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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4437–4452
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Fast and accurate finite element analysis of large-scale three-dimensional photonic devices with a robust domain decomposition method

Ming-Feng Xue, Young Mo Kang, Amir Arbabi, Steven J. McKeown, Lynford L. Goddard, and Jian-Ming Jin  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 4437-4452 (2014)
http://dx.doi.org/10.1364/OE.22.004437


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Abstract

A fast and accurate full-wave technique based on the dual-primal finite element tearing and interconnecting method and the second-order transmission condition is presented for large-scale three-dimensional photonic device simulations. The technique decomposes a general three-dimensional electromagnetic problem into smaller subdomain problems so that parallel computing can be performed on distributed-memory computer clusters to reduce the simulation time significantly. With the electric fields computed everywhere, photonic device parameters such as transmission and reflection coefficients are extracted. Several photonic devices, with simulation volumes up to 1.9× 10 4 (λ/navg)3 and modeled with over one hundred million unknowns, are simulated to demonstrate the application, efficiency, and capability of this technique. The simulations show good agreement with experimental results and in a special case with a simplified two-dimensional simulation.

© 2014 Optical Society of America

1. Introduction

As important applications of electromagnetic (EM) theory, analysis and design of photonic devices have attracted considerable attention in both academia and industry in recent years. For example, microring resonators (MRRs) have become essential building blocks in modern photonic integrated systems to perform a variety of tasks including signal generation [1

1. Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, and N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998). [CrossRef]

], analog and digital processing [2

2. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004). [CrossRef] [PubMed]

6

6. T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12(7), 1437–1442 (2004). [CrossRef] [PubMed]

], wavelength selection [7

7. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]

13

13. Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, and L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014). [CrossRef]

], and sensing [14

14. C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003). [CrossRef]

]. The reader is referred to [15

15. A. Arbabi and L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012). [CrossRef]

] for a recent short summary article of microring applications. In one such application, a microring is designed to serve as a mirror for a laser cavity. A high reflectivity narrow bandwidth spectrum can be achieved by adding a few low refractive-index contrast grating pairs to the microring [10

10. A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, and L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011). [CrossRef]

, 11

11. Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009). [CrossRef]

]. These reflective microrings present an interesting challenge to solve numerically because they are highly resonant structures with large volumes and very small features. Another example of a photonic device with widely contrasting dimensional scales is the palladium coated nano-aperture fiber optic hydrogen sensor [16

16. S. J. McKeown and L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012). [CrossRef]

]. A deep sub-wavelength C-shaped aperture has been shown to have an extraordinarily high transmission for its size [17

17. X. Shi and L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21(7), 1305–1317 (2004). [CrossRef]

]. For the sensing application, this is beneficial because there is a large interaction of the transmitted field with the palladium sensing layer, especially on the sidewall boundaries of the aperture, due to the surface plasmons generated at these metal-air interfaces [16

16. S. J. McKeown and L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012). [CrossRef]

].

To solve the EM problems in these optical applications, several numerical methods, such as the finite-difference time-domain (FDTD) method [17

17. X. Shi and L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21(7), 1305–1317 (2004). [CrossRef]

, 18

18. J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, and P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008). [CrossRef]

], the finite element method (FEM) [19

19. B. Radjenović and M. Radmilović-Radjenović, “Computer-aided design and simulation of optical microring resonators”, Int. J. Numer. Model., Electron. Netw., Devices Fields, available online: http://onlinelibrary.wiley.com/doi/10.1002/jnm.1920/abstract. [CrossRef]

23

23. B. Radjenović and M. Radmilovic-Radjenović, “Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes,” IEEE Sens. J., doi:. [CrossRef]

], and the techniques that make use of rotational symmetry [24

24. A. Arbabi, Y. M. Kang, and L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010). [CrossRef]

], have been proposed during the past few years. Although these methods can effectively model photonic devices with relatively small three-dimensional (3D) electrical volumes, a 3D full-wave numerical simulation of significantly larger structures is difficult due to the scaling of the computational time and necessary resources with the problem size. To reduce the size of the computational domain, an approximate approach is to truncate the original 3D problem into a two-dimensional (2D) problem with the aid of periodic boundary conditions. However, the accuracy of full-wave analysis is sacrificed due to this modeling approximation. One promising solution to this challenging problem is to employ a domain decomposition method (DDM) and perform 3D full-wave simulation on massively parallel clusters. The FDTD method is highly efficient when combined with the divide-and-conquer idea from the DDM because it does not involve matrix operations. However, a FDTD analysis of large-scale optical problems is still very time-consuming because the problem size is too large and the structures are usually highly resonant. Furthermore, its Cartesian grids cannot model curved boundaries accurately. In contrast, the frequency-domain FEM is better suited to deal with optical problems because of its excellent capability to model complex media and delicate geometries. To solve the FEM linear system, an iterative solver is usually very efficient for a parallel implementation. However, the convergence rate may be very slow when the problem is very large or highly resonant. On the other hand, a direct solver is free of iterations but suffers from a low parallel efficiency. Here, we present a novel domain decomposition method hybridizing direct and iterative solvers to achieve a high parallel efficiency and a fast convergence rate.

