A multi-mesh finite element method for phase-field based photonic band structure optimization
Introduction
Photonic crystals are periodic dielectric structures designed to prevent the propagation of electromagnetic waves with specific frequencies. The first studies could be dated back to Rayleigh's work on one-dimensional layered structures in 1887. However, the concept of photonic band gaps in multi-dimensions was introduced by Yablonovitch [1] and John [2] in 1987. With the development on the theory of shape and topology optimization [3], [4], [5], [6], a basic problem of photonic band structure optimization is to find a medium structure that exhibits a band gap as large as possible.
There has been plenty of contributions to maximization problems of band gaps for two-dimensional photonic crystals. In [7], [8], the authors proposed a generalized gradient ascent algorithm to maximize the band gaps iteratively, both for the transverse magnetic field (TM mode) and the transverse electric field (TE mode). In [9], Kao and Osher used the level set methods for maximizing band gaps. In this framework, the Binary Level Set Method (BLSM) is a constantly used strategy for topology and shape optimization in the design of photonic crystal [10] and two-density drum [11]. Various numerical optimization techniques, such as the genetic algorithm, are applied to photonic crystal design by many other researchers [12], [13], [14].
The feasibility of phase-field model for shape and topology optimization was first investigated by Bourdin and Chambolle [15]. In the literature of computational sciences, the phase field method is assumed to be an alternative way to track the interface efficiently [16], [17]. It is always served as a easy-to-use methodology for numerical simulations of the phase transition phenomenon [18], [19], as well as the two phase flow simulations [20], [21], [22]. Recently, Takezawa et al. proposed a phase field model for shape and topology optimization, with applications to structural optimization [23] and design of photonic devices [24]. The main contribution of their work is a new double-well function including the shape sensitive analysis, which is both theoretically featherable and numerically efficient for solving the optimization problem. However, a piecewise constant approximation is assumed for the design problems with two kinds of materials, so that the staircase phenomena could not be avoided. The work of Nguyen T et al. [28] suggests a multi-resolution scheme to obtain high resolution designs for structure, which is a novel idea for topology optimization and significant computational costs are saved.
In this paper, we propose a multi-mesh approach for the finite element calculations of the phase field method [24], a piecewise linear approximation is used for a continuous approximation to the phase field variable. The motivation of multi-mesh strategy is to reduce the overall computational costs without losing the numerical accuracy. The rest of this paper is organized as follows. The phase field models for optimization of photonic band structures are introduced in the next section. Section 3 introduces the details of multi-mesh approach, where the finite element analysis on the coarse mesh and the phase field evolution on the fine mesh are both formulated. In section 4, numerical results on some benchmark problems are reported. Finally, we made some conclusions and discussions.
Section snippets
Photonic band structure optimization
The photonic band structure optimization is a well-known model in computational material science. The work of Cox and Dobson [7] described the problem mathematically as a band gap maximization. For the ease of representation, let us consider the case of TM mode. When a scattered problem is considered, the purpose of the design is to maximize the gap between the j-th band and the -th band . It means to find an optimal shape Ω for the scattered object in the admissible shape set
The multi-mesh finite element methods
The multi-mesh approach can potentially reduce the time costs for solving the partial differential equation system. In [25], Li introduced how the multi-mesh working and built a new adaptive mesh strategy in the framework of multiple meshes settings. It was then proved to be effective in the finite element calculations for the simulations of dendritic growth [26], [29], [30].
Numerical experiments
Let us consider the problem of band gap maximization for photonic crystal design, which is served as the benchmark problem [7], [9], [10], [24]. The computational domain is set to be a square . In all the numerical examples presented in this section, the dielectric constants for the void and material are taken as and respectively. In our calculations, 30 equi-distributed wave number α located at the boundary of the Brillouin zone are calculated for plotting the
Conclusion
We propose a multi-mesh finite element method for the phase field model arising in photonic crystal design. The key contribution is to reduce the cost of finite element analysis on the state equation by solving on a much coarser meshes. The proposed approach has several advantages. The first one is the considerable savings in computational efforts compared with the single mesh scheme. The second advantage is the flexibility of approximation to the phase field variable, where the nodal finite
Acknowledgements
This research work is supported by the Fundamental Research Funds for the Central Universities 2016XZZX005-01, and the National Natural Science Foundation of China (Grant 11471092, 11471284 and 11201153).
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