Conference of Informatics and Management Sciences ICTIC 2013
March, 25. - 29. 2013
Shape Optimization of Two-dimensional Body Utilizing Genetic Algorithms Laisv
,
, Elena
Department of Engineering Mechanics Vilnius Gediminas Technical University Vilnius, Lithuania
[email protected],
[email protected],
[email protected] Abstract The paper proposes a technology for shape optimization of two-dimensional body utilizing genetic algorithm. Main attention is focused on geometry of 2D body, i. e. search for optimal coordinates of body points. Direct analysis of 2D body von Mises stress determination is performed using original program based on finite element method. The set of design parameters contains the coordinates of body points in 2D space. The results of numerical experiments proved the proposed technology to be efficient tool for solution of 2D body shape optimization problem. Keywords - shape optimization, genetic algorithms, finite element method
I.
Body shape optimization problem [1, 2, 3, 4, 5, 6, 7], where main attention is concentrated on body geometry (i. e. search for optimal coordinates of points), is analyzed in this paper. As a rule, the design of body is selected at the beginning and then the default constraints are verified utilizing the appropriate method of calculation. If the requirements are met, then the optimization process is terminated, otherwise modifications are applied and new shape of body is proposed. Obviously, the problem solution requires large computational effort, since the results depend on a number of design variables. The paper aims to obtain the optimal body shape from the given initial design space by allowing fixed length shifts for outer nodes. Optimization problem is solved by stochastic algorithms belonging to the class of genetic algorithms (hereinafter - GA) [8], direct analysis task by von Mises stress determination the finite element method (hereinafter FEM) [9, 10]. IDEALIZATION AND DATA
Constant strain triangular (CST) finite elements [9, 10] were chosen for discretization of the original body shape and the approximate method, when connections between fixed nodes and base are idealized as rigid springs, is utilized for
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Initial geometric shapes, decomposed by finite elements; coordinates and connections. Positions of fixed nodes. Positions of loaded nodes and active forces. Material data of initial body: body thickness t ,
E. Result of a solution the new body shape coordinates and maximal von Mises stress.
INTRODUCTION
Structure design optimization is very common problem involved in engineering practice. Optimal design is defined by different requirements: low material cost, minimal weight of structure, maximal efficiency, minimal manufacturing and operating costs. Stress, deformation, weight of structure and geometry these properties are usually constrained in order to obtain economically based solution. Optimization problem is usually formulated by selecting one or combining several constrains.
II.
assessing of boundary conditions. Initial data for body shape optimization are the following:
III.
MATHEMATICAL FORMULATION
The optimization problem is formulated as von Mises stress minimization problem: min max
i
, X
D,
where objective function is as follows: 2 x
i
x
y
2 y
3
2 xy
.
Possible changes in shape D are described by geometric scheme of body, X denotes the design variables, i the von Mises stresses in the finite elements CST. The problem is subjected to the following constraints: Strength condition: i
max
,i
1...N var ,
here i the i -th element von Mises stress, max the maximal allowable von Mises stress, which corresponds to the maximal of the initial shape of the body tension, N var number of the design variables. All loaded nodes remains after optimization. All fixed nodes do not change its coordinates after optimization.
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Conference of Informatics and Management Sciences ICTIC 2013
March, 25. - 29. 2013
Task of direct analysis is solved utilizing original program, based on FEM, with special mesh pre-processor. The applied finite element CST has three nodes at the top with two degrees of freedom in each of them. The main static equation is
K
a
u
a
F a,
where a the ensemble of elements, displacements, F the active forces, K
u the nodal stiffness matrix.
CST finite element matrix ensemble [10] can be written as:
K
Ki j
K 11 K 12 K 21 K 22 K 31 K 32
a
bi b j
E 4S 1
2
ci b j
1
ci c j 2 1 bi c j 2
In this paper, the selection procedure is realized utilizing tournament selection principle [11, 12]: the best individual is chosen from randomly selected individuals k and such procedure is repeated until sufficient quantity of individuals is obtained. In this method, the absolute fitness value does not decide anything; the key is a single individual higher fitness than other. During the crossover phase individuals exchange genes to create new individuals with good genes according to a certain rule. Here we use single-point crossover [13, 14]. Selected individuals within the population are grouped in pairs, for example:
K 13 K 23 , K 33
bi c j ci c j
X 1,1 , X 1 ,2 ,...,X n1 ,N pop X 2 ,1 , X 2 ,2 ,...,X n2 ,N pop 1 1
2 2
ci b j bi b j
position of crossover is chosen randomly, say equal to 1, and individuals exchange genes: X 1,1 , X 2 ,2 ,...,X n2 ,N pop
here E , S area of a finite element CST, bi , j ,k , ci , j ,k coordinates of nodes of triangular finite element. IV.
