Topology Optimization: A Compact Review and Comparation on Level Set Technique Development in Single and Multi-material

References

Allaire, G., Dapogny, C., & Frey, P. (2014). Shape optimization with a level set based mesh evolution method. Computer Methods in Applied Mechanics and Engineering, 282, 22–53. https://doi.org/10.1016/j.cma.2014.08.028

Allaire, G., Jouve, F., & Michailidis, G. (2013, June 12). Structural and multi-functional optimization using multiple phases and a level-set method. 3rd South-East European Conference on Computational Mechanics.

Allaire, G., Jouve, F., & Toader, A. M. (2004). Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194(1), 363–393. https://doi.org/10.1016/j.jcp.2003.09.032

Andrade, G. C., & Santos, S. A. (2022). A level-set-based topology optimization strategy using radial basis functions and a Hilbertian velocity extension. Applied Mathematical Modelling, 111, 108–125. https://doi.org/10.1016/j.apm.2022.06.026

Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B. S., & Sigmund, O. (2011). Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 43(1), 1–16. https://doi.org/10.1007/s00158-010-0594-7

Azari Nejat, A., Held, A., Trekel, N., & Seifried, R. (2022). A modified level set method for topology optimization of sparsely-filled and slender structures. Struct Multidisc Optim 65, 85. https://doi.org/10.1007/s00158-022-03184-2

Bai, J., & Zuo, W. (2022). Multi-material topology optimization of coated structures using level set method. Composite Structures, 300, 116074. https://doi.org/10.1016/j.compstruct.2022.116074

Belytschko, T., Xiao, S. P., & Parimi, C. (2003). Topology optimization with implicit functions and regularization. International Journal for Numerical Methods in Engineering, 57(8), 1177–1196. https://doi.org/10.1002/nme.824

Bohrer, R., & Kim, I. Y. (2021). Multi-material topology optimization considering isotropic and anisotropic materials combination. Structural and Multidisciplinary Optimization, 64(3), 1567–1583. https://doi.org/10.1007/s00158-021-02941-z

Bruggi, M., Ismail, H., Lógó, J., & Paoletti, I. (2022). Lightweight design with displacement constraints using graded porous microstructures. Computers and Structures, 272, 106873. https://doi.org/10.1016/j.compstruc.2022.106873

Burger, M., Hackl, B., & Ring, W. (2004). Incorporating topological derivatives into level set methods. Journal of Computational Physics, 194(1), 344–362. https://doi.org/10.1016/j.jcp.2003.09.033

Challis, V. J. (2010). A discrete level-set topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 41(3), 453–464. https://doi.org/10.1007/s00158-009-0430-0

Cheng, W., Zhang, X., Hu, T., Li, J., & Wang, Y. (2024). Stress-constrained topology optimization using the velocity field level set method. Computers and Structures, 305, 107577. https://doi.org/10.1016/j.compstruc.2024.107577

Chung, H., Hwang, J. T., Gray, J. S., & Kim, H. A. (2019). Topology optimization in OpenMDAO. Structural and Multidisciplinary Optimization, 59(4), 1385–1400. https://doi.org/10.1007/s00158-019-02209-7

Cui, M., Chen, H., & Zhou, J. (2016). A level-set based multi-material topology optimization method using a reaction diffusion equation. CAD Computer Aided Design, 73, pages 41–52. https://doi.org/10.1016/j.cad.2015.12.002

Cui, Y., Takahashi, T., & Matsumoto, T. (2023). An exact volume constraint method for topology optimization via reaction–diffusion equation. Computers and Structures, 280, 106986. https://doi.org/10.1016/j.compstruc.2023.106986

Doan, Q. H., Lee, D., Lee, J., & Kang, J. (2020). Multi-material structural topology optimization with decision making of stiffness design criteria. Advanced Engineering Informatics, 45, 101098. https://doi.org/10.1016/j.aei.2020.101098

