About numerical realization of the method of subsequent parameters perturbation for calculating a stress-strain state of shallow shells

L. V. Bessonov ; T. A. Kuznetsova ; S. V. Chumakova

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 28-37.

Résumé

The paper investigates a class of nonlinear dynamic shell models, which non-linearity reflects Gaussian curvature of a surface; in the case when loads are smaller than critical ones in every point in time. Moreover, every unknown function from the system of equations, can be uniquely identified through the deflection function. Domain that is defined by the middle shell surface is bounded with piecewise smooth boundary. Such models as Kirchhoff–Love model (that specify Tymoshenko model, defined both in transferences and mixed forma), a model that reflects the bond between deformation fields and temperature and others can represent that equation class. The method of subsequent parameters perturbation developed by professor V. Petrov in 1970s is used as a numerical method for such models. This method brings the solution of nonlinear equations to the solution of a sequence of linear equations. The paper discusses problems connected with the realization of this method. It is known, that method of V. Petrov converges slowly. That is why questions of convergence improvement are examined. The usage of variation methods for solving systems of linear equations requires defined convergence speed and orthogonal system of functions that satisfies the boundary conditions. These questions are investigated in the paper as well.

Détail

Keywords: shell, the stress-strain state, nonlinear shell model, serial parameters perturbation method.

Bibliographie

[1] Petrov V. V., Successive loading method in nonlinear theory of plates and shells, Izd-vo Sarat. un-ta, Saratov, 1975 (in Russian)

[2] Kuznetsov V. N., Method of sequential perturbation of parameters applied to the simulation of dynamic stability thin-walled shell structures, dis. \ldots d-ra tekhn. nauk, Saratov, 2000 (in Russian)

[3] Petrov V. V., Ovchinnikov I. G., Inozemtsev V. K., The deformation of structural elements of the same–module non-linear inhomogeneous material, Izd-vo Sarat. un-ta, Saratov, 1988 (in Russian)

[4] Kuznetsov V. N., Kuznetsova T. A., Chumakova S. V., “About numerical realization of the successive loading method for calculating the geometrically nonlinear shells”, Mezhvuz. sb. nauch. tr., Issledovaniia po algebre, teorii chisel, funktsional'nomu analizu i smezhnym voprosam, 6, Izd-vo Sarat. un-ta, Saratov, 2010, 27–43 (in Russian) | MR | Zbl

[5] Kuznetsov V. N., Kuznetsova T. A., Chumakova S. V., “Operator methods in nonlinear dynamics”, Mezhvuz. sb. nauch. tr., Issledovaniia po algebre, teorii chisel, funktsional'nomu analizu i smezhnym voprosam, 1, Izd-vo Sarat. un-ta, Saratov, 2003, 70–80 (in Russian)

[6] Chumakova S. V., Pshenov D. A., Shabanov L. E., “To the problem of improving the convergence of the Petrovs method — the method of successive perturbation of parameters”, Problemy prochnosti elementov konstruktsii pod deistviem nagruzok i rabochikh sred, Mezhvuz. nauch. sb., Izd-vo SGTU, Saratov, 2002, 61–64 (in Russian)

[7] Mikhlin S. G., Variational methods in mathematical physics, Izdatel'stvo tekhniko-teoreticheskoi literatury, M., 1967 (in Russian)

[8] Kuznetsov E. B., Shalashilin V. I., “The Cauchy problem for mechanical systems with a finite number of degrees of freedom as the problem of continuing on the best parameter”, PMM, 58:6 (1994), 14–21 (in Russian) | MR

[9] Lions Zh. L., Methods of solving nonlinear boundary value problems, Mir, M., 1972, 104 pp. (in Russian)

[10] Bessonov L. V., “Numerical Implementation of Method of Subsequent Perturbation of Parameters for Computation of Stress-Strain State of a Shell Rigidly Fixed on the Boundaries”, Izv. Sarat. un-ta Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:1 (2015), 74–79 (in Russian) | DOI | MR

[11] Bessonov L. V., “The numerical implementation of the algorithm of spectral criteria for the local buckling of the shell structure”, Mezhvuz. sb. nauch. tr., Issledovaniia po algebre, teorii chisel, funktsional'nomu analizu i smezhnym voprosam, 7, Izd-vo Sarat. un-ta, Saratov, 2012, 3–9 (in Russian)

[12] Bessonov L. V., “Numerical realization of the spectral criterion for determining the points of local buckling of the shell structure”, Materialy XIX Mezhdunarodnoi konferentsii po vychislitel'noi mekhanike i sovremennym prikladnym programmnym sistemam, VMSPPS'2015 (Moskva, 2015), 223–225 (in Russian)

[13] Bessonov L. V., “Numerical Realization of The Method of Subsequent Parameters Perturbation for Calculating a Stress-Strain State of The Shell”, Applied Mechanics and Materials, 799–800 (2015), 656–659 | DOI

[14] Bessonov L. V., “Ob operatornom podhode pri raschete napryagenno-deformirovannogo sostoyaniya obolochechnyh konstrukciy”, XI Vserossiiskiy s'ezd po fundamentalnym problemam teoreticheskoy i prikladnoy mehaniki (Kazan, 2015), 467–469 (in Russian)

[15] Bessonov L. V., “Geometriceskie parametry i tochki localnoy poteri ustoychivosti cilindricheskoy obolochki”, Studencheskaya nauka: perekrestki teorii i praktiki, Materialy I Vnutrivuzovskoi nauchno-prakticheskoy konferencii studentov i aspirantov (Saratov, 2013) (in Russian)