Artículos de revistas
An Algorithm For The Topology Optimization Of Geometrically Nonlinear Structures
Registro en:
International Journal For Numerical Methods In Engineering. John Wiley And Sons Ltd, v. 99, n. 6, p. 391 - 409, 2014.
295981
10.1002/nme.4686
2-s2.0-84904070738
Autor
Gomes F.A.M.
Senne T.A.
Institución
Resumen
SUMMARY: Most papers on topology optimization consider that there is a linear relation between the strains and displacements of the structure, implicitly assuming that the displacements of the structure are small. However, when the external loads applied to the structure are large, the displacements also become large, so it is necessary to suppose that there is a nonlinear relation between strains and displacements. In this case, we say that the structure is geometrically nonlinear. In practice, this means that the linear system that needs to be solved each time the objective function of the problem is evaluated is replaced by an ill-conditioned nonlinear system of equations. Moreover, the stiffness matrix and the derivatives of the problem also become harder to compute.The objective of this work is to solve topology optimization problems under large displacements through a new optimization algorithm, named sequential piecewise linear programming. This method relies on the solution of convex piecewise linear programming subproblems that include second order information about the objective function. To speed up the algorithm, these subproblems are converted into linear programming ones. The new algorithm is not only globally convergent to stationary points but our numerical experiments also show that it is efficient and robust. © 2014 John Wiley & Sons, Ltd. 99 6 391 409 Bendsøe, M.P., Optimal shape design as a material distribution problem (1989) Structural Optimization, 1, pp. 193-202. , DOI: 10.1007/BF01650949 Sethian, J.A., Wiegmann, A., Structural boundary design via level set and immersed interface methods (2000) Journal of Computational Physics, 163 (2), pp. 489-528. , DOI: 10.1006/jcph.2000.6581 Jog, C., Distributed-parameter optimization and topology design for nonlinear thermoelasticity (1996) Computer Methods in Applied Mechanics and Engineering, 132, pp. 117-134. , DOI: 10.1016/0045-7825(95)00990-6 Buhl, T., Pedersen, C.B.W., Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization (2000) Structural and Multidisciplinary Optimization, 19, pp. 93-104. , DOI: 10.1007/s001580050089 Bruns, T.E., Tortorelli, D., Topology optimization of non-linear elastic structures and compliant mechanisms (2001) Computer Methods in Applied Mechanics and Engineering, 190, pp. 3443-3459. , DOI:10.1016/S0045-7825(00)00278-4 Gea, C.H., Luo, J., Topology optimization of structures with geometrical nonlinearities (2001) Computers and Structures, 79, pp. 1977-1985. , DOI:10.1016/S0045-7949(01)00117-1 Bruns, T.E., Sigmund, O., Tortorelli, D.A., Numerical methods for the topology optimization of structures that exhibit snap-through (2002) International Journal for Numerical Methods in Engineering, 55, pp. 1215-1237. , DOI:10.1002/nme.544 Ohsaki, M., Nishiwaki, S., Shape design of pin-jointed multistable compliant mechanisms using snapthrough behavior (2005) Structural and Multidisciplinary Optimization, 30, pp. 327-334. , DOI: 10.1007/s00158-005-0532-2 Lazarov, B.S., Schevenels, M., Sigmund, O., Robust design of large-displacement compliant mechanisms (2001) Mechanical Sciences, 2, pp. 175-182. , DOI:10.5194/ms-2-175-2011 Lahuerta, R.D., Simões, E.T., Campello, E.M.B., Pimenta, P.M., Silva, E.C.N., Towards the stabilization of the low density elements in topology optimization with large deformation (2013) Computational Mechanics, 52 (4), pp. 779-797. , DOI: 10.1007/s00466-013-0843-x