A nonconvolutional, split-field, perfectly matched layer, referred to as the multiaxial perfectly matched layer (M-PML), is proposed and implemented. The new formulation is obtained by generalizing the classical perfectly matched layer (PML; as originally proposed by Berenger, [1994]) to a medium in which damping profiles are specified in more than one direction. Under the hypothesis of small damping and using an eigenvalue sensitivity analysis based on first derivatives, we propose a method to study the stability of the M-PML. With this method we demonstrate that the stability of the M-PML is related to the ratios of the specified damping profiles. Recognition of this fact leads to a general procedure for constructing robust, stable M-PML models for anisotropic media. It is also demonstrated that for any anisotropic medium the classical PML exhibits instabilities related to an eigenvalue with zero real part of multiplicity higher than one. Furthermore, we show that exponential growth due to eigenvalues with positive real part can be present in the classical PML for some orthotropic media. The effectiveness of the proposed M-PML and its advantages relative to the classical PML are demonstrated by constructing stable terminations for the aforementioned anisotropic media. The method of stability analysis is developed and demonstrated for two-dimensional elastodynamics problems, but its extension to three-dimensional configurations is straightforward.

A Nonconvolutional, Split-Field, Perfectly Matched Layer for Wave Propagation in Isotropic and Anisotropic Elastic Media: Stability Analysis

http://link.springer.com/content/pdf/10.1007/978-3-642-69569-8_2.pdf

Theory of Elastic Stability

Effects of damping on stability of elastic systems subjected to nonconservative forces

Cantilevered continuous pipe conveying fluid at constant velocity, showing that internal and external damping and Coriolis forces may have destabilizing effect

Destabilizing effect of velocity-dependent forces in nonconservative continuous systems.

In the present paper, a theory is developed qualitatively and quantitatively describing the paradoxical behavior of general non-conservative systems under the action of small dissipative and gyroscopic forces. The problem is investigated by the approach based on the sensitivity analysis of multiple eigenvalues. The movement of eigenvalues of the system in the complex plane is analytically described and interpreted. Approximations of the asymptotic stability domain in the space of the system parameters are obtained. An explicit asymptotic expression for the critical load as a function of dissipation and gyroscopic parameters allowing to calculate a jump in the critical load is derived. The classical Ziegler–Herrmann–Jong pendulum considered as a mechanical application demonstrates the efficiency of the theory.

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A theory of the destabilization paradox in non-conservative systems

Nonconservative loading on linear two-degree of freedom elastic system, discussing damping effects on equilibrium stability

/pdf/on-nonconservative-stability-problems-of-elastic-systems-4il7bb4i7b.pdf

G. Herrmann

Papers

On Nonconservative Stability Problems of Elastic Systems With Slight Damping