**3.3 Homogenization method**

Allaire et al. applied HM to structures made of periodically perforated material in 2D [75] and 3D [76]. Zhang and Khandelwal [77] presented a nonlinear homogenization method to be able to design metamaterials. Lee et al. [78] proposed asymptotic homogenization method to solve topology optimization problem of magnetic composite materials. El-Kahlout and Kiziltas [79] used together MATLAB code to integrate material model derived using homogenization theory with COMSOL and solved several design problems where periodic dielectric materials with desired properties are aimed. Noguchi et al. [80] proposed a level-set-based topology optimization method for the design of hyperbolic acoustic metamaterials using a high-frequency homogenization method. Larsen et al. [81] proposed a new approach based on HM extracting discrete structure from the continuum model. Milani and Bruggi [82] used an adaptive meshing algorithm with HM to optimize multistory masonry wall loaded up to failure. Groen and Sigmund [83] presented a projection method to get better meshes during topology optimization. Xia and Breitkopf [84] implemented a MATLAB code which uses energy-based homogenization approach rather than the asymptotic approach. Bruggi and Milani [85] arranged strut-and-tie models in reinforced concrete structures. Kaminakis et al. [86] used hybrid algorithm based on evolutionary algorithms in the design of microstructures having auxetic behavior.

#### **3.4 Evolutionary structural optimization**

Martínez-Frutos and Herrero-Pérez [87] used evolutionary algorithm to increase the efficiency of GPU and enable to solve with smaller amount of device memory. Daróczy and Jármai [88] proposed a new bidirectional evolutionary structural optimization (BESO) algorithm based on fluid dynamics analogy. Tomšič and Duhovnik [89] discussed simultaneous topology and size optimization of trusses. Abdi et al. [90] used a combination of ESO with XFEM which uses isoline design approach. Ansola et al. [91] used ESO to optimize compliant mechanism under concentrated and thermal loads. Aulig and Olhofer [92] combined a neuroevolution algorithm with a gradient-based optimizer and later proposed another algorithm considering state-based representation [93]. Azamirad and Arezoo [94] combined programming environment with Abaqus FEM software to optimize die components. Bureerat and Sriworamas [95] proposed multiobjective real-code population-based incremental learning (RPBIL) and a hybrid algorithm of RPBIL with differential evolution (DE) (termed RPBIL-DE) to solve water distribution network. Chen et al. [96] used ESO to optimize the rotary lobe of root vacuum pumps. Chen [97] used modified ESO algorithm for the optimization of plate structure under harmonic loading. Cho et al. [98] obtained optimum topology for the inner reinforcement of a vehicle's hood having uncertainties in variables. Finotto et al. [99] used an algorithm combination of ground structure approach, nonlinear finite element analysis, and quantum-inspired evolutionary algorithms. Garcia-Lopez et al. [100] used multiobjective evolutionary algorithm handling uncertainties and also giving the Pareto frontier solutions to let user select the best solution. Greiner and Hajela [101] used multiobjective evolutionary algorithm using reunification criterion to increase search efficiency. Huang and Xie [102] used BESO utilizing an alternative material interpolation scheme. Huang et al. [103] used BESO to optimize the topology of PBC made of two-phase composites. Zuo and Xie [104] used ESO letting limiting displacement. Jantos et al. [105] added a control mechanism for growth factor where at each step Lagrange multiplier is used to find optimum. Jia et al. [106] used hybrid of ESO with LSM. Kaminakis et al. [107]

**79**

material usage.

