Browsing by Author "Webster, Justin"
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Item Dynamic Equations of Motion for Inextensible Beams and Plates(Springer Nature, 2022-04-20) Deliyianni, Maria; McHugh, Kevin; Webster, Justin; Dowell, EarlThe large deflections of cantilevered beams and plates are modeled and discussed. Traditional nonlinear elastic models (e.g., that of von Karman) employ elastic restoring forces based on the effect of stretching on bending, and these are less applicable to cantilevers. Recent experimental work indicates that elastic cantilevers are subject to nonlinear inertial and stiffness effects. We review a recently established (quasilinear and nonlocal) cantilevered beam model, and consider some natural extensions to two dimensions -- namely, inextensible plates. Our principal configuration is that of a thin, isotropic, homogeneous rectangular plate, clamped on one edge and free on the remaining three. We proceed through the geometric and elastic modeling to obtain equations of motion via Hamilton's principle for the appropriately specified energies. We enforce {\em effective} inextensibility constraints through Lagrange multipliers. Multiple plate analogs of the established 1D model are obtained, based on various assumptions. For each plate model, we present the modeling hypotheses and the resulting equations of motion. It total, we present three distinct nonlinear partial differential equation models, and, additionally, describe a class of ``higher order" models. Each model has particular advantages and drawbacks for both mathematical and engineering analyses. We conclude with an in depth discussion and comparison of the various systems and some analytical problems.Item Elastic stabilization of an intrinsically unstable hyperbolic flow–structure interaction on the 3D half-space(World Scientific Publishing, 2023-03-13) Balakrishna, Abhishek; Lasiecka, Irena; Webster, JustinThe strong asymptotic stabilization of 3D hyperbolic dynamics is achieved by a damped 2D elastic structure. The model is a Neumann wave-type equation with low regularity coupling conditions given in terms of a nonlinear von Karman plate. This problem is motivated by the elimination of aeroelastic instability (sustained oscillations of bridges, airfoils, etc.) in engineering applications. Empirical observations indicate that the subsonic wave-plate system converges to equilibria. Classical approaches which decouple the plate and wave dynamics have fallen short. Here, we operate on the model as it appears in the engineering literature with no regularization and achieve stabilization by microlocalizing the Neumann boundary data for the wave equation (given through the plate dynamics). We observe a compensation by the plate dynamics precisely where the regularity of the 3D Neumann wave is compromised (in the characteristic sector).Item Flutter stabilization for an unstable hyperbolic flow-plate interaction(2024-01-21) Lasiecka, Irena; Webster, JustinThe aim of these lectures is to present a mathematical theory of control problems arising in flow-structure interactions. From the modeling point of view, these are coupled problems of two dynamicsat an interface. The equations of interest are an unstable linearization of the compressible Eulerequation with a nonlinear plate dynamics. The latter accounts for the effects of possibly largedisplacements, which are typically modeled by the scalar or vectorial von Karman equations oflarge plate deflections. The coupled model is a hybrid interaction between the flow—defined on a3D flow domain—and the plate or shell—defined on a 2D manifold. The unperturbed flow moves in3D with normalized velocity U > 0and excites the structure through the distributed pressure, actingthrough traces on the lower dimensional manifold. As a consequence, the structural displacementsthen perturb the flow. The communication (feedback) between these two systems lies in the heartof the mathematical problem. The described interaction is ubiquitous in nature, with a multitudeof physical applications. Some central examples arise in Fluid Mechanics, Aerospace Applications,Structural Engineering, and Biological Applications.Item Large Deflections of A Structurally Damped Panel in A Subsonic Flow(Springer Nature, 2020-07-18) Balakrishna, Abhishek; Webster, JustinThe large deflections of panels in subsonic flow are considered. Specifically, a fully clamped von Karman plate accounting for both rotational inertia in plate filaments and structural damping of square root type is considered. The panel is taken to be embedded in the boundary of a linear, subsonic potential flow on the positive halfspace in R³. Solutions