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“Gentlest postbuckling behavior” means many small mode jumps in the postbuckling regime rather than few large and potentially damaging mode jumps.
The authors of the paper write:
“In this study, numerical methods for structural shape optimization proceeded in three major steps (see Fig. 2): (1) A finite element eigenvalue analysis [to find the basis vectors of the imperfection shape] and a nonlinear analysis using the finite element program Abaqus [20] [to compute postbuckling states]; (2) response evaluation through a self-developed code in Matlab [21]; and (3) sensitivity analysis and design updates through the design optimization software HEEDS [22]. Detailed descriptions for each step are given in the following sections [of the paper].”
FROM:
Nan Hu, Rigoberto Burgueno and Nizar Lajnef (Dept. of Civil and Environmental Engineering, Michigan State University, East Lansing, Michigan, USA), “Structural optimization and form-finding of cylindrical shells for targeted elastic postbuckling response”, Proceedings of the ASME 2014 Smart Materials, Adaptive Structures and Intelligent Systems (SMASIS2014), September 8-10, 2014, Newport, Rhode Island, USA
ABSTRACT: This paper presents a finite element based numerical study on controlling the postbuckling behavior of thin-walled cylindrical shells under axial compression. With the increasing interest of various disciplines for harnessing elastic instabilities in materials and mechanical systems, the postbuckling behavior of thin-walled cylindrical shells may have a new role to design materials and structures at multiple scales with switchable functionalities, morphogenesis, etc. In the design optimization approach presented herein, the mode shapes and their amplitudes are linearly combined to generate initial geometrical designs with predefined imperfections. A nonlinear postbuckling finite element analysis evaluates the design objective function, i.e., the desired postbuckling force- displacement path. Single and multi-objective optimization problems are formulated with design variables consisting of shape parameters that scale base eigenvalue shapes. A gradient-based algorithm and numerical sensitivity evaluations are used. Results suggest that an optimized shape for a cylindrical shell can achieve a targeted response in the elastic postbuckling regime with multiple mode transitions and energy dissipation characteristics. The optimization process and the obtained geometry can be potentially used for energy harvesting and other sensing and actuation applications.
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