Dune. Patrick Leidenberger. 13. May Distributed and Unified Numerics Environment. 13. May 2009

Transcription

1 13. May / 25 Dune Distributed and Unified Numerics Environment Patrick Leidenberger 13. May 2009

2 13. May / 25 Table of contents 1 Introduction 2 Programming Techniques 3 Dune Modules 4 Examples 5 Parallel Example 6 Conclusion

3 13. May / 25 What is Dune toolbox for solving PDEs modular for grid-based methods supports FE, FV, FD, DG written in modern C++ slim and common interface to different libraries open source (GPL v2 + runtime exception)

4 13. May / 25 Design Principles Efficiency: low overhead Modularity: maintainability and software reuse Flexibility: separate data structures and algorithms

5 13. May / 25 Developers Group of Peter Bastian, IWR Heidelberg Peter Bastian Markus Blatt Christian Engwer Group of Dietmar Kröner, IAM Freiburg Andreas Dedner Robert Klöfkorn Martin Nolte Group of Ralf Kornhuber, Institut für Mathematik, FU Berlin Oliver Sander Group of Mario Ohlberger, Inst. Num. Angew. Math., Münster Mario Ohlberger

6 13. May / 25 Templates template <typename T> T max(t x, T y ) { i f ( x < y ) return y ; e l s e return x ; } cout << max ( 7, 4 2 ) ; cout << max<double >(7.42, ) ;

7 13. May / 25 Template Metaprogramming template<long B, unsigned long E> s t r u c t pow { s t a t i c long const v a l u e = B*pow<B, E 1 >:: v a l u e ; } ; template<long B> s t r u c t pow<b, 0> { s t a t i c long const v a l u e = 1 ; } ; s t d : : cout << pow<2, 8 >:: v a l u e << s t d : : e n d l ; } s t d : : cout << 256 << s t d : : e n d l ;

8 Dune Modules 13. May / 25

9 13. May / 25 Grid Interfaces AlbertaGrid finite element toolbox ALBERTA ( 1d/2d/3/ simplicial meshes, bisection refinement ALU3dGrid 2d/3d hexa/tetra meshes, adaptive and parallel SGrid structured grid, arbitrary dimension UGGrid 2d/3d unstructured, adaptive and parallel meshes YaspGrid structured grid, parallel and arbitrary dimension periodic boundary conditions...

10 13. May / 25 Grid Traversal Iterate over all elements on the leaf grid: typedef GridType : : Codim <0>:: L e a f I t e r a t o r E l e m e n t L e a f I t e r a t o r ; f o r { } ( E l e m e n t L e a f I t e r a t o r i t=g r i d. template l e a f b e g i n <0>(); i t!= g r i d. template l e a f e n d <0>(); ++i t ) s t d : : cout << element : << i t >type ( ) << s t d : : e n d l ;

11 13. May / 25 Quadrature: Integrate a function f over an element *it Dune : : GeometryType gt = i t >type ( ) ; const Dune : : QuadratureRule <double, dim>& r u l e = Dune : : QuadratureRules <double, dim >:: r u l e ( gt, p ) ; double r e s u l t = 0 ; f o r ( i n t i = 0 ; i < r u l e. s i z e ( ) ; ++i ) { F i e l d V e c t o r <double, dim> g l o b a l P o s i t i o n = i t >geometry ( ). g l o b a l ( r u l e [ i ]. p o s i t i o n ( ) ) ; double f V a l = f ( g l o b a l P o s i t i o n ) ; double weight = r u l e [ i ]. weight ( ) ; double d e t j a c = i t >geometry ( ). i n t e g r a t i o n E l e m e n t ( r u l e [ i ]. p o s i t i o n ( ) ) ; r e s u l t += f v a l * weight * d e t j a c ; }

12 13. May / 25 Iterative Solver Template Library ISTL vector matrix, dense and sparse (BCRS) containers are hierarchic solvers realized via operator concept

13 13. May / 25 Iterative Solver Template Library ISTL vector matrix, dense and sparse (BCRS) containers are hierarchic solvers realized via operator concept Preconditioners Jacobi method successive overrelaxation (SOR) symmetric SOR incomplete LU decomposition (ILU) algebraic multigrid method additive overlapping Schwarz