Du, X., Zhou, Y., Li, L., Persson, C., & Ferguson, S. J. (2023). The porous cantilever beam as a model for spinal implants: Experimental, analytical and finite element analysis of dynamic properties. Mathematical Biosciences and Engineering, 20(4), 6273–6293. https://doi.org/10.3934/mbe.2023270

Dunning, P. D., & Alicia Kim, H. (2013). A new hole insertion method for level set based structural topology optimization. International Journal for Numerical Methods in Engineering, 93(1), 118–134. https://doi.org/10.1002/nme.4384

Feng, J., Fu, J., Lin, Z., Shang, C., & Li, B. (2018). A review of the design methods of complex topology structures for 3D printing. Visual Computing for Industry, Biomedicine, and Art, 1, 5. Springer Science and Business Media B.V. https://doi.org/10.1186/s42492-018-0004-3

Ghasemi, H., Park, H. S., & Rabczuk, T. (2018). A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 332, 47–62. https://doi.org/10.1016/j.cma.2017.12.005

Gibou, F., Fedkiw, R., & Osher, S. (2018). A review of level-set methods and some recent applications. Journal of Computational Physics, 353, 82–109. Academic Press Inc. https://doi.org/10.1016/j.jcp.2017.10.006

Gomes, A. A., & Suleman, A. (2006). Application of spectral level set methodology in topology optimization. Structural and Multidisciplinary Optimization, 31(6), 430–443. https://doi.org/10.1007/s00158-006-0005-2

Han, Y., Xu, B., & Liu, Y. (2021). An efficient 137-line MATLAB code for geometrically nonlinear topology optimization using bi-directional evolutionary structural optimization method. Structural and Multidisciplinary Optimization, 63(5), 2571–2588. https://doi.org/10.1007/s00158-020-02816-9

Hassan, H. Z., & Saeed, N. M. (2024). Advancements and applications of lightweight structures: a comprehensive review. Discover Civil Engineering, 1, 47. https://doi.org/10.1007/s44290-024-00049-z

He, L., Kao, C. Y., & Osher, S. (2007). Incorporating topological derivatives into shape derivatives based level set methods. Journal of Computational Physics, 225(1), 891–909. https://doi.org/10.1016/j.jcp.2007.01.003

Huang, X. (2020). Smooth topological design of structures using the floating projection. Engineering Structures, Volume 208, Article 110330. https://doi.org/10.1016/j.engstruct.2020.110330

Huang, X. (2021). On smooth or 0/1 designs of the fixed-mesh element-based topology optimization. Advances in Engineering Software, 151, 102942. https://doi.org/10.1016/j.advengsoft.2020.102942

Jankovics, D., & Barari, A. (2019). Customization of Automotive Structural Components using Additive Manufacturing and Topology Optimization. IFAC-PapersOnLine, 52(10), 212–217. https://doi.org/10.1016/j.ifacol.2019.10.066

Jauregui, C. M., Hyun, J., Neofytou, A., Gray, J. S., & Kim, H. A. (2023). Avoiding reinventing the wheel: reusable open-source topology optimization software. Structural and Multidisciplinary Optimization, 66, 145. https://doi.org/10.1007/s00158-023-03589-7

Jia, S., Li, Z., Liu, R., Wang, Y., & Kang, Z. (2026). Velocity field level set method for topology optimization on irregular level set grid. Engineering Structures, 349, 121874. https://doi.org/10.1016/j.engstruct.2025.121874

Kumar, H., & Rakshit, S. (2025). Multi-material topology optimization using isogeometric method based reaction–diffusion level set techniques. Mathematics and Computers in Simulation, 233, 530–552. https://doi.org/10.1016/j.matcom.2025.02.010

Laurain, A. (2018). A level set-based structural optimization code using FEniCS. Structural and Multidisciplinary Optimization, 58(3), 1311–1334. https://doi.org/10.1007/s00158-018-1950-2

Li, H., Wang, C., Zhang, X., Li, J., Shen, J., & Zhou, S. (2023). A Mini Review on Fluid Topology Optimization. Materials, 16(18), 1–16. https://doi.org/10.3390/ma16186073

