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On barrier and modified barrier multigrid methods for three-dimensional topology optimization. (English) Zbl 1432.74186
74P15 Topological methods for optimization problems in solid mechanics
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74S99 Numerical and other methods in solid mechanics
35Q93 PDEs in connection with control and optimization
90C51 Interior-point methods
65F08 Preconditioners for iterative methods
65K10 Numerical optimization and variational techniques
PENNON; PENSDP; top88.m; top.m
Full Text: DOI
[1] N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund, Giga-voxel computational morphogenesis for structural design, Nature, 550 (2017), pp. 84-86. ISSN 1476-4687.
[2] O. Amir, N. Aage, and B. S. Lazarov, On multigrid-CG for efficient topology optimization, Struct. Multidiscip. Optim., 49 (2014), pp. 815-829.
[3] E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, Efficient topology optimization in MATLAB using 88 lines of code, Struct. Multidiscip. Optim., 43 (2011), pp. 1-16. · Zbl 1274.74310
[4] R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA, 1994. · Zbl 0814.65030
[5] A. Ben-Tal and M. P. Bendsøe, A new method for optimal truss topology design, SIAM J. Optim., 3 (1993), pp. 322-358.
[6] A. Ben-Tal and M. Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Math. Program., 72 (1996), pp. 51-63. · Zbl 0851.90087
[7] A. Ben-Tal and M. Zibulevsky, Penalty/barrier multiplier methods for convex programming problems, SIAM J. Optim., 7 (1997), pp. 347-366. · Zbl 0872.90068
[8] M. Bendsøe and O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer-Verlag, Heidelberg, 2003.
[9] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31 (1977), pp. 333-390. · Zbl 0373.65054
[10] W. Briggs, V. E. Henson, and S. McCormick, A Multigrid Tutorial, SIAM, Philadelphia, PA, 2000.
[11] I. Ekeland and R. Témam, Convex Analysis and Variational Inequalities, Classics Appl. Math., SIAM, Philadelphia, PA, 1999.
[12] R. W. Freund and F. Jarre, A QMR-based interior-point algorithm for solving linear programs, Math. Program., 76 (1997), pp. 183-210. · Zbl 0881.90094
[13] W. Hackbusch, Multi-Grid Methods and Applications, Springer, Cham, 1985.
[14] F. Jarre, M. Kočvara, and J. Zowe, Optimal truss design by interior-point methods, SIAM J. Optim., 8 (1998), pp. 1084-1107. · Zbl 0912.90231
[15] M. Kočvara and S. Mohammed, Primal-dual interior point multigrid method for topology optimization, SIAM J. Sci. Comput., 38 (2016), pp. B685-B709.
[16] M. Kočvara and M. Stingl, PENNON: A code for convex nonlinear and semidefinite programming, Optim. Methods Softw., 18 (2003), pp. 317-333.
[17] M. Kočvara and M. Stingl, On the solution of large-scale SDP problems by the modified barrier method using iterative solvers, Math. Program., 109 (2007), pp. 413-444. · Zbl 1177.90312
[18] M. Kočvara, Truss topology design by conic linear optimization, in Advances and Trends in Optimization with Engineering Applications, T. Terlaky, M. F. Anjos, and S. Ahmed, eds., SIAM, Philadelphia, PA, 2017, pp. 135-147.
[19] M. Kočvara, M. Zibulevsky, and J. Zowe, Mechanical design problems with unilateral contact, ESAIM Math. Model. Numer. Anal., 32 (1998), pp. 255-281.
[20] B. Maar and V. Schulz, Interior point multigrid methods for topology optimization, Struct. Multidiscip. Optim., 19 (2000), pp. 214-224.
[21] C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629. · Zbl 0319.65025
[22] R. Polyak, Modified barrier functions (theory and methods), Math. Program., 54 (1992), pp. 177-222. · Zbl 0756.90085
[23] R. Polyak and M. Teboulle, Nonlinear rescaling and proximal-like methods in convex optimization, Math. Program., 76 (1997), pp. 265-284. · Zbl 0882.90106
[24] M. Stingl, On the solution of nonlinear semidefinite programs by augmented Lagrangian methods, Ph.D. thesis, University of Erlangen, Erlangen, Germany, 2006.
[25] K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim., 12 (2002), pp. 555-573. · Zbl 1035.90088
[26] M. Teboulle, Entropic proximal mappings with applications to nonlinear programming, Math. Oper. Res., 17 (1992), pp. 670-690. · Zbl 0766.90071
[27] S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1997. · Zbl 0863.65031
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