**3.5 Level-set method (LSM)**

*Topology Optimization Applications on Engineering Structures*

proposed hybrid method of Particle Swarm Optimization and differential evolution in the design of microstructures. Kunakote and Bureerat [108] compared Pareto archive evolution strategy (PAES), population-based incremental learning (PBIL), non-dominated sorting genetic algorithm (NSGA), strength Pareto evolutionary algorithm (SPEA), and multiobjective particle swarm optimization (MPSO). Li et al. [109] used a combination of SIMP and ESO. Li et al. [110] used BESO method in the design of hinge-free compliant mechanisms. Maleki Jebeli and Shariat Panahi [111] used GA as evolutionary algorithm to optimize the material property distribution in FG structures. Okamoto et al. [112] enhanced genetic algorithm, immune algorithm, additional search in the restricted design space with enabling island, and void distribution during FEM analysis to solve a typical magnetic circuit problem. Picelli et al. [113] used BESO to free vibration problems of acoustic-structure systems. Riehl and Steinmann [114] employed the traction method to define descent directions for shape variation. Shi et al. [115] used APDL and UIDL to implement BESO in ANSYS to improve results. Sun et al. [116] applied BESO a cantilever composite laminate under uniform in-plane pressure. Tominaga et al. [117] used GA algorithms for magnetostatic shielding to minimize the magnetic flux intensity in a specified region. Wang et al. [118] used to optimize constrained damping layer structure. Fritzen et al. [119] taken nonlinear elastoviscoplastic microscopic RVE into account at all points of the macroscopic design domain by using BESO. Later, Xia et al. [120] introduced a damping scheme on sensitivity numbers to the same approach. Zhu et al. [121] used bidirectional evolutionary level-set method allowing automatic hole generation. Zuo et al. [27] enhanced the BESO method to multiple constraints of displacement and frequency in addition to the amount of

Allaire et al. [122] applied LSM with enabling local mesh modifications. Chen

and Chen [123] considered geometric uncertainty and related problems. Van Dijk et al. [124] used uses a direct steepest-descent update of the design variables in a LSM. Dunning and Alicia Kim [125] developed a third dimension for 2D problems to adjust new hole positions and to prevent violations with boundaries. Emmendoerfer and Fancello [126] minimized mass under stress constraints using an augmented Lagrangian approach. Gomes et al. [127] interested in the reduction of the design space dimension by the help of a GUI. Guo et al. [128] used LSM in stress-related topology optimization problems. Otomori et al. applied LSM to the design of electromagnetic cloaks using a ferrite material [129] and a light-scattering layer for solar cell applications [130]. Guo et al. [131] developed a local and explicit feature control scheme. James et al. [132] used isoparametric finite element, and James and Martins [133] used a body-fitted, nonuniform finite element mesh to overcome irregular shape problems. Jang et al. [134] considered geometric uncertainties in the production of microsystems. Lim et al. [135] applied to magnetic actuator design problems. Liu et al. [136] adopted extended finite element method (XFEM) with unified structural optimization model help to cover the topology, shape, and sizing optimization at the same time. Luo et al. [137] combined meshless Galerkin method with LSM. Makhija and Maute [138] applied a generalized Heaviside enrichment strategy with XFEM formulation. Mohamadian and Shojaee [139] combined binary level-set method and Merriman-Bence-Osher scheme. Otomori et al. [140] used LSM in the design of negative permeability dielectric metamaterials. Shojaee and Mohammadian [141] combined piecewise constant level-set (PCLS) method with a MBO scheme. Shu et al. [142] used LSM to minimize frequency response which results in the reduction in the vibration of structure.

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

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

*Truss and Frames - Recent Advances and New Perspectives*

microstructures having auxetic behavior.

**3.4 Evolutionary structural optimization**

Allaire et al. applied HM to structures made of periodically perforated material in 2D [75] and 3D [76]. Zhang and Khandelwal [77] presented a nonlinear homogenization method to be able to design metamaterials. Lee et al. [78] proposed asymptotic homogenization method to solve topology optimization problem of magnetic composite materials. El-Kahlout and Kiziltas [79] used together MATLAB