In this paper, we focus on developing a SOTC-TE-based FETI-DP method, apply it to the analysis of computationally complex photonic devices, and investigate its numerical performance. Specifically, we consider a highly resonant structure, namely a reflective microring, and a plasmonic structure, which is a C-shaped nano-aperture etched into a palladium coated fiber tip. The rest of the paper is organized as follows: We first describe the formulation of the proposed method with a PML domain truncation in Section 2.1. Then, we briefly discuss the calculation of device parameters such as transmission and reflection coefficients in Section 2.2 and the parallel implementation on a distributed-memory system by using the message passing interface (MPI) in Section 2.3. Afterwards, we present numerical results in Section 3 to show the efficiency and capability of this method as well as the efficiency of the parallel implementation.

2. Formulation

2.1 FETI-DP formulation with the SOTC-TE

For open-domain simulations, PML is usually employed to truncate the computational domain because it can absorb both propagating and evanescent modes without being placed far away from the structure of interest. However, when used in the frequency-domain FEM, PML increases the condition number of the system matrix significantly and thus increases the number of iterations needed for convergence substantially [35

35. C. T. Wolfe, U. Navsariwala, and S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000). [CrossRef]

, 36

36. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

]. It has been observed that the finite element tearing and interconnecting (FETI) method equipped with a Dirichlet/ Neumann continuity condition can reduce the number of iterations dramatically for the case with a PML truncation [35

35. C. T. Wolfe, U. Navsariwala, and S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000). [CrossRef]

].

To implement a nonoverlapping domain decomposition method, the entire computational domain V is first divided into several nonoverlapping subdomains such that s=1NsVs=V and s=1NsVs=. The governing equation and the boundary condition of the boundary-value problem (BVP) for the sth subdomain can be written as [25

25. Y. J. Li and J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007). [CrossRef]

]
×(μr1×Es)k02εrEs=jk0Z0JimpsinVs
(1)
n^s×(μr1×Es)+αsn^s×(n^s×Es)βs×[n^s(×Es)n]=ΛsonΓs
(2)
where Γs denotes the interface between the subdomain and its neighboring subdomains, Λs is an unknown variable defined on the subdomain interface, n^s is the outward normal unit vector of the subdomain, Z0 is the intrinsic impedance of the free space, and Jimps is an impressed current in the sth subdomain. The αs and βs are complex parameters to be determined to make the subdomain interface as transparent as possible for both propagating and evanescent modes [27

27. M. F. Xue and J. M. Jin, “A hybrid nonconformal FETI/conformal FETI-DP method for arbitrary nonoverlapping domain decomposition modeling,” in Proceedings of IEEE AP-S Int. Symp. (2013). [CrossRef]

, 32

32. Z. Peng, V. Rawat, and J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010). [CrossRef]

]. If we set βs=0, Eq. (2) will be reduced from the SOTC-TE to the FOTC. In this paper, we implement the PML as a diagonally anisotropic artificial medium, with μr=μr[D] and εr=εr[D], where [D] is a tensor given by
[D]=[sysz/sx000szsx/sy000sxsy/sz].
(3)
In Eq. (3), sx, sy, and sz are functions of spatial variables x, y, and z, respectively. This anisotropic medium reduces to an isotropic medium when sx=sy=sz=1. In the PML region, sx, sy, and sz can be expressed as sa=sjs, where a could be x, y, or z, and s and s are real numbers with s1 and s0, which are used to control the attenuation of the evanescent and propagating waves in the PML region [36

36. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

].

From Eq. (4), we can extract two equations. One is for the discrete fields on the subdomain interface, which can be extracted from {Ers}. This equation can be written symbolically as
{EIs}=f({Ecs},{λs},{fs})
(5)
which is called the subdomain interface system. The other equation is for the fields at the corners of the subdomain, which can be written symbolically as
{Ecs}=g({λs},{fs})
(6)
which is called the subdomain corner system. With these, the subdomain interior fields have been fully decoupled and the interaction between neighboring subdomains can be carried out only by the dual unknown {λs}.

