09 15 11 Factorization Laplace Helmholtz Lecture5
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Transcript of 09 15 11 Factorization Laplace Helmholtz Lecture5
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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
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Introduction of Multipoles for
Laplace Equation
Multipole Expansion of LaplaceEquation Solutions
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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
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y
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x
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s1
s2
x
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x
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x
y
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s1
s2
degree
order
Vector form of the addition theorem
S- and R- expansions ofFundamental Solution
Multipole (!)
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r
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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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2D Helmholtz Equation
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
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
r
ir
i
i
x
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O
x
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O
x
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r
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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