Adjoint variational methods in nonconservative stability problems.

Abstract A general nonself-adjoint eigenvalue problem is examined and it is shown that the commonly employed approximate methods, such as the Galerkin procedure, the method of weighted residuals and the least square technique lack variational descriptions. When used in their previously known forms they do not yield stationary eigenvalues and eigenfunctions. With the help of an adjoint system, however, several analogous variational descriptions may be developed and it is shown in the present study that by properly restating the method of least squares, stationary eigenvalues may be obtained. Several properties of the adjoint eigenvalue problem, known only for a restricted group, are shown to exist for the more general class selected for study.

[1]  R. Sani,et al.  THE CONVERGENCE OF THE GALERKIN METHOD FOR THE TAYLOR-DEAN STABILITY PROBLEM. CONVERGENCE OF A GENERALIZED GALERKIN METHOD FOR CERTAIN FLUID STABILITY PROBLEMS, , 1964 .

[2]  E. L. Ince.  Ordinary differential equations , 1927 .

[3]  I. Prigogine,et al.  Sur les propriétés différentielles de la production d'entropie , 1954 .

[4]  P. Roberts.  Characteristic value problems posed by differential equations arising in hydrodynamics and hydromagnetics , 1960 .

[5]  M. Biot.  Lagrangian thermodynamics of heat transfer in systems including fluid motion , 1962 .

[6]  Horst Leipholz.  Anwendung des Galerkinschen Verfahrens auf nichtkonservative Stabilitätsprobleme des elastischen Stabes , 1962 .

[7]  Maurice A. Biot,et al.  Variational Principles in Irreversible Thermodynamics with Application to Viscoelasticity , 1955 .

[8]  P. Glansdorff,et al.  On a general evolution criterion in macroscopic physics , 1964 .

[9]  Über die Konvergenz des Galerkinschen Verfahrens bei nichtselbstadjungierten und nichtkonservativen Eigenwertproblemen , 1963 .

[10]  S. Mikhlin,et al.  Variational Methods in Mathematical Physics , 1965 .

[11]  G. Herrmann,et al.  The usefulness of adjoint systems in solving nonconservative stability problems of elastic continua. , 1969 .

[12]  V. V. Bolotin,et al.  Nonconservative problems of the theory of elastic stability , 1963 .

[13]  Horst Leipholz,et al.  Die Knicklast des einseitig eingespannten Stabes mit gleichmässig verteilter, tangentialer Längsbelastung , 1962 .

[14]  S. Chandrasekhar.  Hydrodynamic and Hydromagnetic Stability , 1961 .

[15]  S. Nemat-Nasser,et al.  Destabilizing effect of velocity-dependent forces in nonconservative continuous systems Technical report no. 65-4 , 1965 .

[16]  B. Friedman.  Principles and Techniques of Applied Mathematics , 1956 .

[17]  L. H. Lee,et al.  ON THE APPROXIMATE AND NUMERICAL SOLUTION OF ORR-SOMMERFELD PROBLEMS , 1967 .