Next, we couple all the subdomains. By employing the SOTC-TE in Eq. (2), we can obtain equivalent interface conditions that relate the interface electric fields and the dual unknown vector Λs on both sides of the interface. Substituting Eq. (5) into the discretized interface conditions at all the subdomain interfaces, we obtain a global interface system, which can be written symbolically as
F({Ec},{λ},{f})=0
(7)
where {λ} denotes the dual unknowns over all the subdomain interfaces, {Ec} denotes the discrete electric fields at all the corners, and {f} is contributed by the excitation source. Similarly, we can assemble Eq. (6) for all the subdomains to obtain a global corner system, which can be written symbolically as
{Ec}=G({λ},{f}).
(8)
From Eqs. (7) and (8), we can eliminate {Ec} to obtain the final system that relates the dual unknown {λ} and the excitation vector {f}. As a result, the original volumetric problem is converted to a surface problem, which has a much smaller number of unknowns. This final interface system can be solved iteratively using a Krylov subspace method such as the generalized minimum residual (GMRES) and the stabilized biconjugate gradient (BiCGStab) method. Once {λ} is computed, it can be used in Eq. (8) to calculate {Ec}. With both {λ} and {Ec}, the electric field inside each subdomain can be evaluated independently by using the first set of equations in Eq. (4).

2.2 Calculation of device parameters using the current sheet excitation

To calculate the transmission coefficient, we extract the electric field on the transmission reference surface ST and then perform numerical integration to obtain [36

36. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

]
Tm=STEemdSSTE0emdS
(9)
where E and E0 are the solutions with and without the embedded device, and em is the tangential electric field of the mth-order mode. Similarly, we can find the reflection coefficient on the reflection reference surface SR through [36

36. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

]

Rm=SR(EE0)emdSSRE0emdS.
(10)

2.3 Consideration for parallel implementation

The parallel implementation strategy adopted in this paper has been investigated previously [29

29. W. Yao, J. M. Jin, and P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

, 34

34. Y. J. Li and J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009). [CrossRef]

]. Regardless of the specific interface continuity condition employed, the FETI-DP algorithm can be divided into the tearing, interconnecting, and subdomain solution recovery stages. In all three stages, computations can be accelerated with the use of parallel computing schemes to reduce the computation time.

In the tearing stage, the computational domain is first automatically decomposed into Ns balanced nonoverlapping subdomains and the subdomain data is distributed across processors. Then, the assembly and factorization of the subdomain matrices can be carried out completely in parallel because they involve only subdomain operations. Next, in the interconnecting stage, the interface system is solved using a Krylov subspace method. The BiCGStab method is usually adopted because it has a faster convergence, a cheaper memory usage, and a better parallel scalability than the conjugate gradient (CG) method and the GMRES method. In each iteration step, the matrix-vector product on the global interface level is obtained as a summation of the forward/backward substitutions and matrix-vector products on the subdomain level. As a result, this step can be parallelized with some communication overhead. Note that before the interconnecting stage, we need to assemble and factorize the coarse grid system matrix. We can choose either a serial or a parallel direct solver such as MUMPS [37

37. MUMPS, A parallel sparse direct solver, available online: http://graal.ens-lyon.fr/MUMPS/.

] to perform the factorization, and the optimal implementation depends on the dimension of the coarse grid system [34

34. Y. J. Li and J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009). [CrossRef]

]. Finally, in the subdomain solution recovery stage, the values of the dual unknowns are broadcasted to each subdomain after the interface system is solved, and the subdomain solutions can then be computed completely in parallel.

3. Numerical results

The proposed SOTC-TE-based FETI-DP method has been implemented on the CISCO Arcetri cluster using the MPI parallel programming scheme. The Arcetri cluster has 24 computation nodes, each of which is equipped with 16 2.70-GHz Intel Xeon E5-2680 processors and 133-GB DDR3 memory. In this section, all examples are modeled and meshed with CUBIT [38

38. CUBIT, available online: https://cubit.sandia.gov/.

]. To automatically partition the computational domain, we employ the graph partitioning package METIS [39

39. METIS, Serial graph partitioning and fill-reducing matrix ordering, available online: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview.

