Anisotropic Diffusion in SPH Sergei Biriukov Supervisor: Daniel - - PowerPoint PPT Presentation

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Anisotropic Diffusion in SPH Sergei Biriukov Supervisor: Daniel - - PowerPoint PPT Presentation

Dec 05, 2022 131 likes •269 views

Anisotropic Diffusion in SPH Sergei Biriukov Supervisor: Daniel Price <latexit

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SLIDE 1

Anisotropic Diffusion in SPH

Sergei Biriukov Supervisor: Daniel Price

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SLIDE 2

Diffusion in SPH

Heat Conduction Radiative Transfer

∂T ∂t = r · (KrT)

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∂2T ∂ri∂rj ≈ X

b

mb ρb T(rb)(5ˆ riˆ rj − δij)Fab

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Gas-Dust Fraction

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SLIDE 3

Anisotropic

! = # ∗ 1 1 ' ( ' ( ! = 1 ' ( ' (

Initial Isotropic

slide-4
SLIDE 4

Isotropic Anisotropic

Question

slide-5
SLIDE 5

Smoothing Length

!"#$%&'() *+(# $ℎ( )ℎ"-( *. /0 − )-'%2( .*# 3%..(#(2$ )4**$ℎ%25 '(25$ℎ.

| – #78 have only

  • ne non-zero

component X – #78 have two non-zero component – all of the #78 components are not zero

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SLIDE 6

Good Outer Part – Good Kernel

3" 2" 1" %&'()* *&+,-./,-& 0ℎ.2&0 (3 4&+)&5 36)'/,()0. 8++(+ /&+90 3(+ :.25.',.) (2&+./(+.

Shapes are different. Shapes are different. Shapes are different… …but errors are the same. Outer part more important!

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SLIDE 7

Understanding Brookshaw Method

r2

aW = 2(r · rW)

(r · r) = 2 Cνhν+2 f 0(q) q

+

−2 Cν hν+2 −2 q exp (−q2) q = 4 h2 exp (−q2) Cν hν = 4Wf h2

=

Brookshaw method With Gaussian The kernel itself is a 2nd derivative

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SLIDE 8

Isotropic Diffusion Operators

1" 2" 3"

%&'()*+,&- &. /0 − 2**&* 3+4ℎ *26)*7, 4& -8'92* &. -2+6ℎ9&*,. Cubic spline as kernel function. All dimensions.

slide-9
SLIDE 9

Direct 2nd derivative Two 1st derivatives

∂T ∂t = X

ij

∂ ∂ri ✓ kij ∂ ∂rj T ◆ = X

b

mb ρb kij

ba

∂ ∂ri

a

Wab ! X

b

mb ρb Tba ∂ ∂rj

a

Wab ! + kij

a

X

b

mb ρb Tba ∂2 ∂ri

a∂rj a

Wab,

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Operators

∂T ∂t = r · (krT)

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∂T ∂t = r · F F = k · rT

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F = X

b

mb ρb TbaraWab ∂T ∂t = X

b

mb (ka · Fa) · raWab Ωaρ2

a

+ (kb · Fb) · rbWab Ωbρ2

b

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⇒

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slide-10
SLIDE 10

Direct 2nd derivative Two 1st derivatives

Diffusion with constant K

slide-11
SLIDE 11

Cylindrical coordinates

Diffusion with variable K

kρφz =   1   =

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= kxyz = 1 x2 + y2   x2 −xy −yx y2  

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SLIDE 12

101 102 103

log N

10−4 10−3 10−2 10−1 100

log L1(T)

2First Second N−2

Kernel bias region 101 102 103

log N

2First Second N−2

Kernel bias region

Convergence

Isotropic Anisotropic

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SLIDE 13
  • 1. The shape of the outer part of the kernel is more important for second derivatives.
  • 2. The idea of Brookshaw method is to mimic the kernel itself.
  • 3. Both direct second derivative method and two first derivatives are stable for

anisotropic diffusion.

  • 4. Two first derivatives is the best method for anisotropic diffusion.

Summary