arXiv subjects

Pelin G. Geredeli

Publications and source records attributed to Pelin G. Geredeli.

15 recordsLinked to original sources

arXiv2025

A novel approach to study the wellposedness of the 3D fluid-2D plate interaction PDE System

We consider a certain fluid-structure interaction (FSI) system with a view of obtaining an alternative methodology for establishing its strongly continuous semigroup wellposedness. (Semigroup generation for this FSI was originally considered in Avalos-Clark (2014).) The FSI model under consideration describes the vibrations of an incompressible fluid within a 3D cavity as it interacts with the elastic membrane on the ``free" upper boundary of the cavity. Such coupled PDE systems appear in variety of natural settings such as biomedicine, aeroelasticity, and fluid dynamics. Our proof of $C_0$-semigroup wellposedness is based on a proper application of Lumer Phillips Theorem. In this regard, our main challenge is to show the maximality of the corresponding semigroup generator. To this end, we develop a ``nonstandard" inf-sup approach which avoids the use of technical nonlocal maps in the associated bilinear forms--unlike the earlier paper Avalos-Clark (2014)--and allows for the solution of the fluid and plate solution variables simultanously. Our new inf-sup strategy will lead to a more efficient mixed finite element method (FEM) for approximating solutions to the FSI problem, inasmuch our novel variational formulation avoids bilinear forms which are free from the computationally-intensive nonlocal solution operators invoked in Avalos-Clark (2014). We also perform numerical tests based on this formulation using a benchmark problem and present numerical results to demonstrate the effectiveness of our approach.

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arXiv2025

Analysis and Numerical Approximation to Interactive Dynamics of Navier Stokes-Plate Interaction PDE System

We consider a Navier-Stokes fluid-plate interaction (FSI) system which describes the evolutions of the fluid contained within a 3D cavity, as it interacts with a deformable elastic membrane on the ``free" upper boundary of the cavity. These models arise in various aeroelastic and biomedical applications as well as in the control of ocular pressure, and sloshing phenomena. We analyze the well-posedness of weak solutions to the stationary ($λ$-parametrized) coupled PDE system by way of invoking the nonlinear generalization of the abstract variational formulations which was introduced in \cite{girault2012finite}, wherein an inf-sup approach is followed to show existence-uniqueness of solutions under a small data assumption. In addition, we provide a numerical approximation scheme of the infinite dimensional coupled system via a finite element method approximation (FEM). The numerical results use a standard conforming scheme and handle the introduced nonlinearities via Picard iterations. Numerical results are obtained for an appropriate test problem satisfying the necessary boundary conditions and coupling. Moreover, error bounds between the FEM and theoretical solution in terms of the characteristic mesh size are supplied in appropriate Sobolev norms which agree with the established literature. These FEM approximations of the coupled system with their associated error bounds validate the theoretical findings.

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arXiv2024

An inf-sup Approach to $C_0$-Semigroup Generation for An Interactive Composite Structure-Stokes PDE Dynamics

In this work, we investigate the existence and uniqueness properties of a composite structure (multilayered) fluid interaction PDE system which arises in multi-physics problems, and particularly in biofluidic applications related to the mammalian blood transportation process. The PDE system under consideration consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics which gives rise to an additional 2D elastic equation on the boundary interface between these 3D PDE systems. By means of a nonstandard mixed variational formulation we show that the PDE system generates a $C_0$-semigroup on the associated finite energy space of data. In this work, the presence of the pressure term in the 3D Stokes equation adds a great challenge to our analysis. To overcome this difficulty, we follow a methodology which is based on the necessarily non-Leray based elimination of the associated pressure term, via appropriate nonlocal operators. Moreover, while we express the fluid solution variable via decoupling of the Stokes equation, we construct the elastic solution variables by solving a mixed variational formulation via a Babuska-Brezzi approach.

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arXiv2022

Numerical Approximations for the Null Controllers of Structurally Damped Plate Dynamics

In this paper, we consider a structurally damped elastic equation under hinged boundary conditions. Fully-discrete numerical approximation schemes are generated for the null controllability of these parabolic-like PDEs. We mainly use finite element method (FEM) and finite difference method (FDM) approximations to show that the null controllers being approximated via FEM and FDM exhibit exactly the same asymptotics of the associated minimal energy function. For this, we appeal to the theory originally given by R. Triggiani [20] for construction of null controllers of ODE systems. These null controllers are also amenable to our numerical implementation in which we discuss the aspects of FEM and FDM numerical approximations and compare both methodologies. We justify our theoretical results with the numerical experiments given for both approximation schemes.

