Applications of Muller’s method and the argument principle to eigenvalue problems in solid mechanics
DOI:
https://doi.org/10.7242/1999-6691/2014.7.3.32Keywords:
algebraic eigenvalue problem, partial eigenvalue problem, Muller’s method, argument principle, complex eigenvalueAbstract
Numerical implementation of the problems of solid mechanics leads to an algebraic problem of real and complex eigenvalues. Using discrete numerical methods, in particular, the finite element technique, it makes sense to solve only the partial eigenvalue problem both from the standpoint of errors of the corresponding numerical method and the mechanical content of the problems under study. This determines the requirement for an algorithm which implies that eigenvalues must be sought in ascending order. In the present paper, we propose an algorithm to solve an algebraic problem of complex eigenvalues using the Muller method. It is shown that even though the algorithm is an efficient one, it has a drawback related to the condition for evaluating roots in ascending order. In order to eliminate this disadvantage, we apply an additional procedure based on the argument principle to the developed algorithm. A technique for calculating eigenvalues that is built upon the Muller method and the argument principle is described. Relevant studies that demonstrate the use of the proposed algorithm for solving the problems of solid mechanics problems are discussed.
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Matveenko V.P., Kligman E.P. Natural vibration problem of viscoelastic solids as applied to optimization of dissipative properties of constructions // J. Vib. Control. - 1997. - Vol. 3, no. 1. - R. 87-102. DOI
2. Matveenko V.P., Kligman E.P., Urlov M.A., Urlova N.A. Modelirovanie i optimizacia dinamiceskih harakteristik smart-struktur s p’ezomaterialami // Fiz. mezomeh. - 2012. - T. 15, No 1. - S. 75-85.
3. Troyanovskii I.Ye., Shardakov I.N., Shevelev N.A. The problem of the eigenvalues and modes of rotating deformable structures // J. Appl. Math. Mech. - 1991. - Vol. 55, no. 5. - P. 733-740. DOI
4. Sevelev N.A., Dombrovskij I.V. Cislennoe modelirovanie dinamiceskogo povedenia prostranstvennyh elementov masinostroitel’nyh konstrukcij // Vycisl. meh. splos. sred.- 2008. - T. 1, No 2. - S. 106-112. DOI
5. Sevelev N.A., Dombrovskij I.V. Cislennyj analiz dinamiceskih harakteristik vrasausihsa deformiruemyh konstrukcij // Vycisl. meh. splos. sred.- 2010. - T. 3, No 1. - S. 93-104. DOI
6. Bochkarev S.A., Matveyenko V.P., Shardakov I.N. Numerical analysis of panel flutter in shells of revolution // J. Vib. Control. - 1997. - Vol. 3, no. 1. - R. 33-54. DOI
7. Matveenko V.P., Nakaryakova T.O., Sevodina N.V., Shardakov I.N. Stress singularity at the vertex of homogeneous and composite cones for different boundary conditions // J. Appl. Math. Mech. - 2008. - Vol. 72, no. 3. - R. 331-337. DOI
8. Faddeev D.K., Faddeeva V.N. Vycislitel’nye metody linejnoj algebry. - M.: Fizmatgiz, 1963. - 655 s.
9. Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators // J. Res. Nat. Bur. Stand. - 1950. - Vol. 45, no. 4. - P. 255-282. DOI
10. Francis J.G.F. The QR transformation - Part 2 // Comput. J. - 1961. - Vol. 4. - P. 332-345.
11. Kublanovskaa V.N. Metody i algoritmy resenia spektral’nyh zadac dla polinomial’nyh i racional’nyh matric // Cislennye metody i voprosy organizacii vycislenij. XII, Zap. naucn. sem. POMI. - CPb.: POMI, 1997. - T. 238. - S. 7-328. DOI
12. Muller D.E. A method for solving algebraic equations using an automatic computer // Mathematical Table and Other Aids to Computation. - 1956. - Vol. 10, no. 5. - P. 208-215. DOI
13. Lavrent’ev M.A., Sabat B.V. Metody teorii funkcij kompleksnogo peremennogo. - M.: Nauka, 1973. - 736 c.
14. Protopopov V.V. Computing first order zeros of analytic functions with large values of derivatives // Numerical Methods and Programming. - 2007. - Vol. 8. - R. 311-316.
15. Berezin I.S., Zidkov N.P. Metody vycislenij. - M.: Nauka, 1966. - T. 2. - 620 c.
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