Topology optimization using the unsmooth variational topology optimization (UNVARTOP) method: an educational implementation in MATLAB

Daniel Yago1, Juan Cante1, Oriol Lloberas-Valls2, Javier Oliver2
1Escola Superior d’Enginyeries Industrial, Aeroespacial i Audiovisual de Terrassa (ESEIAAT), Technical University of Catalonia (UPC/Barcelona Tech), Campus Terrassa UPC, c/ Colom 11, 08222, Terrassa, Spain
2Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Campus Nord UPC, Mòdul C-1 101, c/ Jordi Girona 1-3, 08034 Barcelona, Spain

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Allaire G, Bonnetier E, Francfort G, Jouve F (1997) Shape optimization by the homogenization method. Numer Math 76(1):27–68. https://doi.org/10.1007/s002110050253

Allaire G, Jouve F, Toader A-M (2002) A level-set method for shape optimization. Comptes Rendus Mathematique 334(12):1125–1130. https://doi.org/10.1016/s1631-073x(02)02412-3

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

Anderson N, Björck Å (1973) A new high order method of regula falsi type for computing a root of an equation. BIT 13(3):253–264. https://doi.org/10.1007/bf01951936

Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2010) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43(1):1–16. https://doi.org/10.1007/s00158-010-0594-7

Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1 (4):193–202. https://doi.org/10.1007/bf01650949

Bendsøe MP, Sigmund O (2004) Topology optimization. Springer, Berlin. https://doi.org/10.1007/978-3-662-05086-6

Biyikli E, To AC (2015) Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PLOS ONE 10(12):e0145041. https://doi.org/10.1371/journal.pone.0145041

Bulirsch R, Stoer J (2010) Introduction to numerical analysis. Springer, New York. ISBN 144193006X

Challis VJ (2009) A discrete level-set topology optimization code written in MATLAB. Struct Multidiscip Optim 41(3):453–464. https://doi.org/10.1007/s00158-009-0430-0

Da D, Xia L, Li G, Huang X (2017) Evolutionary topology optimization of continuum structures with smooth boundary representation. Struct Multidiscip Optim 57(6):2143–2159. https://doi.org/10.1007/s00158-017-1846-6

Dowell M, Jarratt P (1971) A modified regula falsi method for computing the root of an equation. BIT 11(2):168–174. https://doi.org/10.1007/bf01934364

Feijoo RA, Novotny A, Taroco E, Padra C (2005) The topological-shape sensitivity method in two-dimensional linear elasticity topology design. Applications of Computational Mechanics in Structures and Fluids

Giusti S, Novotny A, Padra C (2008) Topological sensitivity analysis of inclusion in two-dimensional linear elasticity. Eng Anal Bound Elem 32(11):926–935. https://doi.org/10.1016/j.enganabound.2007.12.007

Guest JK, Prėvost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254. https://doi.org/10.1002/nme.1064

Lazarov BS, Sigmund O (2010) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765–781. https://doi.org/10.1002/nme.3072

Liang QQ, Steven GP (2002) A performance-based optimization method for topology design of continuum structures with mean compliance constraints. Comput Methods Appl Mech Eng 191(13-14):1471–1489. https://doi.org/10.1016/s0045-7825(01)00333-4

Liu K, Tovar A (2014) An efficient 3D topology optimization code written in MATLAB. Struct Multidiscip Optim 50(6):1175–1196. https://doi.org/10.1007/s00158-014-1107-x

Lopes CG, dos Santos RB, Novotny A (2015) Topological derivative-based topology optimization of structures subject to multiple load-cases. Latin Amer J Solids Struct 12(5):834–860. https://doi.org/10.1590/1679-78251252

Luenberger DG, Ye Y (2016) Linear and nonlinear programming. Springer International Publishing. https://doi.org/10.1007/978-3-319-18842-3

Novotny A, Feijȯo R, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Methods Appl Mech Eng 192(7-8):803–829. https://doi.org/10.1016/s0045-7825(02)00599-6

Oliver J, Yago D, Cante J, Lloberas-Valls O (2019) Variational approach to relaxed topological optimization: closed form solutions for structural problems in a sequential pseudo-time framework. Comput Methods Appl Mech Eng 355:779–819. https://doi.org/10.1016/j.cma.2019.06.038

Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49. https://doi.org/10.1016/0021-9991(88)90002-2

Otomori M, Yamada T, Izui K, Nishiwaki S (2014) MATLAB code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidiscip Optim 51(5):1159–1172. https://doi.org/10.1007/s00158-014-1190-z

Patanė G, Falcidieno B (2009) Computing smooth approximations of scalar functions with constraints. Comput Graph 33(3):399–413. https://doi.org/10.1016/j.cag.2009.03.014

Rao SS (2004) The finite element method in engineering. Butterworth-Heinemann. ISBN 0-7506-7828-3

Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75. https://doi.org/10.1007/bf01214002

Sigmund O (2001) A 99 line topology optimization code written in MATLAB. Struct Multidiscip Optim 21(2):120–127. https://doi.org/10.1007/s001580050176

Sokolowski J, Zochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272. https://doi.org/10.1137/s0363012997323230

Suresh K (2010) A 199-line MATLAB code for Pareto-optimal tracing in topology optimization. Struct Multidiscip Optim 42(5):665–679. https://doi.org/10.1007/s00158-010-0534-6

Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718. https://doi.org/10.1016/j.jcp.2009.12.017

Tavakoli R, Mohseni SM (2013) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim 49(4):621–642. https://doi.org/10.1007/s00158-013-0999-1

van Vliet LJ, Young IT, Beckers GL (1989) A nonlinear laplace operator as edge detector in noisy images. Comput Vis Graph Image Process 45(2):167–195. https://doi.org/10.1016/0734-189x(89)90131-x

Wang M, Zhou S (2004a) Synthesis of shape and topology of multi-material structures with a phase-field method. J Comput-Aided Mater Des 11(2-3):117–138. https://doi.org/10.1007/s10820-005-3169-y

Wang M, Chen S, Xia Q (2004b) TOPLSM a 199-line MATLAB program

Wang S, Wang M (2006) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65(12):2060–2090. https://doi.org/10.1002/nme.1536

Wang S, Lim K, Khoo B, Wang M (2007) An extended level set method for shape and topology optimization. J Comput Phys 221(1):395–421. https://doi.org/10.1016/j.jcp.2006.06.029

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

Xie YM, Steven GP (1997) Evolutionary structural optimization, Springer, London. https://doi.org/10.1007/978-1-4471-0985-3

Yago D, Cante J, Lloberas-Valls O, Oliver J (2020) Topology optimization of thermal problems in a nonsmooth variational setting: closed-form optimality criteria. Computational Mechanics 66(2):259–286. https://doi.org/10.1007/s00466-020-01850-0

Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199 (45-48):2876–2891. https://doi.org/10.1016/j.cma.2010.05.013

Yang XY, Xie YM, Steven GP, Querin OM (1999) Bidirectional evolutionary method for stiffness optimization. AIAA J 37:1483–1488. https://doi.org/10.2514/3.14346

Zhang W, Yuan J, Zhang J, Guo X (2015) A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Struct Multidiscip Optim 53 (6):1243–1260. https://doi.org/10.1007/s00158-015-1372-3

Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: its basis and fundamentals. Elsevier LTD, Oxford. ISBN 1856176339

Zuo ZH, Xie YM (2015) A simple and compact python code for complex 3D topology optimization. Adv Eng Softw 85:1–11. https://doi.org/10.1016/j.advengsoft.2015.02.006