], which provides an excellent load balance across subdomains. Curvilinear tetrahedral elements and hierarchical vector basis functions are employed for the FEM analysis. In PML regions, a constant material property is adopted as sa=1j, and the absorption is controlled by the PML thickness. The scalability in terms of number of subdomains and the parallel efficiency are investigated through several numerical examples to demonstrate the performance of the proposed algorithm.

3.1 Medium-scale 2D MRR simulation

Fig. 1 (a) Top view of a ring/bus structure modeled with CUBIT. The ring has the first-order grating along the upper half on the inner side. (b) Enlarged view of the rectangular region in Fig. 1(a) showing the grating.
In the first example, we simulate the TMz mode in a MRR shown in Fig. 1 for the purpose of validation. This structure is invariant along the direction perpendicular to the page and thus it can be modeled as a 2D problem to validate the 3D solution. In our 3D simulation, the ring/bus structure is assumed to have a finite thickness and a perfectly conducting plane is placed at the top and bottom. If the thickness is smaller than a half of a wavelength, a vertically invariant field can be preserved in the 3D configuration. The MRR plays as a band-stopping filter. To enhance reflection at one of the resonant frequencies, a first-order grating is employed along the inner circle of the upper half MRR [12

12. Y. M. Kang, A. Arbabi, and L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010). [CrossRef] [PubMed]

]. If the ring lies in the xy-plane, the parametric function for the inner radius of the ring is given by
{x(ϕ)=(r1+δsin2mϕ)cosϕ,y(ϕ)=(r1+δsin2mϕ)sinϕ,for0ϕ<πx(ϕ)=r1cosϕ,y(ϕ)=r1sinϕ,forπϕ<2π
(11)
where r1=8.267μm is the inner radius, δ=8.3×103μm is the grating size, and m=58 is the azimuthal order. Note that the grating indentations are not observable in Fig. 1 due to their small size. The outer radius parametric function is given by x(ϕ)=r2cosϕ and y(ϕ)=r2sinϕ for 0ϕ<2π with r2=8.68μm. The coupling gap between the ring and the bus waveguide is set to g=0.248μm. The relative permittivities of the core and the cladding are 4.0 and 1.0, respectively. The bus waveguide is placed parallel to the x-axis. Around the wavelength .λ=1550nm., the guided mode in the bus waveguide has the electric field profile in the core and in the cladding as
E(x,y)={z^cos[ky(yy0)]exp(jβx),(|yy0|d/2)z^cos(kyd/2)exp[α(|yy0|d/2)]exp(jβx),(|yy0|>d/2)
(12)
where β=6.8355μm1 is the propagation constant, ky=4.3593μm1 and α=5.5039μm1 describe the y-dependence, d=0.413μm is the waveguide width, and y0=9.135μm denotes the center of the bus waveguide.

For the simulation, the entire ring/bus structure is enclosed by a box, whose dimensions are 24.8, 24.8, and 0.207 μm in the Cartesian coordinate system. Except for the top and bottom boundaries, all four sides of the computational domain are truncated by the PML which has a thickness of 0.827 μm on each side. The core and cladding regions are discretized by a mesh size of 0.0744 and 0.149 μm, respectively. As a result, there are at least 6 layers of elements in the PML for each side truncation. The entire structure is excited by the mode described in Eq. (12) through a current sheet located on the bus. Two contra-directional waves are excited from the sheet and only the forward wave that bypasses the ring is of interest. Accordingly, the reflection and transmission coefficient calculation reference planes are placed on the left and right sides of the current sheet, as shown in Fig. 1. After being meshed with CUBIT, the entire computational domain is decomposed into Ns=512 subdomains, involving 7,522,572 unknowns, 889,488 dual unknowns, and 9,880 corner unknowns when second-order hierarchical vector basis functions are employed. The simulation is carried out from 1.924×1014 to 1.944×1014Hz and the computed reflection and transmission coefficients are plotted in Fig. 2 as functions of frequency. To validate the 3D simulation result, a 2D simulation is carried out by COMSOL Multiphysics [40

