Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with strong damping is considered. We prove that, under suitable assumptions on relaxation functions and certain initial data, the decay rate of the solutions energy is exponential

Dynamics of thermoelastic thin plates: A comparison of four theories

Four distinct theories describing the flexural motion of thermoelastic thin plates are compared. The theories are due to Chadwick, Lagnese and Lions, Simmonds, and Norris. Chadwick's theory requires a 3D spatial equation for the temperature but is considered the most accurate as the others are derivable from it by different approximations. Attention is given to the damping of flexural waves. Analytical and quantitative comparisons indicate that the Lagnese and Lions model with a 2D temperature equation captures the essential features of the thermoelastic damping, but contains systematic inaccuracies. These are attributable to the approximation for the first moment of the temperature used in deriving the Lagnese and Lions equation. Simmonds' model with an explicit formula for temperature in terms of plate deflection is the simplest of all but is accurate only at low frequency, where the damping is linearly proportional to the frequency. It is shown that the Norris model, which is almost as simple as Simmond's, is as accurate as the more precise but involved theory of Chadwick.Comment: 2 figures, 1 tabl

A mixed finite element method for solving coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term

This paper is concerned with the numerical approximation of the solution of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term using a mixed finite element method. The Raviart-Thomas mixed finite element method is one of the most prominent techniques to discretize the second-order wave equations; therefore, we apply this scheme for space discretization. Furthermore, an L2-in-space error estimate is presented for this mixed finite element approximation. Finally, the efficiency of the method is verified by a numerical example. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd

Blow up of Solutions for a Coupled Kirchhoff-type Equations with Degenerate Damping Terms

In this paper, we investigate a system of coupled Kirchhoff-type equations with degenerate damping terms. We prove a nonexistence of global solutions with positive initial energy. Later, we give some estimates for lower bound of the blow up time

On Stability of Hyperbolic Thermoelastic Reissner-Mindlin-Timoshenko Plates

In the present article, we consider a thermoelastic plate of Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absense of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, etc. We present a well-posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Restricting the problem to the rotationally symmetric case, we further prove that a single frictional damping merely for the bending compoment is sufficient for exponential stability. To this end, we construct a Lyapunov functional incorporating the Bogovski\u{i} operator for irrotational vector fields which we discuss in the appendix.Comment: 27 page

Development of a finite volume method for elastic materials and fluid-solid coupled applications

