2. The optimal design problem

Several research papers and text books exist in the field of optimal design of composite structures with a variety of valuable applications in civil, mechanical, ocean, and aerospace engineering. An important stage has now been reached at which an investigation of such

developments and their practical possibilities should be made and presented. Two distinct review papers have been published covering the development of the optimum design of composites over more than 40 years. The first paper by Sonmez [7] presented a comprehensive survey for more than 1000 journal papers, conference papers, textbooks, and web links from the year 1969 to 2009. Sonmez classified the papers according to the type of the composite structure, loading conditions, optimization model, failure criteria, and the utilized search algorithm. The second paper by Ganguli [8] covered a historical review from 1973 to 2013. It provides the growth of the field by including more than 90 references dealing with a variety of optimization methods utilized for tailoring composites to achieve certain design objectives. Applications of several optimization techniques were presented, including feasible direction methods, sequential quadratic programming, and stochastic optimization such as particle swarm and ant colony algorithms. Ganguli classified the published work into five categories named pioneering research for the work published in the 1970s, early research in the 1980s, moving toward design in the 1990s, the new century in the 2010s, and the current research for papers published after 2010.

In general, design optimization seeks the best values of design variables, Xnx1, to achieve, within certain constraints, Gmx1(X) placed on the system behavior, allowable stresses, geometry, or other factors; its goal of optimality is defined by the a vector of objective functions, Fkx1(X), for specified environmental conditions. Mathematically, design optimization may be cast in the following standard form [9]:

Find the set of design variables Xnx1 that will

$$\text{minimize} \quad F(\underline{\mathbf{X}}) = \sum\_{i=1}^{k} w\_{\mathbb{H}} F\_i(\underline{\mathbf{X}}) \tag{1}$$

$$\text{1 subject to} \quad \mathsf{G}\_{\rangle}(\underline{X}) \leq 0, j = 1, 2, \dots \text{I} \tag{2}$$

$$G\_j(\underline{X}) = 0, j = I + 1, I + 2, \dots, m \tag{3}$$

longitudinal stiffeners. The sizes of the constituent elements of the system are measured by such properties as the cross-sectional dimensions, section areas, area moments of inertia, torsional constants, plate's thicknesses, etc. If the skin and/or stiffeners are made of layered composites, the orientation of the fibers and their proportion can become additional variables. If one optimizes for configuration, the design variables will include spatial coordinates. Also, in dynamic problems, the location of nonstructural masses and their magnitudes can be

Introductory Chapter: An Introduction to the Optimization of Composite Structures

http://dx.doi.org/10.5772/intechopen.81165

3

The class of optimization problems described by Eqs. (1)–(3) may be thought of as a search in an n-dimensional space for a point corresponding to the minimum value of the overall objective function and such that it lies within the region bounded by the subspaces representing the constraint functions. Iterative techniques are usually used for solving such optimization problems in which a series of directed design changes (moves) are made between successive points in the design space. Several optimization techniques are classified according to the way of

additional design variables.

Figure 1. Design optimization process.

3. Optimization techniques

where wfi is the weighting factors measuring the relative importance of Fi(x) with respect to the overall design goal:

$$\begin{aligned} 0 \le w\_{\hat{\mathbb{M}}} \le 1\\ \sum\_{i=1}^{k} w\_{\hat{\mathbb{M}}} = 1 \end{aligned} \tag{4}$$

Figure 1 shows the overall structure of an optimization approach to design. Major objectives in mechanical and structural engineering involve minimum fabrication cost, maximum product reliability, maximum stiffness/weight ratio, minimum aerodynamic drag, maximum natural frequencies, maximum critical shaft speeds, etc. Design variables describe configuration, dimensions and sizes of elements, and material properties as well. In the design of structural components, such as those of an automobile structure, the main design variables represent the thickness of the covering skin panels and the spacing, size, and shape of the transverse and Introductory Chapter: An Introduction to the Optimization of Composite Structures http://dx.doi.org/10.5772/intechopen.81165 3

Figure 1. Design optimization process.

