A novel polygonal Fourier spectral plate element was developed using the Rayleigh–Ritz method in this paper. Better than plate elements of traditional spectral element method (SEM), it can be any polygon shape with embed stiffeners and holes. Several case studies, including vibration characteristics analysis of a plate, a plate with hole, a plate with stiffener and an arbitrary shape car side plate, were carried out to test the useability and accuracy of the new element. Compared with results using finite element method (FEM) and traditional SEM, the new element can achieve similar accuracy with a faster computational speed. It can be 10% faster than FEM and 10 times faster than traditional SEM for some cases, and even more if the plate shape is more complicated. This is because the new element uses less elements and couplings than traditional SEM. As part of the SEM family, it also inherited the advantage of smaller system matrix size with less degree of freedoms than FEM. Therefore, it can be a better option for analyzing plate structure vibration characteristics in the middle frequency band gap left by FEM and statistical energy analysis, and can extend the application of SEM significantly.
AmoushahiHAzhariMHeidarpourA (2015) A fully discretized nonlinear finite strip formulation for pre-buckling and buckling analyses of viscoelastic plates subjected to time-dependent loading. Mechanics of Advanced Materials and Structures22(8): 655–669.
2.
BabuškaIMSauterSA (1997) Is the pollution effect of the FEM avoidable for the helmholtz equation considering high wave numbers?SIAM Journal on Numerical Analysis34(6): 2392–2423.
3.
BabuškaISuriM (1994) The p and h - p versions of the finite element method, basic principles and properties. SIAM Review36(4): 578–632.
4.
BiesDAHamidS (1980) In situ determination of loss and coupling loss factors by the power injection method. Journal of Sound and Vibration70(2): 187–204.
5.
BlackK (1997) A spectral element technique with a local spectral basis. SIAM Journal on Scientific Computing18(2): 355–370.
6.
CanutoC (ed) (2007) Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Scientific Computation. Berlin ; New York: Springer.
7.
Chang-HoiB-SYoungS (1991) Efficient hybrid finite element-boundary element method for 3-dimensional open-boundary field problems. IEEE Transactions on Magnetics27(5): 4069–4072.
8.
ChenYJinGZhuM, et al. (2012) Vibration behaviors of a box-type structure built up by plates and energy transmission through the structure. Journal of Sound and Vibration331(4): 849–867.
9.
ChengAH-DChengDT (2005) Heritage and early history of the boundary element method. Engineering Analysis with Boundary Elements29(3): 268–302.
10.
ClementsDL (1998) Fundamental solutions for second order linear elliptic partial differential equations. Computational Mechanics22(1): 26–31.
DuJLiWLJinG, et al. (2007) An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges. Journal of Sound and Vibration306(3–5): 908–927.
13.
DuJLiWLLiuZ, et al. (2011) Free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints. Journal of Sound and Vibration330(4): 788–804.
14.
DubinerM (1991) Spectral methods on triangles and other domains. Journal of Scientific Computing6(4): 345–390.
15.
FortunatoDHaleNTownsendA (2021) The ultraspherical spectral element method. Journal of Computational Physics436: 110087.
16.
HesthavenJS (1998) A stable penalty method for the compressible Navier--Stokes equations: III. Multidimensional domain decomposition schemes. SIAM Journal on Scientific Computing20(1): 62–93.
17.
JamshidiSAzhariMAmoushahiH (2016) Free vibration of plates of various shapes with intermediate point supports by the hp-cloud method and Lagrange multiplier. International Journal of Structural Stability and Dynamics16(09): 1550055.
18.
JinGMaXLiuZ, et al. (2017) Dynamic analysis of general rotationally symmetric built-up structures using a modified fourier spectral element approach. Journal of Vibration and Acoustics139(2): 021012.
19.
KimTLeeU (2016) Modified one-element method for exact dynamic responses of a beam by using the frequency domain spectral element method. International Journal of Mechanical Sciences119: 333–342.
20.
KimTLeeU (2017) Dynamic analysis of a multi-span beam subjected to a moving force using the frequency domain spectral element method. Computers & Structures192: 181–195.
21.
KimYHaSChangF-K (2008) Time-domain spectral element method for built-in piezoelectric-actuator-induced lamb wave propagation analysis. AIAA Journal46(3): 591–600.
22.
KomatitschDTsuboiSTrompJ (2005) The spectral-element method in seismology. In: LevanderANoletG (eds) Geophysical Monograph Series. Washington, D. C.: American Geophysical Union, 205–227. Available at: https://onlinelibrary.wiley.com/doi/10.1029/157GM13
23.
KudelaPOstachowiczW (2009) 3D time-domain spectral elements for stress waves modelling. Journal of Physics: Conference Series181: 012091.
24.
KudelaPŻakAKrawczukM, et al. (2007) Modelling of wave propagation in composite plates using the time domain spectral element method. Journal of Sound and Vibration302(4–5): 728–745.
