Classical and enriched finite element formulations for Bloch-periodic boundary conditions. (English) Zbl 1156.81313
Summary: 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.
MSC:
| 81-08 | Computational methods for problems pertaining to quantum theory |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81T80 | Simulation and numerical modelling (quantum field theory) (MSC2010) |
Keywords:
Floquet waves; quantum mechanics; periodic boundary conditions; constraint equations; higher-order finite elements; partition of unity; enrichmentReferences:
| [1] | Michel, Effective properties of composite materials with periodic microstructure: a computational approach, Computer Methods in Applied Mechanics and Engineering 172 pp 109– (1999) · Zbl 0964.74054 |
| [2] | Porter, Scattered and free waves over periodic beds, Journal of Fluid Mechanics 483 pp 129– (2003) · Zbl 1055.76005 |
| [3] | Yablonovitch, Photonic band-gap crystals, Journal of Physics: Condensed Matter 5 pp 2443– (1993) |
| [4] | Hiett, Application of finite element methods to photonic crystal modelling, IEE Proceedings A: Science Measurement and Technology 149 (5) pp 293– (2002) |
| [5] | Nicolet, Modelling of electromagnetic waves in periodic media with finite elements, Journal of Computational and Applied Mathematics 168 (1-2) pp 321– (2004) · Zbl 1056.78013 |
| [6] | Ashcroft, Solid State Physics (1976) |
| [7] | Davies, Finite-element analysis of all modes in cavities with circular symmetry, IEEE Transactions on Microwave Theory and Techniques 30 pp 1975– (1982) |
| [8] | McGrath, Phased array antenna analysis with the hybrid finite element method, IEEE Transactions on Antennas and Propagation 42 (12) pp 1625– (1994) |
| [9] | McGrath, Periodic structure analysis using a hybrid finite element method, Radio Science 31 (5) pp 1173– (1996) |
| [10] | Mias, Finite element modelling of electromagnetic waves in doubly and triply periodic structures, IEE Proceedings: Optoelectronics 146 pp 111– (1999) |
| [11] | Hermansson, Finite-element approach to band-structure analysis, Physical Review B 33 (10) pp 7241– (1986) |
| [12] | Ferrari, Electronic band structure for two-dimensional periodic lattice quantum configurations by the finite element method, International Journal of Numerical Modelling: Electronic Networks Devices and Fields 6 pp 283– (1993) |
| [13] | Pask, Real-space local polynomial basis for solid-state electronic-structure calculations: a finite-element approach, Physical Review B 59 (19) pp 12352– (1999) |
| [14] | Pask, Finite-element methods in electronic-structure theory, Computer Physics Communications 135 (1) pp 1– (2001) · Zbl 0984.81038 |
| [15] | Pask, Finite element methods in ab initio electronic structure calculations, Modelling and Simulation in Materials Science and Engineering 13 (3) pp R71– (2005) |
| [16] | Jun, Meshfree implementation for the real-space electronic-structure calculation of crystalline solids, International Journal for Numerical Methods in Engineering 59 (14) pp 1909– (2004) |
| [17] | Strang, An Analysis of the Finite Element Method (1973) · Zbl 0356.65096 |
| [18] | Zienkiewicz, The Finite Element Method: Its Basis and Fundamentals (2005) |
| [19] | Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099 |
| [20] | Babuška, The partition of unity method, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) |
| [21] | Arfken, Mathematical Methods for Physicists (2005) |
| [22] | Griffiths, Introduction to Quantum Mechanics (1994) |
| [23] | Gygi, Adaptive Riemannian metric for plane-wave electronic-structure calculations, Europhysics Letters 19 (7) pp 617– (1992) |
| [24] | Tsuchida, Adaptive finite-element method for electronic-structure calculations, Physical Review B 54 (11) pp 7602– (1996) |
| [25] | Pask JE, Sukumar N. Spectral element method for quantum mechanical atomic calculations. 2008; in preparation. |
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