Isogeometric analysis of structural vibrations. (English) Zbl 1119.74024

Summary: This paper begins with personal recollections of John H. Argyris. The geometrical spirit embodied in Argyris’s work is revived in the sequel in applying the newly developed concept of isogeometric analysis to structural vibration problems. After reviewing some fundamentals of isogeometric analysis, application is made to several structural models, including rods, thin beams, membranes, and thin plates. Rotationless beam and plate models are utilized as well as three-dimensional solid models. The concept of \(k\)-refinement is explored and shown to produce more accurate and robust results than corresponding finite elements. Through the use of nonlinear parameterization, “optical” branches of frequency spectra are eliminated for \(k\)-refined meshes. Optical branches have been identified as contributors to Gibbs phenomena in wave propagation problems and the cause of rapid degradation of higher modes in \(p\)-method finite elements. A geometrically exact model of the NASA Aluminum Testbed Cylinder is constructed, and frequencies and mode shapes are computed and shown to compare favorably with experimental results.


74H45 Vibrations in dynamical problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics


Full Text: DOI HAL


[1] J.H. Argyris, The computer shapes the theory. Paper read to the Royal Aeronautical Society on May 18, 1965 (Lecture Summary: Aeronaut. J. Royal Soc. 69 (1965) XXXII).
[2] J.H. Argyris, Energy theorems and structural analysis. Part I. General theory, Aircraft Engrg. 26 (1954) 347-356, 383-387, 394; 27 (1955) 42-58, 80-94, 125-134, 145-158; Part II (with S. Kelsey): Applications to thermal stress analysis and to upper and lower limits of St. Venant torsion constant, Aircraft Engrg. 26 (1954) 410-422.
[3] Argyris, J.H.; Kelsey, S., Energy theorems and structural analysis, (1960), Butterworths London · Zbl 0097.40304
[4] Argyris, J.H.; Kelsey, S., Modern fuselage analysis and the elastic aircraft, (1963), Butterworths London
[5] ARPACK. Available from: <http://www.caam.rice.edu/software/arpack/>.
[6] Benninghof, J.K.; Lehoucq, R.B., An automated multilevel substructuring method for eigenspace computations in linear elastodynamics, SIAM J. sci. comput., 25, 2084-2106, (2004) · Zbl 1133.65304
[7] T.D. Blacker. CUBIT Mesh Generation Environment Users Manual, vol. 1, Technical Report, Sandia National Laboratories, Albuquerque, NM, 1994.
[8] Brillouin, L., Wave propagation in periodic structures, (1953), Dover Publications, Inc. · Zbl 0050.45002
[9] R.D. Buehrle, G.A. Fleming, R.S. Pappa, F.W. Grosveld, Finite element model development for aircraft fuselage structures, in: Proceedings of XVIII International Modal Analysis Conference, San Antonio, TX, 2000.
[10] Bushnell, D., Stress, stability and vibration of complex, branched shells of revolution, Comput. struct., 4, 399-435, (1974)
[11] Bushnell, D., Computerized buckling analysis of shells, (1985), Martinus Nijhoff Publishers Dordrecht
[12] Chopra, A.K., Dynamics of structures. theory and applications to earthquake engineering, (2001), Prentice-Hall Upper Saddle River, New Jersey
[13] Chung, J.; Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. appl. mech., 60, 371-375, (1993) · Zbl 0775.73337
[14] Cirak, F.; Ortiz, M., Fully C1-conforming subdivision elements for finite deformation thin shell analysis, Int. J. numer. methods engrg., 51, 813-833, (2001) · Zbl 1039.74045
[15] Cirak, F.; Ortiz, M.; Schröder, P., Subdivision surfaces: a new paradigm for thin shell analysis, Int. J. numer. methods engrg., 47, 2039-2072, (2000) · Zbl 0983.74063
[16] Cirak, F.; Scott, M.J.; Antonsson, E.K.; Ortiz, M.; Schröder, P., Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision, Computer-aided des., 34, 137-148, (2002)