14 13. May / 25 Iterative Solver Template Library ISTL vector matrix, dense and sparse (BCRS) containers are hierarchic solvers realized via operator concept Preconditioners Jacobi method successive overrelaxation (SOR) symmetric SOR incomplete LU decomposition (ILU) algebraic multigrid method additive overlapping Schwarz Solvers apply preconditioner multiple time preconditioned gradient method preconditioned conjugate gradient method preconditioned biconjugate gradient stabilized method

15 13. May / 25 Laplace Equation with Dune PDE Lab ϕ = 0 fast implementation few code lines define boundary condition define value on boundary (define FE operator) write main program

16 13. May / 25 Laplace Equation with Dune PDE Lab ϕ = 0 fast implementation few code lines define boundary condition define value on boundary (define FE operator) write main program

17 13. May / 25 Laplace Equation: Implement Boundary Condition 1 template<typename GV> c l a s s B : p u b l i c Dune : : PDELab : : BoundaryGridFunctionBase <Dune : : PDELab : : B o u n d a r y G r i d F u n c t i o n T r a i t s <GV, int, 1, Dune : : F i e l d V e c t o r <int,1 >, B<GV> >

18 13. May / 25 Laplace Equation: Implement Boundary Condition 2 template<typename I > i n l i n e void e v a l u a t e ( const I& ig, const typename T r a i t s : : DomainType& x, typename T r a i t s : : RangeType& y ) const { Dune : : F i e l d V e c t o r <typename GV : : G r i d : : ctype,gv : : dimension > xg = i g. geometry ( ). g l o b a l ( x ) ; } y = 0 ; // no D i r i c h l e t // top : D i r i c h l e t i f ( xg [ 1 ] >= (1.0 1 e 6)) { y = 1 ; } // bottom : D i r i c h l e t i f ( xg [ 1 ] <= 1e 6) { y = 1 ; }

19 13. May / 25 Laplace Equation: Value on Boundary y = x [ 1 ] ; i f ( x [ 0 ] > 0. 5 ) { y *= 2 ; }

20 13. May / 25 Laplace Equation Main Program: Define Grid Dune : : F i e l d V e c t o r <double,2> L ( 1. 0 ) ; Dune : : F i e l d V e c t o r <int,2> N( 1 ) ; Dune : : F i e l d V e c t o r <bool,2> P( f a l s e ) ; Dune : : YaspGrid <2> g r i d ( L, N, P, 0 ) ; g r i d. g l o b a l R e f i n e ( 5 ) ; // get view typedef Dune : : YaspGrid <2>:: L e a f G r i d V i ew GV; const GV& gv=g r i d. l e a f V i e w ( ) ;

21 13. May / 25 Laplace Equation Main Program: Execution // make f i n i t e element map typedef GV : : G r i d : : c t y p e DF ; typedef Q1LocalFiniteElementMap <DF, double> FEM; FEM fem ; l a p l a c e d i r i c h l e t <GV,FEM, Q1Constraints > ( gv, fem, 3, l a p l a c e y a s p Q 1 2 d ) ;

22 Laplace Equation: Result on 2d Simplex Mesh 13. May / 25

23 Laplace Equation: Result on 2d Simplex Mesh 13. May / 25

24 Laplace Equation: Result on 2d Quadrilateral Mesh 13. May / 25

25 Laplace Equation: Result on 2d Quadrilateral Mesh 13. May / 25

26 Laplace Equation: Result on 3d Quadrilateral Mesh 13. May / 25

27 13. May / 25 2d Frequency Domain FE solve: ( ) 1 µ E z (x, y) ( ω 2 ɛ iωσ ) E z (x, y) = iωj ext,z higher order scalar base functions arbitrary meshes boundary conditions: PEC, ABC, waveguide port

28 13. May / 25 3d Time Domain FE solve: 1 µ E + σ E t 2 + ɛ 0 ɛ r t 2 E = t J ext vector base functions arbitrary meshes dispersion (Debye and Drude model) parallel (distributed memory)

29 3d Time Domain FE Work by B. Oswald and P. Leidenberger 13. May / 25

30 3d Time Domain FE Work by B. Oswald and P. Leidenberger 13. May / 25

31 Convective flow 13. May / 25

32 Convective flow Density-driven flow (by Peter Bastian) cell-centered finite volume scheme YaspGrid, cells, 384 processors 9000 timesteps, 3 days of computation time 13. May 2009 leidenberger@ifh.ee.ethz.ch 24 / 25

33 13. May / 25 Conclusion modern and efficient C++ framework different mesh backends suitable for large simulations suitable for teaching good developer community