Topology optimization of continuum structures with displacement constraints based on meshless method
Source abstract
Abstract
In this paper, the element free Galerkin method (EFG) is applied to carry out the topology optimization of continuum structures with displacement constraints. In the EFG method, the matrices in the discretized system equations are assembled based on the quadrature points. In the sense, the relative density at Gauss quadrature point is employed as design variable. Considering the minimization of weight as an objective function, the mathematical formulation of the topology optimization subjected to displacement constraints is developed using the solid isotropic microstructures with penalization interpolation scheme. Moreover, the approximate explicit function expression between topological variables and displacement constraints are derived. Sensitivity of the objective function is derived based on the adjoint method. Three numerical examples are used to demonstrate the feasibility and effectiveness of the proposed method.
Source: openaire · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Topology Optimization in Engineering · Composite Structure Analysis and Optimization · Composite Material Mechanics
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (30)
Show source-provided references
1. Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)10.1016/j.jcp.2003.09.032 ↗
2. Atluri, S.N., Shen, S.: The meshless local Petrov-Galerkin (MLPG) method. Crest, UK (2002)
3. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin method. Int. J. Numer. Meth. Eng. 37, 229–256 (1994a)10.1002/nme.1620370205 ↗
4. Belytschko, T., Gu, L., Lu, Y.Y.: Fracture and crack growth by element free Galerkin methods. Model. Simul. Mater. Sci. Eng. 2, 519–534 (1994b)10.1088/0965-0393/2/3A/007 ↗
5. Belytschko, T., Krongauz, Y., Organ, D., et al.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139, 3–47 (1996)10.1016/S0045-7825(96)01078-X ↗
6. Belytschko, T., Krysl, P., Krongauz, Y.: A three-dimensional explicit element-free galerkin method. Int. J. Numer. Meth. Fluids 24(12), 1253–1270 (1997)10.1002/(SICI)1097-0363(199706)24:12<1253::AID-FLD558>3.0.CO;2-Z ↗
7. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988)10.1016/0045-7825(88)90086-2 ↗
8. Bendsoe, M.P., Sigmund, O.: Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69, 635–654 (1999)10.1007/s004190050248 ↗
9. Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods, and Applications. Springer, New York (2003)
10. Cho, S., Kwak, J.: Topology design optimization of geometrically non-linear structures using meshfree method. Comput. Methods Appl. Mech. Eng. 195, 5909–5925 (2006)10.1016/j.cma.2005.08.015 ↗
11. Kang, Z., Wang, Y.Q.: Structural topology optimization based on non-local Shepard interpolation of density field. Comput. Methods Appl. Mech. Eng. 200, 3515–3525 (2011)10.1016/j.cma.2011.09.001 ↗
12. Liu, G.R., Chen, X.L.: A mesh-free method for static and free vibration analyses of thin plates of complicated shape. J. Sound Vib. 241(5), 839–855 (2001)10.1006/jsvi.2000.3330 ↗
13. Liu, G.R., Gu, Y.T.: An introduction to meshfree methods and their programming. Springer, Berlin (2005)
14. Liu, W.K., Jun, S., Li, S.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Eng. 38(10), 1655–1679 (1995)10.1002/nme.1620381005 ↗
15. Luo, Z., Zhang, N., Gao, W., et al.: Structural shape and topology optimization using a meshless Galerkin level set method. Int. J. Numer. Meth. Eng. 90, 369–389 (2012)10.1002/nme.3325 ↗
16. Luo, Z., Zhang, N., Wang, Y., et al.: Topology optimization of structures using meshless density variable approximants. Int. J. Numer. Meth. Eng. 93, 443–464 (2013)10.1002/nme.4394 ↗
17. Matsui, K., Terada, K.: Continuous approximation of material distribution for topology optimization. Int. J. Numer. Meth. Eng. 59, 1925–1944 (2004)10.1002/nme.945 ↗
18. Monaghan, J.J.: Smoothed particle hydrodynamics. Ann. Rev. Astron. Astrophys. 30, 543–574 (1992)10.1146/annurev.aa.30.090192.002551 ↗
19. Rahmatalla, S.F., Swan, C.C.: A Q4/Q4 continuum structural topology optimization implementation. Struct. Multidiscip. Optim. 27, 130–135 (2004)10.1007/s00158-003-0365-9 ↗
20. Rozvany, G., Kirch, U., Bendsoe, M.P., et al.: Layout optimization of structures. Appl. Mech. Rev. 48, 41–119 (1995)10.1115/1.3005097 ↗
21. Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163, 489–528 (2000)10.1006/jcph.2000.6581 ↗
22. Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim. 16(1), 68–75 (1998)10.1007/BF01214002 ↗
23. Sui, Y.K., Peng, X.R.: The ICM method with objective function transformed by variable discrete condition for continuum structure. Acta Mech. Sin. 22, 68–75 (2006)10.1007/s10409-005-0088-9 ↗
24. Wang, M.Y., Wang, X.M., Guo, D.M.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1–2), 227–246 (2003)10.1016/S0045-7825(02)00559-5 ↗
25. Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput Struct. 49, 885–896 (1993)10.1016/0045-7949(93)90035-C ↗
26. Zhang, X., Liu, X.H., Song, K.Z., et al.: Least-squares collocation meshless method. Int. J. Numer. Meth. Eng. 51(9), 1089–1100 (2001)10.1002/nme.200 ↗
27. Zheng, J., Long, S.Y., Xiong, Y.B., et al.: A topology optimization design for the continuum structure based on the meshless numerical technique, CMES. Comput Model. Eng. Sci. 34(2), 137–154 (2008)
28. Zhou, M., Rozvany, G.I.N.: The COC algorithm part II: topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89, 309–336 (1991)10.1016/0045-7825(91)90046-9 ↗
29. Zhou, J.X., Zou, W.: Meshless approximation combined with implicit topology description for optimization of continua. Struct. Multidisci. Optim. 36(4), 347–353 (2008)10.1007/s00158-007-0168-5 ↗
30. Zhou, M., Shyy, Y.K., Thomas, H.L.: Checkerboard and minimum member size control in topology optimization. Struct. Multidiscip. Optim. 21(2), 152–158 (2001)10.1007/s001580050179 ↗