This thesis presents the development of a parallel finite volume numerical method to analyse thermoelastic and hyperelastic materials and applied problems with mutual interaction between a fluid and a structure. The solid problem follows a cell-centred finite volume formulation for three-dimensional unstructured grids under the same framework that is frequently devoted to computational fluid dynamics. Second-order accurate schemes are used to discretise both in time and space. A direct implicit time integration promotes numerical stability when facing vibration and quasi-static scenarios. The geometrical non-linearities, encountered with the large displacements of both Saint Venant-Kirchhoff and neo-Hookean models, are tackled by means of an updated Lagrangian approach. Verification of the method is conducted with canonical cases which involve: static equilibrium, thermal stress, vibration, structural damping, large deformations, nearly incompressible materials and high memory usage. Significant savings in computation time are achieved owing to the acceleration strategies implemented within the system resolution, namely a segregated algorithm with Aitken relaxation and a block-coupled system arrangement. The similarities between the block-coupled method and the displacement-based finite element method, with regards to the matrix form of the resulting equations, allow for including Rayleigh viscous damping within a finite volume solver. The program for structures is to be coupled with the in-house fluid numerical models in order to produce a unified fluid-structure interaction platform, where an arbitrary Lagrangian-Eulerian approach is used to solve the flow in a conforming grid. As a first step, the method for incompressible Newtonian fluids is adapted to deal with structure-coupled problems. To do so, the Lagrangian-Eulerian version of the Navier-Stokes equations is presented, and automatic moving mesh techniques are developed. These techniques are designed to mitigate the mesh quality deterioration and to satisfy the space conservation law. Besides, a semi-implicit coupling algorithm, which only implicitly couples the fluid pressure term to the structure, is implemented. As a result, numerical stability for strongly coupled phenomena at a reduced computational cost is obtained. These new tools are tested on an applied case, consisting of the turbulent flow through self-actuated flexible valves. Finally, a pioneering coupled numerical model for the thermal and structural analysis of packed-bed thermocline storage tanks is developed. This thermal accumulation system for concentrated solar power plants has attracted the attention of the industry due to the economic advantage compared to the usual two-tank system. Dynamic coupling among the thermoelastic equations for the tank shell and the numerical models for all other relevant elements of the system is considered. After validating the model with experimental results, the commercial viability of the thermocline concept, regarding energetic effectiveness and structural reliability, is evaluated under real operating conditions of the power plants.Esta tesis presenta el desarrollo de un método numérico paralelo basado en volúmenes finitos para analizar materiales termoelásticos e hiperelásticos y problemas con una interacción mutua entre un fluido y una estructura. El problema del sólido sigue una formulación de volúmenes finitos centrada en las celdas para mallas no-estructuradas tridimensionales, bajo el mismo marco que se suele emplear en la dinámica de fluidos computacional. Se utilizan esquemas de segundo orden de precisión para discretizar el tiempo y el espacio. Una integración temporal directa implícita asegura estabilidad numérica al afrontar escenarios casi-estáticos o de vibración. Las no linealidades, que aparecen con los amplios desplazamientos de los modelos de Saint Venant-Kirchhoff y de neo-Hookean, son abordadas con un enfoque Lagrangiano actualizado. La verificación del método se realiza a través de casos canónicos que involucran: equilibrio estático, tensiones térmicas, vibración, amortiguación estructural, grandes deformaciones, materiales casi incompresibles y altos requerimientos de memoria. Se registra un ahorro significativo en el tiempo de cálculo gracias a las estrategias de aceleración implementadas dentro de la resolución del sistema, principalmente un algoritmo segregado con relajación Aitken y una disposición acoplada en bloques del sistema. Las similitudes entre este método acoplado en bloques y el método de los elementos finitos basados en el desplazamiento, con respecto a la forma matricial de las ecuaciones resultantes, permiten incluir la amortiguación viscosa tipo Rayleigh dentro de un solucionador de volúmenes finitos. El programa para estructuras se acoplará con los modelos numéricos internos para fluidos con el objetivo de generar una plataforma unificada de interacción fluido-estructura, donde se usa un enfoque arbitrario Lagrangiano-Euleriano sobre una malla conforme para resolver el fluido. Como primer paso, el método para flujos incompresibles Newtonianos se adapta para lidiar con problemas acoplados a una estructura. Para ello, se presenta la versión Lagrangiana-Euleriana de las ecuaciones de Navier-Stokes y se desarrollan técnicas automáticas de movimiento de malla. El diseño de estas técnicas se centra en mitigar el deterioro de la calidad de la malla y satisfacer la ley de conservación del espacio. Además, se implementa un algoritmo de acoplamiento semi-implícito, que sólo acopla implícitamente el término fluido de presión a la estructura. Como resultado, se obtiene estabilidad numérica para fenómenos fuertemente acoplados a un coste computacional reducido. Estas nuevas herramientas se prueban en un caso aplicado, que consiste el flujo turbulento a través de válvulas flexibles autoactivadas. Finalmente, se desarrolla un modelo numérico acoplado pionero para analizar estructuralmente y térmicamente los tanques termoclina de almacenamiento térmico. Este sistema de acumulación para centrales termosolares ha atraído la atención de la industria debido al ahorro económico comparado con el sistema de doble tanque habitual. Se tiene en cuenta el acoplamiento dinámico entre las ecuaciones gobernantes de la pared del tanque y las de todos los elementos relevantes del sistema. Tras validar el modelo con datos experimentales, se evalúa la viabilidad comercial de estos tanques, en cuanto a rendimiento energético y fiabilidad estructural, bajo condiciones reales de operación de las centrales.Postprint (published version

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.Comment: 1 figur