Liu, J., Gaynor, A. T., Chen, S., Kang, Z., Suresh, K., Takezawa, A., Li, L., Kato, J., Tang, J., Wang, C. C. L., Cheng, L., Liang, X., & To, A. C. (2018). Current and future trends in topology optimization for additive manufacturing. Structural and Multidisciplinary Optimization, 57(6), 2457–2483. https://doi.org/10.1007/s00158-018-1994-3

Liu, J., & Ma, Y. (2018). A new multi-material level set topology optimization method with the length scale control capability. Computer Methods in Applied Mechanics and Engineering, 329, 444–463. https://doi.org/10.1016/j.cma.2017.10.011

Liu, P., Luo, Y., & Kang, Z. (2016). Multi-material topology optimization considering interface behavior via XFEM and level set method. Computer Methods in Applied Mechanics and Engineering, 308, 113–133. https://doi.org/10.1016/j.cma.2016.05.016

Lu, H., Lee, T. U., Ma, J., Chen, D., & Xie, Y. M. (2023). Designing 2D stochastic porous structures using topology optimisation. Composite Structures, 321, 117305. https://doi.org/10.1016/j.compstruct.2023.117305

Luo, Z., Tong, L., & Kang, Z. (2009). A level set method for structural shape and topology optimization using radial basis functions. Computers and Structures, 87(7–8), 425–434. https://doi.org/10.1016/j.compstruc.2009.01.008

Luo, Z., Wang, M. Y., Wang, S., & Wei, P. (2008). A level set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering, 76(1), 1–26. https://doi.org/10.1002/nme.2092

Nguyen, K. C., Tran, P., & Nguyen, H. X. (2019). Multi-material topology optimization for additive manufacturing using polytree-based adaptive polygonal finite elements. Automation in Construction, 99, 79–90. https://doi.org/10.1016/j.autcon.2018.12.005

Nguyen-Tran, B. D., Phan, D. H., Nguyen-Thoi, T., & Nguyen, S. H. (2025). Multi-Material Structural Topology Optimization via Dynamic Body-Fitted Multi-Trimmed Mesh Evolution and Reaction–Diffusion-Based Level-Set Method. International Journal for Numerical Methods in Engineering, 126(20), e70167. https://doi.org/10.1002/nme.70167

Noda, M., Noguchi, Y., & Yamada, T. (2022). Extended level set method: A multiphase representation with perfect symmetric property, and its application to multi-material topology optimization. Computer Methods in Applied Mechanics and Engineering, 393, 114742. https://doi.org/10.1016/j.cma.2022.114742

Oka, T., & Yamada, T. (2023). Topology optimization method with nonlinear diffusion. Computer Methods in Applied Mechanics and Engineering, Volume 408, 115940. https://doi.org/10.1016/j.cma.2023.115940

Osher, S., & Sethian, J. A. (1988). Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations. Journal of Computational Physics, 79(1), 12–49. https://doi.org/10.1016/0021-9991(88)90002-2

Otomori, M., Yamada, T., Izui, K., & Nishiwaki, S. (2015). Matlab code for a level set-based topology optimization method using a reaction diffusion equation. Structural and Multidisciplinary Optimization, 51(5), 1159–1172. https://doi.org/10.1007/s00158-014-1190-z

Plocher, J., & Panesar, A. (2019). Review on design and structural optimisation in additive manufacturing: Towards next-generation lightweight structures. In Materials and Design, 183, 108164. Elsevier Ltd. https://doi.org/10.1016/j.matdes.2019.108164

Prathyusha, A. L. R., & Raghu Babu, G. (2022). A review on additive manufacturing and topology optimization process for weight reduction studies in various industrial applications. Materials Today: Proceedings, 62, 109–117. https://doi.org/10.1016/j.matpr.2022.02.604

Rozvany, G. I. N., Zhou, M., & Birker, T. (1992). Generalized shape optimization without homogenization. Structural Optimization, 4, 250–252. https://doi.org/10.1007/BF01742754

