×

A semi-analytical isogeometric analysis for wave dispersion in functionally graded plates immersed in fluids. (English) Zbl 1458.74072

Summary: The semi-analytical finite element (SAFE) method is widely used for studying properties of guided waves along composite waveguides. However, evaluating the modes associated to high wave numbers requires important mesh refinements and may significantly increase the computational cost. This paper presents a semi-analytical isogeometric analysis (SAIGA) to calculate the dispersion relation of functionally graded or multilayer plates coupling with fluids. High-order elements based on non-uniform B-splines (NURBS) basis functions are used. Several numerical examples are then studied for different problems in order to assess the efficiency of proposed method: (i) homogeneous plates; (ii) functionally graded plates; (iii) composite plates (with strong contrast of rigidity between layers); (iv) fluid-immersed plates. The results obtained are compared with the ones derived from analytical approaches and by the conventional SAFE method using Lagrange polynomials. For all cases, the dispersion curves evaluated by using enriched-NURBS basis functions achieve a significant better precision than using conventional Lagrangian functions (for the same number of degrees of freedom or the same order of shape functions), especially for the higher modes. The continuity of the stress shape modes at the interfaces is also shown to be much improved by using SAIGA.

MSC:

74J05 Linear waves in solid mechanics
74K20 Plates
74S22 Isogeometric methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74E30 Composite and mixture properties
74S05 Finite element methods applied to problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bartoli, I.; Marzani, A.; di Scalea, FL; Viola, E., Modeling wave propagation in damped waveguides of arbitrary cross-section, J. Sound Vib., 295, 3, 685-707 (2006)
[2] Bazilevs, Y.; Akkerman, I., Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method, J. Comput. Phys., 229, 9, 3402-3414 (2010) · Zbl 1290.76037
[3] Bazilevs, Y.; Beirao da Veiga, L.; Cottrell, JA; Hughes, TJ; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math. Model. Methods Appl. Sci., 16, 7, 1031-1090 (2006) · Zbl 1103.65113
[4] Bernard, A.; Lowe, M.; Deschamps, M., Guided waves energy velocity in absorbing and non-absorbing plates, J. Acoust. Soc. Am., 110, 1, 186-196 (2001)
[5] Carcione, JM, Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media (2007), Amsterdam: Elsevier, Amsterdam
[6] Castaings, M.; Lowe, M., Finite element model for waves guided along solid systems of arbitrary section coupled to infinite solid media, J. Acoust. Soc. Am., 123, 2, 696-708 (2008)
[7] Cottrell, J.; Reali, A.; Bazilevs, Y.; Hughes, T., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Eng., 195, 41, 5257-5296 (2006) · Zbl 1119.74024
[8] Cottrell, J.; Hughes, T.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. Methods Appl. Mech. Eng., 196, 41, 4160-4183 (2007) · Zbl 1173.74407
[9] Cottrell, JA; Hughes, TJ; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), Hoboken: Wiley, Hoboken · Zbl 1378.65009
[10] Dedé, L.; Jäggli, C.; Quarteroni, A., Isogeometric numerical dispersion analysis for two-dimensional elastic wave propagation, Comput. Methods Appl. Mech. Eng., 284, 320-348 (2015) · Zbl 1423.74094
[11] Duan, W.; Kirby, R., Guided wave propagation in buried and immersed fluid-filled pipes: application of the semi analytic finite element method, Comput. Struct., 212, 236-247 (2019)
[12] Echter, R.; Oesterle, B.; Bischoff, M., A hierarchic family of isogeometric shell finite elements, Comput. Methods Appl. Mech. Eng., 254, 170-180 (2013) · Zbl 1297.74071
