A novel hybrid CA-LP approach for the optimal sizing of planar truss structures

Tags: optimization, optimization problem, structural optimization, the problem, truss structure, CA-LP, structures, iterations, Structural and Multidisciplinary Optimization, optimality criteria, cross sectional area, displacements, objective function, Applied Mechanics, Numerical Methods, International Journal, structure, optimum design, Structural Design, Cellular Automata, internal forces, equilibrium condition, node, min, displacement, truss structures, neighborhood, Journal of Structural Engineering, cell state, approach, Engineering Optimization, Environmental Systems, optimization problems, Journal of Civil Engineering, ASCE, Transition rule, particle swarm optimization algorithm, Particle Swarm optimization, Linear programming, Ee, Evolutionary methods, genetic algorithms, Journal of Optimization Theory and Applications, Mathematical programming methods, Journal of Aerospace Engineering, Non-Linear Programming, Substituting Equation, optimal design, Iran Univ of Science and Tech, Journal of Applied Mechanics, Simplex method, cell states, Structural Mechanics, genetic algorithm
Content: A Novel Hybrid CA-LP Approach for the Optimal Sizing of Planar Truss Structures A.Faramarzi M.Sc. Student, Dept of Civil Engineering .Iran Univ of Science and Tech, Narmak, Tehran, Iran. Email:[email protected] M.H. Afshar associate professor, Dept of Civil Engineering .Iran Univ of Science and Tech, Narmak, Tehran, Iran.Email:[email protected] Abstract This paper presents a novel hybrid Cellular Automata-Linear programming (CA-LP) methodology for the size optimization of planar truss structures. The method uses the nodal displacements and internal forces of the truss as the design variables of the optimization problem. The objective function is the volume of the structure and the constraints of the original optimization problem are recast in terms of the new decision variables in the two-phase step. Starting with an arbitrary set of nodal displacements and internal forces, these are updated in turn in an iterative two-phase process. In the first phase, the internal forces are kept fixed and the nodal displacements are found using a CA approach. In the second phase, a Linear Programming 1
is used to find the set of internal forces of truss elements which minimizes the total volume of the structure. The effectiveness and efficiency of the methodology is also examined with its application to some benchmark truss structures. Keywords: Size optimization; Truss structures; Cellular automata; Newton-Raphson method; Linear programming; Simplex method 1. Introduction "Truss structure" is an extensive category of man-made structures, including bridges, cranes, roof support trusses, building exoskeletons, and temporary construction frameworks. These practical structures are ever-present in the industrialized world and can be tremendously complex and difficult to model due to their size. Naturally, optimal design of these structures by traditional trial and error design method based on engineering judgment can be a time consuming and tedious task. A computerized or an automated method of optimal design is, therefore, desirable. The use of numerical computations in the field of structural design was first introduced by Heyman (1956). Developing computer industries led to the foundation of many numerical based methods for the optimal design of structural systems. Various optimization techniques proposed by different researchers for solving structural optimization problems, in general, can be broadly categorized to: Evolutionary algorithms, Mathematical Programming methods, Optimality Criterion methods (OC), Fully Stressed Design (FSD), and Fully Utilized Design (FUD) methods. Evolutionary algorithms are based on some analogies of physical and biological phenomenon. These approaches are characterized by gradient-free methods which only utilize objective 2
function information. Genetic Algorithms (Adeli and Kumar 1995, Deb and Gulati 2001, Wu and Wang 2002), Ant Colony Optimization (Serra and Venini 2006, Rajasekaran and Chitra 2009), Simulated Annealing (Hasanзebi and Erbatur 2002), Particle Swarm Optimization (Fourie and Groenwold 2002, Li et al. 2007), and Harmony Search Method (Lee and Geem 2004) are some common examples of Evolutionary algorithms used for structural optimizations. evolutionary methods and in particular the population based methods, however, are known for their computational requirement due to the need for a large number of function evaluations. Furthermore, most of the evolutionary search methods involve free parameters which are to be tuned before the main application for the best performance of the methods, contributing to the higher computational requirement of these methods. Mathematical Programming approaches are commonly characterized by the searches using the information on the gradient of the functions used in the definition of the optimization problem and mainly the objective function. Non-Linear Programming (Joseph 1987, Sadek 1984), Dynamic Programming (Howell and Doyle 1978, Distefano and Rath 1975) and Quadratic programming (Belegundu and Aurora 1985) are some common examples of these methods used for structural optimization (Schimit and Miura 1976, Schimit and Farshi 1974, Xingsi and Templeman 1988). The gradient based methods, though generally efficient, are prone to getting stuck in local optima when the problem is non-convex which is the case for most real world design problems. The often used remedy for resolving the optimality issue encountered by gradient based method is to start from different initial designs which dramatically reduce the computational efficiency of the method without guaranteeing the global optimum solution. Optimality Criterion (OC) Methods refer to methods in which the optimality condition for a problem under study is derived by the theory of Karush-Kuhn-Tucker condition and the optimal 3
