Un nouvel element fini du type B-spline pour I' analyse dynamique des coques de revolution
Keywords:
B-spline finite element, free vibrations, shells of revolution, uniform cubic BsplinesAbstract
A new curved axisymmetric B-spline finite element for the free vibrations of thin shells of revolution is presented. The B-spline element, which is based on classical thin shell theory uses the uniform cubic B-spline functions for the interpolation of geometry as well as local displacements. It introduces three new features over the other known B-spline elements: i) instead of the usual control (or spline) variables, only nodal displacements at the internal nodes of a superelement (a group of elements) and their first derivatives with respect to the meridional length at the end nodes are used as degrees of freedom of a superelement; This choice reduces to the minimum the total number of degrees of freedom compared to standard elements. ii) a new approach for the discretization of the meridian, which dissociates the geometry and the behavior is used ; thus, according to the problem in hand either the geometry or the behavior can be favoured by increasing its corresponding number of nodes. iii) all the parameters of the uniform cubic B-spline interpolation (parametrization's spacing, end tangents, ... ) are computed automatically. The efficiency and accuracy of the proposed element are evaluated by four often cited benchmark examples. The results are compared to the analytical, experimental and numerical solutions found in the litterature. A very well behavior of the element in all the examples is observed.
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References
[ALT 86] ALTMAN w., and NETO E. L. Vibration of thin shells of revolution based on a mixed
finite element formulation. Comput. Struct., 1986, 23(3), 291-303.
[BAR 83] BARBE I .,Structures coques : equations generales et stabilite. Ecole nationale
superieure de l'aeronautique et de l'espace, 1983.
(BAT 92] BATOZ I. L., DHATT G., Modelisation des structures par elements finis, volume 3 :
coques. Hermes, Paris, 1992.
(BID 89] BIDMARADDI A., CARR A. I., MOSS P. I., « A shear deformable finite element for the
analysis of general shells of revolution», Comput. Struct., 1989, 31(3), 299-308
[BUS 84] BUSHNEll D., «Computerized analysis of shells -Governing equations». Comput.
Struct., 1984, 18 (3), 471-536.
[CAR 69] CARTER R. L., ROBINSON A. R., SCHNOBRICH W. C., «Free vibrations of hyperboloidal
shells of revolution». J. Engng. Mech. Div., 1969, ASCE 95, 1033-1052.
[coo 74] COOK R. D., Concepts and applications of finite element analysis. NewYork, John
Wiley & sons, 1974.
[DEL 80] DELPAK R. and HAGUE W.M., «An experimental and theoritical investigation of the
frequencies and mode shapes of axisymmetric shell models >>. J. Sound Vibr., 1980,
(2), 235-249.
[DHA 84] DHATI G., TOUZOT G., Une presentation de /a methode des elements finis. 2° ed.
Paris, Maloine, 1984.
[FAN 89] FANs. c., and LUAH M. H., « Spline finite element for axisymmetric free vibrations of
shells of revolution>>. J. Sound Vibr., 1989, 132(1), 61-72.
[FRA 91] FRAMASOFT et C.S.I., Manuel de reference MEF/MOSAIC V2.6., mars 1991.
[GOU 85] GoULD Ph. L., Finite element analysis of shells of revolution. London :Pitman,
[GUE 93] GUERICH M., BENJEDDOU A., HAMDI M.A., « Interpolation par les B3-splines
uniformes : approche elements finis ». In 4e Colloque Maghrebin sur les Modeles
Numeriques de l'/ngenieur. Alger : 22-24 novembre 1993 (a paraitre).
[GUP 86] GUPTA A. Cubic B-spline finite element for axisymmetric shells. Ph. D. Engineering
Mechanics, Pennsylvania State Univ., 1986.
[HAN 86] HANSEN J. S., HEPPLER G. R., « Basis functions for axisymetric CO shell element
which satisfy rigid body requirements>>, in HUGHES T. J. R., HINTON E. (ed.),finite element
method for plate and shell structures, vol. 1 : Element technology. Swansea, Pineridge
Press, 1986.
[HEP 92] HEPPLER G. R., wAHL L., «Finite element analysis of free-free shells of revolution >>.
J. Sound Vibr., 1992, 152(2), 263-283.
[HUA 89] HUANG H. c., Static and dynamic analysis of plates and shells. Berlin : SpringerVerlag,
[JEA 85] JEAN Ph., Une methode variationnelle par equations integrales pour la resolution
numerique de problemes interieurs et exterieurs de couplage elasto-acoustique. Doct.
mecanique appliquee, Compiegne, 1985, annexe A.
[KAR 87] KARDESTUNCERH., (ed.). Finite elemenJ handbook. NewYork: McGraw-Hill, 1987.
[KER 90] KERBER Th., « Revue des elements finis de coques au travers des phenomenes de
verrouillage et de leur remedes ». Rech. Aerosp., 1990, n° 3, p. 45-76.
[KIM 83] KIMJ., B-spline finite elemenJ analysis of arbitrary loaded shells of revolution. Ph.D.
Cornell Univ., 1983.
[KUN 88] KUNIEDA H., « Flexural axisymmetric free vibrations of a spherical dome : exact
results and approximate solutions>>. J. Sound Vibr., 1988, 92(1), 1-10.
[u 90] u w. Y ., THAM L. G., CHEUNG Y. K., «Free vibration analysis of doubly curved shells
by spline finite strip method>>. J. Sound Vibr., 1990, 140(1), 39-53.
[LUA 89] LUAH M. H., FANS. c., « General free vibration analysis of shells of revolution using
the spline finite element method>>. Comput. Struct., 1989, 33(5), 1153-1162.
(LUC 79] LUCAS R. T. A. Problemes associes a Ia modelisation de coques de revolution par
elements finis. Th. 3e cycle: mecanique appliquee a Ia construction, Compiegne, 1979.
(PRO 75] PROST J.P., DHATT G., Analyse de Ia stabilite elastique des Voiles minces de
revolution par Ia methode des elements finis. Rapport GCS-75-07-03. Quebec : Univ.
Laval, 1975.
[SEN 74] SEN K.S., GoULD Ph., « Free vibration of shells of revolution using finite element
method». 1. Engng. Mech. Div., 1974, 100, EM2, 283-303.
[SOE 80] SOEDEL w., «A new frequency formula for closed circular cylindrical shells for a
large variety of boundary conditions.» 1. Sound Vibr., 1980, 70(3), 309-317.
[TOU 87] TOU s. K., WONG K. K., « High-precision finite element analysis of cylindrical
shells». Comput. Struct., 1987, 26(5), 847-854.
[TRO 92] TROMPETTE Ph., Mecanique des structures par la methode des elements jinisStatique
et dynamique. Paris, Masson, 1992.
[VAL 77] VALID R., La mecanique des milieux continus et le calcul des structures. Paris,
Eyrolles, 1977.
[WEB 67] WEBSTER J. J., «Free vibrations of shells of revolution using ring fmite elements».
Int. 1. Mech. Sci., 1967, 9, 559-570.
[ZIE 77] ZIENKmwmcz o. c., The finite element method. 3rd ed., London, McGraw-Hill, 1977