developments and their practical possibilities should be made and presented. Two distinct review papers have been published covering the development of the optimum design of composites over more than 40 years. The first paper by Sonmez [7] presented a comprehensive survey for more than 1000 journal papers, conference papers, textbooks, and web links from the year 1969 to 2009. Sonmez classified the papers according to the type of the composite structure, loading conditions, optimization model, failure criteria, and the utilized search algorithm. The second paper by Ganguli [8] covered a historical review from 1973 to 2013. It provides the growth of the field by including more than 90 references dealing with a variety of optimization methods utilized for tailoring composites to achieve certain design objectives. Applications of several optimization techniques were presented, including feasible direction methods, sequential quadratic programming, and stochastic optimization such as particle swarm and ant colony algorithms. Ganguli classified the published work into five categories named pioneering research for the work published in the 1970s, early research in the 1980s, moving toward design in the 1990s, the new century in the 2010s, and the current research for

In general, design optimization seeks the best values of design variables, Xnx1, to achieve, within certain constraints, Gmx1(X) placed on the system behavior, allowable stresses, geometry, or other factors; its goal of optimality is defined by the a vector of objective functions, Fkx1(X), for specified environmental conditions. Mathematically, design optimization may be

k

i¼1

subject to Gjð Þ X ≤ 0, j ¼ 1, 2, …I (2)

Gjð Þ¼ X 0, j ¼ I þ 1, I þ 2, …m (3)

wfi <sup>¼</sup> <sup>1</sup> (4)

wfiFið Þ X (1)

minimize <sup>F</sup>ð Þ¼ <sup>X</sup> <sup>X</sup>

where wfi is the weighting factors measuring the relative importance of Fi(x) with respect to

0 ≤ wfi ≤ 1

Figure 1 shows the overall structure of an optimization approach to design. Major objectives in mechanical and structural engineering involve minimum fabrication cost, maximum product reliability, maximum stiffness/weight ratio, minimum aerodynamic drag, maximum natural frequencies, maximum critical shaft speeds, etc. Design variables describe configuration, dimensions and sizes of elements, and material properties as well. In the design of structural components, such as those of an automobile structure, the main design variables represent the thickness of the covering skin panels and the spacing, size, and shape of the transverse and

X k

i¼1

papers published after 2010.

2 Optimum Composite Structures

the overall design goal:

cast in the following standard form [9]:

Find the set of design variables Xnx1 that will

longitudinal stiffeners. The sizes of the constituent elements of the system are measured by such properties as the cross-sectional dimensions, section areas, area moments of inertia, torsional constants, plate's thicknesses, etc. If the skin and/or stiffeners are made of layered composites, the orientation of the fibers and their proportion can become additional variables. If one optimizes for configuration, the design variables will include spatial coordinates. Also, in dynamic problems, the location of nonstructural masses and their magnitudes can be additional design variables.

#### 3. Optimization techniques

The class of optimization problems described by Eqs. (1)–(3) may be thought of as a search in an n-dimensional space for a point corresponding to the minimum value of the overall objective function and such that it lies within the region bounded by the subspaces representing the constraint functions. Iterative techniques are usually used for solving such optimization problems in which a series of directed design changes (moves) are made between successive points in the design space. Several optimization techniques are classified according to the way of

selecting the search direction [9]. The most commonly used approaches are the random search, conjugate directions, and conjugate gradients methods. Other algorithms for solving global optimization problems may be classified into heuristic methods that find the global optimum only with high probability and methods that guarantee to find a global optimum with some accuracy. The simulated annealing technique and the genetic algorithms (GAs) belong to the former type, where analogies to physics and biology to approach the global optimum are utilized. The simulated annealing technique is an iterative search method based on the simulation of thermal annealing of critically heated solids. Hasancebi et al. [10] applied it to find the optimum design of fiber composite structures as an efficient method to solve multi-objective optimization models. On the other hand, the GAs [11, 12] are based on the principles of natural genetics and natural selection. GAs do not utilize any gradient information during the searching process. Narayana Naik et al. [12] used GA and various failure mechanisms based on different failure criteria to reach an optimal composite structure. Another robust algorithm in solving complex problems of optimal structural design is named particle swarm optimization algorithm (PSOA). This algorithm is based on the behavior of a colony of living things, such as a swarm of insects like ants, bees, and wasps, a folk of birds, or a school of fish. Omkar et al. [13] applied PSOA to achieve a specified strength with minimizing weight and total cost of a composite structure under different failure criteria. To the author's knowledge, GA has been the most efficient stochastic method for obtaining the global optimum design of composite structures.
