Many structures are composed out of a periodic repetition of smaller units. These units are called "unit cell".

The idea for such structures is inspired by nature (e.g. Honeycombs). Some beneficial characteristics are

  • good light weighting
  • good energy and sound absorbing properties
  • good thermal insulation characteristics

Floquet Theorem

Now we have an infinite periodic structure with traveling waves. According to the Floquet theorem, the relative change of the state variables (displacements and forces) from any location x in the unit cell to the same location in the neighboring unit cell can be expressed by an exponential term.

(1) \begin{align} q(x+L)&=q(x) e^{-ikL} \\ f(x+L)&=f(x) e^{-ikL}, \end{align}

in which L is the length of the unit cell describing the lattice constant and k is a complex quantity, where the real part describes the change in phase and the imaginary part describes the change in amplitude.


The Floquet theoreom (1) relates the state variables at the left and right boundary of a unit cell for a steady state solution

(2) \begin{align} \mathbf{q}_L^{n+1}&=\mathbf{q}_L^n e^{-ikL}\\ \mathbf{f}_L^{n+1}&=\mathbf{f}_L^n e^{-ikL} \label{eq:floquet_bc} \end{align}

Be aware that

(3) \mathbf{q}_R^n=\mathbf{q}_L^{n+1}

but

(4) \mathbf{f}_R^n=-\mathbf{f}_L^{n+1}

Combining the Floquet theorem with a finite element model of the unit cell, we can compute the wave solutions in a periodic structure.

Wave solutions

The exponential term describes the phase change and amplitude decay of a wave traveling from a fixed location in a unit cell to the same location in the neighboring unit cell.

\begin{align} q(x+L)&=q(x) e^{-ikL}\\ &=q(x) e^{-i(k_{re}+i k_{im})L}=q(x) \underbrace{e^{k_{im} L}}_{\text{decay}} \underbrace{e^{-i k_{re}L}}_{\text{phase shift}} \end{align}

Assuming a wave travels through the periodic structure, the displacements at the same location in the neighboring unit cell could be described with

\begin{align} {\color{red} {q(x+L)}}=q(x) e^{-i \frac{3\pi}{L} L} \end{align}

In this example, the phase shift from one cell to another is k_{re} L=3\pi and the change in amplitude is k_{im} L=0 (see figure)



Due to the exponential term e^{-ikL} and the time harmonic approach e^{i\omega t}, there is a space and a time dependency in the exponent. Depending on the combination of the time and space dependency, we distinguish propagating waves, damped waves and evanescent waves.

The type of wave depends on the excitation.

Wave Finite Element Method

The dynamic of the unit cell is described with the dynamic stiffness matrix \mathbf{D} (we assume no damping, \mathbf{C}=0)

\underbrace{\left(\mathbf{K}-\omega^2 \mathbf{M}\right)}_{\mathbf{D}}\mathbf{q}=\mathbf{f}

The matrices \mathbf{K} and \mathbf{M} are the stiffness and mass matrix, \mathbf{f} is the load vector (see article Equation of Motion). The entries in the dynamic stiffness matrix of the unit cell can be ordered with respect to the left, interior and right degrees of freedom

\begin{equation} \begin{bmatrix} \mathbf{D}_{LL} & \mathbf{D}_{LI} & \mathbf{D}_{LR} \\ \mathbf{D}_{IL} & \mathbf{D}_{II} & \mathbf{D}_{IR} \\ \mathbf{D}_{RL} & \mathbf{D}_{RI} & \mathbf{D}_{RR} \end{bmatrix} \begin{bmatrix} \mathbf{q}_{L}\\ \mathbf{q}_{I}\\ \mathbf{q}_{R} \end{bmatrix} = \begin{bmatrix} \mathbf{f}_{L}\\ \mathbf{f}_{I}\\ \mathbf{f}_{R} \end{bmatrix} \label{eq:orderDynamicStiffness} \end{equation}

There are two different approaches to compute the wave solutions:

References

Floquet, Gaston. "Sur les équations différentielles linéaires à coefficients périodiques." Annales scientifiques de l'École normale supérieure. Vol. 12. 1883.