------
61
10
11
Chapter 1.
Introduction
Overview
Operations research is a discipline that deals with the application of advanced analytical
methods to help make better decisions. Structural optimization is about deciding on the
optimal design while complying with certain constraint conditions. This chapter first provides
background for both structural optimization and operations research, and then states the
objective and presents an outline of the work.
12
1.1. Background
During the design process of an engineering project, it is generally the case when all
possibilities of structural design solutions come up. However, to analyze and design all the
alternatives is time-consuming and unrealistic. Usually, one proposal is selected based on
some preliminary analyses and further designed in detail. It is intriguing, however, to have a
generic approach to determine the optimal scheme among all the possible solutions. Such
process, known as structural optimization, has been widely studied for decades, with rich
academic literature showing its potential to dramatically improve the design process as well
as outcomes [1].
More applications of structural optimization can be found in aircraft and automotive
industries, generating lighter and cheaper design proposals. The same motivation can be
spotted in the construction industry, as design efficiency is considered to be one of the major
goals for engineers, architects and developers.
The model described in Figure 1.1 is a simplified block diagram for the overall process of
engineering projects. In actual practice, each block may have to be broken down into several
sub-blocks to arrive at rational decisions. In any case, however, iteration will generally take
place, when optimization concepts and methods are helpful for decision making. Along with
appropriate aids from computer software, structural optimization can be useful in rapidly
selecting various design possibilities.
In brief, structural optimization is an approach using certain algorithms to seek an optimal
solution to a structural engineering design problem represented mathematically. The design of
a structural system can be formulated as a set of optimization problems, where one specific
targeted performance is optimized while all the other requirements are satisfied.
13
Definition of the problem :aeak
Invention of alternatve
--
Figurel.1
-
ova*m
- -- OModfictio
Overall design process for engineering projects
Operations research has a very broad range of application. Any problem, in which a certain
target is optimized while some constraints are satisfied, can be solved using operations
research theories. And deterministic operations research is when all the parameters in the
problem are certain. In this thesis, basic structural optimization problems are presented
formulated and solved using operations research models, and all the discussion is restricted to
deterministic level.
However, the real situations are more complex, where indeterminacy often exists.
Assumptions are important to simplify the problem, making it possible to be formulated
mathematically and applicable for the available methods. Besides, economic considerations
generally play the decisive role in reality. Most projects strive to achieve the most
cost-efficient system, while satisfying the client's needs. In the practice, the entire design is
generally broken down into several sub-problems and then treated independently using
structural optimization.
14
aimulate the problem
as an optimization
problem
CoSect dam
-- cEsfinate inWal desip
Colect data
Estimate inste
Analze system
Check constraints
Check prrmne
Is the design
satisfactory
Analyze system
i
Yes
Stp
Yes
Dt4<
]
I
]
1
Does design satisf
Co
egne criteria
Update design with
optimization method
Update design based
on experience
(b)
(a)
Figurel.2 -Comparison between (a) conventional design approach and
(b) design method with structural optimization
Figure 1.2 compares the conventional design method with the design approach utilizing
structural optimization. Both methods involve iterations and have similar steps during the
whole design process. The key differences lie in the starting step and the criteria for the final
choice.
15
The optimal design method starts with formulating the optimization problem, with an
objective defined and a series of constraints specified. During the iteration process,
conventional method checks the design's performance. Once the performance criteria are met,
the iteration stops and the final design scheme is filed. Termination of iteration for optimal
design, on the contrary, is based on certain convergence criteria. When the termination
criteria cannot be met, the conventional approach updates the design based on the designer's
experience, intuition, or some simple mathematical analyses. Optimum design approach, on
the other hand, updates using optimization concepts and procedures. Comparing the two
approaches, the optimum design method is more quantitative, with a clear objective
measuring the design's merits, and adopting trend information to make design changes.
16
1.2. Thesis Objective
Ever since modem operations research arose during World War II [2], the field of study has
been widely applied in various practical problems, including manufacturing industry, supply
chain and financial engineering. In the civil engineering field, it is mostly studied and utilized
in traffic analysis, transportation engineering and construction management. However, the
problem of structural optimization is rarely approached and analyzed comprehensively
through operations research. The objective of this thesis is to identify and solve structural
optimization problems using operations research theories. All the operations research models
employed in the thesis are deterministic. Simple structures, mainly truss system with discrete
structural parameters, will be listed as a typical and simplified case representing a more
general category of problems.
17
1.3. Thesis Outline
The thesis is divided into five chapters. Chapter 2 covers the general problem formulation for
structural optimization, including the methodology used in deterministic operations research,
the mathematical model for general structural optimization problems, the three categories of
SO problems and the problem formulation procedures. Chapters 3 to 5 then discuss the three
major classes of structural optimization: sizing, shape and topology optimization. Simple
structures are listed as examples to illustrate the application of operations research theories.
Then particular examples are presented and abstract operations research models are
formulated for the more general condition. Finally, conclusions and suggestions for future
work are presented in Chapter 6.
18
Chapter2.
Problem
Formulation
for
Structural
Optimization
Overview
This chapter provides more background information before going into specific problems.
First, the methodology of deterministic operations research is covered. Then the general
mathematical model for structural optimization problems and the three general categories of
SO problems are presented. The general problem formulation procedures are discussed at the
end of the chapter.