design that satisfy these conditions are sought using different forms of resizing rules. These methods are found to be very efficient for large-scale structural problems. Continuum-based Optimality Criteria method (COC) (Rozvani 1989, Prager and Shield 1972) and Discretized Optimality Criteria methods (DOC) (Berke and Khot 1974, Zhou and Haftka 1995) are two examples of different variants of developed OC methods (Gellatly and Berke 1977, Allwood and Chung 1984, Patnaik et al. 1995). Since most Optimality Criterion based methods use KuhnTucker optimality condition, it is required that a good set of Lagrange multipliers corresponding to the active constraints are found at each iteration leading to the inefficiency of these methods. To alleviate the problem, researchers proposed different methods such as linear approximation of constraints, iterative and reciprocal techniques resulting in less robust algorithms (Zhou and Haftka 1995). Fully Stressed Design (FSD) (Gallagher 1973, Al-Salloum 1995), Fully Utilized Design methods (FUD) (Makris 1995, Makris and Provatidis 2002) and Modified Fully Utilized Design (MFUD) (Patnaik et al. 1998) can be recognized as the traditional OC methods. FSD uses the optimality condition that all the members the structure should attain the maximum permissible stress. This method is applicable to the problems with only stress constraints. FUD is a FSD based approach developed to consider other types of constraints such as displacement constraints. The FUD method, however, was realized to over-design and, therefore, was modified to produce MFUD approach. MFUD approach is similar to that of FUD method with the difference that MFUD uses a weighting parameter for each member to obviate the over-design problem. The method, however, requires some additional computational effort to determine the weighting parameters (Patnaik et. al 1998). MFUD has been shown to be a non-effective method compared to the 4
existing approaches in particular for problems with large numbers of design variables (Patnaik et. al 1995). Structural design problems from a physical point of view generally fall into three distinct types: cross-sectional optimization also known as "size optimization", topology optimization and geometry optimization also known as "shape optimization". Optimal sizing of truss structures is concerned with finding cross-sectional area of each element of the structure that will be best for a given set of loads with a fixed topology and geometry. Topology optimization is defined as the task of looking for an optimal material layout. For truss structures, the problem of topological optimization can be simply described as to be or not to be an element of truss structure (bar), in such a way that all optimality and constraint conditions can be satisfied. The goal of geometry optimization is to find the best position of truss joints with fixed topology and size of the structure. The geometry optimization does not have a history as lengthy as size and topology optimization, because these problems are highly non-linear and very much dependent on the other type of optimization defined earlier (Topping, 1983). Figure 1 shows schematic representation of three structural optimization categories for a sample truss structure. A novel two-phase hybrid CA-LP approach is proposed in this paper for the size optimization of truss structures. Instead of cross sectional area of the truss members, the nodal displacements and internal forces of truss members are used here as decision variables of the problem. The size optimization problem is then decomposed into two sub-problems which are subsequently solved in a two phase manner. In the first phase, the internal forces of truss members are kept fixed and then nodal displacements of truss structure are obtained using a CA approach. In the second phase, the internal forces of truss members are calculated using known values of nodal displacements obtained from the first phase. A Linear Programming problem is defined in this 5
stage and solved using a Simplex method of solution. The process is iterated between two phases of the solution strategy until convergence is achieved. The optimal nodal displacements and internal forces are then used to uniquely calculate the optimal cross sectional area of the truss members. The proposed method is used to solve some truss sizing problems and the results are presented and compared with those of other available methods. The results indicate the efficiency and effectiveness of the proposed method to solve truss sizing optimization problems. 2. Cellular Automata The history of CA dates back to year, 1948-1950, and attributed to Von Neumann and Ulam (Von Neumann 1966). The evolution of graphic constructions generated by simple rules was the idea of introduction of CA, originally referred to as automata network or cellular space. The valuable work of these two scientists provided a powerful tool for simulating Complex Systems. Their research and other scientists' works were concentrated on the modeling complex systems, by modeling the behavior of some simpler subsystems. Application of CA to any problem requires that four main components of the CA namely cell structures, cell states, cell neighborhood and the local rule be properly defined. Cell structures In CA, the physical domain of the problem is discretized into simpler individual components, namely cells. The shape of physical domain of the problem under study will determine which kind of cell structures is more suitable than the others. Generally the cell structure can be defined in d-dimensional space. Based on the geometry of the problem under consideration one, two or three-dimensional cell structures are used for CA models. 6
