zbMATH — the first resource for mathematics

Some convergence results in perimeter-controlled topology optimization. (English) Zbl 0947.74050
Summary: We examine the computation of optimal topologies of elastic continuum structures by using a constraint on the ‘perimeter’. Predicting macroscopic ‘black-white’ topologies without the use of homogenization techniques, this approach is presently one of the most attractive approaches in topology optimization. Mathematical justifications are given for both relaxation of discrete-value constraints on the design variable and for the finite element discretizations. It turns out that for the way in which the perimeter was calculated to date, the numerical results will not approximate the intended original problem, but a problem with a ‘taxi-cab’ perimeter which measures lengths of structural edges after projection onto the coordinate axes.

74P15 Topological methods for optimization problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI
[1] Kohn, R.V.; Strang, G.; Kohn, R.V.; Strang, G.; Kohn, R.V.; Strang, G., Optimal design and relaxation of variational problems I-III, Comm. pure appl. math., Comm. pure appl. math., Comm. pure appl. math., 39, 353-377, (1986) · Zbl 0694.49004
[2] Bendsøe, M.P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. methods. appl. mech. engrg., 71, 197-224, (1988) · Zbl 0671.73065
[3] Ambrosio, L.; Buttazzo, G., An optimal design problem with perimeter penalization, Calc. var., 1, 55-69, (1993) · Zbl 0794.49040
[4] Haber, R.B.; Bendsøe, M.P.; Jog, J., A new approach to variable-topology shape design using a constraint on the perimeter, Struct. optim., 11, 1-12, (1996)
[5] Duysinx, P., Layout optimization: A mathematical programming approach, () · Zbl 0924.73158
[6] Beckers, M., Optimisation topologique de structures continues en variable discretes, ()
[7] Beckers, M., Optimisation topologique de structures tridimcnsionelles en variable discrètes, ()
[8] Beckers, M., Méthodes du périmètre et des filtres pour l’optimisation topologique en variable discrètes, ()
[9] Petersson, J.; Sigmund, O., Slope constrained topology optimization, Int. J. num. methods engrg., 41, 1417-1434, (1998) · Zbl 0907.73044
[10] Bendsoe, M.P., ()
[11] Chambolle, A., Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. appl. math., 55, 827-863, (1995) · Zbl 0830.49015
[12] Kohn, R.V., Math. reviews, 98a, 73046, (1998)
[13] Adams, R.A., ()
[14] Evans, L.C.; Gariepy, R.F., ()
[15] Giusti, E., ()
[16] Jog, C.S.; Haber, R.B., Stability of finite element models for distributed-parameter optimization and topology design, Comp. meth. appl. mech. engrg., 130, 203-226, (1996) · Zbl 0861.73072
[17] Zhou, M.; Rozvany, G.I.N., The COC algorithm. part II: topological, geometry and generalized shape optimization, Comp. meth. appl. mech. engrg., 89, 309-336, (1991)
[18] Ciarlet, P.G., ()
[19] Cockburn, B., On the continuity in BV(ω) of the L2-projection into finite element spaces, Math. comput., 57, 551-561, (1991) · Zbl 0736.47006
[20] Sigmund, O.; Petersson, J., Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. optim., 16, 68-75, (1998)
[21] Diaz, A.R.; Sigmund, O., Checkerboard patterns in layout optimization, Struct. optim., 10, 40-45, (1995)
[22] Petersson, J., A finite element analysis of optimal variable thickness sheets, (), (also submitted) · Zbl 0938.74054
[23] Chambolle, A., Finite differences discretizations of the Mumford-Shah functional, () · Zbl 0947.65076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.