19
2.1. Methodology for Deterministic Operations Research
Operations mean the activities carried out in an organization related to the attainment of its
goals, and research is the scientific method to study the operations [3]. Operations research is
the study of how to form mathematical models of complex science, engineering, industrial,
and management problems and how to analyze those using mathematical techniques [4]. A
general model of operations research methodology is illustrated as Figure 2.1. The main
feature for operations research is to make decisions while achieve some objectives, for
example, to maximize the profits or minimize the costs. Operations research is widely applied
to manufacturing industry, logistics and supply chain. In the civil engineering field, more
in-depth studies are often found in traffic analysis and transportation system design and
project management.
modeling
Real-orldOR
problems
sohving
models
Methods
sokuton
interpretation
Figure2.1 - General methodology for operations research
Deterministic operations research is a division of OR, where all parameters are fixed. The
opposite is called stochastic operations research, where some of the problem parameters are
assumed random. Stochastic operations research requires robust modeling, leading to much
more complex models. This thesis deals exclusively with structural optimization using
deterministic operations research models.
20
As Figure2.1 suggested, one of the main step in operations research is to set up the OR model,
which is a mathematical structure where problem choices are represented in decision
variables [4]. The mathematical program looks to optimize some objective functions of
decision variables subject to certain constraints that limit the possible choices. Assuming a
maximization problem, a deterministic operations research model can be written in the most
general form:
max{f(x): x E S}
(2.1)
where,
x is the vector of decision variables,
f(x) is the objective function,
S is the feasible set, where the values for the decision variables satisfy all the constraints.
21
2.2. General Mathematical Form for Structural Optimization
Structural optimization is the subject of making an assemblage of materials sustaining loads
in the best way [5]. Structural optimization is consistent with operations research in essence:
certain structural performance is optimized while certain aspects are constrained. Typical
measurements for structural performance include weight, stiffness, critical load, stress,
displacement and geometry. Quantities usually constrained in structural optimization
problems are stresses, displacements and/or the geometry. A structural optimization problem
is formulated by picking one of these as an objective function that should be maximized or
minimized and using some of the other measurements as constraints.
Consistent with x and f(x) defined in 2.1, the general mathematical form of structural
optimization can be written as
min f (x)
s.t.g (x)
0
(2.2)
where,
x is the decision variables,
f(x) is the one scalar objective function, eliminating multi-criteria structural optimization
from discussion,
g(x) is the constraint functions, both equality and inequality, including design constraints,
behavioral constraints and equilibrium constraints, where they are assumed to be present in
the form of g (x) 5 0.
22
2.3. Classifications of Structural Optimization Problem
The standard model for structural optimization problems can represent many different cases.
Accordingly, the OR models used to represent them include unconstrained, constrained,
linear programming, and nonlinear programming optimization problems. Many times these
problems can be transformed into the standard model.
In the context of SO, x generally represents some sort of geometric features of the structure.
Based on the different decision variables x in a certain problem, SO can be divided into three
typical classes of problem: sizing optimization, shape optimization and topology optimization.
Sizing optimization is when decision variables x represent some structural thickness, for
example, the cross sectional area of the member. In the case of shape optimization, x
indicates the form of the structure, when connectivity and boundary conditions remain
unchanged. Topology optimization is the most general case of structural optimization, where
connectivity of the nodes is the decision variables, cross sectional areas can take the value
zero.
23
XXXXX[X
=>
(a). Sizing Optimization
(b). Shape Optimization
(c). Topology Optimization
Figure2.2 -Three categories of structural optimization (Sigmund, 2004)
Based on the decision variables x, a SO problem can also be categorized as continuous/
discrete-variable optimization problems. If x E R' (x belongs to the space R' of n-tuples
of real numbers), the structure is a discrete parameter system and typical examples are trusses.
For example, x is a finite number of cross sectional areas. On the other hand, if x is a field,
indicating infinite numbers of degrees-of-freedom, it is a continuum problem for a distributed
parameter system. An example would be the shape optimization for a continuous beam
member and x is the cross sectional areas of the beam sections taking any values not less than
zero.
24
2.4. Problem
Formulation
Procedures
of
Structural
Optimization
As indicated previously in Figure 1.2, the structural optimization approach is different from
the conventional approach for its problem formulation as a structural optimization problem
before the design process. The problem formulation involves translating the descriptive
statements into mathematical statements. To set up the deterministic OR model for a
structural optimization model, a general process with five steps can be applied.
Stepi. Problem Description
The problem formulation begins with the problem description, clarifying the overall
objectives and requirements. In cases where the problem description is vague, assumptions
about modeling of the problem need to be made in order to formulate and solve it.
.Step2. Information Collection
More information needs to be collected in order to develop the mathematical formulation of
the problem. This includes information on material properties, performance requirements,
resource limits, cost of raw materials, and so forth.
Step3. Defining Design Variables
The next step is to determine design variables x for the OR model. During identifying of
design variables, a better problem formulation is constructed with a minimum possible
number of, independent design variables.
Step4. Defining Objective Function
The objective function for the model is the function of design variables, being either
maximized or minimized. It measures the merit of a given design.
Step5. Constraints Formulation
The final step in obtaining the OR model is constraints formulation. A meaningful constraint
must be a function of the design variables
25
Chapter3.
Structural Sizing Optimization Problem
Overview
This chapter talks about structural sizing optimization problem. The discussion starts with the
weight minimization problem of the truss system, the discretized version of sizing
optimization problem. Three simple truss structures under different constraint conditions are
listed as examples to illustrate the application of operations research theories. An abstract
operations research model is formulized for the general case of the sizing problem. Following
it is linear programming and simplex algorithm, which is an efficient form of problem
formulation and the corresponding search method for the optimal solution.
