1. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
MODELING OF ELASTIC WAVE PROPAGATION IN PLATE
STRUCTURES USING SPECTRAL ELEMENT METHOD
Mallesh N. G.1
, Ashwin U.2
, S. Raja3
, K. Balakrishna Rao4
1, 4
Department of Civil Engineering, Manipal Institute of Technology, Manipal – 576 104
malleshnenkat@gmail.com and kb.rao@manipal.edu
2, 3
STTD, CSIR - National Aerospace Laboratories, Bangalore – 560 017
ashwin@nal.res.in and raja@nal.res.in
ABSTRACT
Modeling mid and high frequency elastic wave (>10 KHz) in thin / moderately thick elastic
plates using spectral element method has been studied. High frequency elastic waves in plates of
thickness approx. 1–2mm has a wavelength of 38 – 13 mm in a frequency range of 10 – 100
KHz. Modeling such plates excited with such small wavelength waves using Finite Element
Method leads to erroneous results in terms of time of flight and amplitude of response. Hence, in
order to accurately capture elastic wave propagation, spectral element having higher order
polynomial (6th
order) has been studied. This element has 36 nodes and can model in-plane (u, v,
w) and out-of-plane deformations (x, y). The modeled element is then simulated for wave
propagation using a unit load applied on a square plate, clamped on all the sides.
The understanding of the propagation behavior of high frequency elastic wave (Lamb waves) in
thin-walled layered structures is a very important basis of Structural Health Monitoring (SHM)
in large-scale constructions.
INTRODUCTION
The Finite Element Method (FEM) is used to solve complex problem from various fields of
physical science described by partial differential equation or integral equation. A characteristic
property of finite element method is discretization of the analyzed area into a certain number of
smaller subareas known as finite element. The stiffness and mass properties of these finite
elements are well defined, which are assembled to form the global stiffness and mass matrices.
Hence, Finite Element method is widely used to model complex physical systems, for their static
and dynamic simulations.
Spectral Element Method (SEM) is relatively new computational technique, where the element is
defined using higher order polynomials. Such polynomials are usually orthogonal Chebyshev
polynomial or very-high order Lobatto polynomial over non-uniformly spaced nodes. In SEM,
the computation error decreases exponentially as the order of approximating polynomial.
Therefore fast convergence of solution to exact solution is realized with lesser degrees of
freedom of the structure, in comparison with FEM.
The spectral plate element (higher order polynomial) was first developed by Kudela et.al. [1],
who formulated a single layered spectral plate element (five degrees of freedom) and they
studied the wave propagation patterns in orthotropic plate structure. In the present work, a
layered spectral plate element with 36 nodes has been formulated. The displacement and shape
of the element is approximated using Gauss-Lobatto-Legendre polynomial, which defines the
nodes to be non-uniformly distributed within the element. This non-uniform distribution of the

2. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
nodes makes the mass matrix of the element diagonal. Since, the wave propagation is
numerically solved using explicit solvers (Time integration using central difference method),
inverse of diagonal mass matrix becomes very simple to solve, making the computation very
fast. The plate has been modeled using first order shear deformation theory, and the ABD matrix
is appropriately constructed. To demonstrate the spectral element simulation of wave
propagation, a square plate of length to thickness ratio of 266 has been modeled. The plate
clamped from all sides and excited at the centre using amplitude modulated signal.
BACKGROUND STUDIES
Mindlin-Reissner theory
The element is formulated using Mindlin-Reissner theory of plates, which takes into account the
shear deformation. The kinematic relations, constitutive relation and strain – displacement
relations are given in equations 1-2, equation 3 and equations 4-8, respectively.
0   yu u z (1)
0   xv v z (2)
11 12 13
21 22 23
31 32 33
44
55
66
0 0 0
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
 
 
 
 
 
 
    
    
    
    
    
    
    
    
        
x x
x y
x z
yz yz
zx zx
xy xy
Q Q Q
Q Q Q
Q Q Q
Q
Q
Q
(3)
00

 

    
  
y
x x x
uu
z zk
x x x
(4)
00 
 
 
    
  
x
y y y
vv
z zk
y y y
(5)
00 0
 
 
   
       
     
y x
xy xy xy
u vu v
z z zk
y x y x y x
(6)
0
 
 
   
  
xz y
ww u
x z x
(7)
0
 
 
   
  
yz x
ww v
y z y
(8)
ABD Matrix
The layer-wise material stiffness matrices and the force and moment resultants for plane stress
condition are shown in equation 11 and equations 9-10, respectively.

3. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
1 1
_ _ _ _ _ _
0
11 12 16 11 12 16
_ _ _ _ _ _
0
12 22 26 12 22 26
_ _ _ 0 _ _ _
1 1
16 26 66 16 26 66
 
 
 
 
 
       
               
          
            
             
  
k k
k k
n n
xh hx
y xh h
k k
xy xy
x
y
xy
Q Q Q Q Q Q
N
N Q Q Q dz Q Q Q zdz
N
Q Q Q Q Q Q


(9)
1 1
_ _ _ _ _ _
0
11 12 16 11 12 16
_ _ _ _ _ _
0 2
12 22 26 12 22 26
_ _ _ 0 _ _ _1 1
16 26 66 16 26 66
 
 

 
 
   
       
              
          
            
             
k k
k k
n nxx h h
h hy x
k k
xy xy
x
y
xy
Q Q Q Q Q Q
M
M Q Q Q zdz Q Q Q z dz
M
Q Q Q Q Q Q


 
(10)
4 2 2 4
11 11 12 66 22cos 2( 2 )sin cos sin      Q Q Q Q Q
4 2 2 4
22 11 12 66 22sin 2( 2 )sin cos cos      Q Q Q Q Q
3 3
16 11 12 66 12 22 66( 2 )sin cos ( 2 )sin cos        Q Q Q Q Q Q Q
3 3
26 11 12 66 12 22 66( 2 )sin cos ( 2 )sin cos        Q Q Q Q Q Q Q
2 2 4 4
66 11 12 12 66 66( 2 2 )sin cos (sin cos )        Q Q Q Q Q Q
2 2
44 44 55cos sin  Q Q Q
45 55 44( )sin cos  Q Q Q
2 2
55 55 44cos sin  Q Q Q (11)
SPECTRAL ELEMENT FORMULATION
Interpolation function and Gaussian Quadrature
The Legendre polynomial of nth order is defined as,
21
( ) ( 1) 0,1,2...
2 !
 

  
n
n
n n n
d
P where n
n d
(12)
In the current formulation 5th
order Legendre polynomial is chosen, Hence 36 nodes can be
specified in the local coordinate system of the element ,  (shown in Figure 1)
,
,
( ) , 1,2......6,
1 2 1 2 1 2 1 2
1, , , , ,1
3 3 3 33 7 3 7 3 7 3 7
 
 

  
        
  
m n
m n
m n
(13)
The weight Gaussian weight are given as,

4. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
 , 0.0667 0.3785 0.5549 0.5549 0.3785 0.0667m nw
Figure 1 – 36 noded spectral finite element
The interpolation function, when applied unit displacement at different nodes are shown in
Figures 2- 5.
Figure 2, Nodal point 1 (Refer Figure 1) Figure 3, Nodal point 11 (Refer Figure 1)
Figure 4, Nodal point 22 (Refer Figure 1) Figure 5, Nodal point 34 (Refer Figure 1)
Geometry and Displacement model (Iso-parametric)
Shape functions are formed on the specified nodes to approximate the geometry of the element in
the global coordinate system and also to approximate the displacements within the element


1 2 3 4 5 6
31 32 33 34 35 36
7 8 9 10 11 12
25 26 27 28 29 30
19 20 21 22 23 24
13 14 15 16 17 18

5. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
The geometric equations using Lobatto Lagrangian interpolation function can be written as,
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )     
  
  m n k m n k m n mn
k m n
x N x N N x (14)
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )     
  
  m n k m n k m n mn
k m n
y N y N N y (15)
Where )(mN and )(mN are one dimension shape functions, obtained from equation 12, which
is written independently for the two coordinates.
The displacement equations using Lobatto Lagrangian interpolation function is written as,
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )     
  
  m n k m n k m n mn
k m n
u N u N N u (16)
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )     
  
  m n k m n k m n mn
k m n
v N v N N v (17)
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )     
  
  m n k m n k m n mn
k m n
w N w N N w (18)
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )        
  
  x m n k m n xk m n xmn
k m n
N N N (19)
36 6 6
, ,
1 1 1
( ) ( ) ( ) ( )        
  
  y m n k m n yk m n ymn
k m n
N N N (20)
Strain – Displacement Relation
The strain – displacement relation is obtained by substituting equations 16-20 in equations 4-8
as,
    
