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Bratus’, A. S.; Seĭranyan, A. P.

Bimodal solutions in eigenvalue optimization problems. (English. Russian original) Zbl 0569.47004

J. Appl. Math. Mech. 47, 451-457 (1984); translation from Prikl. Mat. Mekh. 47, 546-554 (1983).
The authors study the problem of maximizing the minimum eigenvalue of a self-adjoint operator depending smoothly on a variable. This problem occurs when investigating oscillations of elastic systems. Interesting concrete examples are given in the paper.
Reviewer: W. Arendt

Cited in 7 Documents

MSC:

47A10 Spectrum, resolvent
70J25 Stability for problems in linear vibration theory
49R05 Variational methods for eigenvalues of operators
47B25 Linear symmetric and selfadjoint operators (unbounded)

Keywords:

optimal design of structures; maximizing the minimum eigenvalue of a self-adjoint operator depending smoothly on a variable; oscillations of elastic systems
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References:

[1] Olhoff, N.; Rasmussen, S. H., On single and bimodal optimum buckling loads of clamped columns, Internat. J. Solids and Struct., Vol.13, No.7 (1977) · Zbl 0357.73041
[2] Ol’Khoff, N., Optimal Design of Structures (1981), MIR: MIR Moscow
[3] Arman, Zh.-L.; Lur’e, K. A.; Cherkaev, A. V., On the solution of eigenvalue optimization problems arising in the design of elastic structures, (Mekhan. Tverd. Tela No.5 (1978), Izv. Akad. Nauk SSSR)
[4] Masur, E. F.; Mróz, Z., Non-stationary optimality conditions in structural design, Internat. J. Solids and Struct., Vol.15, No.6 (1979) · Zbl 0411.73073
[5] Prager, S.; Prager, W., A note on optimal design of columns, Internat. J. Mech. Sci., Vol.21, No.4 (1979)
[6] Seyranian, A. P., Sensitivity analysis and optimization of aeroelastic stability, Internat. J. Solids and Struct., Vol.18, No.9 (1982) · Zbl 0503.73070
[7] Hanaoka, M.; Washizu, K., Optimum design of Beck’s column. Comput. and Struct., Vol.11, No.6 (1980) · Zbl 0439.73080
[8] Medvedev, N. G., Some spectral singularities of optimal problems of the stability of shells of variable thickness, Dokl. Akad. Nauk UkrSSR, No.9 (1980), Ser.A · Zbl 0451.49023
[9] Claudon, J.-L.; Sunakawa, M., Optimizing distributed structures for maximum flutter load, AIAA Journal, Vol.19, No.7 (1981) · Zbl 0497.73106
[10] Choi, K. K.; Haug, E. J., Optimization of structures with repeated eigenvalues, (Optimization of Distributed Parameter Structures. Alphen aan den Rijn (1981), Sijthoff & Noordhoff: Sijthoff & Noordhoff Netherlands) · Zbl 0471.49022
[11] Blachot, J.; Gajewski, A., On unimodal and bimodal optimal design of funicular arches, Internat. J. Solids and Struct., Vol.17, No.7 (1981) · Zbl 0454.73078
[12] Larichev, A. D., A method of finding the minimum volume of a beam on an elastic foundation for a given value of the cirtical load, (Applied and Theoretical Investigations of Building Structures (1981), Izd. TsNIISK im. V.A. Kucheronko: Izd. TsNIISK im. V.A. Kucheronko Moscow)
[13] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, Vol.1 (1953), Interscience Publ: Interscience Publ New York), (or 1962). · Zbl 0729.00007
[14] Seyranian, A. P., Homogeneous functionals and structural optimization problems, Internat. J. Solids and Struct, Vol.15, No.10 (1979) · Zbl 0411.73072
[15] Mikhlin, S. G., Variational Methods in Mathematical Physics (1964), Pergamon Press, Book No. 10146 · Zbl 0119.19002
[16] Kolmogorov, A. N.; Fomin, S. V., Elements of the Theory of Functions and Functional Analysis (1961), Graylock Press: Graylock Press Bethesda, MD · Zbl 0103.08801
[17] Timoshenko, S. P., Vibrations in Engineering (1959), FIZMATGIZ: FIZMATGIZ Moscow · Zbl 0201.27501
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