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Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( $${Ra}_{cr}$$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The $${Ra}_{cr}$$ value depends on the non-dimensional frequency $$\omega$$ of the boundary heat-flux modulation. Floquet theory is used to find $${Ra}_{cr}$$ for linear stability, and the energy method is used to find $${Ra}_{cr}$$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $$\omega$$ , with only the latter at large $$\omega$$ . For a given frequency, the linear stability $${Ra}_{cr}$$ is generally higher than the nonlinear stability $${Ra}_{cr}$$ , as expected. For large $$\omega$$ , $${Ra}_{cr} \omega ^{-2}$$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $$\omega$$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.
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Topology optimization of stability‐constrained structures with simple/multiple eigenvalues
Abstract This work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al.Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.
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- Award ID(s):
- 1762277
- PAR ID:
- 10486372
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 125
- Issue:
- 3
- ISSN:
- 0029-5981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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