Browsing by Author "Fuentes, Federico"

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    Fourier analysis of membrane locking and unlocking
    (2023) Hiemstra, Rene R.; Fuentes, Federico; Schillinger, Dominik
    Despite four decades of research membrane locking remains an important issue hindering the development of effective beam and shell finite elements. In this article, we utilize Fourier analysis of the complete spectrum of natural vibrations and propose a criterion to identify and evaluate the severity of membrane locking. To demonstrate our approach, we utilize primal and mixed Galerkin formulations applied to a circular Euler-Bernoulli ring discretized using uniform, periodic B-splines. By analytically computing the discrete Fourier operators, we obtain an exact representation of the normalized error across the entire spectrum of eigenvalues. Our investigation addresses key questions related to membrane locking, including mode susceptibility, the influence of polynomial order, and the impact of shell/beam thickness and radius of curvature. Furthermore, we compare the effectiveness of mixed and primal Galerkin methods in mitigating locking. By providing insights into the parameters affecting locking and introducing a criterion to evaluate its severity, this research contributes to the development of improved numerical methods for thin beams and shells.(c) 2023 Elsevier B.V. All rights reserved.
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    GLOBAL MINIMIZATION OF POLYNOMIAL INTEGRAL FUNCTIONALS
    (2024) Fantuzzi, Giovanni; Fuentes, Federico
    We describe a "discretize-then-relax" strategy to globally minimize integral functionals over functions u in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on u and its derivatives, even if it is nonconvex. The "discretize" step uses a bounded finite element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size h of the finite element mesh. The "relax" step employs sparse moment-sum-of-squares relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order omega . We prove that, as omega ->infinity and h -> 0 , solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain L-p norms) to the global minimizer of the original integral functional if it is unique. We also report computational experiments showing that our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.
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    Global Stability of Fluid Flows Despite Transient Growth of Energy
    (2022) Fuentes, Federico; Goluskin, David; Chernyshenko, Sergei
    Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct nonquadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the the energy stability threshold found by Orr in 1907 [The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid, Proc. R. Ir. Acad. Sect. A 27, 69 (1907)]. This is the first global stability result for any flow that surpasses the energy method.