Cell states To each cell or automata, one or more parameters are associated to represent the state of the cell , referred to as cell state, which can be changed over the time by some specified rule (Kita and Toyoda 2000). The cell state is normally chosen as the unknown variable/variables of the underlying problem and, therefore, is often dependent on the problem being solved. The cell parameters can be defined as discrete, continuous or binary variables depending on the characteristics of the problem. Original CA's used binary on/off values as the cell state by which many physical phenomena like, traffic flow, snowflake generation, diffusion and predator-prey ecosystems were simulated (Nagel, 2002). Cell neighborhood An individual cell always interacts with other cells that lie in certain proximity of it, named as neighbors. Neighborhood is one of the most important components of the CA approach. The neighborhood structure demonstrates the layout of physical domain of the problem under consideration. Transition rules In CA, the communication between a cell and its neighbors is limited to local interaction defined as Local or Transition rule. The transition rule can be defined in an ad-hoc manner based on engineering judgment or can be mathematically derived to reflect the physical interaction between the cell and its surroundings. Once defined, the transition rule is used to update the cell states at each CA iteration. The CA transition rule defines the updated state of the cell in terms of the old state of the cell and its neighborhood. The rule is alike a scalar or vector function depending on the number of cell sates, with the old state of cell and its neighbors as the input and 7
the new state of the cell as its output. For example the Von-Neumann neighborhood which has four neighbors in principal direction, would require five input parameters and returns the updated value of the site C as the output (Tatting and Grdal 2000). The CA transition rule should have the following properties: (Guo et al. 2007) Parallelism: The cells are updated in a synchronous or parallel manner i.e.; cells have to be updated simultaneously and independently of each other. Locality: The new state of a cell is only dependent to its neighbor cell states i.e.; the local interaction relationship has to be between the close neighbors. Homogeneity: The local rule is identical for the whole domain i.e.; the local rule has not to change from one cell to another one. The main steps to implement a CA model can be seen in the flowchart provided in Figure 2. Since the computations in CA are limited to neighborhoods and the local rules are identical for the cells, CAs has been proved to have an inherent massive parallel computation capability (Inou et al. 1998). This capability has led to CA based models to achieve considerable performance of at least several orders of magnitude higher than that of conventional models at comparable cost (Duff and Preston, 1984). Simplicity and the potential for modeling complex systems with simple structures is another important feature of CA making it popular for physical and biological simulator (Resnick, 1994) and language recognizer (Mahajan, 1992). Following the success of CA as a simulation tool, researchers attempted to extend its application to optimization problems. The first application of CA to optimization problems was described by Xie and Steven (1993) for structural design purposes. Tatting and Gьrdal (2000) used CA to design 2-D continuum structures by reducing a cell of continuum structure into a discrete truss structure. CA was also employed for durability analysis of concrete structures in aggressive 8
environment by modeling the diffusion equation (Biondini et al. 2004). Slotta et al. (2002) utilized CA technique for the design of truss structures. They used two set of rules, the first rule was used during simulation as an alternative to the Finite Element Method and the second rule, FSD approach, was employed at design stage of the process. Tetsuya and Eisuke (2000) applied CA for the design of plane truss structures with the objective of minimizing the total weight of the structure and the maximum stress. Gьrdal and Tatting (2000) used CA for fully stressed design of truss structures assuming both linear and non-linear responses. Abdalla and Gьrdal (2000) presented optimal criterion for topology design of two-dimensional continuum media with the minimization of total potential energy over the cell neighborhood to derive displacement updating rules. Kita and Toyoda (2000) developed a CA based topology design method using a Finite Element Method (FEM) based global displacement analysis system. Setoodeh et al. (2006) used CA for the optimal design of curvilinear fiber paths to improve in-plane response of composite laminate. Missoum et al. (2005) proposed an improved local updating rule of CA for structural design. 3. Truss sizing problem The objective of structural design optimization is to reduce the total weight of structure in such a way that all constraints of the problem are satisfied. The truss optimization problem is generally formulated as: Find Ae ; e=1,2,...,n to minimize n w e Aele (1) e1 Subject to: 9
min e
e


