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Project 1: 1D wave equation with finite elements
The purpose of this project is to derive and analyze a finite element method for the 1D wave equation $$ u_ = c^2 u_
The analysis will give insight into how to adjust the default behavior of the finite element method such that its properties are as good as those for the finite difference method for this particular equation. With the necessary adjustments discovered in detailed 1D calculations, one gets a recipe for constructing an accurate finite element method for simulating 2D and 3D waves in complex geometries.
Relevant background material consists of
- Time-dependent finite element discretization, which builds on finite element discretization in space.
- Analysis of wave equations, which builds on analysis of vibration equations.
Introduce a set of ( N_n=N-1 ) nodes numbered from left to right, with coordinates ( x_0,x_1,ldots,x_N ). Associated with these nodes are a set of basis functions ( <asphi_0(x),ldots,asphi_N(x)>).
Let ( uex(x,t) ) be the exact solution of the initial-boundary value problem. After discretization in time by finite differences, we get a set of time-discrete problems for ( uex^n(x) ), where ( uex^n(x)approx uex(x,t_n) ). These functions of space are then approximated by finite element expansions $$ egin
Alternatively, one can use a boundary function to take care of the Dirichlet condition at ( x=0 ): $$ egin
a) Set up equations for the coefficients ( c_j^0 ), ( jinIf ), associated with the initial condition ( u(x,0)=I(x) ). Use two methods: Galerkin and collocation/interpolation.
b) Use a finite difference method in time to formulate a sequence of spatial problems for ( uex^n(x) ), ( n=0,1,ldots,N_t ). A special problem is needed for ( n=1 ) in order to incorporate the boundary condition with ( u_t(x,t) ).
c) Apply Galerkin’s method to derive a variational form of each of the spatial problems. Integrate the term with the second-order derivative by parts. Express the variational form in terms of ( u^ ), ( u^n ), and ( u^ ).
d) Insert eqref or eqref in the variational form and derive the linear system to be solved at each time level. Express the system in the form $$ Mc^ = 2Mc^
Remark. Unless ( M ) is diagonal, a (tridiagonal) linear system must be solved at each time step, whereas the finite difference method leads to an explicit formula for ( u^_i ) at each spatial point at a new time level.
e) Use P1 elements and compute element matrices and vectors corresponding to the ( M ) and ( K ) matrices. Assemble the element contributions to a global matrices.
f) Interpret equation number ( i ) in the linear system as a finite difference approximation of ( u_=c^2u_
Hint. Write out an arbitrary equation in the linear system and group the unknown coefficients to mimic the differences above. Then substitute the coefficients by their corresponding ( u ) values, using a notation ( u^n_i ) as the finite element approximation of ( uex(x_i,t_n) ), and write the finite element equation in the same form as a finite difference scheme.
g) Perform an analysis of the scheme eqref in the same way as is done for the corresponding finite difference scheme in the course notes. That is, investigate a Fourier component ( u^n_p = exp <(i(kpDelta x — ilde omega nDelta t))>). Show that the stability criterion is $$ C leq frac<1><sqrt<3>>,$$ and therefore that any hope for recovering the exact solution for ( C=1 ) becomes impossible.
h) Find the numerical dispersion relation (( ildeomega ) expressed by other parameters) and plot the error in wave velocity ( ilde c/c ) (( ilde c = ilde omega/k ), ( c=omega/c )) as a function of ( kDelta x ) for various Courant numbers. Compare with the corresponding plot for the finite difference method for ( u_=c^2u_
i) Use the Trapezo > j) Instead of using the Trapezo > Filenames: wave1D_fem.py , wave1D_fem.pdf .
Remarks
Say we want to solve the 3D wave equation ( u_=c^2
abla^2u ) with finite elements and get the same stability as in the finite difference method. We can then compute ( M ) and ( K ) in the usual way and thereafter just replace ( M ) by ( mbox
Published on
06-Jul-2016
PAMM Proc. Appl. Math. Mech. 9, 509 510 (2009) / DOI 10.1002/pamm.200910227
Lamb wave propagation using Wave Finite Element MethodZair Asrar Bin Ahmad1,, Juan Miguel Vivar Perez1,, Christian Willberg1,, and Ulrich Gabbert1,1 Institut fr Mechanik, Otto-von-Guericke Universitt, Universittplatz 2, 39106 Magdeburg, Germany
Some of the available techniques for Lamb wave propagation simulation are the Finite Element Method (FEM), the BoundaryElement Method and the Finite Difference Method. The FEM is the best method when complex damage, geometry orboundary is involved. However, high Lamb wave frequency requires very small element size thus high computational costin FEM analysis. By using the existence of periodicity in plates, an attempt to reduce this computational cost is done usingWave FEM. The applicability of this method to model Lamb wave propagation in plate is first assessed in this paper for the1-D wave propagation and compared with FEM explicit method.
