**3. Classification of methodologies**

Topology optimization methods are mathematical techniques/approaches, and they can be programmed using different algorithms. These algorithms could be classified as follows: the criterion algorithm, the mathematical programming algorithm, and the intelligent algorithm.

The criterion algorithm obtains the optimality condition by the perceptual knowledge or the rational derivation. Result geometry will be gained by checking constraint violations and objective function value in an iterative way.

The perceptual criterion is usually the extension of the optimality condition of the full stress criterion of the size optimization. The rational criterion is derived usually by the Lagrange multiplier method of equality constraint. The ESO method is the typical criterion method.

Common mathematical programming algorithms like linear programming (LP) and nonlinear programming methods are also used in topology optimization of structures. The first attempts begin with using LP and successive LP methods later continued with sequential quadratic programming methods. Similar too criterion algorithm, mathematical programming algorithms are solved iteratively. Both stability and sensitivity of the structure are checked in each iteration. Of course it means that more calculation should be done for large-scale systems, and consequently low performance is observed for these cases. Fleury discussed the relationship between the criteria method and the mathematical programming method of size optimization. Fleury found that they both have given approximate results. This study refers still to the basics of the topology optimization [25, 26].

Genetic algorithm, simulated annealing algorithms, and particle swarm are the frequently used algorithms for topology optimization as the intelligent algorithm. The advantage of these algorithms is to keep it from too much calculations. The main idea is to search the optimum topology by checking only the objective function and constraints without calculating any gradients. On the contrary, solution speed can be slow, especially for large-scale system; finding optimum could take longer times [27, 28]. Several algorithms are also developed to combine topology optimization with additive manufacturing [29].

Two classes of approaches, the so-called material or micro-approaches and the geometrical or macro-approaches, are available [30, 31]. For the areas such as MEMS or biomaterial applications, classical continuum mechanics theories sometimes could not give accurate results. So, there are essential conceptual differences between these two types of approaches because of size effect.

Furthermore, another most commonly used classification merit of methodologies is if its discrete elements are used or not. The mainly used methods using discrete elements can be regarded, such as [18] ground structure approach (GSA) [21, 32], solid isotropic material with penalization (SIMP) method [33], homogenization method (HM) [34], evolutionary structural optimization (ESO) [35], and level-set method (LSM) [35]. On the other hand, the mainly used meshless methods are element-free Galerkin (EFG) [36], moving particle [37], and peridynamics [38]. Here, some of the studies post 2010 using these methodologies and their hybrids will be given under different headings.

#### **3.1 Ground structure approach**

Sokol and Rozvany [39] applied a hybrid method of linear programming and GSA to multi-load truss systems. Zhang et al. [40] combined GSA with simulated annealing to apply truss systems. Xu et al. [41] combined GSA with mixed integer

**77**

*Topology Optimization Applications on Engineering Structures*

**3.2 Solid isotropic material with penalization method**

translation coefficients and the cost properties for multiple materials.

linear programming for topology optimization of tensegrity structures. Zhang et al. [42] compared two different ground structure approach (macroelement and macropatch) on a skyscraper and arch bridge. Chun et al. [43] used a discrete filtering scheme in which thin bars are eliminated during reliability-based topology optimization. Gao et al. [44] considered principal stress trajectories to find the suitable nodal points to decrease the computational cost in building ground structure. Ha and Guest [45] applied the method to find the optimum 3D woven material structure and, in a later study, with their colleagues tested this structure [46]. Kosaka et al. [47] applied hybrid method of GSA and ESO to frame structures. Ramos and Paulino [48] considered the materials' nonlinear behavior to solve several topology optimization benchmarking problems. Shakya et al. [49] combined particle swarm optimization (PSO) algorithm with GSA in order to detect and remove useless elements of truss systems. Sokol [50] used GSA in the optimization of large-scale pin-jointed frames considering a new member adding strategy. Wang and Zhang [51] proposed a new approach, parallel optimization tactic, in topology optimization of multi-material compliant mechanism. Zegard and Paulino provided a code for 2D [52] and 3D [53] domains to prevent creating members not intersecting with others. Zhang et al. [54] worked on arranging optimum structure of multi-material composite material using Zhang-Paulino-Ramos design variable update scheme with Karush-Kuhn-Tucker conditions. Zhang et al. [21] used a different filtering scheme for the optimization of multi-materials (hyperelastic Ogden-based and

Shao [55] has combined BESO with SIMP considering 3D printing applications. Lógó [56] has solved a continuum-type topology optimization problem considering uncertainties in load positions. Garcia-Lopez et al. [57] combined simulated annealing with SIMP to eliminate gray areas resulted by SIMP. Gebremedhen et al. [58] used SIMP to solve 3D stress-constrained topology optimization problems. Jantos et al. [59] used a new approach based on thermodynamics material modeling and not containing any filter and compared the results with SIMPs'. Jiao et al. [60] combined ESO with SIMP and used strain energy in their filtering function as sensitivity number. Kandemir et al. [61] proposed a new approach to define intermediate densities (gray areas) with new penalization factor. Marck et al. [62] applied SIMP to solve a multiobjective conductivity problem while using finite volume method (FVM) to solve the energy equation. Ospald and Herzog [63] used projected gradient method with SIMP to solve the structure problem of mold where short-fiber-reinforced polymer material is used in injection molding. Qiao et al. [64] applied the hybrid method of SIMP and BESO to a MBB beam and a cantilever beam and compared the results with literature. Schlinquer et al. [65] applied SIMP to design a mechanism used to amplify the displacement of a piezoelectric actuators. Tsai and Cheng [66] employed SIMP to design flywheel rotor having maximum stiffness. Wang et al. [67] combined topology and size optimization for a folding wing structural design. Yang et al. [68] accomplished topology optimization of an electric vehicle body by SIMP. Yang et al. [69] used SIMP for topology optimization of a hard disk drive. Yunfei et al. [70] applied SIMP to design a robot's upper arm. Zhang and Ren [71] proposed a new optimality criterion method concerning minimum compliance. Zhang et al. [72] presented a new approach to control the length scale of structural members. Zhang et al. [73] presented a method for cellular structures with multiple types of microstructures. Zuo and Saitou [74] introduced power functions with scaling and

