zbMATH — the first resource for mathematics

Numerical study of an intrinsic component mode synthesis method. (English) Zbl 0763.73080
Summary: Component mode synthesis belongs to the class of Galerkin methods and it enables us to compute the normal modes of linearly elastic structures which can be divided into several substructures whose lowest eigenfrequencies and corresponding normal modes are known. Energy transfer between substructures is achieved thanks to the introduction in the Ritz procedure of mode shapes defined on the whole structure, usually called ‘static modes’ or ‘constraint modes’. A new fixed interface method is presented in a continuous framework: it is based on a non-conventional choice of constraint modes tied to the normal modes of the Poincaré- Steklov operator associated with the interface between the substructures. Error bounds are given in the case of three-dimensional elasticity. An efficient domain decomposition algorithm is presented in detail as well as various tests.

74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
[1] M. Aufranc, Junctions between three-dimensional and two-dimensional non-linearly elastic structures, Asymptotic Anal., to appear. · Zbl 0754.73034
[2] Aufranc, M., Sur quelques problèmes de jonctions dans LES multistructures elastiques, Thèse de l’université pierre et marie Curie, (1990), Paris · Zbl 0858.73009
[3] Bernadou, M.; Fayolle, S.; Lene, F., Numerical analysis of junctions between plates, INRIA report nb 865, (1988)
[4] Bourquin, F.; Ciarlet, P.G., Modeling and justification of eigenvalue problems for junctions between elastic structures, J. functional anal., 87, 392-427, (1989) · Zbl 0699.73010
[5] Ciarlet, P.G., Plates and junctions in elastic multi-structures: an asymptotic analysis, () · Zbl 0706.73046
[6] Ciarlet, P.G.; Le Dret, H.; Nzengwa, R., Junctions between three-dimensional and two-dimensional linearly elastic structures, J. math. pures appl., 68, 261-296, (1989) · Zbl 0661.73013
[7] Le Dret, H., Modélisation d’une plaque pliée, Comput. mech., 5, 401-416, (1987) · Zbl 0634.73047
[8] Morand, H.; Ohayon, R., Substructure variational analysis of the vibrations of coupled fluid-structure systems. finite element results, Internat. J. numer. methods engrg., 14, 741-755, (1979) · Zbl 0402.73052
[9] Valid, R.; Ohayon, R., Influence du ballottement dans LES réservoirs des bouts d’ailes sur LES modes propres de vibration d’un avion, La recherche Aérospatiale, 319-325, (1974)
[10] Valid, R., Une méthode de calcul des structures au flambage par sous structuration et synthèse modale, C.R. acad. sci. Paris, Sér. II, 294, 299-302, (1982) · Zbl 0485.73032
[11] Hurty, W.C., Dynamic analysis of structural systems using component modes, Aiaa j., 4, 678-685, (1965)
[12] Craig, R.; Bampton, M.C.C., Coupling of substructures for dynamic analysis, Aiaa j., 6, 1313-1321, (1968) · Zbl 0159.56202
[13] Goldman, R.L., Vibration analysis of dynamic partitioning, Aiaa j., 7, 1152-1154, (1969) · Zbl 0179.55102
[14] MacNeal, R.H., A hybrid method of component mode synthesis, Comput. & structures, 1, 560-581, (1971)
[15] Rubin, S., Improved component mode representation for structural dynamic analysis, Aiaa j., 13, 995-1006, (1975) · Zbl 0334.70014
[16] Gladwell, B.M.L., Branch mode analysis of vibrating systems, J. sound vibration, 1, 41-59, (1964) · Zbl 0124.39303
[17] Jezequel, L., Synthèse modale: théorie et extensions, ()
[18] Destuynder, Ph., Remarks on dynamic substructuring, European J. mech. A/solids, 8, 201-218, (1989) · Zbl 0692.73059
[19] Hale, A.J.; Meirovitch, L., A general substructure synthesis method for the dynamic simulation of complex structures, J. sound vibration, 69, 309-326, (1980) · Zbl 0441.73084
[20] Wang, J.H.; Chen, H.R., A substructure modal synthesis method with high computational efficiency, Comput. methods appl. mech. engrg., 79, 203-217, (1990) · Zbl 0715.73082
[21] Craig, R.R., A review of time domain and frequency domain component mode synthesis methods, ()
