Implementing Floquet Periodicity in FEniCS or FEniCSx
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When analyzing wave propagation inside a periodic structure, one may consider the analysis over a single section or a unit cell instead of the complete domain, by considering the appropriate periodicity at the boundaries. This approach reduces the domain of analysis and simplifies the problem.
By implementing the Floquet-periodicity, we can analyze wave propagation (mechanical waves or electromagnetic waves) within any material. This approach when coupled with FEniCS, will enable the analysis of any periodic structure under wave mechanics physics.
Theory
Bloch theorem describes the behavior of the wave propagation inside a periodic structure are analogous to plane wave modulated by a periodic function, which can be described mathematically as
\[\psi(\mathbf{r})=\mathrm{e}^{\mathrm{ik} \cdot \mathbf{r}} u(\mathbf{r})\]where, \(\psi(\mathbf{r})\) is the overall field {combination of plane wave and periodic function}
\(\mathrm{e}^{\mathrm{ik} \cdot \mathbf{r}}\) is the plane wave {Bloch wave vector}
\(u(\mathbf{r})\) is the envelop function {takes the periodicity and symmetry of the periodic structure}.
Bloch-floquet theory or periodicity provides a methodology to analyze the wave propagation by just considering a single unit cell and applying the following boundary condition as
\[u(\mathbf{r}+\mathbf{R})=u(\mathbf{r}) e^{i \mathbf{k} \mathbf{r}}\]where, \(u()\) is the field variable, \(\mathbf{r}\) is the spatial coordinates of the boundaries, \(R\) is the periodicity, and \(k\) is the wavenumber.
Wavenumber can be defined as the number of waves per unit length and is written as
\[k = \frac{1}{\lambda}\]A structure is considered periodic if its material properties repeat after certain intervals.
Consider a unit cell as shown in below having periodicity \(R\) equal to \(1\) and the boundary value at left and right be \(u_l\) and \(u_r\), respectively. Then above equation reduces to
\[u(\mathbf{r}+1)=u(\mathbf{r}) e^{i \mathbf{k} \mathbf{r}}\]
Considering the value of \(r\) to be 1 at the left most side, then above equation becomes
\[u_r=u_l e^{i \mathbf{k}}\]References
How to implement Bloch-Floquet Boundary condition? - FEniCS Project
Bloch Floquet Boundary condition / Perfect Matching Layer - FEniCS Project
So, the question remains. How to implement Floquet-Periodicity in FEniCS or FEniCSx?