max e
e 1,2,..., n (2)
u min k
uk

u max k
k 1,2,..., m (3)
v min k
vk

v max k
k 1,2,..., m (4)
Amin e

Ae

Amax e
e 1,2,..., n (5)
Where e , Ae ,le are the density of material, cross-sectional area and the length of the eth element
of the structure, respectively. w represents the weight of the structure as the objective function
and n, m denote the total number of elements and nodes of which the structure consists,
respectively. e represents the member stress and (uk , vk ) is the nodal displacements of the structure in the x and y directions, respectively and superscript max,min stands for the maximum and minimum range of corresponding variables. In addition, the displacements and the stresses are to be interrelated through the equilibrium condition and the stress-strain relationship which are often implicitly considered via a matrix or finite element analysis of the structure. 4. Proposed CA-LP method for truss sizing problem As noted in the previous sections, cells, cell states, neighborhood and transition rules are four main components of any CA method. Solution of any optimization problem requires that these components are clearly defined for the problem under consideration. The efficiency and effectiveness of any CA model is very much dependent on the proper choice of these components and in particular the cells or cell states which are closely related. In optimization problem, decision variables may be considered a natural choice for the cell states. This, however, may not be true for the truss sizing optimization problem defined before.
Considering the decision variable of this problem, cross-sectional area of the bars Ai , as the cell state leads to the automatic definition of the truss members as the cell and the patch of members
10
connected to an arbitrary member as the neighborhood of the cell. With the cell and cell state so
defined, it is very difficult and in fact impossible to project the objective function and constraints
of the problem to the cell neighborhood with a local characteristic. This is in particular true for
the stress constraints since the stress of each member is not only a function of the cross sectional
area of the member and its neighboring elements but also all the members of the truss structure.
These results in a lack of locality for any transition rule defined which, in turn, would adversely
affect the performance of the CA approach that might be so defined.
To remedy this problem, the underlying optimization problem defined by the objective function
of Equation (1) and constraints of Equations (2),(3),(4) and (5) is recast in terms of two set of
independent variables, namely nodal displacements of the truss structure and the internal forces
of the truss members. For this consider the definition of member stress as:
e

Re Ae
Ee e
e 1,2,..., n (6)
Where e , Re , Ee , e are the stress, internal force, modulus of elasticity and strain of the member
of
the
structure,
respectively.
Using
the
definition
of
strain, e

le le
,
in
above
equation
leads
to
the following definition of cross sectional area of each member in terms of the member internal
force and the change in the member length as:
Ae

Rele Ee le
e=1,2,...,n (7)
Where le denotes the change in the length of the eth member of the structure.
Substituting Equation (7) in Equation (1) will result in the following form of the objective
function:
11
w n e Rele2 e1 Ee le
(8)
The change in the length of each member le can be written in terms of nodal displacement of the
member as:
le [u j cos(e ) v j sin(e )] [ui cos(e ) vi sin(e )] (9)
Where i,j are the end nodes of the member e with i as the first and j as the second node of the
member, and e is the angle of the member with the positive x direction.
Equation (8) along with Equation (9) represents the objective function of the underlying problem
in terms of the new set of decision variables, namely truss member internal forces Re ;e 1,..., n
and nodal displacement of the structure (uk , vk ) ; k 1,..., m .
Using a penalty approach for the satisfaction of problem constraints defined by Equation (2) to
Equation (4), the original optimization problem is now rewritten as a new optimization problem
with the following penalized objective function:
PO F

