Justin T. Webster

Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

This paper is devoted to a long-time behavior analysis of flow-structure interactions at subsonic and supersonic velocities. An intrinsic component of that analysis is the study of attractors corresponding to von Karman plate equations with delayed terms and without rotational terms. The presence of delay terms in the dynamical system leads to a loss of gradient structure, while the absence of rotational terms in von Karman plates leads to the loss of compactness of the orbits. Both of these features make the analysis of long-time behavior rather subtle, rendering the established tools in the theory of PDE and dynamical systems not applicable. We develop methodology that is capable of addressing this class of problems.

Feedback Stabilization of a Fluttering Panel in an Inviscid Subsonic Potential Flow

Asymptotic-in-time feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [E. Dowell, AIAA, 5 (1967), pp. 1857--1862] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of imposed energy dissipation the plate dynamics converge to a compact and finite dimensional set [I. Chueshov, I. Lasiecka, and J. T. Webster, Comm. Partial Differential Equations, 39 (2014), pp. 1965--1997]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the stationary set in the natural energy topology. To accomplish this task, a novel decomposition of the nonlinear plate dynamics is utilized: a smooth component (globally bounded in a higher topology) and a uniformly exponentially decaying component. Our result implies that flutter (a periodic or chaotic end behavior) can be eliminated (in subsonic flows) with sufficient frictio...

Long-time dynamics and control of subsonic flow-structure interactions

In this paper we present recent results concerning nonlinear (von Karman and Berger) flow-structure interactions with a focus on stability, long-time dynamics, and convergence to equilibria. Flow-structure interactions describe the interaction of a flow of gas over a flexible plate. In particular, we (1) outline well-posedness results via nonlinear semigroup methods, (b) analyze damping mechanisms in the plate (interior and boundary), (c) discuss approaches to the study of long-time behavior of solutions (i.e. global compact attracting sets), and (d) present preliminary results concerning the asympototic smoothness of both the von Karman and Berger flow-structure systems with boundary damping. We conclude with an assessment of several open problems for both the von Karman and Berger flow-structure interactions.

Advection of methane in the hydrate zone: model, analysis and examples

A two-phase two-component model is formulated for the advective-diffusive transport of methane in liquid phase through sediment with the accompanying formation and dissolution of methane hydrate. This free-boundary problem has a unique generalized solution in

A Cantilevered Extensible Beam in Axial Flow: Semigroup Well-posedness and Postflutter Regimes

L^1

Controlling flutter for nonlinear panels in subsonic flows via structural velocity feedback

; the proof combines analysis of the stationary semilinear elliptic Dirichlet problem with the nonlinear semigroup theory in Banach space for an m-accretive multi-valued operator. Additional estimates of maximum principle type are obtained, and these permit appropriate maximal extensions of the phase-change relations. An example with pure advection indicates the limitations of these estimates and of the model developed here. We also consider and analyze the coupled pressure equation that determines the advective flux in the transport model.

Weak and strong solutions of a nonlinear subsonic flow–structure interaction: Semigroup approach

We consider a cantilevered (clamped-free) beam in an axial potential flow. Certain flow velocities may bring about a bounded-response instability in the structure, termed {\em flutter}. As a preliminary analysis, we employ the theory of {\em large deflections} and utilize a piston-theoretic approximation of the flow for appropriate parameters, yielding a nonlinear (Berger/Woinowsky-Krieger) beam equation with a non-dissipative RHS. As we obtain this structural model via a simplification, we arrive at a nonstandard nonlinear boundary condition that necessitates careful well-posedness analysis. We account for rotational inertia effects in the beam and discuss technical issues that necessitate this feature. We demonstrate nonlinear semigroup well-posedness of the model with the rotational inertia terms. For the case with no rotational inertia, we utilize a Galerkin approach to establish existence of weak, possibly non-unique, solutions. For the former, inertial model, we prove that the associated non-gradient dynamical system has a compact global attractor. Finally, we study stability regimes and {\em post-flutter} dynamics (non-stationary end behaviors) using numerical methods for models with, and without, the rotational inertia terms.

Evolution Semigroups in Supersonic Flow-Plate Interactions

Mechanical control of flutter for a thin panel immersed in an inviscid flow is considered. The model arises in aeroelasticity and comprises the interaction between a clamped von Karman plate a surrounding potential flow of gas. Recent results show that the plate dynamics of the model converge to a global compact attracting set of finite dimension [6]. This result was obtained in the absence of mechanical damping of any type. Here, we incorporate a sufficiently large velocity feedback control applied to the structure to show that the full flow-plate system exhibits strong convergence to a stationary state (when flows are subsonic and a “good” energy identity is available). Our method is based on first showing the desired convergence properties when the plate dynamics exhibit additional regularity. We then show a dichotomy for the plate dynamics: they are either asymptotically regular or the plate velocities decay uniformly exponentially. In the case when no additional plate regularity is available, we utilize an approximation by smooth initial data; this requires propagation of initial regularity on the infinite time horizon. The final result complements results previous obtained (for this model and similar models), as we show that there is a strong convergence for the entire dynamics and that the limiting behavior of the flow-plate system is, in fact, stationary. Physically, this implies that flutter (a non-static end behavior) can be eliminated by a velocity feedback control in subsonic flows.