Optimal design of the stability of a minimum surface shell on a rectangular contour with accounting for geometric nonlinearity under thermo-stress loading
DOI:
https://doi.org/10.32347/2410-2547.2026.116.470-480Keywords:
optimization, parametric optimization, multicriteria optimization, objective function optimization, design variables, constraints, minimum surface envelopes, geometric nonlinearity, finite element method, optimal designAbstract
The field of optimal design for spatial thin-walled structures has been developing since the 1950s. In general, when studying optimal design, one must determine all optimized strength characteristics based on the second group of limit states. These strength characteristics include: strength, stability, deflections, and deformations in spatial thin-walled structures. The implementation of optimal design with these strength characteristics occurs simultaneously with the weight of the spatial thin-walled structure.
Multi-criteria parametric optimization is achieved by incorporating two objective functions simultaneously into the mathematical framework and research methodology for spatial thin-walled structures. These include weight or volume, and strength characteristics. Such objective functions include: weight and strength, weight and stability, weight and deflections, and weight and forced or natural frequencies of vibration of a spatial thin-walled structure. The specificity of such studies lies in the fact that when investigating these objective functions, it is necessary to employ three different types of calculations.
Optimal design problems can be formulated in both linear and nonlinear settings. Nonlinear formulations include geometric and physical ones. This scientific article examines the geometric nonlinear formulation, which allows for the consideration of actual displacements and stresses when determining the bifurcation point. The instability coefficient λ is, in fact, the bifurcation point of the minimal surface.
The use of this algorithm leads to a new approach in the design of building structures. This type of optimal design offers significant economic benefits when creating future structures. This approach can be applied not only to shell structures but also to beam and plate structures. An important aspect is the material, which must be isotropic; this includes metals and composites.
The analysis of the objective function is shown in Figure 1.14. We were able to reduce the weight of the minimal surface shell from 52,593 kg to 40,391 kg, which represents a 23.12% reduction. At the same time, we were able to redistribute the thickness of the minimal surface shell to the loaded zones, which made it possible to reduce the buckling coefficient λ from 2.58 to 1.06. This significant result was achieved through a geometrically nonlinear formulation of the problem. Automation, in the optimal design approach, makes it possible to determine the required thickness of the minimal surface shell.
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