Abstract

A look into the literature of the last five years shows a growing interest in the consideration of buckling in the optimization of structures. Buckling is a stability problem. The load factors are determined by solving a generalized eigenvalue problem with the underlying structural stiffness matrix and the geometric stiffness matrix. Similar to structural dynamics, multiple eigenvalues may already exist or arise during optimization due to symmetries. The inherently associated numerical difficulties in calculating the sensitivities increase the complexity of the task compared to standard weight or compliance optimization. In addition, the buckling shapes may change during the optimization. Therefore, a mode tracking method is used to assess the correlation of the eigenmodes. In this paper, first an example of a shape optimization is presented and contrasted with a sampling analysis. Afterwards, a topology optimization using buckling constraints is conducted. Load factors can be used in the objective function or as constraints. Additional result variables that are not used as design constraints in the optimization can also be exported. All calculations are performed using literature examples in PERMAS. References: [1] Wook-han Choi, Jong-moon Kimn Gyung-Jin Park: Comparison study of some commercial structural optimization software systems, Structural and Multidisciplinary Optimization, Vol. 54, (2016), pp. 685--699. [2] Anders Clausen, Niels Age, Ole Sigmund: Exploiting additive manufacturing infill in topology optimization for improved buckling load, Engineering, Vol. 2, (2016), pp. 250--257. [3] Robert Dienemann, Axel Schumacher, Sierk Fiebig: Considering linear buckling for 3D density based topology optimization, Proceedings of the 6th International Conference on Engineering Optimization (2019), pp. 394--406. [4] Quoc Hoan Doan, Dongkyu Lee, Jaehong Lee, Joowon Kong: Design of buckling constrained multiphase material structures using continuum topology optimization, Meccanica, Vol. 54, (2019), pp. 1179--1201. [5] Peter D. Dunning, Evgueni Ovtchinnikov, Jennifer Scott, H. Alicia Kim: Level-set topology optimization with many linear buckling constraints us- ing an ecient and robust eigensolver, International Journal for Numer- ical Methods in Engineering, Vol. 107, (2016), pp. 1029--1053. [6] Xingjun Gao, Yingxiong Li, Haitao Ma, Gongfa, Chen; Improving the overall performance of continuum structures: A topology optimization model considering stiffness, strength and stability, Computer Methods in Applied Mechanics and Engineering, Vol. 359, (2020). [7] Peng Hao, Bo Wang, Kuo Tian, Gang Li, Yu Sun, Chunxiao Zhou: Fast procedure for non-uniform optimum design of stiffened shells under buckling constraint, Structural and Multidisciplinary Optimization, Vol. 55, (2017), pp. 1503--1516. [8] D. Manickarajah, Y. M. Xie, G. P. Steven: Optimisation of columns and frames against buckling, Computers and Structures, Vol. 75, (2000), pp. 45--54. [9] W. Szyskowski: Multimodal optimality criterion for maximum stability, International Journal Non-Linear Mechanics, Vol. 77, (1992), pp. 623--633. [10] Scott Townsend, H. Alicia Kim: A level set topology optimization method for the buckling of shell structures, Structural and Multidisciplinary Optimization, Vol. 60, (2019) pp. 1783-1800. [11] Nils Wagner, Reinhard Helfrich: Einfluss von Parametervariationen auf das Beulen von versteiften Laminatstrukturen, NAFEMS DACH Konferenz, Bamberg, 25-27 April 2016 [12] Dan Wang, Mostafa M. Abdalla, Weihong Zhang: Buckling optimiza- tion design of curved stieners for grid-stiffened composite structures, Composite Structures, Vol. 159, (2017), pp. 656--666.