Periodic boundary conditions and the error controlled fast multipole method / von Ivo Kabadshow. 2010
Content
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