40. COMSOL Multiphysics ver. 4.2, available online: http://www.comsol.com/.

] using the same geometry and material setup in the xy-plane. Two sets of results are in excellent agreement, as shown in Fig. 2.
Fig. 2 Power reflection and transmission coefficients of the ring/bus structure shown in Fig. 1 from 1.924×1014Hz to 1.944×1014Hz.
Fig. 3 Re(Ez) in the z=0μm plane. (a) At 1.9246×1014Hz. (b) At 1.9355×1014Hz.
The field distribution in the z=0μm plane is plotted in Fig. 3 for 1.9246×1014 and 1.9355×1014Hz. It can be observed from the standing-wave field profile in the ring shown in Fig. 3(b) that the MRR has a stronger resonance at 1.9355×1014Hz. Most of the energy guided on the bus is reflected by the grating in the ring instead of being delivered to the receiving port. Usually, an iterative solver takes more iterations to converge at resonant frequencies because the matrix is more ill-conditioned. This prediction can be verified by the convergence history of the iterative solution of the global interface problem given in Fig. 4, which also shows that the FETI-DP method with the SOTC-TE effectively overcomes the convergence difficulty encountered with the FOTC when a PML mesh truncation is employed.
Fig. 4 Convergence history of the iterative solution of the global interface problem using the FETI-DP method with the SOTC-TE and the FOTC.

Next, we examine the parallel efficiency of the proposed algorithm by solving the bus/ring resonator problem using various numbers of processors.

Table 1. Computation times for the bus/ring resonator problem with 7,522,572 unknowns and 512 subdomains.

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Table 1 shows the time for preprocessing, the time for solving the interface problem, and the total computation time. Based on the computation times in Table 1, the speedup, which is defined with respect to the wall-clock time using four processors as
Speedup=T4TNp
(13)
is plotted in Fig. 5.
Fig. 5 Parallel speedup versus the number of processors with Ns=512. The computation time using four processors is taken as the reference.
Note that TNp is the total wall-clock time using Np processors. As can be seen, an excellent speedup has been achieved using up to 128 processors. With 128 processors employed, the peak memory usage is 55.7 GB.

3.2 Full-scale evanescently coupled double-microring resonator simulation

In this example, a fabricated 3D Si3N4/SiO2 evanescently coupled double-microring resonator (ECDMRR) is simulated. The device contains two rings, whose top view is given in Fig. 6(a).
Fig. 6 (a) Top view of an ECDMRR/bus structure modeled with CUBIT. The inner ring has a first-order grating along the upper half on its outer side. (b) Enlarged view of the rectangular region in Fig. 6(a).
The outer ring is a plain ring with no grating. A first-order grating is fabricated only on the outer boundary of the top half of the inner ring, as shown in the enlarged view in Fig. 6(b). The radii of the four concentric circles from the innermost to the outermost are r1=28.948μm, r2=29.548μm, r3=30.148μm, and r4=31.148μm, respectively. The parametric equation has the same format as that in (11) for the circle with grating, where the grating size is δ=0.1μm and the azimuthal order is m=200 in this example. The bus waveguide, which has a width d=1μm, is placed 345nm away from the outer ring at the closest point. The two rings and the bus waveguide have the same thickness t=0.3756μm in the vertical direction. The refractive indices are given by ncore=1.977 for the cores of the two rings and the bus waveguide, which are made of Si3N4, and ncladding=1.437 for the cladding, which is made of SiO2, respectively [41

41. A. Arbabi and L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4 and SiOx using microring resonances,” Opt. Lett. 38(19), 3878–3881 (2013). [CrossRef] [PubMed]

].

3.3 Optical fiber sensor with C-shaped aperture

The photonic hydrogen gas sensor considered in this example is constructed by etching a sub-wavelength aperture into an optically thick palladium film deposited on the facet of an optical fiber [16

16. S. J. McKeown and L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012). [CrossRef]

]. Upon adsorption of hydrogen into the palladium surface, the complex refractive index of the film will change, altering the transmission through the aperture. Due to the plasmonic resonances and enhanced transmission of the C-shaped aperture, the response of a palladium film with a C-shaped aperture to hydrogen is several times larger than that of a plain film or a nonresonant aperture. This aperture is shown to have enhanced transmission and resonant characteristics due to a combination of guided modes and generated surface plasmons at the aperture interfaces [16

16. S. J. McKeown and L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012). [CrossRef]

].

This example is first used to examine the numerical scalability of the proposed method with respect to the number of subdomains. As solving the interface system consumes a major portion of the total computation time, we focus on the number of iterations of the BiCGStab solver. The convergence history of the iterative solution of the global interface problem is plotted in Fig. 10 for different numbers of subdomains at λ=1550nm. It shows clearly that the proposed FETI-DP method scales well with respect to the number of subdomains. In this examination, second-order hierarchical vector basis functions are used, resulting in 16,621,476 unknowns.
Fig. 10 Iterative convergence of the global interface system with respect to the number of subdomains.

