A New Method for Determining the Eigenmodes of Finite Flow-Structure Systems

2006 ◽  
Author(s):  
Anthony D. Lucey ◽  
Mark W. Pitman
Keyword(s):  
Galerkin Method ◽  
Flow Structure ◽  
A Priori ◽  
Computational Modelling ◽  
Classical Problem ◽  
New Method ◽  
Flexible Plate ◽  
Coupled Equations ◽  
Standard Galerkin Method ◽  
Single Matrix

A new method for directly determining the eigenmodes of finite flow-structure systems is presented, using the classical problem of the interaction of a uniform flow with a flexible panel, held at both ends, as an exemplar. The method is a hybrid of theoretical analysis and computational modelling. This new approach is contrasted with the standard Galerkin method that is most often used to study the hydro-elasticity of finite systems. Unlike the Galerkin method, the new method does not require an a priori approximation of perturbations via a finite sum of modes. Instead, the coupled equations for the wall-flow system are cast, using computational methods that, in this exemplar, combine boundary-element and finite-element methods, to yield a single matrix equation for the system that is a second-order differential equation for the panel-displacement variable. Standard state-space methods are then used to extract the eigenmodes of the system directly. We present definitive results for the stability of the case of an unsupported flexible plate, elucidating its divergence and flutter characteristics, and the effect of energy dissipation in the structure. Finally, we present some results for the case of a spring-backed flexible plate that illustrate the complicated dynamics of this type of wall; these dynamics would be poorly modelled by a traditional Galerkin method.

Error Analysis for a Galerkin Finite Element Method Applied to a Coupled Nonlinear Degenerate System of Advection-diffusion Equations

2006 ◽  
Vol 6 (1) ◽  
pp. 3-30 ◽  
Author(s):  
Koffi B. Fadimba
Keyword(s):  
Galerkin Method ◽  
Error Estimates ◽  
Time Discretization ◽  
Diffusion Equations ◽  
Degenerate System ◽  
Two Phase ◽  
Advection Diffusion ◽  
Flow Through ◽  
Standard Galerkin Method ◽  
Regularity Results

AbstractWe consider a standard Galerkin Method applied to both the pressure equation and the saturation equation of a coupled nonlinear system of degenerate advection-diffusion equations modeling a two-phase immiscible flow through porous media. After regularizing the problem and establishing some regularity results, we derive error estimates for a semi-discretized Galerkin Method. A decoupled nonlinear scheme is then proposed for a fully discretized (backward in time) Galerkin Method, and error estimates are derived for that method. We also prove the existence and uniqueness for the nonlinear operator intervening in the backward time discretization.

scholarly journals The lumped mass finite element method for a parabolic problem

1985 ◽  
Vol 26 (3) ◽  
pp. 329-354 ◽  
Author(s):  
C. M. Chen ◽  
V. Thomée
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Equation ◽  
Galerkin Method ◽  
Parabolic Problem ◽  
Piecewise Linear ◽  
Optimal Order ◽  
Lumped Mass ◽  
Two Space Dimensions ◽  
Standard Galerkin Method

AbstractFor the heat equation in two space dimensions we consider semidiscrete and totally discrete variants of the lumped mass modification of the standard Galerkin method, using piecewise linear approximating functions, and demonstrate error estimates of optimal order in L2 and of almost optimal order in L∞.

scholarly journals Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fahe Miao ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Degenerate Parabolic Equation ◽  
Pullback Attractors ◽  
Asymptotic Compactness ◽  
Absorbing Sets ◽  
Degenerate Parabolic ◽  
Standard Galerkin Method

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.

SELECTION OF VISCOUS TERMS APPROXIMATION IN DISCONTINUOUS GALERKIN METHOD

2013 ◽  
Vol 44 (3) ◽  
pp. 327-354
Author(s):  
Aleksey Igorevich Troshin ◽  
Vladimir Viktorovich Vlasenko ◽  
Andrey Viktorovich Wolkov
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Selection Of

Application of an Element-free Galerkin Method to Water Wave Propagation Problems

2014 ◽  
Vol 60 (1-4) ◽  
pp. 87-105 ◽  
Author(s):  
Ryszard Staroszczyk
Keyword(s):  
Wave Propagation ◽  
Galerkin Method ◽  
Present Model ◽  
Free Surface Elevation ◽  
Element Free Galerkin Method ◽  
Plane Flow ◽  
Element Free Galerkin ◽  
Proposed Model ◽  
Sloping Bed ◽  
Element Free

Abstract The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is solved numerically by applying an element-free Galerkin method. First, the proposed model is validated by comparing its predictions with experimental data for the plane flow in water of uniform depth. Then, as illustrations, results of numerical simulations performed for plane gravity waves propagating through a region with a sloping bed are presented. These results show the evolution of the free-surface elevation, displaying progressive steepening of the wave over the sloping bed, followed by its attenuation in a region of uniform depth. In addition, some of the results of the present model are compared with those obtained earlier by using the conventional finite element method.

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