GENETIC ALGORITHM
Classical genetic algorithm [8] is utilized for solution of the optimization problem (1). GA has the following stages: Generation of the initial population.
Crossover. Mutation. At the beginning of GA an individual is obtained, i. e. randomly, depending on constraints of intervals, positions of body shape design variables N var are generated. One design variable is a point of body shape in two-dimensional space R 2 , and individuals are encoded utilizing real numbers. Thus, the design variables are the coordinates of points in x and y directions: x1 , y1 , x2 , y2 ,...x Nvar , y N var
For each generated individual X is calculated the corresponding objective function f X the maximal von Mises stress value. Since the initial population is composed of N pop individuals, a new population is composed in each iteration applying selection, crossover and mutation operations: Pt
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X 1 t , X 2 t ,...X N pop t .
In this method a fixed crossover probability [14] p _ cross determines the crossover procedure. If zero position is obtained crossover procedure is not performed and individuals enter mutation phase without any change. Mutation a way to maintain genetic diversity and prevent premature convergence [13, 14]. During this procedure, a gene of a individual changes its value with a certain probability [14] p _ mut , i. e. very small random number is added or subtracted, so the allowed interval of the design variable is not violated (see next section).
Selection.
X
X 2 ,1 , X 1,2 ,...,X n1 ,N pop
If the strength condition is violated after the crossover or mutation procedure, the objective function value of individual is artificially increased, thereby reducing access to the new population. Algorithm is terminated as soon as the predefined number of populations is achieved. The best individual of the last population is considered as solution of the problem. Since GA belongs to the class of stochastic algorithms [15], calculations must be performed several times, as well as the GA parameters have to be adapted to the problem: population size, values of crossover and mutation probabilities. V.
NUMERICAL EXAMPLE OF BODY SHAPE OPTIMIZATION PROBLEM
Initial body shape and finite element mesh (Fig. 1) are selected. The thickness of plate is constant ( t 1 ), so problem is solved within two-dimensional space R 2 . Body material 0.3 E 2 10 11 N / m 2 .
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Conference of Informatics and Management Sciences ICTIC 2013
March, 25. - 29. 2013
appropriate ellipse, i. e. the validity of the following inequality is verified: x xc a2
2
y yc b2
2
1,
here x, y a point after the mutation, xc , y c coordinates of the centre of an ellipse, a ir b a half-length of the major and minor axis accordingly.
Figure 1. The body to be optimized
All discrete structure (Fig. 1) is considered as individual in this problem, however during optimization changes are allowed only for points P1, P2, P3, P4, P5, P6, P7 with coordinates xi , yi , i 1,...,7 . Point P1 is allowed to change only along ydirection in the range of y 27,33 , point P7 accordingly along x-direction in the range x 27,33 . The changes for the remaining points are allowed within the ellipse (Fig. 2), chosen so, that the minor axes would not interfere each other, and the major axis to another, while dividing the original body with the points obtained.
Figure 2. Possible range of changes of shape of body to be optimized
Fig. 3 demonstrates the distribution of stress in the original body. The maximal von Mises stress of 101.81 MPa have been obtained. If during the run of algorithm objective function value of individual is greater than this obtained number, such individual gets a fine, i. e. objective function value is artificially increased. Coordinates of points P1, P2, P3, P4, P5, P6, P7 (under which random values are generated depending on allowable displacements of points) are submitted to the algorithm in order to compose the initial population. This way objective function values are calculated utilizing FEM program for the individuals (generated in a way mentioned above) within the population. Selection, crossover and mutation are performed. If the mutation is performed for points P2, P3, P4, P5, P6 , then it is additionally verified whether the new point belongs to
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Figure 3. Distribution of von Mises stress in the initial shape of body
When new population is obtained the array of shape nodes is composed again and new values of objective function are calculated. The cycle is executed for a predefined number of iterations. The following values of genetic parameters were chosen experimentally: population size 150 individuals, crossover probability 90%, mutation probability 20%, number of iterations 20. The best result and body changes in different iterations are shown in Fig 4. The biggest von Mises stress of 94.81 MPa is obtained in the new shape of body, i. e. 7.38% better result than in the original shape of body. The changes (which depend on number of iterations) of obtained stress are shown in Fig. 5. The graph (Fig. 5) shows, all population is approaching the solution during the course of iterations, and the difference between the maximal and minimum stress decreases.
Figure 4. Changes of body shape depending on number of iterations
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Conference of Informatics and Management Sciences ICTIC 2013
March, 25. - 29. 2013
[5]
[6]
[7]
[8] Figure 5. Stress of von Mises depending on number of iterations [9]
VI.
CONCLUSIONS
Body shape optimization problem is solved utilizing FEM and genetic algorithms. The best obtained shape of body is 7.38% better than in the original shape of body. In this way suggested technology can be used for solving engineering problems of discussed type.
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[11]
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