[22] Gibert, R.J., Vibrations des structures, interactions avec LES fluides, sources d’excitations aléatoires, ()
[23] Imbert, J.F., Calcul des structures par elements finis, (1979), Cépadues · Zbl 0685.73003
[24] Meirovitch, L., Computational methods in structural dynamics, (1980), Sijthoff & Noordhoff Alphen aan den Ryn · Zbl 0439.73074
[25] Morand, H., Méthodes de détermination approchée des modes propres de vibration en calcul des structures; sous-structuration dynamique, Onera r.t. n 2-3238 ry ooor, (1977)
[26] Valid, R., La Mécanique des milieux continus et le calcul des structures, (1977), Collection de la Direction des Etudes et Recherches d’Electricité de France Eyrolles · Zbl 0454.73003
[27] Bourquin, F., Analysis and comparison of several component mode synthesis methods on one-dimensional domains, Numer. math., 58, 11-34, (1990) · Zbl 0686.34026
[28] Bourquin, F., Synthèse modal d’opérateurs elliptiques du second ordre, C.R. acad. sci., Paris Sér. I, 309, 919-922, (1989) · Zbl 0685.47038
[29] Agoshkov, V.I., Poincaré-Steklov’s operators and domain decomposition methods in finite dimensional spaces, () · Zbl 0683.65097
[30] F. Bourquin, Domain decomposition and eigenvalues of second order operators: Convergence analysis, in preparation. · Zbl 0765.65100
[31] Bourquin, F., Component mode synthesis and eigenvalues of second order operators: discretization and algorithm, RAIRO modél, Math. numer. anal., 26, 3, (1992), to appear · Zbl 0765.65100
[32] Bourquin, F., Synthèse modale et analyse numérique des multistructures élastiques, ()
[33] Bourgat, J.F.; Glowinski, R.; Le Tallec, P., Formulation variationelle et algorithme de décomposition de domaines pour LES problèmes elliptiques, C.R. acad. sci. Paris, Sér. I, 306, 569-572, (1988) · Zbl 0637.65104
[34] Bourgat, J.F.; Glowinski, R.; Le Tallec, P.; Vidrascu, M., Variational formulation and algorithm for trace operator in domain decomposition calculations, INRIA report 804, (1988)
[35] Taylor, A.E., Introduction to functional analysis, (1958), Wiley New York · Zbl 0081.10202
[36] Leguillon, D.; Palencia, E.Sanchez, Computation of singular solutions in elliptic problems and elasticity, () · Zbl 0647.73010
[37] Kondratiev, V.A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow math. soc., 16, 227-313, (1967) · Zbl 0194.13405
[38] Dauge, M., Boundary value problems on corner domains, ()
[39] Grisvard, P., Elliptic problems in non-smooth domains, (1985), Pitman London · Zbl 0695.35060
[40] Dorr, M.R., On the discretization of interdomain coupling in elliptic boundary value problems, () · Zbl 0682.65068
[41] Golub, G.H.; Underwood, R.; Wilkinson, J.H., The Lanczos algorithm for the symmetric \(Au = λBu\) problem, ()
[42] F. d’Hennezel, Opérateur d’interface non symétrique pour les méthodes de décomposition de domain, in preparation.
[43] Lascaux, P.; Theodor, R., Analyse numérique matricielle appliquée à l’art de l’ingénieur, (1986), Masson Paris · Zbl 0601.65016
[44] De Roeck, Y.H.; Le Tallec, P.; Vidrascu, M., Domain decomposition methods for large linearly elliptic three-dimensional problems, INRIA report 1182, (1990) · Zbl 0705.65086
[45] Babuška, I.; Dorr, M.R., Error estimates for the combined \(h and p\) version of the finite element method, Numer. math., 37, 515-545, (1981)
[46] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[47] Bramble, J.H.; Pasciak, J.E.; Schatz, A.H., An iterative method for elliptic problems on regions partitioned into substructures, Math. comp., 46, 361-369, (1986) · Zbl 0595.65111
[48] Bjorstad, P.E.; Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. numer. anal., 23, 1097-1120, (1986) · Zbl 0615.65113
[49] El-Raheb, M.; Wagner, P., Vibration of a liquid with free surface in a spinning spherical tank, J. sound vibration, 76, 83-93, (1981) · Zbl 0486.76032
[50] Grisvard, P., Singularités en elasticité, Arch. rational mech. anal., 107, 157-180, (1989) · Zbl 0706.73013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.