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

Transcription

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

2 Stokes Equations? u i =0 Navier Stokes Input Output Initial Conditions (t) ~u(~, t) Boundary Conditions () Resistance µ Solution Algorithm:, t p(~, t) A fast solver for the Stokes PDE 2

3 Stokes Equations? Continuity ui ) Stokes PDE : Especially useful in small 2 ui + µ 2 No time dependence - completely reversible Linear PDEs :) Integral Equations u : A system of linear PDEs :( P : A coupled system of PDES : (( µ UNM Physics and Astronomy, Sped up 10 A fast solver for the Stokes PDE 3 Boundary Conditions () Solution Algorithm ~u(~) P (~)

4 Present Necessity? Time Goal Stokes Flow Xiaotian et al, Preprint Viscous flow Gazzola et al, 2012 Gazzola et al, Integral Equations & Fast Algorithms A fast solver for the Stokes PDE 4

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

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

7 Constructing a representation - some theory Wikipedia µr 2 u rp + g ( o )=0 u() =D[, q]() Erik Ivar Fredholm Double Layer Potential Source curve Hydrodynamic potential Fluid stress ij = P ij j i Laplace PDE G(, y) = 1 4 r G ij (, y) = Stokeslet u i = G ij g j ij r + i j r 3 Stokes PDE P = µp j g j ij = µt ijk g j p j (, y) =2 j r 3 T ijk (, y) = 6 i j k r 5 Stresslet K D (, y) =ˆn.r y G(, y) K D j (, y) =T ijk (, y)n k (y) Kernels to construct solution eist Stresslet C Pozrikidis, 1992 A fast solver for the Stokes PDE 7

8 Constructing a representation - some theory µr 2 u rp + g ( o )=0 Laplace PDE Multipole G(, y) (, y) Z U ( ' ' ) dv ds (S(ˆn ru) Du)() =u() Stokes PDE For u : Stokeslet, Stokeslet Doublet, j (u 0 i ij u i 0 ij )=u 0 j u 0 j Reciprocal Identity : Strong physical meaning u j () = 1 8 Z 1 G ij (y, )f i (y)ds(y)+ 8 µ Z T ijk (y, )n k (y)u i (y)ds(y) The Stokeslet and Stresslet provide a complete representation C Pozrikidis , 1992 A fast solver for the Stokes PDE 8

9 Boundary Value Problems Typically interested in eternal problems : Dirichlet and Neumann Prescribed velocity Prescribed force Laplace PDE Null space for eternal Dirichlet : Fredholm Alternative from int. Neumann Stokes PDE Null space for eternal Dirichlet : Fredholm Alternative from int. Neumann :( C Pozrikidis, Thm V i () At surface, Z u i () = T jik (y, )n k (y)q j (y)ds(y)+v i () Compensate for deficiency in range Usually Prescribed (or) Single Layer op. Z u d i ()+V i + ijk j X 0,k =4 q i ()+PV T jik (y, )n k (y)q j (y)ds(y)+v i () Deformation Translation Rotation Still (I+ Compact) : Well conditioned BVP straightforward? C Pozrikidis, 1992 A fast solver for the Stokes PDE 9

10 V i () BVPs : Mobility and Resistance complicates the problem, leading to a dichotomy: Z u d i ()+V i + ijk j X 0,k =4 q i ()+PV T jik (y, )n k (y)q j (y)ds(y)+v i () Deformation Translation Rotation Still (I+ Compact) : Well conditioned Resistance problem Mobility problem Linear Linear (V, ) ) (f, t) (f, t) ) (V, ) 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 10

11 Discretization & Solution 1. Surface and function discretisation requirements as required 2. QBX to calculate matri coefficients of discrete system to be solved (accelerated by precomputing/fmm) IE Discretisation Nystrom carries over: Approimate quadrature sufficient for off surface evaluation Now use QBX (with trapz/gauss) to calculate PV of DLP on the surface Nearly singular evaluations: Epansion may fail 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 σ 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 11

12 Conclusions We have an IE method to solve the Stokes flow problem Similarities/ Differences to Laplace PDE Optimal (or) near optimal time Numerical eperiments 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): Pozrikidis, Constantine. Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, 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, af Klinteberg, Ludvig, and Anna-Karin Tornberg. "A fast integral equation method for solid particles in viscous flow using quadrature by epansion." 5. Journal of Computational Physics 326 (2016): Klöckner, Andreas, et al. "Quadrature by epansion: A new method for the evaluation of layer potentials." Journal of Computational Physics 252 (2013): A fast solver for the Stokes PDE 12