n { e Rele2 } e1 Ee le
n p {(csve,1 )2 e1

(csve,2 )2}
(10)
m
p {(csvk,1 )2 (csvk,2 )2 (csvk,3 )2 (csvk,4 )2}
k 1
Subject to the constraints of cross sectional area defined by Equation (5) and the constraints of
equilibrium condition. Here p is the penalty parameter which is assumed with a large enough
value to ensure the satisfaction of the constraints, and
csve,1
e max e
1
(11)
csve,2
e min e
1
(12)
12
csvk ,1
uk
u
max k
1
(13)
csvk , 2

uk u kmi n
1
(14)
csvk ,3
vk vkmax
1
(15)
csvk , 4

vk vkmin
1
(16)
are the normalized values of violation from problem constraints. It should be noted that only the
positive value of these parameters are used in Equation (10) since their negative value indicates
that the corresponding constraint is satisfied. It should be noted that similarity of the normalized
constraint violation for both minimum and maximum constraints is due to the fact that the
minimum values of the stress and displacements are both considered negative to account for the
negative displacements and stresses.
The solution of optimization problem defined by the objective function of Equation (10) and the
corresponding constraints of cross sectional area and the equilibrium condition is carried out here
in a two-stage iterative manner to find the optimal value of the nodal displacements and internal
forces of the structure.
4.1. CA Stage:
In the first stage and starting with an arbitrary set of values for the internal forces Re ; e 1,..., n ,
the value of nodal displacements (uk ,vk ); k 1,..., m are found such that the objective function
of Equation (10) is minimized as an unconstrained optimization problem. The problem so
defined is a nonlinear optimization problem which is solved here using a CA method. For this,
each nodes of the structure is considered as a CA cell with the corresponding displacements as
13
the cell state. The neighborhood of each cell is then simply defined as the patch of truss members connected to the node under consideration as shown in Figure 3. The updating rule of the CA is mathematically derived by simply requiring that the displacements of the node under consideration, (uk ,vk ) minimizes a local objective function defined as the projection of the penalized objective function of Equation (10) on the neighborhood of the node k. For this, the derivative of the local objective function is set to zero with respect to (uk ,vk ) .
(PO F) uk

nn {e e1
cos(e ) Ee le2
Rele2
}

2
p
nn e1
{
Ee
cos( le
e
)
(
csve,1 max e

csve,2 min e
)}

2
p(
csvk ,3 u max k

csvk , 4 u min k
)

0
(17)
(PO F) vk

nn {e e1
sin(e )Rele2 Ee le2
}

2
p
nn e1
{
Ee
sin( le

e
)
(
csve,1 max e

csve,2 min e
)}

2
p(
csvk ,5 v max k

csvk ,6 v min k
)

0 (18)
Equations (17) and (18) are clearly a pair of non-linear equations in terms of the nodal
displacements which can be solved using any of the nonlinear equation solution methods. A
Newton-Raphson method of solution is used here for its efficiency, leading to the following
equations:
nn {e
cos(
e
)
Re
l
2 e
}

2
p
nn
{Ee
cos( e ) ( csve,1

csve,2 )} 2 p( csvk,3

csvk,4 )
uk
nn 2[
e1 {e
Ee le2
cos
2
(
e
)
Re
l
2 e
}

p
e1 nn {Ee 2
le cos 2 ( e ) (
max e 1
min e 1
u
max k
)} p( 1
u
min k
1
(19) )]
e1
Ee le 3
e1
le 2
(
max e
)
2
(
min e
)
2
(u
max k
)
2
(u
min k
)
2
nn { e sin(e )Rele2 } 2 p nn {Ee sin(e ) ( csve,1 csve,2 )} 2 p( csvk,5 csvk,6 )
vk

e1
Ee
l
2 e
2[ nn { e sin 2 ( e )Rele2 }
e1
le
p nn { Ee 2 sin 2 ( e ) (
max e 1
min e 1
vkmax )} p( 1
vkmin 1
(20) )]
e1
Ee le 3
e1
le 2
(
max e
)
2
(
min e
)
2
(v
max k
)
2
(v
min k
)
2
Where uk ,vk are the increments in the current cell state and nn denotes the number of
neighbors of the cell under consideration. In other words, nn represents the number of members
connected to the node k. 14
The updating rule defined by Equations (19) and (20) are used to update the state of each cell in
turn. Once all nodes of the structure are covered, the old values of the nodal displacements of the
truss structure are replaced by the new updated values to conclude the first stage.
4.2. LP Stage:
In the second stage, the internal forces of each member are calculated such that the problem
objective function of Equation (10) is minimized satisfying equilibrium condition and constraints
of the cross sectional area assuming fixed values for the nodal displacements calculated in the
first stage. Since the nodal displacements calculated in the first stage are supposed to satisfy the
displacement and stress constraints, the penalty terms of the objective function of Equation (10)
are identically zero and can, therefore, be eliminated from the objective function. The resulting
objective function for the second stage can, therefore, be seen to be a linear function of the
internal forces and a linear programming (LP) problem can be defined for calculating the internal
forces, as follows:
Min (w)
n e1
e le2 Ee le
Re
(21)
Subject to the constraints:
Re x,k