[13] Fan, Z.; Lowe, M.; Castaings, M.; Bacon, C., Torsional waves propagation along a waveguide of arbitrary cross section immersed in a perfect fluid, J. Acoust. Soc. Am., 124, 4, 2002-2010 (2008)
[14] Gravenkamp, H.; Birk, C.; Van, J., Modeling ultrasonic waves in elastic waveguides of arbitrary cross-section embedded in infinite solid medium, Comput. Struct., 149, 61-71 (2015)
[15] Gravenkamp, H.; Natarajan, S.; Dornisch, W., On the use of NURBS-based discretizations in the scaled boundary finite element method for wave propagation problems, Comput. Methods Appl. Mech. Eng., 315, 867-880 (2017) · Zbl 1439.74484
[16] Hayashi, T.; Inoue, D., Calculation of leaky Lamb waves with a semi-analytical finite element method, Ultrasonics, 54, 6, 1460-1469 (2014)
[17] Hayashi, T.; Tamayama, C.; Murase, M., Wave structure analysis of guided waves in a bar with an arbitrary cross-section, Ultrasonics, 44, 1, 17-24 (2006)
[18] Hughes, TJ, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2012), Chelmsford: Courier Corporation, Chelmsford
[19] Hughes, T.; Reali, A.; Sangalli, G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS, Comput. Methods Appl. Mech. Eng., 197, 49, 4104-4124 (2008) · Zbl 1194.74114
[20] Hughes, TJ; Evans, JA; Reali, A., Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems, Comput. Methods Appl. Mech. Eng., 272, 290-320 (2014) · Zbl 1296.65148
[21] Joseph, R.; Li, L.; Haider, MF; Giurgiutiu, V., Hybrid SAFE-GMM approach for predictive modeling of guided wave propagation in layered media, Eng. Struct., 193, 194-206 (2019)
[22] Kalkowski, MK; Rustighi, E.; Waters, TP, Modelling piezoelectric excitation in waveguides using the semi-analytical finite element method, Comput. Struct., 173, 174-186 (2016)
[23] Kalkowski, MK; Muggleton, JM; Rustighi, E., Axisymmetric semi-analytical finite elements for modelling waves in buried/submerged fluid-filled waveguides, Comput. Struct., 196, 327-340 (2018)
[24] Kaminski, M., On the dual iterative stochastic perturbation-based finite element method in solid mechanics with gaussian uncertainties, Int. J. Numer. Methods Eng., 104, 1038-1060 (2015) · Zbl 1352.74375
[25] Kim, S.; Pasciak, J., The computation of resonances in open systems using a perfectly matched layer, Math. Comput., 78, 267, 1375-1398 (2009) · Zbl 1198.65224
[26] Liu, Y.; Han, Q.; Liang, Y.; Xu, G., Numerical investigation of dispersive behaviors for helical thread waveguides using the semi-analytical isogeometric analysis method, Ultrasonics, 83, 126-136 (2018)
[27] Loveday, PW, Semi-analytical finite element analysis of elastic waveguides subjected to axial loads, Ultrasonics, 49, 3, 298-300 (2009)
[28] Lowe, M., Pavlakovic., B.: DISPERSE. Users Manual (2013)
[29] Mazzotti, M.; Marzani, A.; Bartoli, I.; Viola, E., Guided waves dispersion analysis for prestressed viscoelastic waveguides by means of the SAFE method, Int. J. Solids Struct., 49, 18, 2359-2372 (2012)
[30] Mazzotti, M.; Bartoli, I.; Marzani, A.; Viola, E., A coupled SAFE-2.5 D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary cross-section, Ultrasonics, 53, 7, 1227-1241 (2013)
[31] Mazzotti, M.; Miniaci, M.; Bartoli, I., A numerical method for modeling ultrasonic guided waves in thin-walled waveguides coupled to fluids, Comput. Struct., 212, 248-256 (2019)
[32] Na, WB; Kundu, T., Underwater pipeline inspection using guided waves, J. Press. Vessel Technol., 124, 2, 196-200 (2002)
[33] Nguyen, T.N., Ngo, T.D., Nguyen-Xuan, H.: A novel three-variable shear deformation plate formulation: Theory and isogeometric implementation. Comput. Methods Appl. Mech. Eng. 326, 376-401 (2017) · Zbl 1439.74009