26
3.1. Weight Minimization of a Truss System
The weight minimization problem is a typical sizing optimization problem. The discussion
about structural sizing optimization will start with the discrete parameter systems, where the
variables are finite dimensional. A truss structure is a naturally discrete parameter system [5].
Weight minimization of a truss system concerns finding the cross sectional areas of the bars.
Despite the simplicity, it turns out that this category of problems display several
representative features of structural optimization problems. The discussion of this question
from the operations research perspective is illustrated with three two-bar truss models under
different design constraints.
3.1.1.
Two-Bar Truss System Subject to Stress Constraints
A2,
A1,
L,
L,
E
E
F
Figure3.1 - Two-bar truss system subject to stress constraints
First consider the simple case of a two-bar truss system under a point load F and the goal is to
minimize the weight under the stress constraint. Following the problem formulation
,
procedures as described previously, the decision variables are the cross sectional areas A1 , A 2
and the objective function is the total weight of the system. As the structure is statically
determinate, the forces in the bar can be obtained directly from the equilibrium equations.
Otherwise, a more general approach would be establishing a state constraint with the form
K x u = F. Assume the two bars are of equal length and Young's modulus and the maximum
tensile stress in the bar is ao.
27
The deterministic operations research model for the problem can be identified as,
min f(AA2)= p x (A 1 x L + A 2
s.t.
x
L)
Fcosa
A1
Fsina
A2
As the factor pL in the objective function is constant and does not affect the optimum
answer, it can be left out and the constraints can also be further modified. The simplified OR
model then becomes,
minA 1 +A 2
F cos a
s.t. A 1 >
F sin a
A2
>
G
Go
This simple structural optimization problem is a typical case of Linear Programming where
the requirements are represented in linear relationships. However, it is actually not common
for SO problems to be formulized as LP. The simplicity of this problem results from the
constraints and its statically determinate nature.
The answer is intuitive, and can be interpreted as the stresses are at the maximum level and
the material is fully utilized giving the optimal design state.
F cos a
GO
F sin a
A2*
=
T
The problem can also be interpreted graphically as shown Figure 3.2. The feasible region is
represented in grey for the decision variables A 1 , A 2 . And the objective function is plotted as
28
the linef (A 1 , A 2) = A 1 +A 2 = constant, moving within the feasible region trying to reach the
minimum value. The solution is found when f (A 1 , A2) is given the smallest possible value
that maintains part of the line in the admissible region. This coincides with the general
conclusion of linear programming problem with a convex feasible region: the optimal
solution is located at the edge of the feasible region.
A2
st
A*=F SmG
A1
AIL
wa
Figure3.2 - Graphical representation of the optimization problem for the two-bar truss
system subject to stress constraints
The linear programming problem can also be implemented through MATLAB's Optimization
Toolbox functions linprog.
%%Weight minimization problem of a two-bar truss system
%%Under stress constraint
sigma = 50;
F = 10;
%ksi
%kips
alpha = 30;
%Find x that
%degree
minimizes
f(x)
= x1 + x2
%subject to
%xl >= F cos(alpha)/sigma0
29
%x2 >=
F sin (alpha)/sigmaO
1];
[1;
f =
lb xl = F*cosd(alpha)/sigma;
lbx2 = F*sind(alpha)/sigma;
lb
[lb_xl lbx2];
=
% Returns the value of the objective function fun at the optimal solution x
[x,fval]
= linprog(f,
l,[],[,[],lb)
Optimization terminated.
x
0.1732
=
0.1000
fval
=
0.2732
30
Two-Bar Truss System Subject to Stress and Displacement Constraints
3.1.2.
A,,
L/cos(a), E
a
A2,
L,
60
E
F
Figure3.3 - Two-bar truss system subject to stress and displacement constraints
In addition to the maximum stress constraint in the previous problem, consider another SO
problem with additional constraint with displacement. For the two bar truss system indicated
as in Figure 3.3, the displacement at the tip in the vertical direction needs to be within the
value 60. Assume the system has uniform material properties: the two bars have the same E
2
andp and a=30'. And 1, = TL,
2
= L.
Same as the previous problem, the first step involves identifying the internal forces in the two
bars. As the system is still structurally determinate, the internal force can be obtained
directly.
12F
-VF1
P =JP1
P2
Similarly, the constraint regarding the maximum stress can be formulated as:
A,
-,F
>V3
A2
90
GO
And the objective function is
2
f(A 1, A 2 ) = -A
31
1
+ A2
Then, to construct the deflection constraint, one needs to utilize the Hook's Law,
PL
Substitute the corresponding P and L for barl and bar2, the elongations of the two members
are
V A 1E
1 -2
V-3FL
4FL
A
I
2E
J
The displacement of the free node can be determined based on the truss's geometry,
FL
~=
E
lux
U = tuy
A2
8
3
VA,
Imposing constraint on uY
A21
80,
E60
3
8
V-A 1
A 2 ~ FL
The optimization problem can be summarized as the OR model as follows,
2
min f(A 1 , A 2 ) = -A
1
+ A2
2F
A,,
GO
V3F
s~t.
A2 > Ga 0
8
3
3.