are constructed via a semigroup approach despite the lack of natural dissipativity associated to the generator of the linear dynamics. The flow-plate dynamics are then reduced---via an explicit Neumann-to-Dirichlet (downwash-to-pressure) solver for the flow---to a memory-type dynamical system for the plate. For the non-conservative plate dynamics, a global attractor is explicitly constructed via Lyapunov and quasi-stability methods. Finally, it is shown that via the compactness of the attractor and finiteness of the dissipation integral, that all trajectories converge strongly to the set of stationary states.Item Large Deflections of Inextensible Cantilevers: Modeling, Theory, and Simulation(EDP Sciences, 2020-09-24) Deliyianni, Maria; Gudibanda, Varun; Howell, Jason; Webster, JustinA recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam’s inextensibility—local arc length preservation—rather than traditional extensible effects attributed to fully restricted boundary conditions. Enforcing inextensibility leads to: nonlinear stiffness terms, which appear as quasilinear and semilinear effects, as well as nonlinear inertia effects, appearing as nonlocal terms that make the beam implicit in the acceleration. In this paper we discuss the derivation of the equations of motion via Hamilton’s principle with a Lagrange multiplier to enforce the effective inextensibility constraint. We then provide the functional framework for weak and strong solutions before presenting novel results on the existence and uniqueness of strong solutions. A distinguishing feature is that the two types of nonlinear terms prevent independent challenges: the quasilinear nature of the stiffness forces higher topologies for solutions, while the nonlocal inertia requires the consideration of Kelvin-Voigt type damping to close estimates. Finally, a modal approach is used to produce mathematically-oriented numerical simulations that provide insight to the features and limitations of the inextensible model.Item A Linearized Viscous, Compressible Flow-Plate Interaction with Non-dissipative Coupling(Elsevier, 2019-05-14) Avalos, George; Geredeli, Pelin Guven; Webster, JustinWe address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile U . In contrast to model in [Avalos, Geredeli, Webster, 2017], we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a C₀ -semigroup {eᴬᵗ}ₜ≥₀ on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator A , yielding the flow-structure dynamics.Item Long-time dynamics of a hinged-free plate driven by a non-conservative force(EMS Press, 2022-02-25) Bonheure, Denis; Gazzola, Filippo; Lasiecka, Irena; Webster, JustinA partially hinged, partially free rectangular plate is considered, with the aim to address the possible unstable end behaviors of a suspension bridge subject to wind. This leads to a nonlinear plate evolution equation with a nonlocal stretching active in the span-wise direction. The wind-flow in the chord-wise direction is modeled through a piston-theoretic approximation, which provides both weak (frictional) dissipation and non-conservative forces. The long-time behavior of solutions is analyzed from various points of view. Compact global attractors, as well as fractal exponential attractors, are constructed using the recent quasi-stability theory. The non-conservative nature of the dynamics requires the direct construction of a uniformly absorbing ball, and this relies on the superlinearity of the stretching. For some parameter ranges, the non-triviality of the attractor is shown through the spectral analysis of the stationary linearized (non self-adjoint) equation and the existence of multiple unimodal solutions is shown. Several stability results, obtained through energy estimates under various smallness conditions and/or assumptions on the equilibrium set, are also provided. Finally, the existence of a finite set of determining modes for the dynamics is demonstrated, justifying the usual modal truncation in engineering for the study of the qualitative behavior of suspension bridge dynamics.Item Multilayered Poroelasticity Interacting with Stokes Flow(SIAM, 2021-11-01) Bociu, Lorena; Čanić, Sunčica; Muha, Boris; Webster, JustinWe consider the interaction between an incompressible, viscous fluid modeled by the dynamic Stokes equation and a multilayered poroelastic structure which consists of a thin, linear, poroelastic plate layer (in direct contact with the free Stokes flow) and a thick Biot layer. The fluid flow and the elastodynamics of the multilayered poroelastic structure are fully coupled across a fixed interface through physical coupling conditions (including the Beavers-Joseph-Saffman condition), which