c 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1 IntroductionIn FEM analysis for Lamb wave propagation, the element in the plate model has to be small enough and about the same sizeto ensure no artificial reflection occured due to the change in mesh size. Thus, lots of periodic elements are available in theFEM model. On the other hand, Wave FEM has been used to take advantage of the periodicity condition in the model forstructural vibration analysis [1]. In the following, an initial attempt for Lamb wave propagation in a 1-D waveguide is made toaccess the applicability of Wave FEM method. This method uses the existing FEM procedures to obtain the stiffness matrix,K, the mass matrix, M and damping matrix, C of the basis cell (or elements). Dynamic equation involved is then solved inthe frequency domain using FFT method [2][4].
2 Theory of Wave FEMConsider a waveguide consisting of N periodic elements as shown in Fig.1. The dynamic equation of the basis cell atfrequency is given by
(K + jC 2M)q = f with force, f , displacement, q and dynamic stiffness matrix, D as
D = K + jC 2M. By manipulating matrix D, the transfer matrix, T which relates the displacements and forces incross sections n = 1 (left end of the structure) and n = N +1 (right end of the structure) can be obtained which gives the freewave propagation in the basis cell as described by the eigenproblem
Matrix T is symplectics having right and lefteigenvector, for the same eigenvalue. Thus bysolving the eigenproblems, using the orthogonal-ity relationship of the eigenvectors, and doingsome manipulation, a relation between force anddisplacement at the left end to the right end of theperiodic structure can be obtained as
Fig. 1 Periodic structures with basis cell having the force, (f ) and dis-placement, (q) vectors on the right (r ) and the left-hand (l) sides.
Where dynamic stiffness matrix, DT of the whole structure is
with Pil = Qi(Q)1 and Pir = Q+i(Q+)1. Matrix Q and is the eigenvectors and eigenvalues (where |i | 1)of matrix T respectively. Positive sign of matrix Q correspond to the wave travelling to the right end and vice versa. Eq. (2)is then solved in the frequency domain using FFT methods.
Corresponding author E-mail: zair.ahmad@ovgu.de, Phone: +49 391 67 11632, Fax: +49 391 67 12439 miguel.perez@ovgu.de christian.willberg@ovgu.de ulrich.gabbert@ovgu.de
c 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
510 Short Communications 11: Waves and Acoustics
3 Numerical exampleThe method is applied to the model shown in Fig.2. The model is discretize with 10 plane stress 4-node quadrilateral elements.The loading and measurement is done at the same point. Load F is defined as F = 1000 sin(t)sin2(t/10) N for0 t t/10 and zero otherwise with loading frequency of 500kHz. Analysis is done in period of 15s. Good aggrementis observed when compared to the same analysis using FEM explicit method in Fig.3.
Fig. 2 Analysis model (Youngsmodulus, 20kN/mm2 and density,7.8kg/mm3). Mass proportionaldamping is used.
Time (s)0. 5. 10. 15. [x1.E6]
FEM explicit (ABAQUS)Wave FEM
Fig. 3 Displacement result along x-axis at the loading point.
4 ConclusionThe results show a potential use of this method to reduce the computational cost in FEM. Instead of assembling all elementsin the model, only the element matrix from the basis cell is needed. However, it inherit some limitations from the FFT methodthat needs to be solved i.e. the Gibbs phenomenon (FFT periodicity constraint) that affect the solution at the beginning andthe end of analysis (sampling) period. To overcome this in the current work, analysis period is chosen to be long enough forthe structural response to be almost fully damped at the end of the analysis.
Acknowledgements Financial support by Ministry of Higher Education of Malaysia for the first author is gratefully acknowledged.