A2
E60
FL
The design variables A 1 , A 2 don't appear in the constraint functions in a linear fashion. To
linearize the constraint functions, substitute the variables and rewrite the OR model,
32
2F
x1 =
, X2
NFF
A
=
4
1
+
3x,
-
minf(xix2 ) =
X2
4
S.t-
+V3x2
7=x1
x1,x2
1
<
> 0
the constraint
Represent the problem graphically as in Figure 3.4. It shows that
4
NxX 2
1 is active. By setting Tx1 +
VFx 2 =
1, the problem can be solved analytically.
x2
'x4
Figure3.4 - Graphical representation of the optimization problem for
the two-bar truss system subject to stress and displacement constraints
The optimal solution,
x1
x
-
-,
x2
=
-
=
7
Substitute back to the original design variables,
A*= A*
4
,A4F
__A*
33
=
7F
1
The constrained nonlinear multivariable function can be solved through MATLAB's
Optimization Toolbox functions fmincon.
function f = myfun(x)
f
= 4/3/x(l)
+
1/x(2)
%%Weight minimization problem of a two-bar
%%Under
stress
truss system
and displacement constraint
%Formulate the linear constraints as the matrix inequality A-x <=
sqrt(3);
A =
[4/sqrt(3)
b
[1;0;0];
=
xO =
[10;10];
=
[x,fval]
50;
F
%kips
10;
0; 0 -1];
% Starting guess at the solution
fmincon(@myfun,xO,A,b)
sigma =
=
-1
%ksi
Al
=
2*F/sigma/x(l)
A2
=
sqrt(3)*F/sigma/x(2)
34
b
3.1.3.
Two-Bar Truss System Subject to Stress and Instability Constraints
I
A 1 , L, E
F
A2 ,
L,
E
Figure3.5 - Two-bar truss system subject to stress and instability constraints
Consider a truss with two bars of the same length L and Young's modulus E, placed at right
angle as shown in Figure 3.5. The force F >0 is applied at an angle a = 300. The problem is
to find the circular cross-sectional areas A1 and A 2 such that the weight of the truss is
minimized under constraints both on stresses and Euler buckling.
The weight of the truss is
f(A 1 , A 2 )
=
A1 + A 2
The internal forces in the two bars as
F
P= I
=P1
I-21
With the stress constraints as
VNF
2aO
A2
>
35
F
2-
As the second bar is in compression under the load F, imposing a stability constraint against
Euler buckling with a safety factor of 3.
The buckling load for a column simply supported at both ends is
2
7
PC=
EI
L2
Assume the bar is of circular cross section, the moment of inertia,
A2
-
I=
47
The second bar has to be stable against Euler buckling for a safety factor of 3,
F
-%F2
PC
3
w2 E
w2 E
-3L
=IL2
-
3L 2
x
A2
wE 2
-- = 1
A
4w
12!2
Therefore, the instability constraint regarding design variable A 2 can be formulated as
A2
2
>12F
L2
The OR model for the sizing optimization problem under stress and instability constraints is
min f (A 1 , A 2 )
A1
s.t.
A2
A1 + A 2
V F
2a2co
F
2
=
12FL2
A2 >
According to the OR model, it can be inferred that the optimal answer will depend on the
Fand
relative values of -andF
FL2
r
whether the second or the third constraint will be active.
36
3.2. Abstract OR Model for Sizing Optimization Problems
Sizing optimization problems have a large number of different situations. Typically in
practical design optimization for a discrete parameter system the goal is to find the minimum
cost or weight by selecting the cross-sectional areas of structural members. Meanwhile, the
final design needs to satisfy strength and serviceability requirements determined by standard
design codes. For a given truss structure composed of N members, the deterministic
operations research model can be stated as follows,
(3.1)
N
min
W =piLiAi
s. t. gi, si, di,k
0
In the OR model, the design variable is A = [A 1 , A 2 , ... ,AN ]T represents the cross sectional
arears for N members of the structure. The objective of the problem is to find the optimal
vector A such that it minimizes the weight objective function W = EN, piLiAi, where W
is the overall weight of the structure, pi, Li is the unit weight and length of the i-th member.
The objective function is achieved while satisfying the design constraints consisting of the
overall structural response and behaviors of the individual members. gi, si, dj,k represent
the optimization constraints on stresses, slenderness ratios and displacements;
gi =
-1 <0;
1
si = (
dj,k =
i = 1,..., N
(ci)all
'
(6j,k )aii
0;
- 1 5 0;
37
i =
N
i = 1, ... , N
Where
a-, (a-)all is the computed and allowable axial stress for the i-th member;
Ai, (i)au
6j,k,
is the slenderness ratio and its upper limit for the i-th member;
6
( j,k),ll
is the displacement computed in the k-th direction of the j-th joint and its
allowable value and N is the number of joint.
38
3.3. Linear Programming and Simplex Method
The importance of operations research in business and industry is due to the enormous
numbers of real-world applications that can be modeled as mathematical programs, and in
particular, as linear programs [4]. Linear programming, also called linear optimization, is a
method to achieve the best outcome in a mathematical model whose requirements are
represented by linear relationships.
As shown in the previous examples, LP plays a major role in sizing optimization problem: the
objective and constraint functions can generally be formulized directly as linear functions of
design variables
A = [A 1 , A 2 , ..., AN]T
or indirectly be converted to LP through
linearization.
