Abstract
The paper considers a classic formulation of the topology optimization problem of discrete or discretized structures. The objective function to be maximized is the smallest natural frequency of the structure. We develop non-heuristic mathematical models paying special attention to the situation when some design variables take zero values. These models take into account multiple load conditions, equilibrium of forces, constraints on compliance and volume, and the effect of possible non-structural mass. We discuss serious obstacles for a successful numerical treatment of this formulation such as non-Lipschitzean behavior and even discontinuity of the objective function. As a cure, we present an equivalent reformulation as a bilinear semidefinite programming problem without the pitfalls of the original problem. An algorithm is presented for finding an approximation of a globally optimal solution up to a user-defined accuracy. The key ingredient of this algorithm is the treatment of a sequence of linear semidefinite programs. Numerical examples are provided for truss structures. Examples of both academic and larger size illustrate the theoretical results achieved and demonstrate the practical use of this approach. We conclude with an extension on multiple non-structural mass conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Achtziger W, Kočvara M (2006) Structural topology optimization with eigenvalues. Tech. Rep. No. 315, Institute of Applied Mathematics, University of Dortmund, Germany (to appear in SIAM Journal on Optimization)
Achtziger W, Ben-Tal A, Bendsøe M, Zowe J (1992) Equivalent displacement based formulations for maximum strength truss topology design. Impact Comput Sci Eng 4:315–345
Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia
Bendsøe M, Sigmund O (2002) Topology optimization. Theory, methods and applications. Springer, Heidelberg
Bhatia R (1996) Matrix analysis. Springer, New York
Bhatia R, Li RC (1996) On perturbations of matrix pencils with real spectra. II. Math Comput 65:637–645
Clarke FH (1983) Optimization and nonsmooth analysis. Wiley, New York
Gantmacher FR (1959) The theory of matrices, vol 1. Chelsea, New York
de Klerk E (2002) Aspects of semidefinite programming. Kluwer, Dordrecht
Kočvara M, Stingl M (2003) PENNON—a code for convex nonlinear and semidefinite programming. Optim Methods Softw 18(3):317–333
Kočvara M, Stingl M (2006) PENBMI User’s Guide. Version 2.1. http://www.penopt.com/
Kočvara M, Leibfritz F, Stingl M, Henrion D (2004) A nonlinear SDP algorithm for static output feedback problems in COMPlib. LAAS-CNRS research report no. 04508, LAAS, Toulouse
Lewis AS, Overton ML (1996) Eigenvalue optimization. Acta Numer 5:149–190
Löfberg J (2004) YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD conference, Taipei, Taiwan. Available from http://control.ee.ethz.ch/~joloef/yalmip.php
Luenberger D (1997) Optimization by vector space methods. Wiley, New York
Ohsaki M, Fujisawa K, Katoh N, Kanno Y (1999) Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints. Comp Meth Appl Mech Eng 180:203–217
Olhoff N (1980) Optimal design with respect to structural eigenvalues. In: Rimrott F, Tabarott B (eds) Theoretical and applied mechanics, Proc. XVth Int. IUTAM Congress, North-Holland, pp 133–149
Olhoff N, Rasmussen SH (1977) On single and bimodal optimum bukling loads of clamped columns. Int J Solids Struct 13:605–614
Seyranian AP, Mailybaev AA (2003) Multiparameter stability theory with mechanical applications. World Scientific, Singapore
Stewart GW (1979) Perturbation bounds for the definite generalized eigenvalue problem. Linear Algebra Appl 23:69–86
Stingl M (2005) On the solution of nonlinear semidefinite programs by augmented lagrangian methods. Ph.D. Thesis, Inst. of Appl. Math., University of Erlangen
Zhang T, Law KH, Golub GH (1998) On the homotopy method for perturbed symmetric generalized eigenvalue problems. SIAM J Sci Comput 19:1625–1645
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Achtziger, W., Kočvara, M. On the maximization of the fundamental eigenvalue in topology optimization. Struct Multidisc Optim 34, 181–195 (2007). https://doi.org/10.1007/s00158-007-0117-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-007-0117-3