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Delamination Localization in Sandwich Skin Using Lamb Waves by Finite Element Method
1 LMEET Lab, Department of Applied Physics, FST, Settat 26000, Morocco
2 CRMEF, Settat, Morocco
Received 15 June 2018; Revised 13 October 2018; Accepted 12 November 2018; Published 28 November 2018
Academic Editor: Kim M. Liew
Copyright © 2018 Salah Nissabouri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this work we model by finite element method (FEM) the Lamb waves’ propagation and their interactions with symmetric and asymmetric delamination in sandwich skin. The simulations were carried out using ABAQUS CAE by exciting the fundamental A Lamb mode in the frequency 300 kHz. The delamination was then estimated by analysing the signal picked up at two sensors using two technics: Two-Dimensional Fast Fourier Transform (2D-FFT) to identify the propagating and converted modes, and wavelet transform (WT) to measure the arrival times. The results showed that the mode A is sensible to symmetric and asymmetric delamination. Besides, based on signal changes with the delamination edges, a localization method is proposed to estimate the position and the length of the delamination. In the last section an experimental FEM verification is provided to validate the proposed method.
1. Introduction
A sandwich composite is a materiel made from two thin skins bonded to a thick core. The skin studied in this paper is an orthotropic plate
0]4 with three mutually perpendicular planes of symmetry. The sandwich materials are designed to improve mechanical proprieties of structures. They are widely used in different fields especially in aeronautic industry. Their proprieties are influenced by the proportions of the matrix and the reinforcements. There are other parameters that also affect their characteristics like size, orientation, and distribution of the fibre. However, the heterogeneity of the composites structures leads to their weakness and facilitates the appearance of internal and external damage such as fibre breakage, matrix cracking, through-thickness hole, local delamination. Among these types of damage, delamination is especially easy to appear because the transverse tensile and interlaminar shear strengths are weak compared to the in-plane strength [1]. The delamination causes wave scattering, mode conversion, and multiple reflections. To understand these mechanisms, theoretical, numerical, and experimental studies are conducted.
Feng, Lopes Ribeiro, and Geirinhas Ramos [2] analysed the interaction of symmetric S and antisymmetric A mode with the delamination using finite element simulations. The Lamb waves propagation in a 4-layer
laminate were compared with propagations obtained in 1-layer 0] and in 3-layer 90/90/0] sublaminates. Chiua, Roseb, and Nadarajaha [3] investigated the scattering of the S mode by a delamination in quasi-isotropic fibre-composite laminate. Guo and Cawley [4] studied by finite element analysis and by experiment the interaction of the S Lamb mode with delamination. Nadarajah, Vien, and Chiu [5] presented results for the scatter field for various angles of incidence and for varying defect sizes. Hayashi and Kawashima [6] studied the reflections of Lamb waves at a delamination by semianalytical finite element method. Ching-Tai and Veidta [7] investigated the scattering characteristics of A mode Lamb wave at a delamination in a quasi-isotropic composite laminate. Bin [8] suggested an algorithm to localize and identify the damage in Woven Glass Fibre reinforced epoxy (WGF/epoxy). Mustapha [9] characterized fundamental symmetric and antisymmetric Lamb modes in terms of their velocity and magnitude variation as they change gradually in the thickness of a composite sandwich plate with a high density foam core. Ng [10] presented a theoretical and finite element (FE) investigation of the scattering characteristics of A at delaminations in a quasi-isotropic composite laminate. Veidt and Ng [11] studied the influence of stacking sequence on fundamental antisymmetric Lamb wave (A0) scattering characteristics through holes in composite laminates. Luca [12] developed a valid finite element model to simulate Lamb waves’ propagation in a Carbon Fibre Reinforced Plastic (CFRP) laminate for damage detection purpose and investigated the effects of the wave interaction with respect to damage parameters such as size and orientation. Yang [13] investigated some aspects of numerical simulation of excitation and detection of Lamb waves using piezoelectric disks in plate-like composite laminates.
In this paper, a numerical approach is proposed to localize a delamination in sandwich skin as presented in Figure 1. The aim of this work is to study the conversion by identifying the propagating modes and also to evaluate the symmetric and asymmetric delamination.
2. Lamb Waves Theory
Lamb waves are most used for many reasons; they can propagate long distances alongplates and shells so they permit quick inspection of large structures, and also they are sensitive to the small variations either in material proprieties or in structure of the plate. However, they are dispersive which means that the interpretation of received signals can be complicated. So the key is to choose one pure mode to excite and to analyse its reflection, conversion, and transmission.
2.1. Equation of Lamb Waves Propagation in Unidirectional Lamina
The characteristic equations of the symmetrical and antisymmetric waves are [14]