Invented by George Dantzig in 1947 [6], simplex method is a specialized version of the
general search algorithm that is designed to take advantage of the properties of linear
programs. As the previous graphical methods of sizing optimization problems suggested, the
optimal answer only happens at the corner of the feasible region. The basic idea of the
simplex method is to start at a corner of the feasible region followed by visiting neighboring
corners that improve the objective. The corner points are referred to as basic feasible
solutions (BFS) in OR problems.
Figure3.6 - Graphical representation of simplex algorithm as a 3D polyhedron
(Image from Wikipedia, Simplex Algorithm)
39
The MATLAB function 'linprog' implements the simplex algorithm to solve linear
programming. By determining the A and B matrix, it solves linear programming problem
minimize f(x) = cT x subject to Ax
B. The simplex algorithm operates on linear
programming in standard form
max cTX
ts.t. Ax = B,x
(3.2)
0
with
x = [x 1, x 2 , --- xn] as the variables of the problem,
c
=
[c 1 , c 2 , ---
rall
A =
...
ami
,
cn] as the coefficients of the objective function,
1i
...
-..
...
...
,
the m x n matrix as the coefficients of the constraint functions,
amn
B = [bl, b 2 , ... , bm], bi
0, non-negative constants.
40
The general searching algorithm of the simplex methods is
Identify an initial BFS
Construct simplex direction
No
Is simplex direction
Yes
Check ifunbounded
improving?
Yes
No
Stop
LP is unbounded
Stop
Current solution is
optimal
Figure3.7 - Flowchart for the simplex algorithm
The sizing optimization problems generally fall into the category of linear programming by
nature: the objective function as well as the constraint functions are mostly linear functions of
the design variables A = [A 1 , A 2 , ..., AN ]T. Therefore, formulating the problem as an LP and
solving with simplex method can greatly improve the computational efficiency, especially for
high-dimensional problems.
41
Chapter4.
Structural Shape Optimization Problem
Overview
Shape optimization is part of the field of optimal control. The typical problem is to find
the shape which is optimal minimizing a certain cost function
and satisfying a few
given constraints. In many cases, the function being solved depends on the solution of a given
partial differential equation defined on the variable domain. This chapter discusses the shape
optimization problem by presenting how to represent the shape mathematically and formulize
the constraint functions about continuity on the curve. And the end of this chapter will talk
about how to formulize the OR model for the general shape optimization problem.
42
4.1. Shape Representation
Dealing with shape optimization problem through OR method requires representing the shape
of the structure mathematically. A typical way to achieve this is to adopt polynomial
functions of the design variables to describe the boundary. A complex geometry can be
represented from a high-order polynomial function or with a low-order piecewise polynomial
function. It shows that better results are obtained with low-order splines for shape
optimization, while boundaries from higher-order polynomial tend to become highly
oscillatory [7].
Similar to drawing a spline in AutoCAD by determining a series of points, the idea of shape
representation is to define the spline using a number of control vertices, of which their
coordinates will change during the iteration process. The control vertices are shown in
Figure4. 1.
Vi = t~j,
i = 0, 1, 2, ...
V2
V0
Figure4.1 - Control vertices of a spline curve
Besides using control vertices to represent the spline curve for the shape function, design
variables ai are introduced to determine the coordinates of the control vertices. To limit the
control vertices to move within a certain space, propose two end points
Vin =
[x '"i
yff
V max
[[<ax
y!ax T
43
]T
Therefore, the control vertices will move within the span as indicated in Figure 4.2.
Sa-=1
V.I
ai=0
Figure4.2 - Moving space for the control vertex Vi
Define the constant vector
Li = Vimax
-
vmin
The location of vertex Vi can be represented with regard of design variable ai as
V, = Vmin + aiLi, 0 : ati 1
(4.1)
Bezier and B-splines are two types of piecewise low-order splines commonly used for shape
representation. Detailed introduction is followed in the next session. r(u) is adopted
representing the shape function in terms of design variables u.
44
4.1.1.
Bezier Spline
Following the idea in (4.1), define design variable u as 0 5 u ! 1, a straight line can be
described with two control vertices and this is the 1 " degree Bezier Spline,
r(u) = (1 - u)V + uV
(4.2)
For a more complex curve, a higher order Bezier Spline [8] is required for shape
representation. As implied from (4.2), a Bezier Spline with n degrees has n+1 control vertices.
The coefficients for the control vertices are called Bernstein polynomials and defined as (for
an n-degree Bezier Spline):
Bi,,,(u) =
n)
u(1-U)n-i =.
__
n__
.U'
t(n - t
.
(1 - U)n-i
(4.3)
1
(44)
An n-th degree Bezier Spline is
n
r(u) =
Bi,n(u)Vi,
i=O
0 5 u
The design variable u is a scalar variable ranges from 0 to 1 and one u value corresponds
to a specific point on the spline.
45
Examples for the first three degree B'zier Splines: (0
u
1)
For n = 1,
Bu,1 =
UO(1 - U)1-0 = 1 - u
B1!=u'(1
- u)1-1 = u
r(u)= (1 - u)VO + uV
For n = 2,
BO,2
2!
B1,2
2!
u(1- u)2-- = (1 - u) 2
B2 ,2 -
r(u) = (1
u'(1
2!
-
2
u 1(1 -u)
-
- 1 = 2u(1
- u)
U)2-2 = U 2
u) 2 V0 + 2u(1
u)V 1 + u 2V 2
-
For n = 3,
BO 3
u (1 - u) 3 -0 = (1 - u) 3
B 1,3 =
u (1 - u) 3 -1= 3u(1 - u) 2
B 2,3 =
u 2 (1 - U 3
2!3!
