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CMSC 858M/AMSC 698R

Fast Multipole Methods

Nail A. Gumerov & Ramani Duraiswami

Lecture 5

Outline

• Laplace equation in 3D (continued)

• Helmholtz equation in 2D

• Helmholtz equation in 3D

2

3D Laplace Equation

Translation and Differentiation

Properties for Laplace Equation

3

Introduction of Multipoles for

Laplace Equation

Multipole Expansion of Laplace

Equation Solutions

4

Legendre Polynomials

Legendre Polynomials (2)

First six polynomials (n = 0,…,5):

5

Legendre Polynomials (3)

A lot of other nice properties!

Expansion of Fundamental

Solution

6

Addition Theorem for Spherical

HarmonicsSpherical Harmonics

x

y

z

x

y

z

x

y

z

s1

s2

x

y

z

x

y

z

x

y

z

s1

s2

degree

order

Vector form of the addition theorem

S- and R- expansions of

Fundamental Solution

Multipole (!)

x

y

z

O

x

y

z

O

x

y

z

O

r

rir

i

i

x

y

z

O

x

y

z

O

x

y

z

O

r

rir

i

i

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`Multipole expansion` means

S-expansion

Associated Legendre Functions

Orthogonal!

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Spherical Harmonics

n m

n

m

Orthonormality of Spherical Harmonics

9

R- and S- expansions of arbitrary

solutions of the 3D Laplace equation

Helmholtz Equation

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Helmholtz Equation

• Wave equation in frequency domain– Acoustics

– Electromagneics (Maxwell equations)

– Diffusion/heat transfer/boundary layers

– Telegraph, and related equations

– k can be complex

• Quantum mechanics– Klein-Gordon equation

– Shroedinger equation

• Relativistic gravity (Yukawa potentials, k is purely imaginary)

• Molecular dynamics (Yukawa)

• Appears in many other models

2D Helmholtz Equation

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2D Helmholtz Equation

Helmholtz equation:

Wave equation:

Time harmonic

solution

Helmholtz equation:

Fourier transform:

circular

frequency

wave speed

wavenumber

Construction of basis for 2D

Helmholtz equation

Polar coordinates

Helmholtz

equation

Elementary solution

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Construction of basis for 2D Helmholtz

equation (2): Angular dependence

Periodicity condition

Discontinuous at the ends

(use Analytical continuation)

Construction of basis for 2D Helmholtz

equation (3): Radial dependence

Bessel equation:

Bessel functions

of order n

First kind Second kind

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Bessel functions of integer order

Learn properties of Bessel functions! Abramowitz & Stegun (1964)

(Handbook on special functions: available online!)

Construction of basis for 2D

Helmholtz equation (4)

Usual basis functions

Regular expansion

Singular (radiating)

expansion

Hankel functionsfirst kind

second kind

Sommerfeld radiation

condition

Radiating

functions

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Factorization of Green’s function

Addition

theorem

3D Helmholtz Equation

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Reference

N.A. Gumerov & R. Duraiswami

Fast Multipole Methods for the Helmholtz Equation in Three

Dimensions

Academic Press, Oxford, UK (2004).

Spherical Basis Functions

x

y

z

O

x

y

z

O

x

y

z

O

r

rir

i

i

x

y

z

O

x

y

z

O

x

y

z

O

r

rir

i

i

Spherical Coordinates

Regular Basis Functions

Singular Basis Functions

Spherical Harmonics

Associated Legendre Functions

Spherical Bessel Functions

Spherical Hankel Functions

of the First Kind

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Spherical Harmonics

n

m

Spherical Bessel Functions

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Isosurfaces For Regular Basis Functions

n

m

Isosurfaces For Singular Basis Functions

n

m

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Factorization of Green’s function

Addition

theorem