[34] Nguyen, KL; Treyssede, F.; Hazard, C., Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods, J. Sound Vib., 344, 158-178 (2015)
[35] Nguyen, LB; Thai, CH; Zenkour, AM; Nguyen-Xuan, H., An isogeometric Bézier finite element method for vibration analysis of functionally graded piezoelectric material porous plates, Int. J. Mech. Sci., 157-158, 165-183 (2019)
[36] Nguyen, VH; Naili, S., Simulation of ultrasonic wave propagation in anisotropic poroelastic bone plate using hybrid spectral/finite element method, Int. J. Numer. Methods Biomed. Eng., 28, 8, 861-876 (2012)
[37] Nguyen, VH; Naili, S., Ultrasonic wave propagation in viscoelastic cortical bone plate coupled with fluids: a spectral finite element study, Comput. Methods Biomech. Biomed. Eng., 16, 9, 963-974 (2013)
[38] Nguyen, VH; Abdoulatuf, A.; Desceliers, C.; Naili, S., A probabilistic study of reflection and transmission coefficients of random anisotropic elastic plates, Wave Motion, 64, 103-118 (2016) · Zbl 1469.74029
[39] Nguyen, VH; Tran, TN; Sacchi, MD; Naili, S.; Le, LH, Computing dispersion curves of elastic/viscoelastic transversely-isotropic bone plates coupled with soft tissue and marrow using semi-analytical finite element (SAFE) method, Comput. Biol. Med., 87, 371-381 (2017)
[40] Park, CB; Miller, RD; Xia, J., Multichannel analysis of surface waves, Geophysics, 64, 3, 800-808 (1999)
[41] Pereira, D.; Haiat, G.; Fernandes, J.; Belanger, P., Simulation of acoustic guided wave propagation in cortical bone using a semi-analytical finite element method, J. Acoust. Soc. Am., 141, 4, 2538-2547 (2017)
[42] Piegl, L.; Tiller, W., The NURBS book (2012), Berlin: Springer, Berlin · Zbl 0828.68118
[43] Rose, JL, Ultrasonic Guided Waves in Solid Media (2014), Cambridge: Cambridge University Press, Cambridge
[44] Su, Z.; Ye, L.; Lu, Y., Guided Lamb waves for identification of damage in composite structures: a review, J. Sound Vib., 295, 3-5, 753-780 (2006)
[45] Teixeira, F.; Chew, WC, Complex space approach to perfectly matched layers: a review and some new developments, Int. J. Numer. Model. Electron. Netw. Devices Fields, 13, 5, 441-455 (2000) · Zbl 1090.78534
[46] Thai, CH; Nguyen-Xuan, H.; Nguyen-Thanh, N.; Le, TH; Nguyen-Thoi, T.; Rabczuk, T., Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach, Int. J. Numer. Meth. Eng., 91, 6, 571-603 (2012) · Zbl 1253.74007
[47] Thai, CH; Kulasegaram, S.; Tran, LV; Nguyen-Xuan, H., Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach, Comput. Struct., 141, 94-112 (2014)
[48] Tjahjowidodo, T., A direct method to solve optimal knots of b-spline curves: an application for non-uniform B-spline curves fitting, PLoS ONE, 12, 3, e0173857 (2017)
[49] Treyssede, F., Spectral element computation of high-frequency leaky modes in three-dimensional solid waveguides, J. Comput. Phys., 314, 341-354 (2016) · Zbl 1349.74201
[50] Wang, D.; Liu, W.; Zhang, H., Novel higher order mass matrices for isogeometric structural vibration analysis, Comput. Methods Appl. Mech. Eng., 260, 92-108 (2013) · Zbl 1286.74051
[51] Willberg, C.; Duczek, S.; Perez, JV; Schmicker, D.; Gabbert, U., Comparison of different higher order finite element schemes for the simulation of Lamb waves, Comput. Methods Appl. Mech. Eng., 241, 246-261 (2012) · Zbl 1353.74077
[52] Zuo, P.; Fan, Z., SAFE-PML approach for modal study of waveguides with arbitrary cross sections immersed in inviscid fluid, J. Sound Vib., 406, 181-196 (2017)
[53] Zuo, P.; Yu, X.; Fan, Z., Numerical modeling of embedded solid waveguides using SAFE-PML approach using a commercially available finite element package, NDT E Int., 90, 11-23 (2017)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.