09 15 11 factorization laplace helmholtz lecture5

7/24/2019 09 15 11 Factorization Laplace Helmholtz Lecture5

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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


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3D Laplace Equation

Translation and DifferentiationProperties for Laplace Equation


3/18

Introduction of Multipoles for

Laplace Equation

Multipole Expansion of LaplaceEquation Solutions


4/18

Legendre Polynomials

Legendre Polynomials (2)

First six polynomials (n = 0,,5):


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Legendre Polynomials (3)

A lot of other nice properties!

Expansion of FundamentalSolution


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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 ofFundamental Solution

Multipole (!)

x

y

z

O

x

y

z

O

x

y

z

O

r

r

ir

i

i

x

y

z

O

x

y

z

O

x

y

z

O

r

r

ir

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


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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

kcan 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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Helmholtz equation:

Wave equation:

Time harmonic

solution

Helmholtz equation:

Fourier transform:

circular

frequency

wave speed

wavenumber

Construction of basis for 2DHelmholtz 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 Helmholtzequation (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 2DHelmholtz 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 Greens function

Addition

theorem



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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

r

ir

i

i

x

y

z

O

x

y

z

O

x

y

z

O

r

r

ir

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 Greens function

Addition

theorem

09 15 11 factorization laplace helmholtz lecture5

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