Schmidt, M. P., Pedersen, C. B. W., & Gout, C. (2019). On structural topology optimization using graded porosity control. Structural and Multidisciplinary Optimization, 60(4), 1437–1453. https://doi.org/10.1007/s00158-019-02275-x

Seong, H. K., Shin, H., Yoo, J., Yamada, T., & Nishiwaki, S. (2018). Reaction-diffusion equation based topology optimization combined with the modified conjugate gradient method. Finite Elements in Analysis and Design, 140, 84–95. https://doi.org/10.1016/j.finel.2017.11.009

Sethian, J. A., & Wiegmann, A. (2000). Structural Boundary Design via Level Set and Immersed Interface Methods. Journal of Computational Physics, 163(2), 489–528. https://doi.org/10.1006/jcph.2000.6581

Sha, W., Xiao, M., Gao, L., & Zhang, Y. (2021). A new level set based multi-material topology optimization method using alternating active-phase algorithm. Computer Methods in Applied Mechanics and Engineering, 377, 113674. https://doi.org/10.1016/j.cma.2021.113674

Shah, V., Pamwar, M., Sangha, B., & Kim, I. Y. (2023). All-terrain vehicle chassis design using multi-material topology optimization. Structural and Multidisciplinary Optimization, 66(3). https://doi.org/10.1007/s00158-023-03489-w

Shen, Y., & Feng, J. (2018). Parametric Level Set-Based Multimaterial Topology Optimization of Heat Conduction Structures. Mathematical Problems in Engineering, 2018. https://doi.org/10.1155/2018/9804123

Sigmund, 0, & Torquato, S. (1997). Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids, 45(6), 1037–1067. https://doi.org/10.1016/S0022-5096(96)00114-7

Sigmund, O. (2001). A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21, 21, 120–127. https://doi.org/10.1007/s001580050176

Sigmund, O. (2007). Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 33(4–5), 401–424. https://doi.org/10.1007/s00158-006-0087-x

Sobczak, P., & Sokół, T. (2024). Level-set based shape optimization for plane elastic structures using radial basis functions and Hilbertian descent direction. Structural and Multidisciplinary Optimization, 67, 174. https://doi.org/10.1007/s00158-024-03868-x

Subedi, S. C., Singh Verma, C., & Suresh, K. (2020). A Review of Methods for the Geometric Post-Processing of Topology Optimized Models. ASME. J. Comput. Inf. Sci. Eng., 20, 060801. https://doi.org/10.1115/1.4047429

Sugimoto, S., Sakane, S., & Takaki, T. (2024). Multi-phase-field framework for multi-material topology optimization. Computational Materials Science, 244, 113201. https://doi.org/10.1016/j.commatsci.2024.113201

Svanberg, K. (1987). The method of moving asymptotes - A new method for structural optimization. International Journal for Numerical Methods in Engineering, 24(2), 359–373. https://doi.org/10.1002/nme.1620240207

Tajs-Zielińska, K., & Bochenek, B. (2021). Multi-domain and multi-material topology optimization in design and strengthening of innovative sustainable structures. Sustainability, 13(6), 3435. https://doi.org/10.3390/su13063435

Tian, J., Li, M., Han, Z., Chen, Y., Gu, X. D., Ge, Q. J., & Chen, S. (2022). Conformal topology optimization of multi-material ferromagnetic soft active structures using an extended level set method. Computer Methods in Applied Mechanics and Engineering, 389, 114394. https://doi.org/10.1016/j.cma.2021.114394

Van Dijk, N. P., Langelaar, M., & Van Keulen, F. (2012). Explicit level-set-based topology optimization using an exact Heaviside function and consistent sensitivity analysis. International Journal for Numerical Methods in Engineering, 91(1), 67–97. https://doi.org/10.1002/nme.4258

Van Dijk, N. P., Maute, K., Langelaar, M., & Van Keulen, F. (2013). Level-set methods for structural topology optimization: A review. Structural and Multidisciplinary Optimization, 48(3), 437–472. https://doi.org/10.1007/s00158-013-0912-y