Martínez-Frutos and Herrero-Pérez [87] used evolutionary algorithm to increase the efficiency of GPU and enable to solve with smaller amount of device memory. Daróczy and Jármai [88] proposed a new bidirectional evolutionary structural optimization (BESO) algorithm based on fluid dynamics analogy. Tomšič and Duhovnik [89] discussed simultaneous topology and size optimization of trusses. Abdi et al. [90] used a combination of ESO with XFEM which uses isoline design approach. Ansola et al. [91] used ESO to optimize compliant mechanism under concentrated and thermal loads. Aulig and Olhofer [92] combined a neuroevolution algorithm with a gradient-based optimizer and later proposed another algorithm considering state-based representation [93]. Azamirad and Arezoo [94] combined programming environment with Abaqus FEM software to optimize die components. Bureerat and Sriworamas [95] proposed multiobjective real-code population-based incremental learning (RPBIL) and a hybrid algorithm of RPBIL with differential evolution (DE) (termed RPBIL-DE) to solve water distribution network. Chen et al. [96] used ESO to optimize the rotary lobe of root vacuum pumps. Chen [97] used modified ESO algorithm for the optimization of plate structure under harmonic loading. Cho et al. [98] obtained optimum topology for the inner reinforcement of a vehicle's hood having uncertainties in variables. Finotto et al. [99] used an algorithm combination of ground structure approach, nonlinear finite element analysis, and quantum-inspired evolutionary algorithms. Garcia-Lopez et al. [100] used multiobjective evolutionary algorithm handling uncertainties and also giving the Pareto frontier solutions to let user select the best solution. Greiner and Hajela [101] used multiobjective evolutionary algorithm using reunification criterion to increase search efficiency. Huang and Xie [102] used BESO utilizing an alternative material interpolation scheme. Huang et al. [103] used BESO to optimize the topology of PBC made of two-phase composites. Zuo and Xie [104] used ESO letting limiting displacement. Jantos et al. [105] added a control mechanism for growth factor where at each step Lagrange multiplier is used to find optimum. Jia et al. [106] used hybrid of ESO with LSM. Kaminakis et al. [107]

code to integrate material model derived using homogenization theory with COMSOL and solved several design problems where periodic dielectric materials with desired properties are aimed. Noguchi et al. [80] proposed a level-set-based topology optimization method for the design of hyperbolic acoustic metamaterials using a high-frequency homogenization method. Larsen et al. [81] proposed a new approach based on HM extracting discrete structure from the continuum model. Milani and Bruggi [82] used an adaptive meshing algorithm with HM to optimize multistory masonry wall loaded up to failure. Groen and Sigmund [83] presented a projection method to get better meshes during topology optimization. Xia and Breitkopf [84] implemented a MATLAB code which uses energy-based homogenization approach rather than the asymptotic approach. Bruggi and Milani [85] arranged strut-and-tie models in reinforced concrete structures. Kaminakis et al. [86] used hybrid algorithm based on evolutionary algorithms in the design of

**3.3 Homogenization method**

**78**

proposed hybrid method of Particle Swarm Optimization and differential evolution in the design of microstructures. Kunakote and Bureerat [108] compared Pareto archive evolution strategy (PAES), population-based incremental learning (PBIL), non-dominated sorting genetic algorithm (NSGA), strength Pareto evolutionary algorithm (SPEA), and multiobjective particle swarm optimization (MPSO). Li et al. [109] used a combination of SIMP and ESO. Li et al. [110] used BESO method in the design of hinge-free compliant mechanisms. Maleki Jebeli and Shariat Panahi [111] used GA as evolutionary algorithm to optimize the material property distribution in FG structures. Okamoto et al. [112] enhanced genetic algorithm, immune algorithm, additional search in the restricted design space with enabling island, and void distribution during FEM analysis to solve a typical magnetic circuit problem. Picelli et al. [113] used BESO to free vibration problems of acoustic-structure systems. Riehl and Steinmann [114] employed the traction method to define descent directions for shape variation. Shi et al. [115] used APDL and UIDL to implement BESO in ANSYS to improve results. Sun et al. [116] applied BESO a cantilever composite laminate under uniform in-plane pressure. Tominaga et al. [117] used GA algorithms for magnetostatic shielding to minimize the magnetic flux intensity in a specified region. Wang et al. [118] used to optimize constrained damping layer structure. Fritzen et al. [119] taken nonlinear elastoviscoplastic microscopic RVE into account at all points of the macroscopic design domain by using BESO. Later, Xia et al. [120] introduced a damping scheme on sensitivity numbers to the same approach. Zhu et al. [121] used bidirectional evolutionary level-set method allowing automatic hole generation. Zuo et al. [27] enhanced the BESO method to multiple constraints of displacement and frequency in addition to the amount of material usage.