To obtain an accurate numerical result, third-order basis functions are employed in the final computation and the entire computational domain is divided into 960 subdomains, which yields 70,420,434 unknowns, 8,207,586 dual unknowns, and 124,971 corner unknowns. The BiCGStab iterative solver takes 255 iterations to converge to the residue of 10−6. The real part of the dominant component in the z=6.5μm plane for both the x- and y-polarized excitations is plotted in Fig. 11.
Fig. 11 (a) Re(Ex) in the plane 1.5μm above the aperture, with x-polarized excitation. (b) Re(Ey) in the plane 1.5μm above the aperture, with y-polarized excitation.
Fig. 12 (a) |Ex| in the aperture, with x-polarized excitation. (b) |Ey| in the aperture, with y-polarized excitation.
Two polarizations produce different responses due to the asymmetry of the aperture. It can be seen from Fig. 11(a) that the x-polarized excitation yields a C-shaped transmitted field profile due to the aperture resonance. The electric field profiles for the dominant component in the aperture are enlarged in Fig. 12 for a better view. The two arms of the aperture (aligning with the x-axis) support a stronger resonance for the y-polarized excitation than does the connecting portion between the two arms for the x-polarized excitation. To show the wave propagation along the z-axis, the real part of the dominant component in the y=0μm plane is plotted in Fig. 13 for both polarizations. The computed reflection and transmission coefficients are 0.8968 and 4.871×104 for the x-polarized excitation, and 0.8902 and 2.738×103 for the y-polarized excitation.
Fig. 13 (a) Re(Ex) in the y=0μm plane, with x-polarized excitation. (b) Re(Ey) in the y=0μm plane, with y-polarized excitation.

Finally, to investigate the parallel efficiency of the optical fiber sensor example, simulations using 40, 80, 160, and 320 processors are carried out and their wall-clock computation times are recorded for a fixed number of subdomains Ns = 960. The 40-processor case is taken as the reference.

Table 2. Computation times for the fiber tip sensor problem with 70,420,434 unknowns and 960 subdomains.

table-icon
View This Table
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Table 2 shows the time for preprocessing, the time for solving the interface problem, and the total computation time. The speedup is also plotted in Fig. 14based on Table 2. For up to 320 processors, the parallel efficiency can be as good as 62.5%.
Fig. 14 Parallel speedup versus the number of processors with Ns = 960.

4. Conclusion

This paper described the application of a newly developed domain decomposition solver for the analysis of large-scale photonic devices. As a nonoverlapping DDM, the presented FETI-DP method decomposes an original large-scale problem into smaller subdomain problems and distributes the subdomain data across processors so that parallel computing schemes can be employed to reduce the computation time significantly. With the aid of a higher-order transmission condition on the subdomain interfaces and a global coarse grid correction at the geometrical crosspoints, the method achieves a quite good numerical scalability. Two realistic examples, a reflective microring resonator and a palladium coated optical fiber sensor, are presented to demonstrate the capability of the method for the fast and accurate full-wave simulation of 3D photonic devices. Both devices have a large electrical size and contain small structural details; therefore, curvilinear tetrahedral elements are employed to provide accurate modeling of the complicated geometry. Higher-order hierarchical basis functions are also used to ensure the simulation accuracy. Transmission and reflection coefficients are computed over the frequency band of interest and the prediction is validated by the result from the measurement or other simulation tools in a special case. Field distributions are plotted to illustrate the underlying electromagnetic mechanism. Although the examples are resonant structures at the operating frequencies and have PML truncations, the SOTC-TE FETI-DP method exhibits a fast convergence rate and achieves a good parallel efficiency on a distributed-memory system with over three hundreds processors.

Finally, we note that to make a finite element simulation highly robust, one can implement a self-adaptive mesh refinement technique [19

19. B. Radjenović and M. Radmilović-Radjenović, “Computer-aided design and simulation of optical microring resonators”, Int. J. Numer. Model., Electron. Netw., Devices Fields, available online: http://onlinelibrary.wiley.com/doi/10.1002/jnm.1920/abstract. [CrossRef]

] to refine finite element meshes and increase the element orders judiciously to achieve an optimal convergence with respect to mesh density and order of basis functions. This self-adaptive scheme would require a repetitive solution of the problem and estimation of local field errors. The proposed FETI-DP algorithm, because of its high efficiency and implementation of hierarchical basis functions, provides an ideal forward solution that can be combined with a self-adaptive algorithm for the development of a highly accurate and robust FEM full-wave simulation for engineering applications. To perform a fast frequency sweep, the FETI-DP algorithm can be further hybridized with the asymptotic waveform evaluation method based on the complex frequency hopping technique and the rational Padé approximation.