Fx,k
ke
k 1,..., m (22)
Re y,k
Fy,k
ke
k 1,..., m (23)
R min e

Re

R max e
e 1,..., n (24)
Where Fx,k and Fy,k represent the x and y component of the external force applied at node k,
R
e x,k
,
R
e y,k
are
the components
of
the internal
force exerted
by member
e at
node
k,
respectively.
It
is noted that Equations (22) and (23) represent the equilibrium condition at structure nodes and, 15
therefore, the summations ranges over those members which are connected to node k. Equation (24) defines the box constraint on the cross sectional area defined by Equation (5) which is now rewritten in terms of the internal forces using the definition of Equation (7). A simplex method of solution is used here to solve the LP problem for the truss internal forces. The two-phase process defined by the CA and LP problems is iterated until convergence is achieved. The set of optimal internal forces and nodal displacement are then used to uniquely define the cross sectional area of the truss elements representing the optimal solution to the problem under consideration. The main steps for the numerical implementation of the proposed method are as follows: 1: Randomly define a set of internal forces satisfying box constraints of Eq. (24) and a set of nodal displacements satisfying the box constraints of Eqs. (3) and (4). 2 : Using the known internal forces, iteratively update the nodal displacement (cell state) for the network nodes as follows: 2.1: Use Eq. (9) to calculate le for all members connected to node k assuming known nodal displacements. 2.2: Calculate the stresses in all the members connected to node k using calculated le in 2.1. 2.3: Calculate all the constraint violations and then the change in the nodal displacements using Eqs. (19) and (20). 2.4: Update the nodal displacements using the old displacements and the displacements increment calculated above. 2.5: Repeat steps 2.1 to 2.4 for the current node until convergence is met, i.e; the displacement increments are too small. 16
2.6. Repeat steps 2.1 to 2.5 for all truss nodes leading to updated new values of the nodal displacements. 3. Replace the old displacements with the new displacements. 4: Calculate le for all truss members using new displacements and solve the LP problem represented by Eqs. (21) to (24) to find the new internal forces. 5: Replace the old internal forces with the new ones. 6 : Calculate the cross sectional area of each member via Eq. 7 using updated displacements and internal forces. 7 : Repeat steps 2 to 6 until convergence is achieved, i.e; the cross sectional areas obtained in two consecutive iteration remain nearly constant. The proposed CA-LP method is seen to be free of the limitations and shortcomings of the existing methods including both conventional and evolutionary search methods. The method does not involve any free parameters such as those present in most of the evolutionary methods and, therefore, does not need any parameter tuning except for the penalty parameter which can also be easily determined by physical/engineering jugdement. The proposed method takes advantage of the computational efficiency of LP and CA which are both known for their efficiency in contrast to the population based evolutionary methods known for higher computational requirement. While the CA method proposed here uses a gradient based search method at the local level, the interaction between the local searches reduces the possibility of getting stuck in the local optimums as observed by most of the conventional NLP methods. This possibility is further reduced by the interaction provided between CA and LP method, with the LP acting as a driving mechanism toward the global optimum. 17
A note should be made regarding the computational effort required by the proposed method compared to alternative methods. Each iteration of the proposed CA-LP method requires the solution of an LP problem with a size of n equal to the number of truss members, and the solution of 2Чm scalar weakly nonlinear equation with m denoting the number of nodes in the system. Assuming that the computational effort required by each iteration of the proposed CALP method is at most equal to that required for the simulation of the corresponding truss structure, normally termed as the function evaluation in the optimization literature, means that the number of CA-LP iterations can be compared to the number of function evaluations of other methods. This assumption is reasonable as each simulation of a truss system requires calculation of n stiffness matrix, their assembly and more importantly the solution of an algebraic system of equation with a size of 2Чm, and finally some post-processing to calculate the member stresses.
5. Numerical assessment
In this section, the efficiency and effectiveness of the proposed methodology is verified by
solving four benchmark truss structures and comparison of the results with those of the other
methods in the literatures. All the problems considered here are solved using a 1.83 GHz
Pentium IV PC equipped with 1 GB RAM memory. A value of 1010 is used for the penalty
parameter in all the examples which was obtained by some preliminary trial and error runs.
5.1. Example 1: 3-bar truss structure
Figure 4 shows the 3-bar truss structure as the first illustrative example. The members are made
of material with E 500 and 1. The structure is subjected to two loads of the magnitude 8
and 1 in the vertical and horizontal directions, respectively at node 1. The allowable stresses of
(
max 1
,
min 1
)