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arXiv2022

Improved convergence of the Arrow-Hurwicz iteration for the Navier-Stokes equation via grad-div stabilization and Anderson acceleration

We consider two modifications of the Arrow-Hurwicz (AH) iteration for solving the incompressible steady Navier-Stokes equations for the purpose of accelerating the algorithm: grad-div stabilization, and Anderson acceleration. AH is a classical iteration for general saddle point linear systems and it was later extended to Navier-Stokes iterations in the 1970's which has recently come under study again. We apply recently developed ideas for grad-div stabilization and divergence-free finite element methods along with Anderson acceleration of fixed point iterations to AH in order to improve its convergence. Analytical and numerical results show that each of these methods improves AH convergence, but the combination of them yields an efficient and effective method that is competitive with more commonly used solvers.

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arXiv2022

Rational Decay of A Multilayered Structure-Fluid PDE System

In this work, we consider a certain multilayered (thick layer) wave--(thin layer) wave--heat (fluid) interactive PDE system. Such coupled PDE systems have been used in the literature to describe the blood transport process in mammalian vascular systems. In particular, the deformations of the boundary interface (thin layer) are described via the two dimensional elastic equation. The present work constitutes an investigation of the extent of the stabilizing effects of the underlying fluid dissipation -- across the boundary interface -- upon both the thick and thin structural components. (All three PDE components evolve on their respective geometries.) In this regard, our main result is the derivation of uniform decay rates for classical solutions of this multilayered PDE model. To obtain these estimates, necessary a priori inequalities for certain static multilayered PDE models are generated here to ultimately allow an application of a wellknown resolvent criterion for rational decay.

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arXiv2021

A Resolvent Criterion Approach to Strong Decay of a Multilayered Lamé-Heat System

We consider a multilayer hyperbolic-parabolic PDE system which constitutes a coupling of 3D thermal - 2D elastic - 3D elastic dynamics, in which the boundary interface coupling between 3D fluid and 3D structure is realized via a 2D elastic equation. Our main result here is one of strong decay for the given multilayered - heat system. That is, the solution to this composite PDE system is stabilized asymptotically to the zero state. Our proof of strong stability takes place in the "frequency domain" and ultimately appeals to the pointwise resolvent condition introduced by Tomilov [45]. This very useful result, however, requires that the semigroup associated with our multilayered FSI system be completely non-unitary (c.n.u). Accordingly, we firstly establish that the semigroup $\{e^{\mathcal {A}t}\}_{t\geq 0}$ is indeed c.n.u., in part by invoking relatively recent results of global uniqueness for overdetermined Lamé systems on nonsmooth domains. Although the entire proof also requires higher regularity results for some trace terms, this \textit{"resolvent criterion approach"} allows us to establish a "classially soft" proof of strong decay. In particular, it avoids the sort of technical PDE multipliers invoked in [9].

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arXiv2020

Bounded Semigroup Wellposedness for a Linearized Compressible Flow Structure PDE Interaction with Material Derivative

We consider a compressible flow structure interaction (FSI) PDE system which is linearized about some reference rest state. The deformable interface is under the effect of an ambient field generated by the underlying and unbounded material derivative term which further contributes to the non-dissipativity of the FSI system, with respect to the standard energy inner product. In this work we show that, on an appropriate subspace, only one dimension less than the entire finite energy space, the FSI system is wellposed, and is moreover associated with a continuous semigroup which is \emph{uniformly bounded} in time. Our approach involves establishing maximal dissipativity with respect to a special inner product which is equivalent to the standard inner product for the given finite energy space. Among other technical features, the necesssary PDE estimates require the invocation of a multiplier which is intrinsic to the given compressible FSI system.