0
0
0
36 36
1 1
0 0 0 0
0 0 0 0
0 0 0
0 0 0 0
0 0 0 0
0 0 0
0 0 0
0 0 0




  
 


 
 
 
 
 
 
    
    
                
  
           
 
 
 
  
 
i
i
i ix
y
ixy
x
i i
iy i i x
xy
ys
i ixy
s
xy
i
i
i
i
N
x
N
y
N N
y x
uN
v
x wB q N
y
N N
x y
N
N
x
N
N
y

  
 
 
  i
(21)

6. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
Element Stiffness and Mass Matrix
The element stiffness is then computed from equations 21 and 10 as,
        
T
AK B ABD B dxdy (22)
Similarly, the mass matrix is written as,
       
T
AM N N dxdy (23)
Where,
0 0 0 0
0 0 0 0
0 0 0 0 , 1 36
0 0 0 0
0 0 0 0
 
 
 
  
 
 
  
i
i
i
i
i i
N
N
N N where i to
N
N
The element stiffness and mass matrix in equations 22-23 is assembled to obtain the global
stiffness and mass matrix.
Central Difference Method
The central difference method computes the time integration on elastic wave propagation and the
displacement in the next time step (  x t t ) is computed using the displacement information of
the current (  x t ) and previous (  x t t ) time steps as follows,
2 2
2
2 ( )1
( ) ( ) ( ) ( )
2
2
  
          
         
g g g
g
g g
M x t M C
x t t f t K x t x t t
M C t t t
t t
(24)
Where, gK , gM , and gC are the global stiffness, mass and damping matrices. In the present
work, the damping has been taken as zero, which makes the solution of equation 24, very simple
and computationally very fast.
RESULTS AND DISCUSSION
To demonstrate the simulation using spectral element, a square plate of dimension 400 × 400
mm2
and 1.5mm has been modeled. The plate is clamped on all edges and excited with a point
load of 1N at the centre. The excitation signal 3.5 cycle amplitude modulated. The simulation of
the wave propagation at different propagation time is presented. Figure 6 shows the excitation
point of the wave, Figure 7 the wave propagation through the plate, Figure 8 shows the start of
wave reflection at the plate boundary, Figure 9 shows the reflected wave from the plate boundary
and Figures 10 and 11 shows the wave interacting reflecting the different boundaries of the plate.

7. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
Figure 6, Wave propagation (0.4 µs) Figure 7, Wave propagation (0.266 µs)
Figure 8, Wave propagation (0.2 µs) Figure 9, Wave propagation (0.16 µs)
Figure 10, Wave propagation (0.133 µs) Figure 11, Wave propagation (0.1143 µs)
CONCLUSIONS
The forumulated element is very efficient in simulating elastic wave propagation (lamb wave)
and is computationally very efficient in comparison to finite element method. The element needs
to be benchmarked by comparsing the group velocity with that esimated theoretically from the
dispersion curves. Further, we also intend to extend the procedure to model piezoelectric actuator
and sensor.

8. NATIONAL CONFERENCE ON NEW HORIZONS IN
CIVIL ENGINEERING – NHCE 2013
April 12-13 at M.I.T, Manipal,
REFERENCES
[1] Pawel Kudela, Arkadiusz Zak, Marek Krawczuk, Wieslaw Ostachowicz, Modeling of wave
propagation in composite plates using the time domain spectral element method, Journal of
sound and vibration 302(2007) 728-745.
[2] C. S. Krishnamoorthy, Finite Element Analysis, Tata McGraw Hill Education private
Limited, New Delhi, 2010
[3] Wieslaw Ostachowicz, Pawel Kudela, Marek Krawczuk, Arkadiusz Zak, Guided waves in
Structures for SHM, Wiley, United Kingdom, 2012.
[4] Autar K. Kaw, Mechanics of Composite Material, Taylor & Francis, New York, 2006
[5] Giora Maymon, Structural Dynamics and Probabilistic Analyses for Engineer, Elsevier,
Jordan Hill, 2008
[6] J.N Reddy, Mechanics of Laminated Composite Plates, CRC press, New York, 1997