[17] Clough, R.W.; Penzien, J., Dynamics of structures, (1993), McGraw-Hill New York
[18] Couchman, L.; Dey, S.; Barzow, T., ATC eigen-analysis, STARS/ARPACK/NRL solver, (2003), Naval Research Laboratory Englewood Cliffs, NJ
[19] Engel, G.; Garikipati, K.; Hughes, T.J.R.; Larson, M.G.; Mazzei, L., Continuous /discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. methods appl. mech. engrg., 191, 3669-3750, (2002) · Zbl 1086.74038
[20] Farin, G.E., NURBS curves and surfaces: from projective geometry to practical use, (1995), A.K. Peters, Ltd. Natick, MA · Zbl 0848.68112
[21] Fried, I.; Malkus, D.S., Finite element mass matrix lumping by numerical integration with no convergence rate loss, Int. J. solids struct., 11, 461-466, (1975) · Zbl 0301.65010
[22] Gallagher, R.H., A correlation study of methods of matrix structural analysis, (1964), Pergamon Oxford · Zbl 0135.37404
[23] F.W. Grosveld, J.I. Pritchard, R.D. Buehrle, R.S. Pappa, Finite element modeling of the NASA Langley Aluminum Testbed Cylinder. in: 8th AIAA/CEAS Aeroacoustics Conference, Breckenridge, CO, 2002, AIAA 2002-2418.
[24] Hilber, H.; Hughes, T.J.R., Collocation, dissipation, and overshoot for time integration schemes in structural dynamics, Earthquake engrg. struct. dyn., 6, 99-117, (1978)
[25] Hilber, H.; Hughes, T.J.R.; Taylor, R., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake engrg. struct. dyn., 5, 283-292, (1977)
[26] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Mineola, NY
[27] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comput. methods appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[28] Meirovitch, L., Analytical methods in vibrations, (1967), The MacMillan Company New York · Zbl 0166.43803
[29] Miranda, I.; Ferencz, R.M.; Hughes, T.J.R., An improved implicit – explicit time integration method for structural dynamics, Earthquake engrg. struct. dyn., 18, 643-653, (1989)
[30] Montgomery, D.C.; Runger, G.C.; Hubele, N.F., Engineering statistics, (2003), Wiley New York, NY
[31] Oñate, E.; Cervera, M., Derivation of thin plate bending elements with one degree of freedom per node: a simple three node triangle, Engrg. comput., 10, 543-561, (1993)
[32] Oñate, E.; Zarate, F., Rotation-free triangular plate and shell elements, Int. J. numer. methods engrg., 47, 557-603, (2000) · Zbl 0968.74070
[33] Phaal, R.; Calladine, C.R., A simple class of finite-elements for plate and shell problems: 1. elements for beams and thin flat plates, Int. J. numer. methods engrg., 35, 955-977, (1992) · Zbl 0775.73285
[34] Phaal, R.; Calladine, C.R., A simple class of finite-elements for plate and shell problems: 2. an element for thin shells with only translational degrees of freedom, Int. J. numer. methods engrg., 35, 979-996, (1992) · Zbl 0775.73286
[35] Piegl, L.; Tiller, W., The NURBS book (monographs in visual communication), (1997), Springer-Verlag New York
[36] K.S. Pister, R.R. Reynolds, K. Willam (Eds.), Proceedings of Fenomech’78 - 1st International Conference on Finite Elements in Nonlinear Mechanics, University of Stuttgart, August 30-September 1, 1978. Computer Methods in Applied Mechanics and Engineering, vol. 17-18, 1979, 720 pp.
[37] Prudhomme, S.; Pascal, F.; Oden, J.T.; Romkes, A., A priori error estimate for the baumann – oden version of the discontinuous Galerkin method, Comptes rendus de l’academie des sciences I, numer. anal., 332, 851-856, (2001) · Zbl 1007.65084
[38] Rogers, D.F., An introduction to NURBS with historical perspective, (2001), Academic Press San Diego, CA
[39] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[40] Truesdell, C., The computer: ruin of science and threat to mankind, (), 594-631
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.