Multipole expansion of the coulomb potential now consider the coulomb potential of some continuous charge distribution. Exponential convergence for multipole expansion and translation to local expansions for sources in layered media. Given a number n of sources in a unit box with potential in the form phiziqzizj, and given the desired level of accuracy for the multipole expansion, these scripts solve all pairwise interactions using of the fast multipole method. Vector spherical harmonics now take the curl of the third equation, substitute for r h from the fourth, to give. A heterogeneous fmm for 2d layered media helmholtz equation. The basic idea behind speeding up the m2l translation is to convert the multipole expansion m n m and the transfer function g n m r to fourier space and replace the convolution with the elementwise product. Multipole definition and meaning collins english dictionary. The key components of the fmm are expansion multipole and local, translation multipole to multipole, multipole tolocal and localtolocal and space partition. How do multipole moments relate to a taylor expansion.
A short course on fast multipole methods rickbeatson departmentofmathematicsandstatistics,universityofcanterbury lesliegreengard. Fast multipole bem for simulation of 2d solids containing. While the performance of the fftm relative to the fmm should be more carefully analyzed in terms of. In elliotts notation the expression for the m2l translation takes a very simple form. The operator contains all the coulombic chargecharge interactions, where one of the point charges belongs to the subsystem a, the second to b. The most important operation in the fmm is the multipole to local, or m2l, translation by a vector r. Prerequisites for the use of the fast multipole method. See also for a recent description of the link algorithm. Commonly the conversion from the fft circular convolution to the m2l linear convolution is achieved with zero padding.
Trond saue lcpq, toulouse electric and magnetic multipoles virginia tech 2017 14 22. What is the physical meaning of the terms in the multipole. Jan 01, 2008 the most important operation in the fmm is the multipole to local, or m2l, translation by a vector r. Tensors are useful in all physical situations that involve complicated dependence on directions. It is shown that translations of elementary solutions of the biharmonic equation. From my understanding, a multipole expansion is used to approximate the potential from a cluster of points, and can be evaluated at any point. The aggregation is just the multipoletomultipole shift from a current expansion situated at this level to its parent. The key components of the fmm are expansion multipole and local, translation multipoletomultipole, multipoletolocal and localtolocal and space partition.
The first order term or the second term of the multipole expansion of a function. M 2 l, which converts a multipole expansion to a local expansion with a new center. Box mg6, bucharest, romania abstractwe establish the exact formulas of multipole expansion in cartesian coordinates for the most general distribution of charges and currents including toroidal sources. To handle much larger particle ensembles, to enable the iterative use of the subroutines in a solver, and to remove. We have developed software implementing this solution. The runtime of the fmm is significantly less than that of the pairwise calculation if the particle number, n, is sufficiently large. A multipole expansion is a mathematical series representing a function that depends on anglesusually the two angles on a sphere.
Several studies that are dedicated to scattering by two spheres marnevskaya 1969, 1970, gaunaurd and huang 1994, gaunaurd et al 1995 use representations of the translation coe. Multipole expansions in radiation theory of quantum systems. Multipole translation theory for the threedimensional. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function.
The use of both multipole and local expansions reduces both the computational complexity and the memory requirement to on. Multipole expansion approach is the first book to introduce micromechanics researchers to a more efficient and accurate alternative to computational micromechanics, which requires heavy computational effort and the need to extract meaningful data from a multitude of numbers produced by finite element software code. This algorithm proceeds by performing the necessary integration recursively within a speci. The new version of fast multipole method fmm that diagonalizes the multipoletolocal translation operator is applied with the tree structure adaptive to sample particle locations. A heterogeneous fmm for 2d layered media helmholtz. We employ recently developed procedures of fast and stable computation of general multipole translation coe. Physics 214 20, electricity and magnetism multipole expansion for radiation. Computation of scattering from n spheres using multipole. Optimization of the multipole to local translation. In the perturbational theory of intermolecular interactions see chapter, the perturbation operator v plays an important role. Automatic generation of fft for translations of multipole.