Lee, H., Park, G., Topology optimization for structures with nonlinear behavior using the equivalent static loads method (2012) Journal of Mechanical Design, 134 (3). , 031004:1-031004:14, DOI: 10.1115/1.4005600 Luo, Z., Tong, L., A level set method for shape and topology optimization of large-displacement compliant mechanisms (2008) International Journal for Numerical Methods in Engineering, 76, pp. 862-892. , DOI:10.1002/nme.2352 Cho, S., Ha, S.H., Kim, M.G., Level set based shape optimization of geometrically nonlinear structures (2006) IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials Solid Mechanics and Its Applications, 137. , In, Bendsøe M, Olhoff N, Sigmund O (eds),Springer-Verlag, Netherlands. DOI: 10.1007/1-4020-4752-5_22 Kegl, M., Harl, M., Topology optimization using nonlinear finite elements and control-point-based parametrization (2013) International Journal of Nonlinear Sciences and Numerical Simulation, 14 (5), pp. 247-322. , DOI:10.1515/ijnsns-2012-0041 Dijk, N.P., Yoon, G.H., Keulen, F., Langelaar, M., A level-set based topology optimization using the element connectivity parameterization method (2010) Structural and Multidisciplinary Optimization, 42 (2), pp. 269-282. , DOI:10.1007/s00158-010-0485-y Svanberg, K., The method of moving asymptotes - a new method for structural optimization (1987) International Journal for Numerical Methods in Engineering, 24, pp. 359-373. , DOI: 10.1002/nme.1620240207 Svanberg, K., A class of globally convergent optimization methods based on conservative convex separable approximations (2002) SIAM Journal on Optimization, 12 (2), pp. 555-573. , DOI: 10.1137/S1052623499362822 Bruyneel, M., Duysinx, P., Fleury, C., A family of MMA approximations for structural optimization (2004) Structural and Multidisciplinary Optimization, 24 (4), pp. 263-276. , DOI: 10.1007/s00158-002-0238-7 Zillober, C., Global convergence of a nonlinear programming method using convex approximations (2001) Numerical Algorithms, 27, pp. 256-289. , DOI:10.1023/A:1011841821203 Zillober, C., Schittkowski, K., Moritzen, K., Very large scale optimization by sequential convex programming (2004) Optimization Methods and Software, 19 (1), pp. 103-120. , DOI:10.1080/10556780410001647195 Gomes, F.A.M., Senne, T.A., An SLP algorithm and its application to topology optimization (2011) Computational and Applied Mathematics, 30, pp. 53-89. , DOI:10.1590/S1807-03022011000100004 Sigmund, O., On the design of compliant mechanisms using topology optimization (1997) Mechanics of Structures and Machines, 25, pp. 493-524. , DOI:10.1080/08905459708945415 Kikuchi, N., Nishiwaki, S., Ono, J.S.F., Silva, E.C.S., Design optimization method for compliant mechanisms and material microstructure (1998) Computer Methods in Applied Mechanics and Engineering, 151, pp. 401-417. , DOI:10.1016/S0045-7825(97)00161-8 Nishiwaki, S., Frecker, M.I., Seungjae, M., Kikuchi, N., Topology optimization of compliant mechanisms using the homogenization method (1998) International Journal for Numerical Methods in Engineering, 42, pp. 535-559. , DOI: 10.1002/(SICI)1097-0207(19980615)42:3<535::AID-NME372>3.0.CO;2-J Etman, L.F.P., Groenwold, A.A., Rooda, J.E., First-order sequential convex programming using approximate diagonal QP subproblems (2012) Structural and Multidisciplinary Optimization, 45, pp. 479-488. , DOI: 10.1007/s00158-011-0739-3 Groenwold, A.A., Etman, L.F.P., A quadratic approximation for structural topology optimization (2010) International Journal for Numerical Methods in Engineering, 82, pp. 505-524. , DOI: 10.1002/nme.2774 Ciarlet, P.G., (1988) Mathematical Elasticity - Vol. 1: Three-Dimensional Elasticity, , North-Holland: Netherlands Simo, J.C., Hughes, T.J.R., (1998) Computational