Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems Jan Mayer Hendryk Bockelmann, Vincent Heuveline Universität Karlsruhe GAMM Linear Algebra Workshop, Sept. 2008 Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Outline Introduction 1 Problem Formulation and Discretization 2 Multilevel ILU Preconditioners 3 Software 4 Numerical Results 5 Conclusion 6 Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Two CFD Benchmark Problems Streamlines Flow Field Lid Driven Cavity in 3D, Heat Driven Cavity, Rayleigh-Number 100 000 Reynolds-Number 100 Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Heat Driven Cavity: A Highly Challenging Problem Pseudo time-step approach: highly time consuming (several days/weeks computational time assuming HPC) Stationary approach: stiff problem (less than one hour computational time on a workstation) [Paillère et al. 2005] more then 30 institutes involved. Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Heat Driven Cavity: Low Mach Model Asymptotic model (pressure splitting) ∂ t ρ + ∇ · ( ρ v ) = 0 , ∂ t ( ρ v ) + [ ∇ · ρ vv ] + ∇ p + ∇ · τ = ρ g , c p ρ D t T − D t p − ∇ · ( κ ∇ T ) + ( τ : ∇ v ) = 0 . 2 p ( x , t ) = p th ( t ) + p hyd ( x , t ) µ ( ∇ v + ∇ v T ) + := µ ( ∇ · v ) I τ 3 p th « − 1 ρ = „Z 1 = P th m 0 dx RT RT Ω where τ , c p , κ , g , and µ represent the shear-stresstensor , the heat capacity , the heat conductivity , the volume forces and the shear vis- cosity , respectively. Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Lid Driven Cavity: Navier-Stokes Equations Model: incompressible Navier-Stokes equations − ν ∆ u + ( u · ∇ ) u + 1 v = 1 ρ ∇ p = 0 in Ω ∇ · u = 0 in Ω v = 0 v = 0 u = b on ∂ Ω v = 0 Here, we chose Ω = [ 0 , 1 ] d , u top = 1 and u | ∂ Ω = 0 otherwise. ν = 0 . 01, Re = ρ u top L = 100 guarantees a stationary solution. ν Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Variational Formulation: Stokes Model − ∆ u + ∇ p = f in Ω , ∇ · u = 0 in Ω , (1) u = 0 on ∂ Ω . 0 (Ω)) d × L 2 Find ( u , p ) ∈ ( H 1 0 (Ω) such that ∀ ϕ ∈ H 1 a ( u , ϕ ) − b ( p , ϕ ) = ( f , ϕ ) 0 (Ω) , ∀ q ∈ L 2 b ( q , u ) = 0 0 (Ω) , where a ( u , ϕ ) = ( ∇ u , ∇ ϕ ) , b ( q , ϕ ) = ( q , ∇ · ϕ ) . Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion FEM Discretization: Conformal Stable Elements FEM: Find ( u N , p N ) ∈ V N × M N such that a ( u N , ϕ N ) − b ( p N , ϕ N ) = ( f , ϕ N ) ∀ ϕ N ∈ V N , b ( q N , u N ) = 0 ∀ q N ∈ M N . (Discrete) Inf-Sup condition (LBB-condition): ( ∇ · v N , q N ) ≥ β N > 0 , inf sup | u N | 1 , Ω || q N || 0 , Ω q N ∈ M N ∩ L 2 0 (Ω) v N ∈ V N ∩ H 1 0 (Ω) where β N = β N ( V N , M N ) . Taylor-Hood Element ( Q 2 , Q 1 ) no pressure stabilization needed! Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Discretization: Algebraic Formulation Linearization of Navier-Stokes equations by a (damped) 1 Newton iteration yields linear saddle point problem. Discretization by Galerkin FE method yields a linear 2 system with coefficient matrix � N � B A = B T 0 Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Overview of the Solution Process Algorithm choose initial x = ( u , p ) and tolerance ε , set k = 0 compute residual r of nonlinear system while ( � r � > ε and k < MaxIter) compute Jacobian (linearization) A solve A ∆ x = r with preconditioned GMRES update solution x = x + ω k ∆ x compute residual r of non-linear system set k = k + 1 end while Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Preconditioning the Linear System The Algebraic Multilevel Incomplete Factorization: Practical View Algorithm Preprocess matrix 1 U (permute/scale rows and columns to make more k suitable for incomplete L LU-factorization). Factor until absolute value of 2 S pivot less than 0.01. Compute approximate Schur 3 k complement. Goto 1, proceed recursively. 4 Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Preconditioning the Linear System The Algebraic Multilevel Incomplete Factorization: Summary Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Preprocessing the Linear System Options Preprocessing: permute/scale row and columns to make A more suitable for incomplete factorization, e.g. Normalize columns and then rows of A 1 → lid driven cavity I-matrix, i.e. such that | a ij | ≤ 1 and | a ii | = 1 for all i , j 2 I-matrix + symmetric permutation (preserves I-matrices) 3 → heat driven cavity Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion Symmetric Permutations for I-Matrices Let A be an I-matrix of dimension n . One option is: Algorithm Assign a weight 1 w k = nz ( A : , k ) · || A : , k || 1 + nz ( A k , : ) · || A k , : || 1 for each index k = 1 , . . . , n . Reorder rows and columns of A by increasing weights. 2 Note: low weights indicate sparse rows and columns having good diagonal dominance Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion HiFlow HiFlow is an FE package featuring CFD (incompressible Navier-Stokes, Low-Mach flows, heat convection) reactive flows eigenvalue lab for stability analysis conformal hp-FEM a posteriori error estimation moving boundaries More information under www.hiflow.de Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion ILU++ ILU++ is a software package for solving large sparse linear systems. It provides several: preprocessing techniques multilevel incomplete LU-factorisations dropping techniques iterative solvers ILU++ is fully templated, object-oriented self-contained C++ code. Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
Introduction Problem Formulation and Discretization Multilevel ILU Preconditioners Software Numerical Results Conclusion ILU++ in HiFlow #include <vector> #include "iluplusplus_interface.h" void GMRES( const iluplusplus::multilevel_preconditioner& Pr, . . . ) { std::vector<double> v; . . . Pr.apply_preconditioner(v); . . . } Jan Mayer Universität Karlsruhe Algebraic Multilevel ILU-Based Preconditioning for Stationary Flow Problems GAMM LA Workshop 2008
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