Nonlinear Finite-Amplitude Dynamics

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Linear Model ◽  
Finite Amplitude ◽  
Numerical Code ◽  
Spectral Space ◽  
Simple Geometry ◽  
Nonlinear Simulations ◽  
The Galerkin Method

This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After explaining the modifications to the linear model, the chapter shows how to add the nonlinear terms to the code. It also discusses the Galerkin method, the strategy of computing the contribution to the nonlinear terms for each mode due to the binary interactions of many other modes. The Galerkin method works fine as far as calculating the nonlinear terms is concerned because of the simple geometry and convenient boundary conditions. The chapter concludes by showing how to construct a nonlinear code and performing nonlinear simulations.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

Cattaneo–LTNE porous ferroconvection

2019 ◽  
Vol 15 (4) ◽  
pp. 779-799 ◽  
Author(s):  
Ravisha M. ◽  
I.S. Shivakumara ◽  
Mamatha A.L.
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Boundary Conditions ◽  
Galerkin Method ◽  
Free Boundaries ◽  
Content Type ◽  
Darcy Model ◽  
Different Types ◽  
Saturated Porous Layer ◽  
The Galerkin Method

Purpose The onset of convection in a ferrofluid-saturated porous layer has been investigated using a local thermal nonequilibrium (LTNE) model by allowing the solid phase to transfer heat via a Cattaneo heat flux theory while the fluid phase to transfer heat via usual Fourier heat-transfer law. The flow in the porous medium is governed by modified Brinkman-extended Darcy model. The instability of the system is discussed exactly for stress-free boundaries, while for rigid-ferromagnetic/paramagnetic boundaries the results are obtained numerically using the Galerkin method. The presence of Cattaneo effect introduces oscillatory convection as the preferred mode of instability contrary to the occurrence of instability via stationary convection found in its absence. Besides, oscillatory ferroconvection is perceived when the solid thermal relaxation time parameter exceeds a threshold value and increase in its value is to hasten the oscillatory onset. The effect of different boundary conditions on the instability of the system is noted to be qualitatively same. The paper aims to discuss these issues. Design/methodology/approach The investigators would follow the procedure of Straughan (2013) to obtain the expression for Rayleigh number. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The investigators have used a Galerkin method to obtain the numerical results for rigid-ferromagnetic/paramagnetic boundaries, while the instability of the system is discussed exactly for stress-free boundaries. Findings The Cattaneo–LTNE porous ferroconvection has been analyzed for different velocity and magnetic boundary conditions. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The effect of different types of velocity and magnetic boundary conditions on the instability of the system has been highlighted. The instability of the system is discussed exactly for stress-free boundaries, while for rigid-ferromagnetic/paramagnetic boundaries the results are obtained numerically using the Galerkin method. Originality/value The novelty of the present paper is to combine LTNE and second sound effects in solids on thermal instability of a ferrofluid-saturated porous layer by retaining the usual Fourier heat-transfer law in the ferrofluid. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The effect of different types of velocity and magnetic boundary conditions on the instability of the system is discussed.

A DYNAMICAL SYSTEM APPROACH FOR THE BONDI PROBLEM

2009 ◽  
Vol 24 (08n09) ◽  
pp. 1700-1704 ◽  
Author(s):  
H. P. DE OLIVEIRA ◽  
E. L. RODRIGUES
Keyword(s):  
Galerkin Method ◽  
System Approach ◽  
High Accuracy ◽  
Computational Effort ◽  
Numerical Code ◽  
Field Equations ◽  
Numerical Tests ◽  
Stability And Accuracy ◽  
The Galerkin Method ◽  
Convergence Stability

We present the first numerical code based on the Galerkin method to integrate the field equations of the Bondi problem. The Galerkin method is a spectral method whose main feature is to provide high accuracy with moderate computational effort. Several numerical tests were performed to verify the issues of convergence, stability and accuracy with promising results. This code opens up several possibilities of applications in more general scenarios for studying the evolution of spacetimes with gravitational waves.

Nonlinear Stability of Circular Cylindrical Shells in Annular and Unbounded Axial Flow

2001 ◽  
Vol 68 (6) ◽  
pp. 827-834 ◽  
Author(s):  
M. Amabili ◽  
F. Pellicano ◽  
M. A. Pai¨doussis
Keyword(s):  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Cylindrical Shells ◽  
Axial Flow ◽  
Longitudinal Direction ◽  
Finite Amplitude ◽  
External Flow ◽  
Linear Potential ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

The stability of circular cylindrical shells with supported ends in compressible, inviscid axial flow is investigated. Nonlinearities due to finite-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory; the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied at the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear potential flow theory is applied to describe the fluid-structure interaction. Both annular and unbounded external flow are considered by using two different sets of boundary conditions for the flow beyond the shell length: (i) a flexible wall of infinite extent in the longitudinal direction, and (ii) rigid extensions of the shell (baffles). The system is discretized by the Galerkin method and is investigated by using a model involving seven degrees-of-freedom, allowing for traveling-wave response of the shell and shell axisymmetric contraction. Results for both annular and unbounded external flow show that the system loses stability by divergence through strongly subcritical bifurcations. Jumps to bifurcated states can occur well before the onset of instability predicted by linear theory, showing that a linear study of shell stability is not sufficient for engineering applications.

scholarly journals Existence of Solutions of a Nonlocal Elliptic System via Galerkin Method

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Alberto Cabada ◽  
Francisco Julio S. A. Corrêa
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Galerkin Method ◽  
Positive Solution ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Dimensional System ◽  
The Galerkin Method

By means of the Galerkin method and by using a suitable version of the Brouwer fixed-point theorem, we establish the existence of at least one positive solution of a nonlocal ellipticN-dimensional system coupled with Dirichlet boundary conditions.

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