Inelasticity, , Springer-Verlag: New York, DOI: 10.1007/b98904 Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity (1977) Archive for Rational Mechanics and Analysis, 63, pp. 337-403. , DOI: 10.1007/BF00279992 Crisfield, M.A., (2001) Non-Linear Finite Element Analysis of Solids and Structures, 1. , John Wiley & Sons: Chichester Gomes, F.A.M., Maciel, M.C., Martínez, J., Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters (1999) Mathematical Programming, 84, pp. 161-200. , DOI:10.1007/s10107980014a Senne, T.A., (2013), Topology optimization of structures under geometric nonlinearity (in portuguese), PhD Thesis, State University of Campinas, Campinas, BrazilByrd, R., Nocedal, J., Waltz, R., Wu, Y., On the use of piecewise linear models in nonlinear programming (2013) Mathematical Programming A, 137, pp. 289-324. , DOI: 10.1007/s10107-011-0492-9 Byrd, R., Nocedal, J., Waltz, R., Wu, Y., (2011), An implementation of an algorithm for nonlinear programming based on piecewise linear models, Technical Report, Optimization Center, Northwestern University, Evanston, Illinois, USAGoulart, E., Herskovits, J., (2005), pp. 1-11. , Sparse quasi-Newton matrices for large scale nonlinear optimization, Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, BrazilFadel, G.M., Riley, M.F., Barthelemy, J.M., Two point exponential approximation method for structural optimization (1990) Structural Optimization, 2, pp. 117-124. , DOI: 10.1007/BF01745459 Diaz, A.R., Sigmund, O., Checkerboard patterns in layout optimization (1995) Structural and Multidisciplinary Optimization, 10, pp. 40-45. , DOI:10.1007/BF01743693 Bruns, T.E., Tortorelli, D.A., An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms (2003) International Journal for Numerical Methods in Engineering, 57, pp. 1413-1430. , DOI: 10.1002/nme.783 Guest, J.K., Prevost, J.H., Belytschko, T., Achieving minimum length scale in topology optimization using nodal design variables and projection functions (2004) International Journal for Numerical Methods in Engineering, 61, pp. 238-254. , DOI: 10.1002/nme.1064 Sigmund, O., Morphology-based black and white filters for topology optimization (2007) Structural and Multidisciplinary Optimization, 33, pp. 401-424. , DOI: 10.1007/s00158-006-0087-x Wempner, G.A., Discrete approximation related to nonlinear theories of solids (1971) International Journal of Solids and Structures, 17, pp. 1581-1599. , DOI:10.1016/0020-7683(71)90038-2 Riks, E., An incremental approach to the solution of snapping and buckling problems (1979) International Journal of Solids and Structures, 15, pp. 524-551. , DOI: 10.1016/0020-7683(79)90081-7 Batoz, J.L., Dhatt, G., Incremental displacement algorithms for nonlinear problems (1979) International Journal for Numerical Methods in Engineering, 14, pp. 1262-1267. , DOI: 10.1002/nme.1620140811 Crisfield, M.A., A fast incremental/iterative solution procedure that handles snap-through (1981) Computers and Structures, 13, pp. 55-62. , DOI:10.1016/0045-7949(81)90108-5 Chen, Y., Davis, T.A., Hager, W.W., Rajamanickam, S., Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate (2008) ACM Transactions on Mathematical Software, 35 (3). , 22:1-22:14, DOI:10.1145/1391989.1391995 Jung, D., Gea, H.C., Topology optimization of nonlinear structures (2004) Finite Element in Analysis and Design, 40, pp. 1417-1427. , DOI:10.1016/j.engstruct.2008.01.009 Dolan, E.D., Moré, J.J., Benchmarking optimization software with performance profiles (2002) Mathematical Programming A, 91, pp. 201-213. , DOI:10.1007/s101070100263 Amir, O., Bendsøe, M.P., Sigmund, O., Approximate reanalysis in topology optimization (2009) International Journal for Numerical Methods in Engineering, 78, pp. 1474-1491. , DOI: 10.1002/nme.2536