# Topology Optimisation in Structural Steel Design for Additive Manufacturing
## 4 Approaches, Methods, Criteria, and Software in Topology Optimisation
### Approaches and Optimisation Methods
#### 4.1.2 Framing New Developments into the Body of Knowledge
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Approach** & **References** & **Non-Exhaustive List of Methods** & **References** & **Recent Developments** \\ \hline \multirow{6}{*}{\begin{tabular}{c} Density-based \\ (also Material \\ Distribution) \\ \end{tabular} } & \multirow{6}{*}{Homogenisation} & Optimal Microstructure with Penalisation (OMP) & [163; 164] \\ \cline{3-4}  & & Near-optimal Microstructure (NOM) & [21; 201] \\ \cline{3-4}  & & Rational Approximation of Material Properties (RAMP) & [202] \\ \cline{3-4}  & & Solid Isotropic Microstructure (or Material) with Penalisation (SIMP) & [22; 35; 34; 23] \\ \cline{3-4}  & & SINH (due to employing the hyperbolic sine function) & [203] \\ \cline{3-4}  & & Sum of the Reciprocal Variables (SRV) & [204] \\ \cline{3-4}  & & Reliability-Based Topology Optimisation (RBTO) & [205; 206; 207; 208; 209; 210; 211; 212; 213; 214] \\ \hline \multirow{6}{*}{\begin{tabular}{c} Level-set (LS) \\ methods \\ \end{tabular} } & \multirow{6}{*}{[18; 215; 216; 217; 218; 219; 220; 221; 222]} & Conventional LS for solving the Hamilton-Jacobi Equation & [164; 217; 223] \\ \cline{3-4}  & & Radial-Basis Functions (RBF) for solving the Hamilton-Jacobi Equation & [234; 235; 236; 237; 238; 239] \\ \cline{3-4}  & & Spectral LS & [240] \\ \cline{3-4}  & & Non-Linear Programming & [234] \\ \hline \multirow{2}{*}{\begin{tabular}{c} Topological \\ Derivatives \\ \end{tabular} } & \multirow{2}{*}{[241; 242; 243]} & Bubble Method & [1; 244; 245] \\ \cline{3-4}  & & Topological Sensitivity & [243; 248; 249] \\ \cline{3-4}  & & Cahn-Hilliard Method & [250; 251; 252; 253; 254] \\ \cline{3-4}  & & Allen-Cahn Method & [257] \\ \cline{3-4}  & & Relaxed Phase-Field Methods & [250; 252; 258] \\ \hline \multirow{6}{*}{\
\begin{tabular}{c} Heuristic (also \\ Evolutionary) \\ \end{tabular} } & \multirow{6}{*}{[35; 123; 141; 259; 260; 261; 262; 263]} & Evolutionary Structural Optimisation (ESO), also Sequential Elements Rejection and Admission (SERA) & [31; 35; 68; 264] \\ \cline{3-4}  & & \multicolumn{2}{c}{(Hard-kill) Bidirectional ESO (BESO)} & [35; 121; 288; 289; 290; 291; 292; 293; 294] \\ \cline{3-4}  & & Additive ESO (AESO) & [123; 131; 295] \\ \cline{3-4}  & & Soft-kill BESO & [121; 296; 297] \\ \cline{3-4}  & & Swarms, including Particle Swarm Optimisation (PSO), Fish Swarm Optimisation (FSOA), Ant Colony Optimisation (ACO), Stochastic Diffusion Search (SDS), Artificial Swarm Intelligence (ASI), Multi-Swarm Optimisation, Artificial Bee Colony Algorithm (ABC) & [165; 166; 267; 168; 169; 170; 171; 172; 173; 174; 175; 176; 177; 178; 179; 180; 181; 182; 183; 184; 185; 186; 187; 188; 189; 190; 191; 192; 193; 194; 195; 196; 197; 198; 199; 299] \\ \cline{3-4}  & & Genetic Algorithms (GA), including Genetic ESO (GESO), Lindenmayer (also map-L) Method & [300; 301; 302; 303; 304; 305; 306; 307; 308; 309; 310; 311] \\ \hline \multirow{6}{*}{
\begin{tabular}{c} Hybrid \\ approaches \\ \end{tabular} } & \multirow{6}{*}{[122]} & Combination of several features and techniques & [312; 313] \\ \cline{3-4}  & & & & \\ \cline{3-4}  & & & & \\ \hline \hline \end{tabular}  
* Another common name for this approach is the “Hard-kill” method. However, it does not account for its current diversity, which includes the “Soft-kill” option.  
\end{table} 
Table 2: Topology optimisation (TO) approaches, methods, and recent developments. Among the newer approaches, Level-Set Methods appear to be in accelerated development. This can be explained by the migration of density-based, and especially SIMP, researchers and practitioners to an approach with so many standard features. On the other hand, Phase-Field Approaches are yet to gain momentum and Topological Derivatives, even if with a significant history, have not much recent research output. The latter, however, continues to provide a background for many developments and comparisons in mainstream methods.