# Poeplau ECLOUD07

Published on January 3, 2008

Author: WoodRock

Source: authorstream.com

3D Space Charge Routines: Multigrid and FFT Compared:  3D Space Charge Routines: Multigrid and FFT Compared Gisela Pöplau ECLOUD’07, Daegu, April 12, 2007 Overview:  Overview Algorithms for 3D space charge calculations Properties of FFT and iterative Poisson solvers Optimal iterative Poisson solver: multigrid technique Numerical investigations: ASTRA: Tracking example with FFT and multigrid Numerical studies of cylindrical shaped bunches Multigrid Poisson solver in tracking codes Computation of Space-Charge Fields:  Computation of Space-Charge Fields Particle-Mesh Method Solve Poisson´s equation Good accuracy for „smooth“ particle distributions Fast with best solver - O(N): multigrid methods Particle-Particle Method No mesh required Straightforward summation O(N2) Due to Hockney, Eastwood Particle-Particle Particle-Mesh Method Particle-Mesh Method:  Particle-Mesh Method FFT Poisson solvers Boundary conditions Free space boundary Periodic boundary Rectangular PEC box (with Fast Cosine Transformation) Iterative Poisson solvers Boundary conditions Free space boundary Perfect conducting rectangular box PEC beam pipe pipe with elliptical cross section Direct solution Finite difference discretization G: Green‘s function, PEC: Perfect Electric Conductor Solvers & Properties:  Solvers & Properties FFT Poisson solvers FFT FFT based algorithms (FCT) Step size: equidistant Numerical effort: O(M logN) Number of grid points: N=2t Iterative Poisson solvers Multigrid (MG) Multigrid Preconditioned Conjugate Gradients (MG-CG) Jacobi Preconditioned CG Successive overrelaxation (SOR) BiCG, BiCGSTAB Step size: non-equidistant Numerical effort: O(M) (per iteration step) Convergence: MG: O(1) CG,SOR: O(N2) with h=1/N Total number of grid points Gauss-Seidel Relaxtion and SOR(w):  Gauss-Seidel relaxation is part of multigrid Simple implementation Convergence acceptable for low number of mesh lines Convergence: w=1, Gauss-Seidel: 1-p2h2 w=wopt: 1-ph, problem: find wopt Slows down for meshes with large aspect ratios (will be demonstrated on a later slide) Gauss-Seidel Relaxtion and SOR(w) Multigrid & MG-PCG:  Multigrid & MG-PCG Convergence O(1) on non-equidistant grids Convergence O(1) on grids with high aspect ratios Implementation is more complicated Implementation has always to be adapted to the problem Multigrid Technique:  Multigrid Technique Fine grid interp.: eH->eh cgc: vhnew=vh+eh interpolation + coarse grid correction History: 1961 R. P. Fedorenko - first MG 1972 A. Brandt - adaptive grid refinement 1976 W. Hackbusch - First MG program - New proofs for convergence 1985 W. Hackbusch - First monograph 1989 U. Langer - MG-PCG interp.: eH->eh cgc: vhnew=vh+eh Coarsening Strategy:  Coarsening Strategy Developed for space charge calculations The coarsening strategy is essential for a good convergence Objective of the coarsening is a sequence of coarser grids with a mesh spacing of descending aspect ratio MG-PCG stabelizes convergence (Langer et al., 1989) hy =10hx Stretched grids occur often in space charge simulations! after 10 MG steps (0.3%) Why Multigrid?:  Why Multigrid? Other Poisson solvers are much easier to implement But, they slow down considerably on non-equidistant meshes equidistant mesh non-equidistant mesh bunch with Gaussian distribution in a sphere Discretization of a spherical bunch Simulations with ASTRA:  Simulations with ASTRA EPAC 2006 Gaussian particle distribution: sx=sy=0.75 mm , sz=1.0 mm 10,000 macro particles charge: -1 nC energy: 2 MeV tracking distance: 3 m quadrupol at z=1.2 m number of steps (Poisson solver): Nz=32 FFT MG Numerical Investigations:  Numerical Investigations Parameters for simulations Cylindrical bunches Uniform partical distribution 20,000 macro particles Charge -1 nC Aspect ratio of the bunch sx/sz FFT MG ICAP 2006 Bunch with Aspect Ratio 1:  Bunch with Aspect Ratio 1 FFT Multigrid Bunch size # of grid points: (32)3 = 32,768 Field at the edges of the bunch is not approximated correctly # of grid points: (28)3 = 21,952 Bunch with Aspect Ratio 1:  Bunch with Aspect Ratio 1 FFT Multigrid # of grid points: (64)3 = 262,144 Field at the edges of the bunch is better approximated # of grid points: (60)3 =216,000 Resolution too high: more particles required for smoother fields Short Bunch with Aspect Ratio 10:  FFT Short Bunch with Aspect Ratio 10 Multigrid # of grid points: (32)3 = 32,768 Incorrect approximation of the field at the edges of the bunch # of grid points: (28)3 = 21,952 Long Bunch with Aspect Ratio 0.01:  Long Bunch with Aspect Ratio 0.01 FFT Multigrid # of grid points: (32)3 = 32,768 Incorrect approximation of the field at the edges of the bunch # of grid points: (28)3 = 21,952 Fast Poisson Solvers in Tracking Codes:  Fast Poisson Solvers in Tracking Codes Software package MOEVE 2.0 Tracking with MOEVE-Poisson solvers (Aleksandar Markovik, Rostock) Part of ASTRA, test phase (Klaus Flöttmann, DESY) Part of the tracking code GPT 2.7 (General Particle Tracer, Pulsar Physics) Initial and final projections of the charge density of an expanding `pancake´ bunch into the (x,z)-plane. One million particles are used on a 65x65x65 mesh. GPT simulation Simultions with GPT:  Simultions with GPT COMPUMAG 2003 Graphics: Pulsar Physics DC/RF gun @ TU Eindhoven Space charge simulations with 3D Multigrid Poisson Solver MOEVE 1.0 2D: 1,000 particles 3D: 100,000 particles Summary:  Summary Poisson solver based on multigrid Particle mesh method For non-equidistant tensor product meshes MG Poisson solvers are more flexible than FFT Poisson solvers (boundary, discretization) MG Poisson solvers enable a better approximation Software package MOEVE 2.0 Part of the tracking codes ASTRA and GPT 2.7

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