Classical and enriched finite element formulations for Bloch‐periodic boundary conditions

Abstract

AbstractIn this paper, classical and enriched finite element (FE) formulations to impose Bloch‐periodic boundary conditions are proposed. Bloch‐periodic boundary conditions arise in the description of wave‐like phenomena in periodic media. We consider the quantum‐mechanical problem in a crystalline solid and derive the weak formulation and matrix equations for the Schrödinger and Poisson equations in a parallelepiped unit cell under Bloch‐periodic and periodic boundary conditions, respectively. For such second‐order problems, these conditions consist of value‐ and derivative‐periodic parts. The value‐periodic part is enforced as an essential boundary condition by construction of a value‐periodic basis, whereas the derivative‐periodic part is enforced as a natural boundary condition in the weak formulation. We show that the resulting matrix equations can be obtained by suitably specifying the connectivity of element matrices in the assembly of the global matrices or by modifying the Neumann matrices via row and column operations. The implementation and accuracy of the new formulation is demonstrated via numerical examples for the three‐dimensional Poisson and Schrödinger equations using classical and enriched (partition‐of‐unity) higher‐order FEs. Copyright © 2008 John Wiley & Sons, Ltd.