Vogiatzis, P., Chen, S., Wang, X., Li, T., & Wang, L. (2017). Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method. CAD Computer Aided Design, 83, 15–32. https://doi.org/10.1016/j.cad.2016.09.009

Wang, C., Xie, Y. M., Lin, X., & Zhou, S. (2022). A reaction diffusion-based B-spline level set (RDBLS) method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 398, 115252. https://doi.org/10.1016/j.cma.2022.115252

Wang, C., Zhao, Z., Zhou, M., Sigmund, O., & Zhang, X. S. (2021). A comprehensive review of educational articles on structural and multidisciplinary optimization. Structural and Multidisciplinary Optimization, 64(5), 2827–2880. Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/s00158-021-03050-7

Wang, F., Lazarov, B. S., & Sigmund, O. (2011). On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 43(6), 767–784. https://doi.org/10.1007/s00158-010-0602-y

Wang, M. Y., & Wang, X. (2004). “Color” level sets: A multi-phase method for structural topology optimization with multiple materials. Computer Methods in Applied Mechanics and Engineering, 193(6–8), 469–496. https://doi.org/10.1016/j.cma.2003.10.008

Wang, M. Y., Wang, X., & Guo, D. (2003). A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 192(1–2), 227–246. https://doi.org/10.1016/S0045-7825(02)00559-5

Wang, S., & Wang, M. Y. (2006). Radial basis functions and level set method for structural topology optimization. International Journal for Numerical Methods in Engineering, 65(12), 2060–2090. https://doi.org/10.1002/nme.1536

Wang, Y., & Kang, Z. (2018). A velocity field level set method for shape and topology optimization. International Journal for Numerical Methods in Engineering, 115(11), 1315–1336. https://doi.org/10.1002/nme.5845

Wang, Y., & Kang, Z. (2021). MATLAB implementations of velocity field level set method for topology optimization: an 80-line code for 2D and a 100-line code for 3D problems. Structural and Multidisciplinary Optimization, 64(6), 4325–4342. https://doi.org/10.1007/s00158-021-02958-4

Wang, Y., Luo, Z., Kang, Z., & Zhang, N. (2015). A multi-material level set-based topology and shape optimization method. Computer Methods in Applied Mechanics and Engineering, 283, 1570–1586. https://doi.org/10.1016/j.cma.2014.11.002

Wegert, Z. J., Manyer, J., Mallon, C. N., Badia, S., & Challis, V. J. (2025). GridapTopOpt.jl: a scalable Julia toolbox for level set-based topology optimisation. Structural and Multidisciplinary Optimization, 68, 22. https://doi.org/10.1007/s00158-024-03927-3

Wei, P., Li, Z., Li, X., & Wang, M. Y. (2018). An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Structural and Multidisciplinary Optimization, 58(2), 831–849. https://doi.org/10.1007/s00158-018-1904-8

Wei, P., & Paulino, G. H. (2020). A parameterized level set method combined with polygonal finite elements in topology optimization. Structural and Multidisciplinary Optimization, 61, 1913–1928. https://doi.org/10.1007/s00158-019-02444-y/Published

Wu, J., Aage, N., Westermann, R., & Sigmund, O. (2018). Infill Optimization for Additive Manufacturing-Approaching Bone-Like Porous Structures. IEEE Transactions on Visualization and Computer Graphics, 24(2), 1127–1140. https://doi.org/10.1109/TVCG.2017.2655523

Xie, X., & Mirmehdi, M. (2011). Radial basis function based level set interpolation and evolution for deformable modelling. Image and Vision Computing, 29(2–3), 167–177. https://doi.org/10.1016/j.imavis.2010.08.011

Xie, Y. M., & Steven, G. P. (1993). A simple evolutionary procedure for structural optimization. Computers and Structures, 49(5), 885–896. https://doi.org/10.1016/0045-7949(93)90035-c

Xu, S., Liu, J., & Ma, Y. (2022). Residual stress constrained self-support topology optimization for metal additive manufacturing. Computer Methods in Applied Mechanics and Engineering, 389, 114380. https://doi.org/10.1016/j.cma.2021.114380