25.
LeeU (2009) Spectral Element Method in Structural Dynamics. Singapore ; Hoboken, NJ: J. Wiley & Sons Asia.
26.
LiWL (2000) Free vibrations of beams with general boundary conditions. Journal of Sound and Vibration237(4): 709–725.
27.
LiWL (2002) Comparison of fourier sine and cosine series expansions for beams with arbitrary boundary conditions. Journal of Sound and Vibration255(1): 185–194.
28.
LiWLZhangXDuJ, et al. (2009) An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. Journal of Sound and Vibration321(1–2): 254–269.
29.
LiYWangL-LLiH, et al. (2011) A new spectral method on triangles. In: HesthavenJSRønquistEM (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 237–246. Available at: https://link.springer.com/10.1007/978-3-642-15337-2_21
30.
LiW-YLvBOuyangH, et al. (2014) A hybrid finite element-fourier spectral method for vibration analysis of structures with elastic boundary conditions. Mathematical Problems in Engineering2014: 1–11.
31.
LiuTHuGWangA, et al. (2019) A unified formulation for free in-plane vibrations of arbitrarily-shaped straight-sided quadrilateral and triangular thin plates. Applied Acoustics155: 407–422.
32.
LyonRH (1975) Statistical Energy Analysis of Dynamical Systems: Theory and Applications. Cambridge, Mass: MIT Press.
33.
LyuPDuJWangY, et al. (2019) Free in-plane vibration analysis of rotating annular panels with elastic boundary restraints. Journal of Sound and Vibration439: 434–456.
34.
MaXJinGXiongY, et al. (2014) Free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions. International Journal of Mechanical Sciences88: 122–137.
35.
MachadoMRKhalijLFabroAT (2019) Dynamic analysis of a composite structure under random excitation based on the spectral element method. International Journal of Nonlinear Sciences and Numerical Stimulation20(2): 179–190.
36.
MavriplisC (1994) Adaptive mesh strategies for the spectral element method. Computer Methods in Applied Mechanics and Engineering116(1–4): 77–86.
37.
MullenRBelytschkoT (1982) Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation. International Journal for Numerical Methods in Engineering18(1): 11–29.
38.
ParkILeeU (2012) Dynamic analysis of smart composite beams by using the frequency-domain spectral element method. Journal of Mechanical Science and Technology26(8): 2511–2521.
39.
ParkHWKimEJLimKL, et al. (2010) Spectral element formulation for dynamic analysis of a coupled piezoelectric wafer and beam system. Computers & Structures88(9–10): 567–580.
40.
PateraAT (1984) A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of Computational Physics54(3): 468–488.
41.
PathriaDKarniadakisGE (1995) Spectral element methods for elliptic problems in nonsmooth domains. Journal of Computational Physics122(1): 83–95.
42.
PetersenSDreyerDVon EstorffO (2006) Assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics. Computer Methods in Applied Mechanics and Engineering195(44–47): 6463–6478.
43.
PozrikidisC (2005) Introduction to Finite and Spectral Element Methods Using MATLAB. Boca Raton: Chapman & Hall/CRC.
44.
RuiminTWangE (2001) A new boundary element method for calculating the magnetic field of permanent magnetic machine. ICEMS’2001. Proceedings of the Fifth International Conference on Electrical Machines and Systems (IEEE Cat. No.01EX501)2: 1181–1183.
45.
SherwinSJKarniadakisGE (1995) A triangular spectral element method; applications to the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering123(1–4): 189–229.
46.
ThompsonLLPinskyPM (1995) A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation. International Journal for Numerical Methods in Engineering38(3): 371–397.
47.
TzouHSTsengCI (1990) Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. Journal of Sound and Vibration138(1): 17–34.
48.
WuZ-JLiF-MWangY-Z (2013) Study on vibration characteristics in periodic plate structures using the spectral element method. Acta Mechanica224(5): 1089–1101.
49.
XuHLiWL (2008) Dynamic behavior of multi-span bridges under moving loads with focusing on the effect of the coupling conditions between spans. Journal of Sound and Vibration312(4–5): 736–753.
50.
XuHDuJLiWL (2010) Vibrations of rectangular plates reinforced by any number of beams of arbitrary lengths and placement angles. Journal of Sound and Vibration329(18): 3759–3779.
51.
YaoLQLuL (2003) Hybrid-stabilized solid-shell model of laminated composite piezoelectric structures under non-linear distribution of electric potential through thickness. International Journal for Numerical Methods in Engineering58(10): 1499–1522.
52.
YuKHKadarmanAHDjojodihardjoH (2010) Development and implementation of some BEM variants—a critical review. Engineering Analysis with Boundary Elements34(10): 884–899.
53.
ZhangXLiWL (2010) A unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions. Journal of Sound and Vibration329(25): 5307–5320.
