Periodic boundary conditions and the error controlled fast multipole method / von Ivo Kabadshow. 2010
Inhalt
- Abstract
- Table of Contents
- List of Figures
- List of Tables
- 1 Introduction
- 2 FMM in Three Dimensions
- 2.1 Informal Description of the FMM
- 2.2 Mathematical Preliminaries
- 2.2.1 Expansion of the Inverse Distance
- 2.2.2 Spherical Harmonic Addition Theorem
- 2.2.3 Expansion of Particle-Particle Interactions
- 2.2.4 Addition Theorem for Regular Solid Harmonics
- 2.2.5 Addition Theorem for Irregular Solid Harmonics
- 2.2.6 Formal Double Sum Manipulations
- 2.3 Mathematical Operators
- 2.3.1 Translation of a Multipole Expansion (M2M)
- 2.3.2 Conversion of a Multipole Expansion into a Local Expansion (M2L)
- 2.3.3 Translation of a Local Expansion (L2L)
- 2.3.4 Rotation-Based Operators
- 2.3.5 Further Operator Compression
- 2.4 O(NlogN) Algorithm
- 2.5 O(N) Algorithm
- 2.6 Implementation Details
- 2.6.1 Fractional Tree Depth
- 2.6.2 Data Structures
- 2.6.3 Minimization of Near Field and Far Field Computation Time
- 2.6.4 Limitations of the Algorithm
- 2.7 Error Analysis
- 3 FMM and Periodic Boundary Conditions
- 3.1 Minimum Image Convention
- 3.2 Definition of the Boundary Condition
- 3.2.1 Three Dimensional Periodicity
- 3.2.2 Two Dimensional Periodicity
- 3.2.3 One Dimensional Periodicity
- 3.2.4 Periodic Potential and Energy
- 3.3 Convergence of Lattice Sums
- 3.4 Ewald-Based Summation Schemes
- 3.5 Parameter-Free Renormalization Approach
- 3.6 Implementation Details
- 3.6.1 Additional FMM Pass for the Lattice Operator
- 3.6.2 Modifications of FMM Pass 1–5
- 3.6.3 Fractional Tree Depth and Periodic Boundary Conditions
- 3.6.4 Dipole Correction
- 3.7 Periodic Boundaries and Error Control
- 4 Performance Details of the FMM Implementation
- 5 Conclusion and Future Directions
- A Appendix
- Bibliography