zbMATH — the first resource for mathematics

State space Newton’s method for topology optimization. (English) Zbl 1423.74744
Summary: We introduce a new algorithm for solving certain classes of topology optimization problems, which enjoys fast local convergence normally achieved by the full space methods while working in a smaller reduced space. The computational complexity of Newton’s direction finding subproblem in the algorithm is comparable with that of finding the steepest descent direction in the traditional first order nested/reduced space algorithms for topology optimization. That is, the space reduction is computationally inexpensive, and more importantly it does not ruin the sparsity of the full-space system of optimality conditions. The fast local convergence of the algorithm allows one to efficiently solve a sequence of optimization problems for varying parameters (numerical continuation). This can be utilized for eliminating the errors introduced by the approximate enforcement of the boundary conditions or \(0 / 1\)-type constraints on the design field through penalties in many topology optimization approaches.
We test the algorithm on the benchmark problems of dissipated power minimization for Stokes flows, and in all cases the algorithm outperforms the traditional first order reduced space/nested approaches by a factor varying from two to almost twenty in terms of the number of iterations while attaining an almost unprecedented accuracy in solving the discretized topology optimization problem. Finally we present a few extensions to the algorithm, one involving computations on adaptively refined meshes and another related to solving topology optimization problems for non-Newtonian fluids.

74P15 Topological methods for optimization problems in solid mechanics
49M15 Newton-type methods
65K10 Numerical optimization and variational techniques
65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[1] Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 2, 197-224, (1988) · Zbl 0671.73065
[2] Bendsøe, M. P.; Sigmund, O., Topology optimization: theory, methods, and applications, (2003), Springer-Verlag Berlin · Zbl 1059.74001
[3] Allaire, G., Shape optimization by the homogenization method, (2002), Springer · Zbl 0990.35001
[4] Sigmund, O.; Maute, K., Topology optimization approaches, Struct. Multidiscip. Optim., 48, 1031-1055, (2013)
[5] Maar, B.; Schulz, V., Interior point multigrid methods for topology optimization, Struct. Optim., 19, 3, 214-224, (2000)
[6] Hoppe, R.; Petrova, S.; Schulz, V., A primal-dual Newton-type interior-point method for topology optimization, J. Optim. Theory Appl., 114, 3, 545-571, (2002) · Zbl 1026.90108
[7] Schulz, V., Simultaneous solution approaches for large optimization problems, J. Comput. Appl. Math., 164-165, 629-641, (2004) · Zbl 1038.65055
[8] Biros, G.; Ghattas, O., Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. part I. the Krylov-Schur solver, SIAM J. Sci. Comput., 27, 2, 1-27, (2005) · Zbl 1091.65061
[9] Borrvall, T.; Petersson, J., Topology optimization of fluids in Stokes flow, Internat. J. Numer. Methods Fluids, 41, 1, 77-107, (2003) · Zbl 1025.76007
[10] Sigmund, O., On the usefulness of non-gradient approaches in topology optimization, Struct. Multidiscip. Optim., 43, 5, 589-596, (2011) · Zbl 1274.74390
[11] Svanberg, K., The method of moving asymptotes—a new method for structural optimization, Internat. J. Numer. Methods Engrg., 24, 2, 359-373, (1987) · Zbl 0602.73091
[12] Svanberg, K., A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim., 12, 2, 555-573, (2002) · Zbl 1035.90088
[13] Hinze, M., A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30, 45-63, (2005) · Zbl 1074.65069
[14] Maute, K.; Ramm, E., Adaptive topology optimization, Struct. Optim., 10, 2, 100-112, (1995)
[15] Maute, K.; Schwarz, S.; Ramm, E., Adaptive topology optimization of elastoplastic structures, Struct. Optim., 15, 2, 81-91, (1998)
[16] Stainko, R., An adaptive multilevel approach to the minimal compliance problem in topology optimization, Comm. Numer. Methods Engrg., 22, 2, 109-118, (2006) · Zbl 1095.74023
[17] Wang, Y.; Kang, Z.; He, Q., An adaptive refinement approach for topology optimization based on separated density field description, Comput. Struct., 117, 10-22, (2013)
[18] Wang, Y.-Q.; He, J.-J.; Luo, Z.; Kang, Z., An adaptive method for high-resolution topology design, Acta Mech. Sin., 29, 6, 840-850, (2013) · Zbl 1346.74156
[19] Wang, Y.; Kang, Z.; He, Q., Adaptive topology optimization with independent error control for separated displacement and density fields, Comput. Struct., 135, 50-61, (2014)
[20] Bennett, J. A.; Botkin, M. E., Structural shape optimization with geometric description and adaptive mesh refinement, AIAA J., 23, 3, 458-464, (1985)
