三维各向异性光子晶体的快速仿真算法

Fast simulation algorithm for three-dimensional anisotropic photonic crystals

为获得三维各向异性光子晶体的带隙,基于Lebedev网格设计了一套求解其能带结构的快速仿真算法.首先,采用有限差分方法对Maxwell方程组进行离散,通过结合早期在Yee氏网格上的工作,对离散所得的Maxwell特征值问题的系数矩阵结构进行分析,给出其显式奇异值分解,并利用零空间压缩方法给出无零空间的标准特征值问题形式;在结合求逆Lanczos方法和共轭梯度法以及利用快速傅里叶变换大幅加速系数矩阵与向量乘法的基础上,设计出针对三维各向异性光子晶体能带结构的快速数值仿真算法.数值实验表明,相比于商业软件COMSOL,该套算法不仅数值结果准确,迭代算法所需的平均次数低于360次,且总计算时间少于1.25h,展现了算法在结合图形处理单元(GPU)高性能计算技术后的有效性与高效性.

To obtain the bandgap of three-dimensional anisotropic photonic crystals,a set of fast simulation al-gorithms for solving its band structure is designed based on the Lebedev grid.First,the finite difference meth-od is used to discretize the Maxwell equations.Based on the work on the Yee's grid,the coefficient matrix structure of the discrete Maxwell eigenvalue problem is analyzed,and its explicit singular value decomposition is given,and the standard form of the eigenvalue problem without null space is given by using the null space compression method.Based on the inverse Lanczos method and the conjugate gradient method,as well as the fast Fourier transform(FFT)acceleration coefficient matrix and vector multiplication,a fast numerical simula-tion algorithm for the energy band structure of three-dimensional anisotropic photonic crystals is designed.Nu-merical experiments show that compared with the commercial software COMSOL,the algorithm is not only ac-curate in numerical results,but also requires less than 360 times of iteration and less than 1.25 h of total calcu-lation time,which shows the effectiveness and efficiency of the algorithm combined with graphics processing unit(GPU)high-performance computing technology.