[PPT] - A Fast Solver for the Stokes Equations Tejaswin Parthasarathy CS PowerPoint Presentation

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A Fast Solver for the Stokes Equations

Tejaswin Parthasarathy CS 598APK, Fall 2017

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Stokes Equations?

2

wallpapers-xs.blogspot.com http://www.earthtimes.org/ http://testingstufftonight.blogspot.com/

Initial Conditions (t) Boundary Conditions (x) Solution Algorithm: x, t

µ

Input

~ u(~ x, t) p(~ x, t)

Output

Resistance

Navier Stokes Continuity

∂ρ ∂t + ∂(ρui) ∂xi = 0

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Stokes Equations?

3

µ → ∞

https://www.promegaconnections.com/ https://www.youtube.com/watch?v=p08_KlTKP50 UNM Physics and Astronomy, Sped up 10x

0 = − ∂p ∂xi + µ∂2ui ∂x2

j

+ fi

Stokes PDE : Especially useful in small scales

∂(ρui) ∂xi = 0

Continuity

Integral Equations No time dependence - completely reversible Linear PDEs :) u : A system of linear PDEs :( P : A coupled system of PDES :’((

Boundary Conditions (x) Solution Algorithm

ρ µ

~ u(~ x)

P(~ x)

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Necessity ?

4

http://www.elveflow.com/ http://techgenmag.com/

Time Goal

Gazzola et al, 2012 Gazzola et al, 2012

Present

Xiaotian et al, Preprint

Stokes Flow Viscous flow

+

Integral Equations & Fast Algorithms

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Why Integral Equation techniques?

5

µr2u rP + f = 0 r · u = 0

C Pozrikidis, 1992 Malhotra et al, 2014 Klinteberg et al, 2016

Challenges 4 unknowns @ x Continuity Discretisation spaces Conditioning Time to Solution Higher order FDM FEM BEM x x x x

Needs projection

x

Needs projection

√

Identically satisfied

x

What spaces?

x

inf-sup restriction

√

Any meaningful rep.

x

Very Bad

x

Bad: Preconditioners

√

Good κ indep.

f problem size

x

Slow

x

Slow

√

Fast O(n)

x

Difficult

x

Difficult

√

Not difficult

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Procedure

6

Construct representation : Integral Operators + Potential Theory BVP & IE solution existence/uniqueness IE discretization Quadrature Rule Time progression

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Constructing a representation - some theory

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µr2u rP + gδ(x xo) = 0

u(x) = D[Γ, q](x)

Double Layer Potential Source curve Hydrodynamic potential

σij = −Pδij + µ ✓ ∂ui ∂xj + ∂uj ∂xi ◆

Fluid stress Laplace PDE Stokes PDE

σij = µTijkgj

ui = Gijgj

P = µpjgj

Gij(x, y) = δij r + xixj r3

G(x, y) = 1 4πr

pj(x, y) = 2 ˜ xj r3 Tijk(x, y) = −6 ˜ xi˜ xj ˜ xk r5

Stokeslet Stresslet

KD(x, y) = ˆ n.ryG(x, y)

KD

j (x, y) = Tijk(x, y)nk(y)

Stresslet

C Pozrikidis, 1992

Kernels to construct solution exist

Erik Ivar Fredholm

Wikipedia

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Constructing a representation - some theory

8

µr2u rP + gδ(x xo) = 0

Laplace PDE

G(x, y) ⇒ ˆ a · ∂G ∂x (x, y)

Z

U

(ψ ∆ϕ − ϕ ∆ψ) dV = I

∂U

✓ ψ ∂ϕ ∂n − ϕ∂ψ ∂n ◆ dS

(S(ˆ n · ru) Du)(x) = u(x)

Stokes PDE Multipole Stokeslet, Stokeslet Doublet, Stokeslet Quadrupole For u :

