Dune-Fem School DUNE-FEM School Adaptive solution of Poisson equation. Outline. Notes. Notes. Notes. Local Grid Adaptivitiy in DUNE-FEM

Transcription

1 Dune-Fem School 2016 DUNE-FEM School 2016 Local Grid Adaptivitiy in DUNE-FEM laus-justus Heine, Robert Klöfkorn, Andreas Dedner, Martin Nolte, hristoph Gersbacher Institute for Applied Analysis and Numerical Simulation University of Stuttgart September 26-30, 2016 Stuttgart, Germany International Research Institute of Stavanger Adaptive solution of Poisson equation Figure: Adaptive P 1 Finite-Element solution of the Poisson Equation. DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 Outline 1 2 Adaptive Simulations Using DUNE-FEM 3 DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20

2 Marking Entities for Refinement / oarsening For marking entities the Grid class provides two methods: bool mark ( int rc, const odim< 0 >::Entity &e ); int getmark ( const odim< 0 >::Entity &e ) const; The method mark sets the refinement count rc for an entity e. It returns true if the counter has been updated successfully. The method getmark returns the current refinement count for an entity. Interpretation of the refinement count: rc > 0 the entity shall be refined (at least) rc times. rc = 0 the entity need not be refined. rc < 0 the entity may be coarsened (at most) rc times. DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 R R grid.mark(1,entity); grid.mark(1,entity); DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 R R R grid.mark(1,entity); (for a grid with conform bisection) DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20

3 R R R R grid.mark(1,entity); grid.mark(-1,entity); grid.mark(-1,entity); grid.mark(-1,entity); (for a grid with non-conform bisection) DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 R R R R Note: the result of a marking sequence depends on the grid implementation: quatering instead of bisection hanging nodes instead of conforming... DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 The Grid Modification ycle For grid modification three methods on the Grid have to be called: bool preadapt (); bool adapt (); void postadapt (); The grid modification in initiated by preadapt. It returns true if entities might vanish during the adaptation. The method adapt actually modifies the grid. It returns true, if new entities were created by the adaptation. The grid modification is finalized by calling postadapt. Warning: The method adapt will rebuild the index sets. DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20

4 The Grid Modification ycle A basic adaptation cycle consists of the following steps: 1 mark (leaf) entities for refinement or coarsening, 2 call preadapt, 3 if entities might vanish, (hierarchally) restrict the data of (possibly) vanishing entities, 4 call adapt, 5 if new entities have been create, (hierarchically) prolong the data to newly generated entities, 6 call postadapt. DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 odim 0: Extended Entity Interface Entities of codimension 0 (elements) provide additional methods to traverse the grid hierarchy: class Entity { //... public: bool hasfather () const; Entity father () const; LocalGeometry geometryinfather () const; bool isleaf () const; HierarchicIterator hbegin ( int maxlevel ) const; HierarchicIterator hend ( int maxlevel ) const; //... }; DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 Adaptive Simulations Using DUNE-FEM Outline 1 2 Adaptive Simulations Using DUNE-FEM 3 DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20

5 Adaptive Simulations Using DUNE-FEM AdaptationManager in DUNE-FEM Adaptation with prolongation and restriction of data is possible through the DUNE-GRID interface but quite involved. DUNE-FEM provides a singleton class typedef Dune::Fem::RestrictProlongDefault< DiscreteFunctionType > RestrictionProlongationType; typedef Dune::Fem::AdaptationManager< GridType, RestrictionProlongationType > AdaptationManagerType; restrictprolong_( solution_ ); adaptationmanager_( gridpart_.grid(), restrictprolong_ ); adaptationmanager_.adapt(); The call to the AdaptationManager will refine/coarsen the grid according to the marks set and will prolong and restrict the data using a provided operation. In addition, the dof storages for all instanciated discrete functions are set to the correct size. Furthermore, load balancing is carried out in parallel. DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 Outline 1 2 Adaptive Simulations Using DUNE-FEM 3 DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 A Posteriori Error Estimator For Poisson s problem we have implemented the following residual error estimator: u u h 2 H 1 (Ω) e G η 2 e. Local error estimator η e : ηe 2 = he f 2 + u h 2 L 2 (e) + h e [ u h ] n e 2 L 2 (i) i I e The jump is computed by [τ] := (τ inside τ outside ). The local mesh width h e can, e.g., be computed as h e = e 1 dim G. DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20

6 Refinement Strategy For a given tolerance TOL we set, for example, TOL 2 e = TOL 2 / G. 1 Solve: ompute solution of problem 2 Estimate: e G compute η 2 e 3 Mark: Refine: if η 2 e > TOL 2 e oarsen: if θη 2 e < TOL 2 e, θ (0, 1) 4 Refine: refine mesh due to marking and restrict and prolong solution DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 Intersection Iterators For a given entity e E 0 the intersections i I e can be accessed via an iterator: IntersectionIteratorType ibegin( const typename odim<0>::entitytype &entity ) const; IntersectionIteratorType iend( const typename odim<0>::entitytype &entity ) const; intersection outside n inside Again, the usage of the intersection iterator mimicks the STL: const IntersectionIteratorType iend = gridpart.ibegin(entity); for(intersectioniteratortype iit = gridpart.iend(entity); iit!= iend; ++iit) { typedef typename IntersectionIteratorType::Intersection IntersecionType; const IntersectionType &intersecion = *iit; } DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 Intersection Interface Methods An intersection is either an inner or a boundary intersection: // returns true, if inside elemen has a neighbor bool neighbor() const; // returns true, if inside element is on boundary bool boundary() const; Inner intersections can return a pointer to the neighboring element: // return pointer to neighbor element, assert neighbor() is true! Entity outside() const; Geometric realizations: // return geometry of this intersection in global coordinates Geometry geometry() const; // return geometry in local coordinates of the inside element LocalGeometry geometryininside () const; // return geometry in local coordinates of the outside element LocalGeometry geometryinoutside () const; // assert neighbor() is true! DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20

7 Intersection Interface Methods (ont.) The intersection provides outer normals to the inside element: // return outer normal in local coordinate x Globaloordinate outernormal (const Localoordinate& local) const; // as outernormal(), scaled with geometry().integrationelement(local) Globaloordinate integrationouternormal (const Localoordinate& local) const; // return unit outer normal in center Globaloordinate centerunitouternormal() const; DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20 TODO: 06-afem-1 TODO In estimator.hh implement the local error contribution for the jump of the gradient i I e [ u h ] n e 2 L 2 (i). TODO Test the different problems and adaptation strategies TODO Add appropriate model parameter to the estimator TODO Add the mass term and non-constant diffusion DUNE Group (IANS & IRIS) DUNE-FEM School 2016 September 26-30, / 20