A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma

1975 ◽  
Vol 60 (1) ◽  
pp. 51-73 ◽  
Author(s):  
R. Temam
Keyword(s):  
Eigenvalue Problem ◽  
Linear Eigenvalue Problem ◽  
Non Linear

Positive solutions of a non-linear eigenvalue problem with discontinuous non-linearity

1979 ◽  
Vol 83 (1-2) ◽  
pp. 133-145 ◽  
Author(s):  
Paolo Nistri
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Nonlinear Eigenvalue Problem ◽  
Linear Term ◽  
Linear Eigenvalue Problem ◽  
Existence And Multiplicity ◽  
Non Linear ◽  
Nonlinear Eigenvalue

SynopsisWe seek non-trivial solutions (u,λ)∈C1([0,1])×[0,∞ with u(x)≧0 for all x ∈[0,1], of the nonlinear eigenvalue problem –u″(x)=λf(u(x)) for x ∈ (0,1) and u(0)=u(1)=0,where f:[0,∞)→[0,∞) is such that f(p) = 0, for p ∈ [0,1), and f(p) = K(p), for p ∈ (1,∞), and K: [1, ∞)→(0, ∞) is assumed to be twice continuously differentiable. (The value ƒ(1) is only required to be positive.)Existence and multiplicity theorems are given in the cases where ƒ is asymptotically sub-linear and ƒ is asymptotically super-linear. Moreover if strengthened assumptions are made on the growth of the non-linear term ƒ we obtain the precise number of non-trivial solutions for given values of λ ∈ [0, ∞).

LOW-FREQUENCY APPROXIMATIONS FOR ACOUSTIC POROUS MATERIALS: LINEARISATION OF THE POROELASTIC EIGENVALUE PROBLEM

1997 ◽  
Vol 21 (4) ◽  
pp. 401-413 ◽  
Author(s):  
R. Panneton ◽  
N. Atalla
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Porous Materials ◽  
Acoustic Waves ◽  
Low Frequency ◽  
Poroelastic Media ◽  
Linear Eigenvalue Problem ◽  
Non Linear ◽  
Frequency Independent ◽  
Poroelastic Material

Recently in acoustics, it was shown that a finite element discretisation of Biot’s dynamic equations — for poroelastic media — leads to a non-linear eigenvalue problem. This non-linearity comes from the complex dissipation mechanisms of the elastic and acoustic waves prevailing within the poroelastic material. These complex dissipation mechanisms are related to viscous and thermal effects. The main drawback of the non-linear eigenvalue problem is that it prevents the use of classical modal analysis techniques for efficient solution of the corresponding matrix system. Since the finite element method is mostly used at low-freuqencies, the objective of this paper is to derive low-frequency approximations on the viscous and thermal disssipation mechanisms that will be used to linearise the poroelastic eigenvalue problem. To achieve the linearisation, it will be shown that the first Lamé coefficient of the poroelastic medium can be considered frequency-independent for most acoustic porous materials.

A Variant Algorithm for Solving Eigenvalue Equation in Linear Stability Problems

2011 ◽  
Author(s):  
Zongchun Qiao ◽  
P. N. Kaloni
Keyword(s):  
Eigenvalue Problem ◽  
Linear Stability ◽  
Numerical Solutions ◽  
Search Algorithm ◽  
New Approach ◽  
Linear Eigenvalue Problem ◽  
Non Linear ◽  
Qz Algorithm ◽  
Optimization Routine ◽  
Principal Advantage

We present a new approach, based upon the chebyshev tau-QZ algorithm, quesi-Newton technique, and the minimum optimization routine, to study the numerical solutions of the non-linear eigenvalue problem, arising in the linear stability analysis of non-Newtomian fluid. One principal advantage of our approach is that the resolution of the non-linear eigenvalue problem is changed into a search algorithm, and the calculation leading to the determination by both natural stability and overstability can be unified in a single code. Numerical results show that our results, obtained by this approach, are in excellent agreement with the worthy previous authors.

Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

2020 ◽  
Author(s):  
Camilo F. Silva ◽  
Laura Prieto ◽  
Maximiliano Ancharek ◽  
Pablo Marigliani ◽  
Georg A. Mensah
Keyword(s):  
Eigenvalue Problem ◽  
Helmholtz Equation ◽  
Combustion Chamber ◽  
Industrial Test ◽  
Linear Eigenvalue Problem ◽  
Large Perturbation ◽  
Non Linear ◽  
Flame Response ◽  
Micro Turbine ◽  
Two Parameters

Abstract The aim of the present study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed micro-turbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigen frequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding non-linear eigenvalue problem, one time per parameter combination. With that approach, the non-linear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, the present work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function — the Padé approximant — is obtained to estimate the eigen-frequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

A NON-LINEAR EIGENVALUE PROBLEM ASSOCIATED WITH INEXTENSIBLE WHIRLING STRINGS

2001 ◽  
Vol 239 (5) ◽  
pp. 969-982 ◽  
Author(s):  
J. COOMER ◽  
M. LAZARUS ◽  
R.W. TUCKER ◽  
D. KERSHAW ◽  
A. TEGMAN
Keyword(s):  
Eigenvalue Problem ◽  
Linear Eigenvalue Problem ◽  
Non Linear
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