(1,1)
,
(
max 2
,
min 2
)

(0.5,0.5)
,
and
(
max 3
,

mi 3
n
)

(0.5,0.5)
are
considered
and
18
a lower side constraint of Amin 0.01 is used for all members. No units are used for the parameters of this example in previous works. This problem is solved using the proposed CA-LP method and the results are presented and compared with some of the existing results in the literature (Zhou and Haftka 1995). Zhou and Haftka (1995) investigated different methods of optimality criteria and closely related dual method to the truss optimization in their work. Table 1 shows the optimized weight and the number of iterations required to solve the problem using different methods. It is seen that the proposed CA-LP method was able to produce the final solution within 3 iterations. The optimum values of the truss cross sectional area are also shown in Table 2. The results show that the proposed method was capable of producing the best existing result with improved efficiency compared to the existing methods used to solve this problem. This is particularly evident from the number of iterations required to solve the problem by the proposed CA-LP method. 5.2. Example 2: 10-bar truss structure Second example is one of the known benchmark example with 10 members shown in Figure 5, which has been attended by many researchers. The minimum and maximum allowable cross sectional areas are considered to be 0.1 and 35 in2, respectively. The members are subjected to stress limitation of 25000 psi . All nodes are subjected to the displacement limitation of 2 in in x and y directions. The density of the material is 0.1lb/in3 with the modulus of elasticity E 104 ksi. The vertical loads are applied at node 2, 4 with the magnitude of 100 kips downward. This problem has been solved by Schimit and Miura (1976) using mathematical programming method, by Patnaik et al. (1998) employing Fully Utilized Design (FUD) and Modified Fully 19
Utilized Design (MFUD) methods, Wu and Wang (2002) and Ghasemi et al. (1997) using Parallel and Rebirthing Genetic Algorithm, respectively, by Rizzi, (1976) and Allwood and Chung (1984) employing Optimality Criteria methods and Li et al. (2007) using Particle Swarm optimization algorithm. The problem is solved here using the proposed hybrid CA-LP method and the results for the cross sectional areas and the total weight of the structure is given in Table 3 along with those obtained using other methods. The solution was obtained in 0.525 second of CPU time requiring 52 iterations (function evaluations) compared to 15,000 function evaluations required by the method of by Li et al.(2007). No information regarding the CPU time required by other methods of Table 3 was available for comparison purposes but the very small CPU time of 0.525 seconds can be considered as a measure of the efficiency of the proposed method. Figure 6 illustrates the convergence curves obtained for 10 runs using different initial designs. The figure shows that the performance of the method is not sensitive to the initial design used to start the optimization process. Ghasemi et al. (1997) considered a modified form of this problem referred to as case 2 to differentiate it from the first case defined above in which only stress constraints with the same magnitude were used for all members except member 9 for which the value of 75000 psi was considered. The solution to this problem using the proposed CA-LP method is compared with the solution of Ghasemi et al. (1997) and the analytically calculated optimal design in Table 4. The solution shown in this Table is the best solution out of ten solution obtained using different initial guesses. It is seen that the proposed method outperforms the Rebirthing Genetic Algorithm (RGA) of Ghasemi et al. (1997) by nearly producing the exact optimal solution. It should be noted that the CA-LP solution was obtained in 60 iterations requiring only 0.5 second of CPU 20
time. This can be compared with the 24600 function evaluations required by the GA of Ghasemi et al. (1997). 5.3. Example 3: 17-bar truss structure This example considers the case of a 17-bar truss structure shown in Figure 7. The density of material is 0.268 lb / in3 , the modulus of elasticity is E 30,000 ksi and the minimum crosssectional area of each member is 0.1 in2. The members are subjected to stress limitation of 50000 psi . All nodes in x,y directions are subjected to the displacement constraint of 2 in. The structure is subjected to an external load with the magnitude of 100 kips at node 9. This problem has also been solved by Adeli and kumar (1995) using distributed Genetic Algorithm and Li et al. (2007) employing the particle swarm optimization (PSO) algorithm. Table 5 provides a comparison of the best designs produced by the proposed method with those of alternative methods emphasizing on the superiority of the proposed method. It should be noted that the proposed method has been able to produce a better design within 71 iterations requiring 0.62 seconds of CPU time which can be compared to the computational effort required by 6000 function evaluations of Adeli and kumar (1995) and 15000 function evaluations of Li et al. (2007). Figure 8 shows the convergence history of optimization for the problem and Table 7 presents the statistical evaluation for this example. 