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Unfitted Nitsche’s Method for Computing Wave Modes in Topological Materials

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Abstract

In this paper, we propose an unfitted Nitsche’s method for computing wave modes in topological materials. The proposed method is based on the Nitsche’s technique to study the performance-enhanced topological materials which have strongly heterogeneous structures (e.g., the refractive index is piecewise constant with high contrasts). For periodic bulk materials, we use Floquet-Bloch theory and solve an eigenvalue problem on a torus with unfitted meshes. For the materials with a line defect, a sufficiently large domain with zero boundary conditions is used to compute the localized eigenfunctions corresponding to the edge modes. The interfaces are handled by the Nitsche’s method on an unfitted uniform mesh. We prove the proposed methods converge optimally. Several numerical examples are presented to validate the theoretical results and demonstrate the capability of simulating topological materials.

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Data Availability

The authors affirm that this manuscript submitted to Journal of Scientific Computing is an honest, accurate, and transparent account of the study being reported; The data was generated by the authors using the numerical schemes proposed in this manuscript. Data will be available on reasonable requests. Code Availability All codes were written by the authors and will be available on reasonable requests.

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Acknowledgements

The authors thank Professor Michael I. Weinstein for useful discussions. H.G. was partially supported by Andrew Sisson Fund of the University of Melbourne, X.Y. was partially supported by the NSF grant DMS-1818592, and Y.Z. was partially supported by NSFC grant 11871299.

Funding

Hailong Guo was partially supported by Andrew Sisson Fund of the University of Melbourne. Xu Yang was partially supported by the NSF grant DMS-1818592. Yi Zhu was partially supported by NSFC grant 11871299.

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Proof of the Lemma 2

Proof of the Lemma 2

1.1 A Technical Lemma

Before giving the proof of Lemma 2, we present a lemma that we shall use.

Lemma 5

Let \(\mathbf{x}^j= (x_1^j, x_2^j), ~j = 1, 2, 3\), be the three vertices triangle K and \(b_j(\mathbf{x})\) be the standard nodal basis function associated with \(\mathbf{x}^j\). Then the following relationship holds

$$\begin{aligned} |b_j(\mathbf{x})| \le 2h|\nabla b_j|, \quad \forall \mathbf{x}\in K, \end{aligned}$$
(A.1)

for \(j = 1, 2, 3\).

Proof

Without loss of generality, we only prove (A.1) for \(j = 1\). Using the area coordinates [10], we have

$$\begin{aligned} b_1(\mathbf{x}) = \frac{(x_2-x_2^3)(x_1^3-x_1^2)- (x_1-x_1^2)(x_2^3 - x_2^2) }{2|K|}, \end{aligned}$$
(A.2)

and

$$\begin{aligned} \nabla b_i= \left( \frac{-(x_2^3-x_2^2)}{2|K|}, \frac{(x_1^3-x_1^2)}{2|K|} \right) . \end{aligned}$$
(A.3)

From the above two expressions, we can deduce that

$$\begin{aligned} |b_i(\mathbf{x})| \le h\frac{|x_1^3-x_1^2| + |(x_2^3 - x_2^2)| }{2|K|} \le 2h\frac{\sqrt{|x_1^3-x_1^2|^2 + |(x_2^3 - x_2^2)|^2} }{2|K|} = 2h |\nabla b_i| \end{aligned}$$

where we have used the fact \( |(x_2-x_2^3|\le h\) and \(|x_1-x_1^2|\le h\) for any point \(\mathbf{x}= (x_1, x_2)\) in the triangle K. \(\square \)

1.2 Proof of Lemma 2

It is sufficient to show the lemma for the basis functions \(b_j\) since \(\phi _h\) is a linear combination of \(b_j\). Using Lemma 5, we can deduce that

$$\begin{aligned} \Vert b_j\Vert ^2_{0, \varGamma _T}&\le |\varGamma _T| \Vert b_j\Vert ^2_{0, \infty , \varGamma _T} \le |\varGamma _T| \Vert b_j\Vert ^2_{0, \infty , K_i} \le 4h^2|\varGamma _T| |\nabla b_i|^2 =\frac{ 4h^2|\varGamma _T|}{|K_i|} \Vert \nabla b_i\Vert _{0, K_i}^2; \end{aligned}$$

which completes the proof of (3.24). The inequality (3.25) is implied in the above proof.

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Guo, H., Yang, X. & Zhu, Y. Unfitted Nitsche’s Method for Computing Wave Modes in Topological Materials. J Sci Comput 88, 24 (2021). https://doi.org/10.1007/s10915-021-01540-w

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