Solution Methods of Eigenvalue Problems

2004 ◽  
pp. 161-182
Keyword(s):  
Eigenvalue Problems ◽  
Solution Methods

Modal Stability Theory

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Matthew P. Juniper ◽  
Ardeshir Hanifi ◽  
Vassilios Theofilis
Keyword(s):  
Stability Theory ◽  
Eigenvalue Problems ◽  
Stokes Equations ◽  
Temporal Stability ◽  
Collocation Methods ◽  
Navier Stokes ◽  
Parallel Flows ◽  
Numerical Solution Methods ◽  
Solution Methods ◽  
Spatio Temporal

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier–Stokes equations, the Orr–Sommerfeld equation, the Rayleigh equation, the Briggs–Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability. It also covers the parabolized stability equation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigenvalue problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes.

scholarly journals Subspace Reduction for Stochastic Planar Elasticity

2021 ◽  
Vol 3 (1) ◽  
pp. 1-13
Author(s):  
Harri Hakula ◽  
Mikael Laaksonen
Keyword(s):  
Static Loading ◽  
Eigenvalue Problems ◽  
Computational Mechanics ◽  
Model Problem ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Material Parameters ◽  
Planar Elasticity ◽  
Solution Methods ◽  
Rank Approximation

Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the spectrum. If the inputs, domain or material, are perturbed, the cluster breaks and tracing of the eigenpairs become difficult due to possible crossing of the modes. In this paper we have shown that the eigenvalue crossing can occur within clusters not only by perturbations of the domain, but also of material parameters. What is new is that in this setting, the crossing can be controlled; that is, the effect of the perturbations can actually be predicted. Moreover, the basis of the subspace is shown to be a well-defined concept and can be used for instance in low-rank approximation of solutions of problems with static loading. In our industrial model problem, the reduction in solution times is significant.

scholarly journals New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xinran Zheng ◽  
Mingqi Huang ◽  
Dongqi An ◽  
Chao Zhou ◽  
Rui Li
Keyword(s):  
Hamiltonian System ◽  
Free Vibration ◽  
Eigenvalue Problems ◽  
Analytic Solutions ◽  
Symplectic Space ◽  
Superposition Method ◽  
Mathematical Solution ◽  
Simply Supported ◽  
Solution Methods ◽  
Symplectic Eigenvalue

AbstractNew analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

scholarly journals Low-Rank Solution Methods for Stochastic Eigenvalue Problems

2019 ◽  
Vol 41 (4) ◽  
pp. A2657-A2680
Author(s):  
Howard C. Elman ◽  
Tengfei Su
Keyword(s):  
Eigenvalue Problems ◽  
Low Rank ◽  
Solution Methods
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