∂ ∂xj (u0

iσij − uiσ0 ij) = u0 i

∂σij ∂xj − ui ∂σ0

ij

∂xj

Reciprocal Identity : Strong physical meaning

C Pozrikidis 2.3.10, 1992

uj(x) = − 1 8πµ Z

Γ

Gij(y, x)fi(y)dS(y)+ 1 8π Z

Γ

Tijk(y, x)nk(y)ui(y)dS(y) The Stokeslet and Stresslet provide a complete representation

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Boundary Value Problems

9

Typically interested in external problems : Dirichlet and Neumann Laplace PDE Null space for external Dirichlet : Fredholm Alternative from int. Neumann Stokes PDE Null space for external Dirichlet : Fredholm Alternative from int. Neumann :(

ui(x) = Z

Γ

Tjik(y, x)nk(y)qj(y)dS(y) + Vi(x)

C Pozrikidis, 1992

Usually Prescribed (or) Single Layer op. Compensate for deficiency in range

Vi(x)

∴ At surface,

ud

i (x) + Vi + ✏ijkΩjX0,k = 4⇡qi(x) + PV

Z

Γ

Tjik(y, x)nk(y)qj(y)dS(y) + Vi(x)

Deformation Translation Rotation Still (I+ Compact) : Well conditioned

BVP straightforward?

www.exposureguide.com

Prescribed velocity Prescribed force

www.livescience.com C Pozrikidis, Thm 4.7.1 1992

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

BVPs : Mobility and Resistance

10

C Pozrikidis, 1992

complicates the problem, leading to a dichotomy:

Vi(x)

ud

i (x) + Vi + ✏ijkΩjX0,k = 4⇡qi(x) + PV

Z

Γ

Tjik(y, x)nk(y)qj(y)dS(y) + Vi(x)

Deformation Translation Rotation Still (I+ Compact) : Well conditioned

Mobility problem

(f, t) ⇒ (V, Ω)

If prescribed forces, find motion Linear Resistance problem

(V, Ω) ⇒ (f, t)

If prescribed motion, find forces Linear This leads to (additional) constraints in some cases:

Bubble Rigid bodies Gazzola et al, 2013

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Discretization & Solution

11

C Pozrikidis, 1992

1. Surface and function discretisation requirements as required

Nearly singular evaluations: Expansion may fail

2. QBX to calculate matrix coefficients of discrete system to be solved (accelerated by

precomputing/FMM)

3. Enforce BC at quadrature points to solve linear system Aq = b by GMRES (const iter.)
4. With q obtained, get u on domain using DLP (FMM accelerated)
5. Calculate p or 𝞽 as a post processing step, as needed
6. Get new particle positions using force history and some time stepping scheme

Now use QBX (with trapz/gauss) to calculate PV of DLP on the surface Nystrom carries over: Approximate quadrature sufficient for off surface evaluation IE Discretisation

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A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017

Conclusions

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We have an IE method to solve the Stokes flow problem Optimal (or) near optimal time Numerical experiments to be conducted Any questions?

1. Gazzola, Mattia, Wim M. Van Rees, and Petros Koumoutsakos. "C-start: optimal start of larval fish." Journal of Fluid Mechanics 698 (2012): 5-18.
2. Pozrikidis, Constantine. Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, 1992.
3. Malhotra, Dhairya, Amir Gholami, and George Biros. "A volume integral equation stokes solver for problems with variable coefficients." Proceedings of the International

Conference for High Performance Computing, Networking, Storage and Analysis. IEEE Press, 2014.

4. af Klinteberg, Ludvig, and Anna-Karin Tornberg. "A fast integral equation method for solid particles in viscous flow using quadrature by expansion."
5. Journal of Computational Physics 326 (2016): 420-445.
6. Klöckner, Andreas, et al. "Quadrature by expansion: A new method for the evaluation of layer potentials." Journal of Computational Physics 252 (2013): 332-349.

Similarities/ Differences to Laplace PDE