Translation of multipole expansion in spherical harmonics is the most important operation of the fast multipole method and the fast fourier transform fft acceleration of this operation is among. The fast multipole method has been called one of the ten most signi. Numerical experiments show that our algorithm is asymptotically linear in operations for nearly uniform particle. Elliot and board 1996 in the renormalized basis translation matrices are simple these are structured matrices 2d toeplitzhankel type fast translation procedures are possible e. The dielectric constant of a solvent is the measure of its capacity to. Compute multipole expansion me for each box if leaf box, compute from exact sources in that box if nonleaf box, compute from its childrens me using m2m 2. The on method translation of multipole expansion conversion of a multipole expansion into a local expansion translation of local expansion fmm 7 matrix version of fmm matrix vector product quad tree. If there are n particles, then the time required is proportional to n, whereas the simple method of doing a double loop and computing the interaction of each pair requires time. Translate dipole moment in english online and download now our free translator to use any time at no charge. We have described a software package for evaluating the screened coulomb interactions. The fast multipole method was used to solve the traction boundary integral equation for 2d crack analysis. The on method translation of multipole expansion conversion of a multipole expansion into a local expansion translation of local expansion fmm. A method to calculate the spherical multipole expansion of. The cost of applying a translation operator to a multipole or taylor expansion is op2 in the current implementation, but methods exist that can reduce this to op log p 17, 1, 11.
In this paper, we will first give a derivation of the multipole expansion me and local expansion le for the far field from sources in general 2d layered media and the multipole tolocal translation m2l operator by using the generating function for bessel functions. The fast multipole method f mm is an accurate and fast way to calculate potentialsfields created by a very large number of particles. Two major parts in the fmm are the upwa rd pass and the downward pass. Need for translation theorem in fast multipole method. New results for the laplace equation include an elementary proof of the innertoinner translation theorem, from which follows the definition of a farfield signature function analogous to that of the. Maybe this is not an independent question, but i am wondering if there is something like a geometricalpictorial meaning of every term in the multipole expansion. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential for clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density. Multiple expansion is a fundamental finance concept and impacts lbo returns. How do multipole moments relate to a taylor expansion, with.
The mathematical theory of multipole translation operators for the threedimensional laplace and helmholtz equations is summarized and extended. Generic cartesianbased adaptive multipole calculator. Then, we present a rigorous proof of the exponential convergence of the me, le, and m2l for 2d helmholtz equations in. Essentially, multiple expansion is the concept of buying low and selling high. Compute local expansion le for each box use m2l to translate the me of the boxes in the interaction list and add to the le of the current box. This is called the taylor series expansion of fx about x. M ossbauer spectroscopy isomer shift modi cation of nuclear transition by electrostatic interaction. I have rephrased my original question to be clearer.
A finite field method for calculating spherical tensor molecular polarizability tensors. For each parent box, instead of communicating with the particles, the multipole expansion is derived by shifting and merging its childrens multipole expansions, using the following \multipoletomultipole tmm translation. It uses an adaptive new version of fmm based on a diagonal form for the multipoletolocal translation operator, which extends the modern fmm for the laplace equation. Computing a bestfit of discrete points from a multipole expansion, i. Implementation of the local expansion implementation of the multipole expansions coefficients implementation of the translation local to local implementation of the translation multipole to local implementation of the translation multipole to multipole from numpy import. Apr 06, 2016 mod07lec46 multipole expansion, continuous charge distriution, and assembly of point charges duration. Mathematical methods of physicsthe multipole expansion. The aggregation is just the multipole to multipole shift from a current expansion situated at this level to its parent. Usually if the partitioning is based on an octree then each 8 children are summed after translation to their corresponding parent. Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance r to a source, i. Fx physics november 25, 2009 4d,0,3d z x x q r curly. The multipole expansion uses a complex taylor series expansion to reduce the number of multipole moments.
A single level fast multipole method solver file exchange. Multipole expansion an overview sciencedirect topics. Pdf the spherical multipole expansion of a triangle. Here, we consider one such example, the multipole expansion of the potential of a charge distribution. Trond saue lcpq, toulouse electric and magnetic multipoles virginia tech 2017 16 22. When the expansion point of the multipole expansion is outside the. If we relabel the indices of the local expansion matrix to l j k, the m2l translation can be written as. Appendix x multipole expansion what is the multipole expansion for. Fast multipole method for 3d helmholtz equation in layered media, bo wang, wenzhong zhang, wei cai, arxiv. Generic cartesianbased adaptive multipole calculator multipole methods allow the rapid computation of pairwise interactions among a large number of particles. Multipole expansions in quantum radiation theory m. Compute local expansion le for each box use m2l to translate the me of the boxes in the.
Nov 30, 2011 a finite field method for calculating spherical tensor molecular polarizability tensors. Multiple expansion is a form of arbitrage where a buyer pays an entry valuation multiple that is lower than the exit valuation multiple. Lets start by calculating the exact potential at the. We develop a complete translation theory for these equations. Go back through our intermediate lbo template and see how much multiple expansion affects projected returns. In chapter 2, we present the fast multipole method based on 4, 5. Agre national university of kyivmohyla academy, 04655 kyiv, ukraine email.
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