Analytical Model Of Spectral Element For Two-dimensional Plates With Symmetrical Boundary And Guided Wave Numerical Simulations

Compared to the Finite Element Method,Spectral Element Method,which is based on Fourier transform theory and spectral analysis method,has the advantages of high precision,few elements and high frequency sensitivity.With the development of guided-wave-based fault detection technology and increasing requirements for numerical simulation in terms of accuracy and efficiency,spectral element method gains more attention.Existing spectral element model is mainly developed for simple structral systems,such as one-dimensional systems or equivalent one-dimensional systems.This paper is aimed to propose a two-dimensional plate’s spectral element model that can be applied to different boundary based on a dynamic stiffness model of two-dimensional rectangular plate with arbitrary symmetrical boundary.This work will lay the basis for numerical simulation of guided wave in plate structures for default detection.A dynamic stiffness model with arbitrary symmetric boundary of two-dimensional rectangular plates is deduced in detail and a complete solution is presented.Based on previous research on a frequency-dependent two-dimensional analytical model,the displacement field,boundary displacements and boundary forces are divided into four parts by taking advange of the symmetry of the plate.The relations of Fourier coefficients of boundary forces and boundary displacements with the unknown coefficients in the displacement field are obtained by Fourier series expansion.And then,infinite equations of Fourier coefficients of boundary forces and boundary displacements are obtaind,so are the dynamic stiffness matrix and dynamic stiffness equations of two-dimensional plates.MATHEMATICA is applied for the deduction of the dynamic stiffness model of two-dimensional plate.Based on the dynamic stiffness model with arbitrary symmetric boundary,the natural frequencies and modal shapes of rectangular two-dimensional plates is numerically analyzed.After boundary displacement and boundary force are imposed,the eigen equation and dimensionless natural frequencies are obtained.For each natural frequency,its modal shapes can be obtained by substituting the Fourier coefficient vectors of boundary forces and boundary displacements into the displacement field.Natural frequencies and modal shapes of two-dimensional plates with the boundary of Free-Free-Free-Free,Clamped-ClampedClamped-Clamped,Clamped-Free-Clamped-Free and Simple-Free-Simple-Free are obtained.It is noticed that the spectral element method is more accurate than the other methods.Based on the dynamic stiffness model with arbitrary symmetric boundary of a two-dimensional rectangular plate,Lamb wave in a default-free two-dimensional plate is numerically simulated.By Fourier series expansion to the boundary loads in the space domain and Fourier transform in the time domain,the boundary excitation Fourier coefficient vector is obtained.For each frequency,the boundary displacement Fourier coefficient vector is calculated by the dynamic stiffness equations.Furthermore,boundary displacements in tim e domain are obtained by using Fourier transform in time and Fourier series expansion summation in space.The boundary displacements of the Clamped-Free-Clamped-Free plate under the impedance uniform symmetrical loads are simulated.The results coincide with thoses of the Finite Elemet Method using ANSYS,which prove the accuracy and the convergence of the spectral element method.