S21(3-2)1
B 3,3 =
r(u) =(
-- )3
3!
3!(3-3)!
u 3 (1
+ 3u(1I
-
)V
46
2
=
3u 2 (1
u) 3 - 3
=
-
u)
u3
+ 3&21-u
u33
4.1.2.
B-Spline
An alternative method of shape representation is B-Splines [9]. Different from Bezier Splines,
the degree and the number of control vertices of B-Splines are independent. And in Bezier
Splines, a change of one control vertex will result in an entirely-different curve. B-Splines, on
the other hand, have better local control: moving one control vertex will only change a part of
the spline.
An n-degree B-Spline with m+1 control vertices is defined as
m
r(u) =
Bi,(u)Vt,i 0 u<1
(4.5)
i=O
There are also a series of given scalar called knots: t1 , t2 , t 3 -...
. .. . The knot sequence T
containing the knots is defined as
--- ,1.
tm, 1, ...
...
,I
0 tn+1, ...
,.
T = {0,. ... ...
(4.6)
And tn+1 5 .. !5 tm
The number of the knots is (n + m + 1) and the fist (n + 1) knots are 0 and the last
(n + 1) knots are 1. A uniform knot vector means that the intermediate part of the knots
{ tn+1 , ... ... , tm, 1
}
is evenly spaced.
47
The coefficients for the control vertices are called B-spline basis functions and defined as (for
an n-degree B-Spline,n > 1):
U - t ( Bix
1 (u) + ti+n+1 - U
Bi,n(u) =ti+n - ti
ti+n+1 - ti+1
if ti :5 U < ti+1
10
(4.7)
otherwise
For ti+n - ti = 0,
ti+n+1 - U
B i ,n(u) =
ti+n+1 -
Bi,n-1(U)
ti+1
For ti+n+l - ti+1 = 0,
Bix,(u) =
ti+n - ti
Bi,,Ju)
The definition is recursive, involving Bi,_ 1 (u)and Bi+1,n-1(u). One special case to be
noted is when n = 0,
Bj, 0(u) = 1
if ki
u< k 1
otherwise
(4.8)
MATLAB has function spmax in the Curve Fitting Toolbox putting together spline in
B-form. Assume a
3 rd
degree B-Spline has 8 control vertices (n = 3, m = 7), and the number
of knots is n + m + 2 = 12,
Vo= {,
V1
=
{ },
V2 =
{4),
V3 = f 5, V4 = f 5, V5 = { 6, V7 =
48
t(3,
V8 = to8
Construct the B-Spline with spmax using four different knot sequences,
%%Construct
c =
[0 0;
a 3rd degree B-Spline using different
1 2; 2 1;
5 1; 5 4;
6 6;
7 3;
8
01.';
knots sequences
%Control Vertices
=
[0,
0,
0,
0,
1/5,
2/5,
3/5,
4/5,
1,
1,
1,
1];
%Single knot
t2 =
[0,
0,
0,
0,
1/4,
1/4,
2/4,
3/4,
1,
1,
1,
1];
%Double knot
t3
=
[0,
0,
0,
0,
1/3,
1/3,
1/3,
2/3,
1, 1, 1, 1]; %Triple knot
t4
=
[0,
0,
0,
0,
1/2,
1/2,
1/2,
1/2,
1,
ti
spl
=
spmak(tl,c);
sp2
=
spmak(t2,c);
sp3
=
spmak(t3,c);
sp4
=
spmak(t4,c);
figure
subplot (2,2,1)
fnplt (spl)
hold on
plot(c(1,:) ,c(2, :),':ok')
title
('B-Spline
with
single
knots')
subplot (2,2,2)
49
1,
1,
1];
%Quadruple knot
fnplt (sp2)
hold on
plot(c(1,:),c(2,:),':ok')
title('B-Spline
with double knots')
subplot (2, 2, 3)
fnplt (sp3)
hold on
plot(c(1,:),c(2,:),':ok')
title
('B-Spline with triple
knots')
subplot (2, 2, 4)
fnplt (sp4)
hold on
plot(c(1,:),c(2,:),':ok')
title
('B-Spline
with quadruple knots')
50
The figures constructed from MATLAB,
B-Spline with single knots_
6
4
4
Ck
2
01 W1
2
B-Spline with double knots
6
-f
.
4
6
2
L
8
0
2
4
6
8
B-Spline with quadruple knots
B-Spline with triple knots
7
6
4
4
2
2
A
~J
0
2
4
6
~;.v-
0
8
2
4
6
Figure4.3 - 3 rd degree B-splines constructed with different knots sequences
As indicated in Figure 4.3, the knot sequence has influenced the shape of the curve. Knot
sequence with multiplicity r is continuous at the points on B-Spline corresponding to control
vertices Vn-r. B-Spline with single and double knots are continuous for the whole spline, but
B-Spline with quadruple knots is discontinuous.
51
4.2. Geometric Design Constraints
After defining the splines mathematically, certain constraints on the control vertices have to
be imposed so that the boundary of the structure can achieve a desired degree of smoothness.
Continuity of the curve is a very important criterion and two constraints of continuity are
discussed as follows.
4.2.1.
Nodal Sensitivity and Spline Continuity
Solving SO problem generally involves differentiating the objective and constraint functions
with respect to the design variables. The derivatives are referred to as sensitivities. Nodal
sensitivity is the derivative at the spline's endpoints. It is important for shape optimization
problem as it is related to the continuity of the boundary curve.
