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Calculation of band structures of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalues. In this paper, we try to address this problem in such a broad context that band structures of three dimensional isotropic photonic crystals in all $14$ Bravais lattices can be efficiently computed in a unified framework. In this work, we uncover the delicate machinery behind several key results of our framework and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments. Particular effort is made on reformulating the Bloch condition for all $14$ Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by identifying the hierarchical structure of the underlying normal (block) companion matrix, and reducing the eigen-decomposition of the double-curl operator to a simple factorization of a $3$-by-$3$ complex skew-symmetric matrix. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm to solve Maxwell eigenvalue problems.

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This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter $\gamma$. We show that all the eigenvalues are real and finite for a small chirality $\gamma$. For a critical value $\gamma = \gamma^*$, the GEP has $2 \times 2$ Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when $\gamma$ increases from $\gamma^*$, the $2 \times 2$ Jordan block will first split into a complex conjugate eigenpair, then rapidly collide with the real axis and bifurcate into positive (resonance) and negative eigenvalues with modulus smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.

Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected 3-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle- and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.