Xu, X., Gu, X. D., & Chen, S. (2022). Shape and topology optimization of conformal thermal control structures on free-form surfaces: A dimension reduction level set method (DR-LSM). Computer Methods in Applied Mechanics and Engineering, 398. https://doi.org/10.1016/j.cma.2022.115183

Xu, X., Gu, X. D., & Chen, S. (2023). Topology optimization of thermal cloaks in euclidean spaces and manifolds using an extended level set method. International Journal of Heat and Mass Transfer, 202, 123720. https://doi.org/10.1016/j.ijheatmasstransfer.2022.123720

Yago, D., Cante, J., Lloberas-Valls, O., & Oliver, J. (2022). Topology Optimization Methods for 3D Structural Problems: A Comparative Study. Archives of Computational Methods in Engineering, 29(3), 1525–1567. https://doi.org/10.1007/s11831-021-09626-2

Yamada, T., Izui, K., Nishiwaki, S., & Takezawa, A. (2010). A topology optimization method based on the level set method incorporating a fictitious interface energy. Computer Methods in Applied Mechanics and Engineering, 199(45–48), 2876–2891. https://doi.org/10.1016/j.cma.2010.05.013

Yang, W., Chen, H., Bai, H., Sun, Y., Zhang, A., Liu, Y., Song, Y., Han, Q., & Wang, J. (2024). Additive manufactured osseointegrated screws with hierarchical design. In Bio-Design and Manufacturing, 7(2), 206–235. Zhejiang University. https://doi.org/10.1007/s42242-024-00269-3

Yi, G., & Kim, N. H. (2017). Identifying boundaries of topology optimization results using basic parametric features. Structural and Multidisciplinary Optimization, 55(5), 1641–1654. https://doi.org/10.1007/s00158-016-1597-9

Zhao, Z., & Zhang, X. S. (2021). Design of graded porous bone-like structures via a multi-material topology optimization approach. Structural and Multidisciplinary Optimization, 64(2), 677–698. https://doi.org/10.1007/s00158-021-02870-x

Zheng, J., Zhu, S., & Soleymani, F. (2024). A new efficient parametric level set method based on radial basis function-finite difference for structural topology optimization. Computers and Structures, 297, 107364. https://doi.org/10.1016/j.compstruc.2024.107364

Zheng, R., Yi, B., Peng, X., & Yoon, G. H. (2024). An Efficient Code for the Multi-Material Topology Optimization of 2D/3D Continuum Structures Written in Matlab. Applied Sciences, 14(657), 1–23. https://doi.org/10.3390/app14020657

Zhou, Y., Li, H., Li, X., & Gao, L. (2022). Design of multiphase auxetic metamaterials by a parametric color level set method. Composite Structures, 287, 115385. https://doi.org/10.1016/j.compstruct.2022.115385

Zhu, J. H., Zhang, W. H., & Xia, L. (2016a). Topology Optimization in Aircraft and Aerospace Structures Design. Archives of Computational Methods in Engineering, 23, 595–622. https://doi.org/10.1007/s11831-015-9151-2

Zhuang, Z., Xie, Y. M., Li, Q., & Zhou, S. (2023). A 172-line Matlab code for structural topology optimization in the body-fitted mesh. Structural and Multidisciplinary Optimization, 66, 11. https://doi.org/10.1007/s00158-022-03464-x

Zhuang, Z., Xie, Y. M., & Zhou, S. (2021). A reaction diffusion-based level set method using body-fitted mesh for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 381, 113829. https://doi.org/10.1016/j.cma.2021.113829

Zhuang, Z., Xu, F., Ye, J., Tong, W., Chen, Z., & Weng, Y. (2024). A 262-line Matlab code for the level set topology optimization based on the estimated gradient field in the body-fitted mesh. Structural and Multidisciplinary Optimization, 67(9), 1–23. https://doi.org/10.1007/s00158-024-03891-y