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

bilinear materials).

### *Topology Optimization Applications on Engineering Structures DOI: http://dx.doi.org/10.5772/intechopen.90474*

*Truss and Frames - Recent Advances and New Perspectives*

Topology optimization methods are mathematical techniques/approaches, and they can be programmed using different algorithms. These algorithms could be classified as follows: the criterion algorithm, the mathematical programming

The criterion algorithm obtains the optimality condition by the perceptual knowledge or the rational derivation. Result geometry will be gained by checking

The perceptual criterion is usually the extension of the optimality condition of the full stress criterion of the size optimization. The rational criterion is derived usually by the Lagrange multiplier method of equality constraint. The ESO method

Common mathematical programming algorithms like linear programming (LP) and nonlinear programming methods are also used in topology optimization of structures. The first attempts begin with using LP and successive LP methods later continued with sequential quadratic programming methods. Similar too criterion algorithm, mathematical programming algorithms are solved iteratively. Both stability and sensitivity of the structure are checked in each iteration. Of course it means that more calculation should be done for large-scale systems, and consequently low performance is observed for these cases. Fleury discussed the relationship between the criteria method and the mathematical programming method of size optimization. Fleury found that they both have given approximate results. This

Genetic algorithm, simulated annealing algorithms, and particle swarm are the frequently used algorithms for topology optimization as the intelligent algorithm. The advantage of these algorithms is to keep it from too much calculations. The main idea is to search the optimum topology by checking only the objective function and constraints without calculating any gradients. On the contrary, solution speed can be slow, especially for large-scale system; finding optimum could take longer times [27, 28]. Several algorithms are also developed to combine topology

Two classes of approaches, the so-called material or micro-approaches and the geometrical or macro-approaches, are available [30, 31]. For the areas such as MEMS or biomaterial applications, classical continuum mechanics theories sometimes could not give accurate results. So, there are essential conceptual differences

Furthermore, another most commonly used classification merit of methodolo-

Sokol and Rozvany [39] applied a hybrid method of linear programming and GSA to multi-load truss systems. Zhang et al. [40] combined GSA with simulated annealing to apply truss systems. Xu et al. [41] combined GSA with mixed integer

gies is if its discrete elements are used or not. The mainly used methods using discrete elements can be regarded, such as [18] ground structure approach (GSA) [21, 32], solid isotropic material with penalization (SIMP) method [33], homogenization method (HM) [34], evolutionary structural optimization (ESO) [35], and level-set method (LSM) [35]. On the other hand, the mainly used meshless methods are element-free Galerkin (EFG) [36], moving particle [37], and peridynamics [38]. Here, some of the studies post 2010 using these methodologies and their hybrids

constraint violations and objective function value in an iterative way.

study refers still to the basics of the topology optimization [25, 26].

optimization with additive manufacturing [29].

will be given under different headings.

**3.1 Ground structure approach**

between these two types of approaches because of size effect.

**3. Classification of methodologies**

algorithm, and the intelligent algorithm.

is the typical criterion method.

**76**

linear programming for topology optimization of tensegrity structures. Zhang et al. [42] compared two different ground structure approach (macroelement and macropatch) on a skyscraper and arch bridge. Chun et al. [43] used a discrete filtering scheme in which thin bars are eliminated during reliability-based topology optimization. Gao et al. [44] considered principal stress trajectories to find the suitable nodal points to decrease the computational cost in building ground structure. Ha and Guest [45] applied the method to find the optimum 3D woven material structure and, in a later study, with their colleagues tested this structure [46]. Kosaka et al. [47] applied hybrid method of GSA and ESO to frame structures. Ramos and Paulino [48] considered the materials' nonlinear behavior to solve several topology optimization benchmarking problems. Shakya et al. [49] combined particle swarm optimization (PSO) algorithm with GSA in order to detect and remove useless elements of truss systems. Sokol [50] used GSA in the optimization of large-scale pin-jointed frames considering a new member adding strategy. Wang and Zhang [51] proposed a new approach, parallel optimization tactic, in topology optimization of multi-material compliant mechanism. Zegard and Paulino provided a code for 2D [52] and 3D [53] domains to prevent creating members not intersecting with others. Zhang et al. [54] worked on arranging optimum structure of multi-material composite material using Zhang-Paulino-Ramos design variable update scheme with Karush-Kuhn-Tucker conditions. Zhang et al. [21] used a different filtering scheme for the optimization of multi-materials (hyperelastic Ogden-based and bilinear materials).