present mathematical challenges related to the regularity of associated velocity traces. We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe's method, and using energy methods and a version of Aubin-Lions compactness lemma (in the nonlinear case) to recover the weak solution as the limit of approximate subsequences. We also provide uniqueness criteria and show that constructed weak solutions are indeed strong solutions to the coupled problem if one assumes additional regularity.Item Nonlinear Quasi-static Poroelasticity(Elsevier, 2021-06-10) Bociu, Lorena; Webster, JustinWe analyze a quasi-static Biot system of poroelasticity for both compressible and incompressible constituents. The main feature of this model is a nonlinear coupling of pressure and dilation through the system's permeability tensor. Such a model has been analyzed previously from the point of view of constructing weak solutions through a fully discretized approach. In this treatment, we consider simplified Dirichlet type boundary conditions in the elastic displacement and pressure variables and give a full treatment of weak solutions. Our construction of weak solutions for the nonlinear problem is natural and based on a priori estimates, a requisite feature in addressing the nonlinearity. This is in contrast to previous work which exploits linearity or monotonicity in the permeability, both of which are not available here. We utilize a spatial semi-discretization and employ a multi-valued fixed point argument in for a clear construction of weak solutions. We also provide regularity criteria for uniqueness of solutions.Item Theory of Solutions for An Inextensible Cantilever(Springer Nature, 2021-07-12) Deliyianni, Maria; Webster, JustinRecent equations of motion for the large deflections of a cantilevered elastic beam are analyzed. In the traditional theory of beam (and plate) large deflections, nonlinear restoring forces are due to the effect of stretching on bending; for an inextensible cantilever, the enforcement of arc-length preservation leads to quasilinear stiffness effects and inertial effects that are both nonlinear and nonlocal. For this model, smooth solutions are constructed via a spectral Galerkin approach. Additional compactness is needed to pass to the limit, and this is obtained through a complex procession of higher energy estimates. Uniqueness is obtained through a non-trivial decomposition of the nonlinearity. The confounding effects of nonlinear inertia are overcome via the addition of structural (Kelvin-Voigt) damping to the equations of motion. Local well-posedness of smooth solutions is shown first in the absence of nonlinear inertial effects, and then shown with these inertial effects present, taking into account structural damping. With damping in force, global-in-time, strong well-posedness result is obtained by achieving exponential decay for small data.Item Weak and Strong Solutions for A Fluid-Poroelastic-Structure Interaction via A Semigroup Approach(2024-01-08) Avalos, George; Gurvich, Elena; Webster, JustinA filtration system, comprising a Biot poroelastic solid coupled to an incompressible Stokes free-flow, is considered in 3D. Across the flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. In the inertial, linear, and non-degenerate case, the hyperbolic-parabolic coupled problem is posed through a dynamics operator on an appropriate energy space, adapted from Stokes-Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. C0-semigroup generation for the dynamics operator is obtained with a non-standard maximality argument. The latter employs a mixed-variational formulation in order to invoke the Babuška-Brezzi theorem. The Lumer-Philips theorem yields semigroup generation, and thereby, strong and generalized solutions are obtained. As the dynamics are linear, a standard argument by density obtains weak solutions; we extend this argument to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial Biot-Stokes filtration, we provide a clear elucidation of strong and weak solutions, as well as their regularity through associated estimates.Item Weak Solutions for a Poro-elastic Plate System(Taylor & Francis, 2021-07-21) Gurvich, Elena; Webster, JustinWe consider a recent plate model obtained as a scaled limit of the three dimensional Biot system of poro-elasticity. The result is a "2.5" dimensional linear system that couples traditional Euler-Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We alow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions to implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate, and exposits a variety of interesting related models and associated analytical investigations.