5.4. Example 4: n-bay truss structures This example, shown in Figure 9, is considered here to test the efficiency of the proposed method for large-scale problems. Each bay includes five bars with the same material properties and constraints as used in example 2. The vertical loads of p1 50 kips and p2 100kips are the applied at the last upper and lower nodes of the structures. This benchmark example was also solved by Joseph (1987) using reduced gradient method and by Makris and Provatidis (2002) 21
utilizing an evolutionary method based on the strain-energy-density criterion. The comparison of the best results obtained by the proposed CA-LP method and those of other methods are summarized in Table 6 for different number of bays. The proposed method was able to get the optimum solution within 103, 381, 650, 776 iterations with the corresponding run time of 1, 2.11, 4.74, 7.70 seconds for the 5, 10, 15 and 20 bay problems, respectively. It is clearly seen that the proposed method has been able to produce solutions with the weight equal or less than the existing design in all cases considered. The convergence history of the method is shown in Figure 10 for the case of 10-bay problem. 6. statistical analysis The maximum, minimum, and average solution costs of the results obtained in ten runs using different randomly produced initial solutions along with the normalized standard deviation of the cost calculated as the ratio of the standard deviation to the average cost and the magnitude of the total constraint violation of the best solution are shown in Table 7 for all the examples considered in this paper. The Table clearly shows that the final solution reported for all the problems considered in this work are clearly feasible. Furthermore, the results show that the method is virtually insensitive to the initial designs used to start the solution process. In fact, the 10-bar truss structure example with displacements and stress constraints referred to as 10-bar truss structure (case 1) is the most sensitive problem to the initial guesses for which the normalized standard deviation of the solutions obtained in ten runs is the very small value of 3 percent. This may be attributed to the larger number of constraints or the topology of the problem. It should also be noted that no assumption about the loading condition is used in deriving the proposed CA-LP method, hence its performance would not be affected by different loading conditions. 22
Conclusion In this paper a hybrid CA-LP method was presented for size optimization of planar truss structures. While most of the methods used for the size optimization of truss structures consider the cross sectional area of the members as the decision variables, the nodal displacements and internal forces of the truss were used here as the design variables of the optimization problem. The objective function represented by the total volume of the material used in the structure and the constraints of the original optimization problem were recast in terms of the new decision variables and an iterative two-phase algorithm was defined for its solution. In the first phase, the nodal displacements were found using a CA approach while fixing the internal forces of the structure. Each node of the structure was considered as a cell with the nodal displacements as the corresponding cell state. The CA updating rule was derived by requiring that the total volume of the structure was minimized for a set of fixed internal forces. In the second phase, a Linear Programming (LP) problem was defined and solved using the Simplex method to find the set of internal forces of truss elements using the fixed nodal displacements found in the first phase. The proposed two-phase process was iterated until convergence was achieved or the maximum number of iteration was exhausted. The effectiveness and efficiency of the proposed methodology was examined with its application to some benchmark examples. Comparison of the results with those of other methods showed improved efficiency and effectiveness of the method for truss sizing optimization. References Abdalla, M. and Gьrdal Z., 2002. Structural design using optimality based cellular automata. In Proceedings of the 43rd AIAA/ASME/ASCE/AHS Structures, Structural Mechanics and Materials Conference, AIAA, Denver. 23
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