Differentiating (4.4) leads to the nodal sensitivity of the Bezier Spline
dr(O)
= n(V1 - VO)
du
=
n(V,
-
V_ 1
)
dr(1)
du
(4.9)
(4.10)
This indicates that Bezier Spline's nodal sensitivity only depends on the first two control
1
.
vertices V1 , Vo and the last two control vertices Vn, V,_
Similarly, differentiating (4.5) leads to the nodal sensitivity of the B-Spline, which is also
only related to the first and the last two control vertices.
dr(O)
n
(V 1 -V 0
d =
)
du
tn4 i
n
dr(1)
du
(4.11)
- l
tm (V-Vn- 1 )
52
(4.12)
4.2.2.
Continuity between the Splines
As discussed above, generally for complex shape boundary representation, lower-order
piecewise polynomial functions are adopted rather than one single high-order polynomial
function. Therefore, the continuity between the two different splines at the joint control
vertices should be formulated as a constraint function for the problem.
Yr
VIV
Figure4.4 -Two linked
3 rd
degree Bezier splines
Figure4.4 shows two 3 rd degree Bezier Spline joining at control vertex V. The parameters for
the left spline are denoted with "I" and right spline with "r".
According to (4.9) and (4.10), the curve is continuous at the joint if
nl(V - VI) = nr(Vr - Vj)
(4.13)
Particularly for two splines with the same degree joining together, the joint control vertex is
located right between the two control vertices from the two splines.
V =
Vi + Vr
2
53
(4.14)
4.2.3.
Continuity on the Point of Symmetry
link
Symmetry is a common feature for the structure. As shown in Figure 4.5, the two splines
together on V, which is located on the axis of symmetrical. Besides, as the left and right
splines are symmetrical, they should be of the same degree.
Figure4.5 -Two symmetrical
3 rd
degree Bezier splines
Assume x1 = 0, therefore
To achieve both continuity and symmetry, the location of V and V should satisfy the two
requirements at the same time,
+ Vr
Vi
V= V;=
2
xi
=
-xr,
Y1 = Yr
This indicates that the y coordinates for V, , Vr and V
54
are the same
YI = Yr = yj
Vi =
Y.
V,
,
,
V;= {.
-xrIj
Similar to (4.1), define L, , LY for both x and y coordinates but with the same design
variable ai,
x, =x
+ a1 Li,x, 0
ai
1
(4.15)
y,= yi
+ aiLi,y, 0
a
1
(4.16)
Therefore,
Xi
= xlnin +
Liy
(y-i_),
0
0!
1
a
(4.17)
1
According to the relationship established previously, V = [Xr
yr]T can be expressed in
terms of aj for V.
= yjTin + a Ljy, 0
Yr = Y
x=
r
xrmin
r
= x;"?l
+ LX (Yr
Lr,y
+ ~ (y 7
55
(4.18)
a 5 1
(4.19)
-
yrif)
+ aj L1 , - yrVu'), 0 ! a1
1
4.3. Abstract OR Model for Optimal Shape Design Problems
The typical problem of shape optimization is to find the shape which is optimal and minimize
certain cost functions while satisfying given constraints. An abstract optimal shape design
problem can be modeled as a deterministic operations research model,
Find a* E U
fs.t. J(fl(a*), u(a*)) ;J(Q(a),u(a))
(4.20)
Va E U
In the OR model, a represents the design variables which has the admissible domain U.
And fl(a) represents the curve defined by the design variables a, which is the optimization
objective.
Following the definition U, fl(a) E S stands for are all the curves within a
certain boundary of variation. And since fl(a) is solely characterized by a, there exists a
one-to-one correspondence between S and U.
S= ffl(a) I a E U}
Uniquely determined for a given
,
(4.21)
u(a) is defined as the state variable, representing the
structure's response including displacements, stress, strain and force.
V = fu(a) I a E U}
(4.22)
And the cost function is determined as
J(f, u) E R,
fl E S, u E V
(4.23)
As described previously, fl represents the form or contour of some part of the structural
boundary. It is noted that the state problems for a solid shape are described by a set of partial
differential equations. The shape optimization consists in choosing the integration domain for
the differential equations in an optimal way. Therefore, different from sizing optimization
problems controlling the parameters, shape optimization is concerned with the control of the
equation domain.
56
Chapter5.
Structural Topology Optimization Problem
Overview
This chapter gives a brief introduction to the formulations and solving techniques for topology
optimization problems of elastic structures. As a starting point, the problem is focused on the
structures with discretized parameters. Then a general case for continuum is discussed and the
corresponding OR model is proposed.
57
5.1. Problem Formulation for Minimum Compliance Truss
Design
Different from sizing and shape optimization problems, topology optimization problems seek
to find the optimal layout of a structure within a specified region: structural features including
numbers of elements, shape of voids and connectivity of the domain [10]. Generally, the
problem is formulized under some given loading, support conditions, total volume of the
material and other possible design constraints.
One major type of topology problem is minimum compliance design optimizing the structure
for the maximum global stiffness. Compliance is defined as the property of a material
undergoing elastic deformation and is equal to the reciprocal of stiffness [10]. Instead of
formulating the objective function directly, using compliance FTu has the advantage of
being a convex function of the design variables.
Topology optimization problem can be treated as a special case of sizing optimization, when
the sizes can take zero values. In a truss structure with a fixed set of nodal points, the lay-out
can be determined by regarding all connections between the nodes as potential or
non-existent members. This approach to topology optimal design is known as the ground
structure approach [10].