[21] Noboru, K.; Chung, K. Y.; Toshikazu, T.; Taylor, J. E., Adaptive finite element methods for shape optimization of linearly elastic structures, Comput. Methods Appl. Math., 57, 1, 67-89, (1986) · Zbl 0578.73081
[22] Bugeda, G.; Oliver, J., A general methodology for structural shape optimization problems using automatic adaptive remeshing, Int. J. Numer. Methods Eng., 36, 18, 3161-3185, (1993) · Zbl 0780.73049
[23] Schleupen, A.; Maute, K.; Ramm, E., Adaptive FE-procedures in shape optimization, Struct. Multidiscip. Optim., 19, 4, 282-302, (2000)
[24] Keulen, F. V.; Toropov, V. V., New developments in structural optimization using adaptive mesh refinement and multipoint approximations, Eng. Optim., 29, 1-4, 217-234, (1997)
[25] Désidéri, J.-A.; Majd, B. A.E.; Janka, A., Nested and self-adaptive Bézier parameterizations for shape optimization, J. Comput. Phys., 224, 1, 117-131, (2007) · Zbl 1123.65303
[26] Berrone, S.; Verani, M., A new marking strategy for the adaptive finite element approximation of optimal control constrained problems, Optim. Methods Softw., 26, 4-5, 747-775, (2011) · Zbl 1228.65095
[27] Kaland, L.; Los Reyes, J. De; Gauger, N., One-shot methods in function space for PDE-constrained optimal control problems, Optim. Methods Softw., 29, 2, 376-405, (2014) · Zbl 1283.49036
[28] Theillard, M.; Djodom, L. Fokoua; Vie, J.-L.; Gibou, F., A second-order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids: application to shape optimization, J. Comput. Phys., 233, 430-438, (2013) · Zbl 1286.74109
[29] Morin, P.; Nochetto, R. H.; Pauletti, M. S.; Verani, M., Adaptive finite element method for shape optimization, ESAIM Control Optim. Calc. Var., 18, 4, 1122-1149, (2012) · Zbl 1259.49046
[30] Hess, W.; Ulbrich, S., An inexact \(\ell\)1 penalty SQP algorithm for PDE-constrained optimization with an application to shape optimization in linear elasticity, Optim. Methods Softw., 28, 5, 943-968, (2013) · Zbl 1278.49040
[31] Talischi, C.; Paulino, G. H., An operator splitting algorithm for Tikhonov-regularized topology optimization, Comput. Methods Appl. Math., 253, 599-608, (2013) · Zbl 1297.74090
[32] Othmer, C., A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows, Internat. J. Numer. Methods Fluids, 58, 8, (2008) · Zbl 1152.76025
[33] Amir, O.; Stolpe, M.; Sigmund, O., Efficient use of iterative solvers in nested topology optimization, Struct. Multidiscip. Optim., 42, 1, 55-72, (2010) · Zbl 1274.74308
[34] Renardy, M.; Rogers, R. C., An introduction to partial differential equations, vol. 13, (2004), Springer
[35] Kurdila, A. J.; Zabarankin, M., Convex functional analysis, (2005), Birkhäuser · Zbl 1077.46002
[36] Moré, J. J.; Sorensen, D. C.; Hillstrom, K. E.; Garbow, B. S., The MINPACK project, (Cowell, W. J., Sources and Development of Mathematical Software, (1984), Prentice-Hall), 88-111
[37] Ortega, J. M., The Newton-Kantorovich theorem, Amer. Math. Monthly, 75, 6, 658-660, (1968) · Zbl 0183.43004
[38] Strang, G.; Fix, G., An analysis of the finite element method, (2008), Wellesley-Cambridge Press · Zbl 0278.65116
[39] Karniadakis, G.; Sherwin, S., Spectral/hp element methods for computational fluid dynamics, (2013), Oxford University Press · Zbl 1256.76003
[40] Logg, A.; Wells, G.; Hake, J., DOLFIN: a C++/python finite element library, (Automated Solution of Differential Equations by the Finite Element Method, Lecture Notes in Computational Science and Engineering, vol. 84, (2012), Springer), (Chapter 10)
[41] Bangerth, W.; Hartmann, R.; Kanschat, G., Deal.II—A general purpose object oriented finite element library, ACM Trans. Math. Software, 33, 4, 24/1-24/27, (2007) · Zbl 1365.65248
[42] Hood, P.; Taylor, C., Numerical solution of the Navier-Stokes equations using the finite element technique, Comput. & Fluids, 1, 1-28, (1973) · Zbl 0328.76020
[43] Crouzeix, M.; Raviart, P., Conforming and nonconforming finite-element methods for solving stationary Stokes equations I, Rev. Fr. Autom. Inform., 7, 33-75, (1973) · Zbl 0302.65087
[44] Arnold, D.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344, (1984) · Zbl 0593.76039
[45] Challis, V. J.; Guest, J. K., Level-set topology optimization of fluids in Stokes flow, Internat. J. Numer. Methods Engrg., 79, 1284-1308, (2009) · Zbl 1176.76039
[46] Davis, T. A., Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30, 2, 196-199, (2004) · Zbl 1072.65037
[47] Aage, N.; Poulsen, T. H.; Gersborg-Hansen, A.; Sigmund, O., Topology optimization of large scale Stokes flow problems, Struct. Multidiscip. Optim., 35, 2, 175-180, (2008) · Zbl 1273.76094
[48] Carstensen, C.; Peterseim, D.; Rabus, H., Optimal adaptive nonconforming FEM for the Stokes problem, Numer. Math., 123, 2, 291-308, (2013) · Zbl 1316.76046
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.