scholarly journals Bifurcation from interval at infinity for discrete eigenvalue problems which are not linearizable

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
2004 ◽  
Vol 71 (1) ◽  
pp. 109-119 ◽  
Author(s):  
Lingyuan Kong ◽  
Robert G. Parker

A method is developed to evaluate the natural frequencies and vibration modes of serpentine belt drives where the belt is modeled as a moving beam with bending stiffness. Inclusion of bending stiffness leads to belt-pulley coupling not captured in moving string models. New dynamic characteristics of the system induced by belt bending stiffness are investigated. The belt-pulley coupling is studied through the evolution of the vibration modes. When the belt-pulley coupling is strong, the dynamic behavior of the system is quite different from that of the string model where there is no such coupling. The effects of major design variables on the system are discussed. The spatial discretization can be used to solve other hybrid continuous/discrete eigenvalue problems.


Author(s):  
J. H. Kuang ◽  
M. H. Hsu

The eigenvalue problems of grouped turbo blades were numerically formulated by using the generalized differential quadrature method (GDQM). Different boundary approaches accompanying the GDQM to transform the partial differential equations of grouped turbo blades into a discrete eigenvalue problem are discussed. Effects of the number of sample points and the different boundary approaches on the accuracy of the calculated natural frequencies are also studied. Numerical results demonstrated the validity and the efficiency of the GDQM in treating this type of problem.


2003 ◽  
Vol 11 (03) ◽  
pp. 341-350 ◽  
Author(s):  
Guido Bartsch ◽  
Christian Wulf

Solving Helmholtz problems for low frequency sound fields by a truncated modal basis approach is very efficient. The most time-consuming process is the calculation of the undamped modes. Using traditional FE solvers, the user has to provide a mesh which has at least six nodes per wavelength in each spatial direction to achieve acceptable results. Because the mesh size increases with the 3rd power of the highest frequency of interest, this uniform dense mesh approach is a very expensive way of creating a modal space. However, the number of modes and the accuracy of the modal basis directly influences the solution quality. It is well known that the representation of sound fields by modal basis functions φi is optimal with respect to the L2 error norm. This means that having a modal basis Φ := {φi, i = 1⋯n}, the distance between true and approximated sound field takes its minimum in the mean square. So, it is necessary to have a FE basis which also minimizes the discretization error when computing the modal basis. One can reach this goal by applying adaptive mesh refinements. Additionally, this yields the opportunity of using fast multigrid methods to solve discrete eigenvalue problems. In context of this presentation we will discuss the results of our adaptive multigrid algorithms.


2002 ◽  
Vol 124 (4) ◽  
pp. 1011-1017 ◽  
Author(s):  
J. H. Kuang ◽  
M. H. Hsu

The eigenvalue problems of grouped turbo blades were numerically formulated by using the generalized differential quadrature method (GDQM). Different boundary approaches accompanying the GDQM to transform the partial differential equations of grouped turbo blades into a discrete eigenvalue problem are discussed. Effects of the number of sample points and the different boundary approaches on the accuracy of the calculated natural frequencies are also studied. Numerical results demonstrated the validity and the efficiency of the GDQM in treating this type of problem.


2019 ◽  
Vol 53 (3) ◽  
pp. 749-774 ◽  
Author(s):  
Francesca Gardini ◽  
Gianmarco Manzini ◽  
Giuseppe Vacca

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of theL2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.


2001 ◽  
Vol 11 (01) ◽  
pp. 43-56 ◽  
Author(s):  
FRANK R. de HOOG ◽  
ROBERT S. ANDERSSEN

In the analysis of both continuous and discrete eigenvalue problems, asymptotic formulas play a central and crucial role. For example, they have been fundamental in the derivation of results about the inversion of the free oscillation problem of the Earth and related inverse eigenvalue problems, the computation of uniformly valid eigenvalues approximations, the proof of results about the behavior of the eigenvalues of Sturm–Liouville problems with discontinuous coefficients, and the construction of a counterexample to the Backus–Gilbert conjecture. Useful formulas are available for continuous eigenvalue problems with general boundary conditions as well as for discrete eigenvalue problems with Dirichlet boundary condition. The purpose of this paper is the construction of asymptotic formulas for discrete eigenvalue problems with general boundary conditions. The motivation is the computation of uniformly valid eigenvalue approximations. It is now widely accepted that the algebraic correction procedure, first proposed by Paine et al.,13 is one of the simplest methods for computing uniformly valid approximations to a sequence of eigenvalues of a continuous eigenvalue problem in Liouville normal form.8 This relates to the fact that, for Liouville normal forms with Dirichlet boundary conditions, it is not too difficult to prove that such procedures yield, under quite weak regularity conditions, uniformly valid O(h2) approximations. For Liouville normal forms with general boundary conditions, the corresponding error analysis is technically more challenging. Now it is necessary to have, for such Liouville normal forms, higher order accurate asymptotic formulas for the eigenvalues and eigenfunctions of their continuous and discrete counterparts. Assuming that such asymptotic formulas are available, it has been shown1 how uniformly valid O(h2) results could be established for the application of the algebraic correction procedure to Liouville normal forms with general boundary conditions. Algorithmically, this methodology represents an efficient procedure for determining uniformly valid approximations to sequences of eigenvalues, even though it is more complex than for Liouville normal forms with Dirichlet boundary conditions. As well as giving a brief review of the subject for general (Robin) boundary conditions, this paper sketches proofs for the asymptotic formulas, for Robin boundary conditions, which are required in order to construct the mentioned O(h2) results.


2003 ◽  
Vol 33 (4) ◽  
pp. 1233-1260 ◽  
Author(s):  
Martin Bohner ◽  
Ondřej Došl'y ◽  
Werner Kratz

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Hua Luo

This paper discusses bifurcation from interval for the elliptic eigenvalue problems with nonlinear boundary conditions and studies the behavior of the bifurcation components.


Author(s):  
Lingyuan Kong ◽  
Robert G. Parker

A method is developed to evaluate the natural frequencies and vibration modes of serpentine belt drives where the belt is modeled as a moving beam with bending stiffness. Inclusion of bending stiffness leads to belt-pulley coupling not captured in moving string models. New dynamic characteristics of the system induced by the belt bending stiffness are investigated. The belt-pulley coupling is studied through the evolution of the vibration modes. When the belt-pulley coupling is strong, the dynamic behavior of the system is quite different from that of the string model where there is no such coupling. The effects of major design variables on the system are discussed. The spatial discretization can be used to solve other hybrid continuous/discrete eigenvalue problems.


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