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

A Fast Solver for the Stokes Equations Tejaswin Parthasarathy CS 598APK, Fall 2017

Stokes Equations? http://www.earthtimes.org/ wallpapers-xs.blogspot.com http://testingstufftonight.blogspot.com/ ∂ t + ∂ ( ρ u i ) ∂ρ Continuity = 0 ∂ x i Navier Stokes Input Output u ( ~ x, t ) ~ Initial Conditions (t) Boundary Conditions ( x ) p ( ~ x, t ) µ Resistance Solution Algorithm: x , t A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 2

Stokes Equations? µ → ∞ https://www.promegaconnections.com/ ∂ ( ρ u i ) = 0 Continuity ∂ x i Stokes PDE : Especially useful in small scales + µ ∂ 2 u i 0 = − ∂ p + f i ∂ x 2 ∂ x i j No time dependence - completely reversible Linear PDEs :) Integral Equations u : A system of linear PDEs :( P : A coupled system of PDES :’(( u ( ~ x ) ~ Boundary Conditions (x) ρ µ Solution Algorithm P ( ~ x ) https://www.youtube.com/watch?v=p08_KlTKP50 UNM Physics and Astronomy, Sped up 10x A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 3

Necessity ? Time Present Goal Stokes Flow http://www.elveflow.com/ http://techgenmag.com/ Xiaotian et al, Preprint Viscous flow + Integral Equations & Fast Algorithms Gazzola et al, 2012 Gazzola et al, 2012 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 4

Why Integral Equation techniques? µ r 2 u � r P + f = 0 r · u = 0 FDM FEM BEM Challenges x x x 4 unknowns @ x x x √ Continuity Needs projection Needs projection Identically satisfied x x √ Discretisation spaces What spaces? inf-sup restriction Any meaningful rep. √ x x Conditioning Good κ indep. Very Bad Bad: Preconditioners of problem size x x √ Time to Solution Slow Slow Fast O(n) x x √ Higher order Difficult Difficult Not difficult C Pozrikidis, 1992 Malhotra et al, 2014 Klinteberg et al, 2016 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 5

Procedure Construct representation : Integral Operators + Potential Theory BVP & IE solution existence/uniqueness IE discretization Quadrature Rule Time progression A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 6

Constructing a representation - some theory Wikipedia µ r 2 u � r P + g δ ( x � x o ) = 0 u ( x ) = D [ Γ , q ]( x ) Double Layer Potential Source curve Hydrodynamic potential ✓ ∂ u i Erik Ivar Fredholm ◆ + ∂ u j Fluid stress σ ij = − P δ ij + µ ∂ x j ∂ x i Stokes PDE Laplace PDE u i = G ij g j P = µp j g j σ ij = µT ijk g j 1 p j ( x , y ) = 2 ˜ T ijk ( x , y ) = − 6 ˜ x i ˜ x j ˜ x k G ij ( x , y ) = δ ij r + x i x j x j G ( x , y ) = r 3 r 3 r 5 4 π r Stokeslet Stresslet K D j ( x , y ) = T ijk ( x , y ) n k ( y ) K D ( x , y ) = ˆ n. r y G ( x , y ) Stresslet Kernels to construct solution exist C Pozrikidis, 1992 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 7

Constructing a representation - some theory µ r 2 u � r P + g δ ( x � x o ) = 0 Laplace PDE a · ∂ G G ( x , y ) ⇒ ˆ ∂ x ( x , y ) Multipole ✓ ◆ ( S (ˆ n · r u ) � D u )( x ) = u ( x ) Z I ψ ∂ϕ ∂ n − ϕ∂ψ ( ψ ∆ ϕ − ϕ ∆ ψ ) dV = dS ∂ n U ∂ U Stokes PDE For u : Stokeslet, Stokeslet Doublet, Stokeslet Quadrupole u j ( x ) = − 1 Z G ij ( y , x ) f i ( y ) dS ( y )+ ∂σ 0 ∂ ∂σ ij 8 π µ ij ( u 0 i σ ij − u i σ 0 ij ) = u 0 Γ − u i i ∂ x j ∂ x j ∂ x j 1 Z T ijk ( y , x ) n k ( y ) u i ( y ) dS ( y ) 8 π Reciprocal Identity : Strong physical meaning Γ The Stokeslet and Stresslet provide a complete representation C Pozrikidis 2.3.10, 1992 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 8

Boundary Value Problems Typically interested in external problems : Dirichlet and Neumann Prescribed force Prescribed velocity www.livescience.com www.exposureguide.com Laplace PDE Null space for external Dirichlet : Fredholm Alternative from int. Neumann Stokes PDE Null space for external Dirichlet : Fredholm Alternative from int. Neumann :( Z u i ( x ) = T jik ( y , x ) n k ( y ) q j ( y ) dS ( y ) + V i ( x ) C Pozrikidis, Thm 4.7.1 1992 Γ V i ( x ) Compensate for deficiency in range Usually Prescribed (or) Single Layer op. ∴ At surface, Z u d i ( x ) + V i + ✏ ijk Ω j X 0 ,k = 4 ⇡ q i ( x ) + PV T jik ( y , x ) n k ( y ) q j ( y ) dS ( y ) + V i ( x ) Γ Deformation Translation Rotation Still (I+ Compact) : Well conditioned BVP straightforward? C Pozrikidis, 1992 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 9

BVPs : Mobility and Resistance V i ( x ) complicates the problem, leading to a dichotomy: Z u d i ( x ) + V i + ✏ ijk Ω j X 0 ,k = 4 ⇡ q i ( x ) + PV T jik ( y , x ) n k ( y ) q j ( y ) dS ( y ) + V i ( x ) Γ Deformation Translation Rotation Still (I+ Compact) : Well conditioned Resistance problem Mobility problem Linear Linear ( V , Ω ) ⇒ ( f , t ) ( f , t ) ⇒ ( V , Ω ) If prescribed motion, find forces If prescribed forces, find motion This leads to (additional) constraints in some cases: Bubble Rigid bodies Gazzola et al, 2013 C Pozrikidis, 1992 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 10

Discretization & Solution 1. Surface and function discretisation requirements as required 2. QBX to calculate matrix coefficients of discrete system to be solved (accelerated by precomputing/FMM) IE Discretisation Nystrom carries over: Approximate quadrature sufficient for off surface evaluation Now use QBX (with trapz/gauss) to calculate PV of DLP on the surface Nearly singular evaluations: Expansion may fail 3. Enforce BC at quadrature points to solve linear system A q = 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 C Pozrikidis, 1992 A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 11

Conclusions We have an IE method to solve the Stokes flow problem Similarities/ Differences to Laplace PDE 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. A fast solver for the Stokes PDE Tejaswin : CS598, Fall 2017 12