**2. Topology optimization**

During the twentieth century, architects and engineers have used innovative and novel methods to develop optimum forms of structures and sculptures. While the techniques employed by these innovators generated efficient and aesthetic forms, they shared a common limitation: reaching optimum structure. Although the purpose of applying topology optimization has never been a standard procedure, developments in finding optimum structure form let the researchers and designers be free to constructing better designs [13, 14].

Topology optimization offers conceptual design for lighter and stiffer structures. It helps to reach to efficient and aesthetic designs within a small time interval (**Figure 1**). The benefits are:

	- Determining feasible design range.
	- Accurate checking for different loads and conditions.
	- Considering design and manufacturing constraints [15].

By the time, TO has shown its power and efficiency in the design of structures by the increase in advances on computational speed and power. Changes in computer hardware and software technology have also changed the approach to topology formation of structures. Nowadays, you could use a drawing software in forming different topologies as if it is a standard task, and so, you are able to alter

**73**

*Topology Optimization Applications on Engineering Structures*

The problem statement includes the following:

constraint) or maximum stress values.

have a known analytical solution [3].

The conceptual process is shown in **Figure 3**.

**2.2 Structural topology optimization**

can be realized in **Figure 2** [18].

old designs and produce new alternative designs in virtual environment. Designers

A topology optimization problem can be written in the general form of an

*Gj*(*u*(ρ), ρ) ≤ 0 *with j* = 1,….,*m*

• An objective function *F*(*u*(,. Even though each problem could have different objective functions, generally the most used one is minimizing compliance, or in another word, maximizing the stiffness of the structure.

• Main design variable: material distribution. Here material density at each point of the members (*u)* could be this variable. 1 represents the places where density is described, and 0 is for the places where the material is deleted or there is none. On the other hand, *u* defines if the state is linear or nonlinear [11].

• The design space *u*(. This points out how much volume exists in design.

mined, then no need to change these places in the optimization stage.

There are many design factors such as manufacturing and handling that should be taken into account in determination of this value. Once this value is deter-

• m constraints is a characteristic that the solution must satisfy *G j*(*u*(, *≤* **0**. The examples are the maximum amount of material to be distributed (volume

• Evaluating *u*( often includes solving a differential equation. This is most commonly done using the finite element method since these equations do not

The topology of a structure is defined as a spatial arrangement of structural members and joints or internal boundaries. For both discrete and continuum structures, topology optimization helps to arrange association form of members as

Structural optimization is concerned with maximizing the utility of a fixed quantity of resources to fulfill a given objective. In structural optimization the best "structural" design is selected regarding three categories: size optimization, shape optimization, and topology optimization [19]. The application of topology optimization to structures to reveal the best position and size of the parts in a continuum is the most favorite one. Michell presented the first solutions as seen in **Figure 4**. Today much more advanced techniques are used, and by the help of finite element method, it could be applied to complex problems. Weight savings are managed by engineers in several structures as a consequence of utilization of these

∫

<sup>Ω</sup>*dV* <sup>−</sup>*V*0

∫

Ω *f*(*u*(ρ), ρ)*dV*

(1)

and engineers are pleased to have such a powerful tool in their work [16].

*minimize*<sup>ρ</sup> *F* = *F*(*u*(ρ), ρ) =

*subject to G*0(ρ) =

*DOI: http://dx.doi.org/10.5772/intechopen.90474*

**2.1 General form**

optimization problem as [3, 17]:

**Figure 1.** *Optimized unit by using topology optimization approach (Dassault) [15].*

old designs and produce new alternative designs in virtual environment. Designers and engineers are pleased to have such a powerful tool in their work [16].
