Extension of the variational theory of complex rays to orthotropic shallow shell structures

- Journal title : Advances in aircraft and spacecraft science
- Volume 3, Issue 3, 2016, pp.317-330
- Publisher : Techno-Press
- DOI : 10.12989/aas.2016.3.3.317

Title & Authors

Extension of the variational theory of complex rays to orthotropic shallow shell structures

Cattabiani, Alessandro; Barbarulo, Andrea; Riou, Herve; Ladeveze, Pierre;

Cattabiani, Alessandro; Barbarulo, Andrea; Riou, Herve; Ladeveze, Pierre;

Abstract

Nowadays, the interest of aerospace and automotive industries on virtual testing of medium-frequency vibrational behavior of shallow shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e., FEM, SEA) are not fully suitable for the medium-frequency bandwidth. In this context the Variational Theory of Complex Rays (VTCR) is taking place as an ad-hoc technique to address medium-frequency problems. It is a Trefftz method based on a weak variational formulation. It allows great flexibility because any shape function that satisfies the governing equations can be used. This work further develops such theory. In particular, orthotropic materials are introduced in the VTCR formulation for shallow shell structures. A significant numerical example is proposed to show the strategy.

Keywords

VTCR;medium-frequency;orthotropic materials;shallow shells;

Language

English

References

1.

Barbarulo, A., Ladeveze, P., Riou, H. and Kovalevsky, L. (2014), "Proper generalized decomposition applied to linear acoustic: a new tool for broad band calculation", J. Sound Vib., 333(11), 2422-2431.

2.

Bouillard, P., Suleaub, S. and Suleau, S. (1998), "Element-free galerkin solutions for helmholtz problems: formulation and numerical assessment of the pollution effect", Comput. Meth. Appl. Mech. Eng., 162(14), 317-335.

3.

Cattabiani, A., Riou, H., Barbarulo, A., Ladeveze, P., Bezier, G. and Troclet, B. (2015), "The Variational Theory of Complex Rays applied to the shallow shell theory", Comput. Struct., 158, 98-107.

4.

Cessenat, O. and Despres, B. (1998), "Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem", SIAM J. Numer. Anal., 35(1), 255-299.

5.

Chevreuil, M., Ladeveze, P. and Rouch, P. (2007), "Transient analysis including the low- and the medium-frequency ranges of engineering structures", Comput. Struct., 85(17-18), 1431-1444.

6.

De Rosa, S. and Franco, F. (2010), "On the use of the asymptotic scaled modal analysis for time-harmonic structural analysis and for the prediction of coupling loss factors for similar systems", Mech. Syst. Signal Pr., 24(2), 455-480.

7.

Deraemaeker, A. (1999), "Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions", Int. J. Numer. Meth. Eng., 46(4), 471-499.

8.

Desmet, W., Van Hal, B., Sas, P., Vandepitte, D. and Hal, B.V. (2002), "A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems", Adv. Eng. Softw., 33(7), 527-540.

9.

Farhat, C. and Roux, F.X. (1991), "A method of finite element tearing and interconnecting and its parallel solution algorithm", Int. J. Numer. Meth. Eng., 32(6), 1205-1227.

10.

Farhat, C., Harari, I. and Franca, L.P. (2001), "The discontinuous enrichment method", Comput. Meth. Appl. Mech. Eng., 190, 6455-6479.

11.

Genechten, B.V., Vandepitte, D. and Desmet, W. (2011), "A direct hybrid finite element wave based modelling technique for efficient coupled vibro-acoustic analysis", Comput. Meth. Appl. Mech. Eng., 200(5), 742-764.

12.

Hall, W. S. (1994), Boundary Element Method, Springer.

13.

Hughes, T.J.R. (2012), The finite element method: linear static and dynamic finite element analysis, Courier Dover Publications.

14.

Ihlenburg, F. (1998), Finite Element Analysis of Acoustic Scattering, Springer.

15.

Kovalevsky, L., Ladeveze, P., Riou, H. and Bonnet, M. (2012), "The variational theory of complex rays for three-dimensional helmholtz problems", J. Comput. Acoust., 20(4), 1-25.

16.

Kovalevsky, L., Riou, H. and Ladeveze, P. (2014), "A Trefftz approach for medium-frequency vibrations of orthotropic structures", Comput. Struct., 143, 85-90.

17.

Ladeveze, P. and Riou, H. (2005), "Calculation of medium-frequency vibrations over a wide frequency range", Comput. Meth. Appl. Mech. Eng., 194(27-29), 3167-3191.

18.

Ladeveze, P., Arnaud, L., Rouch, P. and Blanze, C. (2001), "The variational theory of complex rays for the calculation of medium-frequency vibrations", Eng. Comput., 18(1-2), 193-214.

19.

Ladeveze, P., Rouch, P., Riou, H. and Bohineust, X. (2003), "Analysis of medium-frequency vibrations in a frequency range", J. Comput. Acoust., 11(2), 255-283.

20.

Liu, Y. (2009), Fast Multipole Boundary Element Method: Theory and Applications in Engineering, Cambridge, University Press.

21.

Lyon, R.H. (1975), Statistical Energy Analysis of Dynamical Systems: Theory and Applications, MIT Press Cambridge, MA.

22.

Mace, B. (2003), "Statistical energy analysis, energy distribution models and system modes", J. Sound Vib., 264(2), 391-409.

23.

Monk, P. and Wang, D.Q.Q. (1999), "A least-squares method for the helmholtz equation", Comput. Meth. Appl. Mech. Eng., 175(1), 121-136.

24.

Perrey-Debain, E., Trevelyan, J. and Bettess, P. (2004), "Wave boundary elements: a theoretical overview presenting applications in scattering of short waves", Eng. Anal. Bound. Elem., 28, 131-141.

25.

Riou, H., Ladevèze, P. and Rouch, P. (2004), "Extension of the variational theory of complex rays to shells for medium-frequency vibrations", J. Sound Vib., 272(1), 341-360.

26.

Soize, C. (1998), "Reduced models in the medium frequency range for general external structural-acoustics Systems", J. Acoust. Soc. Am., 103(6), 3393-3406.

27.

Strouboulis, T. and Hidajat, R. (2006), "Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases", Appl. Math., 51(2), 181-204.

28.

Tezaur, R., Kalashnikova, I. and Farhat, C. (2014), "The discontinuous enrichment method for medium- frequency Helmholtz problems with a spatially variable wavenumber", Comput. Meth. Appl. Mech. Eng., 268, 126-140.

29.

van der Heijden, A.M.A. (2009), WT Koiter's Elastic Stability of Solids and Structures, Cambridge University Press Cambridge.

30.

Ventsel, E. and Krauthammer, T. (2001), Thin Plates and Shells: Theory, Analysis, and Applications, CRC press, Basel.