Acknowledgments

The authors would like to express their appreciation to CISCO for the financial support to this project and for access to its Arcetri cluster. This work was also supported in part by National Science Foundation (CAREER award ECCS-1055941). The authors would also like to thank Prof. Luke Olson for useful feedback.

References and links

1.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, and N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998). [CrossRef]

2.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004). [CrossRef] [PubMed]

3.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

4.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003). [CrossRef]

5.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004). [CrossRef] [PubMed]

6.

T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12(7), 1437–1442 (2004). [CrossRef] [PubMed]

7.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]

8.

Y. Ding, M. Pu, L. Liu, J. Xu, C. Peucheret, X. Zhang, D. Huang, and H. Ou, “Bandwidth and wavelength-tunable optical bandpass filter based on silicon microring-MZI structure,” Opt. Express 19(7), 6462–6470 (2011). [CrossRef] [PubMed]

9.

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004). [CrossRef]

10.

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, and L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011). [CrossRef]

11.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009). [CrossRef]

12.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010). [CrossRef] [PubMed]

13.

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, and L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014). [CrossRef]

14.

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003). [CrossRef]

15.

A. Arbabi and L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012). [CrossRef]

16.

S. J. McKeown and L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012). [CrossRef]

17.

X. Shi and L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21(7), 1305–1317 (2004). [CrossRef]

18.

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, and P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008). [CrossRef]

19.

B. Radjenović and M. Radmilović-Radjenović, “Computer-aided design and simulation of optical microring resonators”, Int. J. Numer. Model., Electron. Netw., Devices Fields, available online: http://onlinelibrary.wiley.com/doi/10.1002/jnm.1920/abstract. [CrossRef]

20.

M. M. El Gowini and W. A. Moussa, “A finite element model of a MEMS-based surface acoustic wave hydrogen sensor,” Sensors (Basel) 10(2), 1232–1250 (2010). [CrossRef] [PubMed]

21.

A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express 19(21), 20405–20419 (2011). [CrossRef] [PubMed]

22.

B. Milanović, B. Radjenović, and M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012). [CrossRef]

23.

B. Radjenović and M. Radmilovic-Radjenović, “Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes,” IEEE Sens. J., doi:. [CrossRef]

24.

A. Arbabi, Y. M. Kang, and L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010). [CrossRef]

25.

Y. J. Li and J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007). [CrossRef]

26.

M. F. Xue and J. M. Jin, “Nonconformal FETI-DP methods for large-scale electromagnetic simulation,” IEEE Trans. Antenn. Propag. 60(9), 4291–4305 (2012). [CrossRef]

27.

M. F. Xue and J. M. Jin, “A hybrid nonconformal FETI/conformal FETI-DP method for arbitrary nonoverlapping domain decomposition modeling,” in Proceedings of IEEE AP-S Int. Symp. (2013). [CrossRef]

28.

Y. J. Li and J. M. Jin, “Fast full-wave analysis of large-scale three-dimensional photonic crystal devices,” J. Opt. Soc. Am. B 24(9), 2406–2415 (2007). [CrossRef]

29.

W. Yao, J. M. Jin, and P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

30.

W. Yao, J. M. Jin, and P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012). [CrossRef]

31.

A. Alonso-Rodriguez and L. Gerardo-Giorda, “New nonoverlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput. 28(1), 102–122 (2006). [CrossRef]

32.

Z. Peng, V. Rawat, and J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010). [CrossRef]

33.

Z. Peng and J.-F. Lee, “Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys. 229(16), 5615–5629 (2010). [CrossRef]

34.

Y. J. Li and J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009). [CrossRef]

35.

C. T. Wolfe, U. Navsariwala, and S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000). [CrossRef]

36.

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

37.

MUMPS, A parallel sparse direct solver, available online: http://graal.ens-lyon.fr/MUMPS/.

38.

CUBIT, available online: https://cubit.sandia.gov/.

39.

METIS, Serial graph partitioning and fill-reducing matrix ordering, available online: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview.

40.

COMSOL Multiphysics ver. 4.2, available online: http://www.comsol.com/.

41.

