9 The Semi-Analytical Finite Element Method
9.2 SAFE Formulation for Plate Structures
The SAFE method adopts a harmonic exponential term, ei kx( -ωt), to describe the wave behavior in the wave propagation direction, where x represents the wave propagation direction, k represents the wave number, ω is the radial frequency, and t is time.
For anisotropic waveguides, it is possible to have different propagation directions for phase velocity and energy velocity. To avoid confusion, the wave propagation direction of phase velocity is hereafter named the wave vector direction. The finite element discretization of the SAFE method takes place at the cross section of the waveguide that is perpendicular to the wave vector direction. For the problem of plane wave propagation in a plate, a one-dimensional discretization across the plate thickness is sufficient. The coordinate system and the finite element discretization for the SAFE calculation are shown in Figure 9.1. The plate width in the y direction is assumed to be infinite. Three-node line elements are employed here. The shape functions for the three-node line element are:
N1 2
=ξ 2-ξ ,
N2= -1 ξ , 2 (9.1)
N3 2
=ξ 2+ξ ,
where ξ is the variable in the local coordinate system for the element itself. For a given point in the local coordinate described by ξ, the global coordinate of the point can be calculated from the global coordinates of the three nodes (z1, z2, and z3) and the shape functions in Equation (9.1).
x z
… …
… …
……
= 1
= 0
= –1
Figure 9.1. The coordinate system and the finite element discretization for the problem of wave propagation in plates. The insert shows the local coordinates of the three nodes of an element.
137 9.2 SAFE Formulation for Plate Structures
z N N N as shown in the insert of Figure 9.1.
Combining the time harmonic assumption and the finite element discretization, one can write the particle displacements of any point in an element in terms of the shape functions and the nodal displacements as follows:
ue α direction. The corresponding strain and stress vectors can then be calculated from the following equations:
and C( )e is the material stiffness matrix of the element in the global coordinate system.
For anisotropic materials, stiffness matrix transformations from the principal axes to the global coordinate system are necessary (Auld 1990). Substituting Equation (9.3) into Equation (9.6) results in:
ε( )e =
(
B1+ikB Q2)
( )eei kx( -ωt), (9.9)where
B1=L Ny ,y+L Nz ,z, B2=L Nx . (9.10) In Equation (9.10), N,y and N,z are the derivatives of the shape function matrix given in Equation (9.4) with respect to the y and z directions, respectively.
A governing equation for the wave motion of each element can be obtained through the virtual work principle (Hayashi, Song, and Rose 2003):
δue Tte δue T ρeue δ ε
where δu and δε represent the virtual displacement and virtual strain, respectively, t( )e represents the external traction vector that can also be expressed using shape functions and nodal external tractions T(e) (Equation (9.12)), •T denotes a complex conjugate transpose, ρ( )e is density, • is a second derivative with respect to time, Γ is the surface of the element, and V is the volume of the element.
t( )e =N
( )
ξ T( )eei kx( -ωt) (9.12) Substituting Equations (9.13), (9.7), (9.9), and (9.12) into Equation (9.11) yields:δ δ ρ
Equation (9.13) is satisfied for any arbitrarily chosen virtual displacement. Therefore, the virtual displacement term δQ( )eT can be eliminated from the equation:
N NT N NQ
For the line elements used here, the integrands in both sides of Equation (9.14) are functions of the variable ξ only. Equation (9.14) can thus be further simplified to:
F( )e =
(
K( )1e +ikK( )2e +k2K( )3e)
Q( )e -ω2M Q( ) ( )e e, (9.15)139 9.2 SAFE Formulation for Plate Structures
K3 B C B2 2
Adopting a convertional finite element assembly methodology for all the elements and applying the traction-free boundary conditions on the top and bottom surfaces of the plate, one can form an eigenvalue problem in the global coordinate system:
K1+ K2+ 2K3- 2M Q 0
(
ik k ω)
= . (9.21)The size of matrices K1, K2, K3, and M is 3N × 3N, where N is the total number of nodes.
Q is a 3N × 1 vector representing the particle displacements at the node positions.
It has been shown that the imaginary unit in Equation (9.21) can be eliminated by introducing a 3N × 3N unitary transformation matrix T (Damljanović and Weaver 2004; Bartoli et al. 2006). All off-diagonal elements of the matrix T are zero. All diagonal elements of T are equal to one except for the elements corresponding to the particle displacements in the wave vector direction that are equal to the imaginary unit i. Because T T IT = (I is a 3N × 3N unit matrix), by replacing Q in Equation (9.21) with T TQT and then multiplying the equation with T from the left side followed by applying the properties of the matrices, one can obtain a new eigenvalue equation:
K1 K2 2K M
For elastic materials, that is, materials with real stiffness matrices, the matrices A and B are 6N × 6N real matrices.
In total, 6N eigenvalues for wavenumber k can be solved at each frequency ω from Equation (9.23). Among all of the eigenvalues, there are real eigenvalues for propagating guided wave modes and complex eigenvalues including pure imaginary eigenvalues for the evanescent modes. The eigenvalues are also solved in pairs from Equation (9.23), that is, if kµ is a solution of Equation (9.23), -kµ is also a solution. Let kreal denote the real solutions from Equation (9.23). Positive wavenumbers among kreal are wavenumbers of wave modes propagating in the +x direction, while their negatives are wavenumbers of the corresponding wave modes propagating in the –x
direction. For complex solutions kcomplex, if the imaginary parts Im k
{
complex}
> 0, the wave modes whose wavenumbers are kcomplex attenuate in the +x direction. Otherwise, kcomplex are the wavenumbers of the wave modes that attenuate in the –x direction.For each eigenvalue, an eigenvector is solved from Equation (9.23) as well. It is explicit that the corresponding wave structure information is contained in the eigenvector. Based on the wave structures, the strain and stress fields can be obtained using Equation (9.9) and Equation (9.7).