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arXiv2019

Wellposedness, Spectral Analysis and Asymptotic Stability of a Multilayered Heat-Wave-Wave System

n this work we consider a multilayered heat-wave system where a 3-D heat equation is coupled with a 3-D wave equation via a 2-D interface whose dynamics is described by a 2-D wave equation. This system can be viewed as a simplification of a certain fluid-structure interaction (FSI) PDE model where the structure is of composite-type; namely it consists of a \textquotedblleft thin\textquotedblright\ layer and a \textquotedblleft thick\textquotedblright\ layer. We associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay. Our first result is semigroup well-posedness for the (FSI) PDE dynamics. Utilizing here a Lumer-Phillips approach, we show that the fluid-structure system generates a $C_0$-semigroup on a chosen finite energy space of data. As our second result, we prove that the solution to the (FSI) dynamics generated by the $C_0$-semigroup tends asymptotically to the zero state for all initial data. That is, the semigroup of the (FSI) system is strongly stable. For this stability work, we analyze the spectrum of the generator $\mathbf A$ and show that the spectrum of $\mathbf A$ does not intersect the imaginary axis. \vskip.3cm \noindent \textbf{Key terms:} Fluid-structure interaction, heat-wave system, well-posedness, semigroup, strong stability.

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arXiv2017

Stability Analysis of Coupled Structural Acoustics PDE Models under Thermal Effects and with no Additional Dissipation

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber $% Ω$. This acoustic wave equation is coupled on a boundary interface ($% Γ_{0}$) to a two dimensional system of thermoelasticity: this thermoelastic PDE comprises a structural beam or plate equation, which governs the vibrations of flexible wall portion $Γ_{0}$ of the chamber $Ω$; the elastic dynamics is coupled to a heat equation which also evolves on $Γ_{0}$, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface $% Γ_{0}$, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of \ any additive feedback dissipation on the hard walls $Γ_{1}$ of the boundary $\partial Ω$. Under a certain geometric assumption on $Γ_{1}$, an assumption which has appeared in the literature in conection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.

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arXiv2017

Exponential Stability and Supporting Spectral Analysis of a Linearized Compressible Flow-Structure PDE Model

In this work, a result of exponential stability is obtained for solutions of a compressible flow-structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and its associated state equation for the associated pressure variable, each evolving within a three dimensional domain $\mathcal{O}$, are coupled to a fourth order plate equation which holds on a flat portion $% Ω$ of the boundary $\partial \mathcal{O}$. Moreover, since this coupled PDE model is the result of a linearization of the compressible Navier-Stokes equations about an arbitrary state, the flow PDE component contains a generally nonzero ambient flow profile $\mathbf{U}$. By way of obtaining the aforesaid exponential stability, a \textquotedblleft frequency domain\textquotedblright\ approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow-structure semigroup generator.

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arXiv2017

Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile $ \mathbf{U}$. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically \cite {Chu2013-comp}---the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a $C_{0}$-semigroup $\left\{ e^{ \mathcal{A}t}\right\} _{t\geq 0}$ on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator $\mathcal{A}$ which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field $ \mathbf{U}\in \mathbf{H}^{3}(\mathcal{O})$ has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.

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arXiv2016

Qualitative Results on the Dynamics of a Berger Plate with Nonlinear Boundary Damping

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogenous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly non- linear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary [22]). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler-Bernoulli plate in [24]. We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities.

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arXiv2014

Exponential attractors for abstract equations with memory and applications to viscoelasticity

We consider an abstract equation with memory of the form $$\partial_t \boldsymbol{x}(t)+\int_{0}^\infty k(s) \boldsymbol{A}\boldsymbol{x}(t-s){\rm d} s+\boldsymbol{B}\boldsymbol{x}(t)=0$$ where $\boldsymbol{A},\boldsymbol{B}$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{tt} u - h(0)Δu - \int_{0}^\infty h'(s) Δu(t-s){\rm d} s+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.

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arXiv2012

Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer

In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well-known, and the long-time behavior of nonlinear plates has been a topic of recent interest. Since the von Karman plate system is of "hyperbolic type" with critical nonlinearity (noncompact with respect to the phase space), this latter topic is particularly challenging in the case of geometrically constrained and nonlinear damping. In this paper we first show the existence of a compact global attractor for finite-energy solutions, and we then prove that the attractor is both smooth and finite dimensional. Thus, the hyperbolic-like flow is stabilized asymptotically to a smooth and finite dimensional set. Key terms: dynamical systems, long-time behavior, global attractors, nonlinear plates, nonlinear damping, localized damping

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