A. Arbabi and L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4 and SiOx using microring resonances,” Opt. Lett. 38(19), 3878–3881 (2013). [CrossRef] [PubMed]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(130.3120) Integrated optics : Integrated optics devices
(260.2110) Physical optics : Electromagnetic optics
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Integrated Optics

History
Original Manuscript: December 20, 2013
Revised Manuscript: February 7, 2014
Manuscript Accepted: February 10, 2014
Published: February 19, 2014

Citation
Ming-Feng Xue, Young Mo Kang, Amir Arbabi, Steven J. McKeown, Lynford L. Goddard, and Jian-Ming Jin, "Fast and accurate finite element analysis of large-scale three-dimensional photonic devices with a robust domain decomposition method," Opt. Express 22, 4437-4452 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4437


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References

  1. Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998). [CrossRef]
  2. V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004). [CrossRef] [PubMed]
  3. Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
  4. T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003). [CrossRef]
  5. M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004). [CrossRef] [PubMed]
  6. T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12(7), 1437–1442 (2004). [CrossRef] [PubMed]
  7. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
  8. Y. Ding, M. Pu, L. Liu, J. Xu, C. Peucheret, X. Zhang, D. Huang, H. Ou, “Bandwidth and wavelength-tunable optical bandpass filter based on silicon microring-MZI structure,” Opt. Express 19(7), 6462–6470 (2011). [CrossRef] [PubMed]
  9. J. K. S. Poon, J. Scheuer, A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004). [CrossRef]
  10. A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011). [CrossRef]
  11. Y. M. Kang, A. Arbabi, L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009). [CrossRef]
  12. Y. M. Kang, A. Arbabi, L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010). [CrossRef] [PubMed]
  13. Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014). [CrossRef]
  14. C.-Y. Chao, L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003). [CrossRef]
  15. A. Arbabi, L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012). [CrossRef]
  16. S. J. McKeown, L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012). [CrossRef]
  17. X. Shi, L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21(7), 1305–1317 (2004). [CrossRef]
  18. J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008). [CrossRef]
  19. B. Radjenović and M. Radmilović-Radjenović, “Computer-aided design and simulation of optical microring resonators”, Int. J. Numer. Model., Electron. Netw., Devices Fields, available online: http://onlinelibrary.wiley.com/doi/10.1002/jnm.1920/abstract . [CrossRef]
  20. M. M. El Gowini, W. A. Moussa, “A finite element model of a MEMS-based surface acoustic wave hydrogen sensor,” Sensors (Basel) 10(2), 1232–1250 (2010). [CrossRef] [PubMed]
  21. A. Einat, U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express 19(21), 20405–20419 (2011). [CrossRef] [PubMed]
  22. B. Milanović, B. Radjenović, M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012). [CrossRef]
  23. B. Radjenović, M. Radmilovic-Radjenović, “Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes,” IEEE Sens. J., doi:. [CrossRef]
  24. A. Arbabi, Y. M. Kang, L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010). [CrossRef]
  25. Y. J. Li, J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007). [CrossRef]
  26. M. F. Xue, J. M. Jin, “Nonconformal FETI-DP methods for large-scale electromagnetic simulation,” IEEE Trans. Antenn. Propag. 60(9), 4291–4305 (2012). [CrossRef]
  27. M. F. Xue, J. M. Jin, “A hybrid nonconformal FETI/conformal FETI-DP method for arbitrary nonoverlapping domain decomposition modeling,” in Proceedings of IEEE AP-S Int. Symp. (2013). [CrossRef]
  28. Y. J. Li, J. M. Jin, “Fast full-wave analysis of large-scale three-dimensional photonic crystal devices,” J. Opt. Soc. Am. B 24(9), 2406–2415 (2007). [CrossRef]
  29. W. Yao, J. M. Jin, P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).
  30. W. Yao, J. M. Jin, P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012). [CrossRef]
  31. A. Alonso-Rodriguez, L. Gerardo-Giorda, “New nonoverlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput. 28(1), 102–122 (2006). [CrossRef]
  32. Z. Peng, V. Rawat, J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010). [CrossRef]
  33. Z. Peng, J.-F. Lee, “Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys. 229(16), 5615–5629 (2010). [CrossRef]
  34. Y. J. Li, J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009). [CrossRef]
  35. C. T. Wolfe, U. Navsariwala, S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000). [CrossRef]
  36. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  37. MUMPS, A parallel sparse direct solver, available online: http://graal.ens-lyon.fr/MUMPS/ .
  38. CUBIT, available online: https://cubit.sandia.gov/ .
  39. METIS, Serial graph partitioning and fill-reducing matrix ordering, available online: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview .
  40. COMSOL Multiphysics ver. 4.2, available online: http://www.comsol.com/ .
  41. A. Arbabi, L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4 and SiOx using microring resonances,” Opt. Lett. 38(19), 3878–3881 (2013). [CrossRef] [PubMed]

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