Assume n possible connections between a set of chosen node points, and that all bars are
made of the same linear elastic material with Young's modulus E. The deterministic OR
model for minimum compliance truss design with a limit on the total amount of material is,
min F u
(5.1)
Ku= F
Vmax
liAi
S. t.
<A<=1
n
1ATn" < Ai !! Ain,
58
i=
i =,,..,n
2
where,
u is the displacement vector,
F is the given force vector for the external loadings,
K is the system stiffness matrix and K = Z' ki
ki is the element stiffness matrix in the global coordinate system,
n is the number of elements,
A = [A 1, ... , An] is the design variables, with Ai as the cross-sectional area of the i-th
element and is within a certain range [Af n,
Afax],
1i is the length for the i - th element,
and Knax is the maximum volume of the material for the truss system.
The above OR model is the discretized version for topology optimization problem. If the
lower bound of the design variables Af n is non-zero, the system stiffness matrix K
determined from A = [A 1, ... , An] is positive definite and the existence of the solution is
assured [10]. Allowing zero lower bounds may complicate the analysis. Besides, it is noted
that every load case F will give rise to a distinct structure topology lay-out. Then topology
optimal design under multiple load cases should be reformulated as the problem of worst case
minimum compliance design.
59
5.2. Abstract OR Model for Minimum Compliance Design
Problems
The OR model for the minimum compliance truss design is the discretized version of the
topology optimization problem. For the general problem formulation with continuous
structural parameters, consider a structure existing on a reference domain fl occupying the
domain fo. Define the boundary of the structure as &I = F, and Ft to be the part of the free
boundary and F, to be where the structure is connected to the support with no deflection.
The structure experiences external loadings and the force per unit length is t(x) for x E Ft.
The material layout is represented by the density distribution p(x) for x E fl, which is also
the problem's design variable.
Figure5.1 -Two-dimensional elastic domain fl
Similar to FTu representing compliance for the discretized parameter case, the compliance
of the continual system is given by,
C(u) = fr
tTu
The total material volume can be obtained by integrating over the whole domain fl.
6o
(5.2)
f
(5.3)
p(x) dfl = Vmax
The strain energy of a continuum is defined as
E(p, u, u)=2f, p e(u)dfl
(5.4)
where
e (u) is the specific strain energy: if fl is two-dimensional, e (u) is the strain energy per area;
if f is three-dimensional, e(u) is the strain energy per volume,
E(p, u, u) represents the total strain energy in the entire structure.
And the set of admissible displacements is defined as
U={u:*I>R2|u=0onu}
(5.5)
Utilizing the virtual displacement method, the OR model of minimum compliance design for
continuous parameters can be formulized as,
min C(u)
(5.6)
u e U s.t. E(p,u,v) = C(v) for all v E U
S. t.
f
p(x) dl <V ax
for all x E E
p(x) > 0,
A typical approach to solve the problem is to discretize the structure using finite elements
[11]. Assume the material property E is constant in each element. The discrete form of OR
model for topology optimization problem can be written as
{
min F u
Ee E Ead
is. t. K( E,)u = F,
61
(5.7)
where,
u is the displacement vector,
F is the force vector,
Ee is the stiffness of the element (e = 1,2, ... , N),
Ead
is the admissible stiffness tensors set,
K is the system stiffness matrix and K
=
EN
ke
ke is the global-level element stiffness matrix, which depends on E,,
N is the number of finite elements.
62
Chapter6.
Conclusions
Overview
This chapter summarizes the discussion from the previous chapters with the conclusions for
the thesis. Besides, suggestions for future studies are provided for potential continuing
exploration and extension of this study.
63
6.1. Conclusions
Following the discussion in the previous chapters, a few conclusions can be drawn from this
study of application of deterministic operations research for structural optimization:
1. Operations research provides mathematical insights for structural optimization problems.
The theories are of significant value for solving high-dimensional structural optimization
problems.
2. Appropriate mathematical model formulization is crucial for solving the problem
efficiently.
3. It is very important to utilize the properties of convex set and convex function. If the
feasible region can be formulized as a convex set, or the objective function as a convex
function, the search for the optimal solution will be much more efficient.
4. Linear programming can be solved very efficiently for its embedded properties of having
a convex feasible region. Simplex algorithm can be utilized accordingly to find the
optimal answer.
5. Linearization is a common technique to simplify the operations research models. The goal
for representing the structural optimization problems mathematically is to try to formulize
the problem directly or indirectly as linear programming models.
6. To solve the structural optimization problems with continuous properties, one typical
approach is to discretize the structures using finite elements.
64
6.2. Suggestions for Future Studies
The entire discussion of structural optimization through operations research here is limited to
deterministic level. While in real situations, uncertainty always takes place and the
mathematical formulations need to be altered accordingly. The operations research including
indeterminacy is referred to as stochastic operations research and solving requires robust
modeling. Robustness means the ability of a system to resist change without adapting its
initial stable configuration [12].
Julia is a high-level, high-performance dynamic programming language for technical
computing [13]. Development of Julia began in 2009 and an open-source version was
publicized in February 2012 [14]. Its syntax is similar to other technical computing
environments, like MATLAB or Python. For potential future studies about structural
optimization through robust modeling, the mathematical models can be fed into Julia,
utilizing its built-in robust modeling solving algorithms.
65
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