Variational principle of elasticity of quasicrystals, numerical analysis and applications

2011 ◽  
pp. 257-272
Author(s):  
Tianyou Fan
Keyword(s):  
Numerical Analysis ◽  
Variational Principle

scholarly journals Tokamak elongation – how much is too much? Part 1. Theory

2015 ◽  
Vol 81 (6) ◽  
Author(s):  
J. P. Freidberg ◽  
A. Cerfon ◽  
J. P Lee
Keyword(s):  
Numerical Analysis ◽  
Variational Principle ◽  
Stability Problem ◽  
Feedback System ◽  
Stability Parameter ◽  
Real Experiment ◽  
Conducting Wall ◽  
Vertical Field ◽  
Resistive Wall ◽  
Resistive Wall Modes

In this and the accompanying paper, the problem of the maximally achievable elongation${\it\kappa}$in a tokamak is investigated. The work represents an extension of many earlier studies, which were often focused on determining${\it\kappa}$limits due to (i) natural elongation in a simple applied pure vertical field or (ii) axisymmetric stability in the presence of a perfectly conducting wall. The extension investigated here includes the effect of the vertical stability feedback system which actually sets the maximum practical elongation limit in a real experiment. A basic resistive wall stability parameter,${\it\gamma}{\it\tau}_{w}$, is introduced to model the feedback system which although simple in appearance actually captures the essence of the feedback system. Elongation limits in the presence of feedback are then determined by calculating the maximum${\it\kappa}$against$n=0$resistive wall modes for fixed${\it\gamma}{\it\tau}_{w}$. The results are obtained by means of a general formulation culminating in a variational principle which is particularly amenable to numerical analysis. The principle is valid for arbitrary profiles but simplifies significantly for the Solov’ev profiles, effectively reducing the 2-D stability problem into a 1-D problem. The accompanying paper provides the numerical results and leads to a sharp answer of ‘how much elongation is too much’?

Numerical Simulation of Drill-String Dynamics with an Initial Curvature

2015 ◽  
Vol 799-800 ◽  
pp. 523-527 ◽  
Author(s):  
Almaz Sergaliyev ◽  
Aliya Umbetkulova
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Numerical Analysis ◽  
Variational Principle ◽  
Cross Section ◽  
Drill String ◽  
Finite Deformations ◽  
One Dimensional ◽  
Initial Curvature ◽  
Transverse Vibrations

This work is devoted to modeling of nonlinear dynamics of the rotating drill-string taking into account initial curvature and finiteness of deformations. The drill-string is compressed by variable axial force. The case of flat bending of the drill-string with an initial curvature is studied, where drill-string considered as a one-dimensional rotating rod of a symmetric cross-section. A nonlinear model is developed on the basis of the variational principle Ostrogradsky-Hamilton and the theory of finite deformations. The numerical analysis of the model is carried out. The influence of bending forms and initial curvature of the drill-string on the amplitude of transverse vibrations is established.

Dynamic Analysis of Flexible Rotors Subjected to Torque and Force

1993 ◽  
Author(s):  
Jong-Seop Yun ◽  
Chong-Won Lee
Keyword(s):  
Numerical Analysis ◽  
Variational Principle ◽  
Dynamic Analysis ◽  
Natural Frequency ◽  
Stability Criterion ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Flexible Rotors ◽  
The Stability ◽  
Admissible Functions

Abstract The effect of the applied direction and magnitude of loads on the stability and natural frequency of flexible rotors is analyzed, when the rotors are subject to nonconservative torque and force. The stability criterion derived from the energy and variational principle is discussed and a general Galerkin’s method which utilizes admissible functions is employed for numerical analysis. Illustrative examples are treated to demonstrate the analytical developments.

Analysis of developable shells with special reference to the finite element method and circular cylinders

1976 ◽  
Vol 281 (1300) ◽  
pp. 113-170 ◽  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Analysis ◽  
Variational Principle ◽  
Rigid Body ◽  
The Finite Element Method ◽  
Body Movements ◽  
Numerical Examples ◽  
Lines Of Curvature ◽  
Element Method

The paper begins by noting that the practical and efficient numerical analysis of thin walled shells is far from a reality. Groundwork for the investigation starts with an examination of existing sufficiency conditions for convergence of the finite element method of analysis with refinement of mesh size; new and more practical conditions are then given specifically for shells. Working formulae of a suitable first approximation theory for the linear small deflexion behaviour are then given for arbitrary shells in lines of curvature and in geodesic coordinates. A variational principle is introduced which is more general than that for the well known assumed stress hybrid finite element model; its purpose is to provide a means to overcome the excessive rank deficiency which is sometimes encountered in the derive element stiffness matrix. , The formulae are next specialized to general developable shells for they are tne simplest to analyse and frequently occur in technology. Emphasis is given to the derivation of general formulae governing inextensional deformation, membrane action and rigid body movement because these constitute important factors in any adequate numerical analysis. . . , Specific application is made to circular cylindrical shells by first considering the interpolation of the kinematic continuity conditions along an arbitrary geodesic line. Details and numerical examples are provided for the first known fully compatible lines of curvature rectangular finite element which directly recovers arbitrary rigid body movements as well as inextensional deformations and membrane actions. The paper concludes with details and numerical examples of an arbitrarily shaped triangular finite element which employs the above mentioned variational principle m conjunction with linearly varying stress fields. All the rigid body movements are directly recovered as well as inextensional deformations and membrane actions. It is anticipated that this finite element and its derivatives will find widespread application.

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