High Performance Computing on 3D Maxwell’s equations in Chiral Media

• Main Authors: Tsai, Yi-Lin, 蔡翊琳
• Other Authors: Lin, Wen-Wei
• Format: Others
• Language: en_US
• Published: 2019
• Online Access: http://ndltd.ncl.edu.tw/handle/u48dtw

碩士 === 國立交通大學 === 應用數學系數學建模與科學計算碩士班 === 107 === The issue of photonic crystals has been gradually emphasized. How- ever, how to deal with such sparse large-scale eigenvalue problems is always a key point to derive the wanted properties. As a result, we implement the numerical computation in parallel through GPU and observe the performance of electromagnetism in crystals under the chiral medium. Since the light belongs to electromagnetic waves, it depends on the three-dimensional source free Maxwell’s equations to model the properties of light propagation in crystals. In order to find the expected consequences, we need to solve a series of eigenvalue problems to get the frequencies. At first, we apply the periodic structure of photonic crystals, time-harmonic assumption, and the Bloch’s theorem to derive concise formulas from source free Maxwell’s equations. Then, by using Yee’s finite difference scheme to get discrete grid points. In this discretization method, the divergence-free is naturally satisfied. In the process of acquiring the frequencies, we use some mathematical techniques, such as singular value decomposition, invariant subspace and Fast Fourier Transformation (FFT) to help us reduce the cost of large scale matrix computation. Finally, for the chirality parameter γ is larger than the critical number γ∗, we solve the problem by exploiting the inexact Shift-Invert Residual Arnoldi (SIRA) methods rather than the inverse Lanczos method with Conjugate Gradient(CG). In order to accomplish the numerical computation in parallel on GPU, several libraries are chosen to reach this target. It contains cuFFT, cuBLAS, LAPACK and MAGMA. In this paper, we focus on the null-space free generalized eigenvalue problem (NFGEP) in the different chirality parameter γ. As γ > γ∗, some high-frequency eigenvalues created by the Jordan form would change into complex eigenpairs. As the γ increases, those complex eigenpairs would bifurcate at the real axis and become some small and real eigenvalues. Corresponding to the newly added eigenvalues, we perform the lock of the energy in the crystals by the eigenmode of the electric fields. On the other hand, we compare the elapsed time and iteration numbers between different linear solvers in the SIRA.