Classes | |
class | ExcNegativeCriteria |
class | ExcInvalidParameterValue |
Functions | |
template<int dim, class Vector , int spacedim> | |
void | refine_and_coarsen_fixed_number (Triangulation< dim, spacedim > &tria, const Vector &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const unsigned int max_n_cells=std::numeric_limits< unsigned int >::max()) |
template<int dim, class Vector , int spacedim> | |
void | refine_and_coarsen_fixed_fraction (Triangulation< dim, spacedim > &tria, const Vector &criteria, const double top_fraction, const double bottom_fraction, const unsigned int max_n_cells=std::numeric_limits< unsigned int >::max()) |
template<int dim, class Vector , int spacedim> | |
void | refine_and_coarsen_optimize (Triangulation< dim, spacedim > &tria, const Vector &criteria) |
template<int dim, class Vector , int spacedim> | |
void | refine (Triangulation< dim, spacedim > &tria, const Vector &criteria, const double threshold) |
template<int dim, class Vector , int spacedim> | |
void | coarsen (Triangulation< dim, spacedim > &tria, const Vector &criteria, const double threshold) |
The functions in this namespace are in two classes. There are the auxiliary functions refine() and coarsen(). More important for users are the other functions, which implement refinement strategies, as being found in the literature on adaptive finite element methods. For mathematical discussion of these methods, consider works by Dörfler, Morin, Nochetto, Rannacher, Stevenson and many more.
void GridRefinement::refine_and_coarsen_fixed_number | ( | Triangulation< dim, spacedim > & | tria, | |
const Vector & | criteria, | |||
const double | top_fraction_of_cells, | |||
const double | bottom_fraction_of_cells, | |||
const unsigned int | max_n_cells = std::numeric_limits< unsigned int >::max() | |||
) | [inline] |
This function provides a refinement strategy with predictable growth of the mesh.
The function takes a vector of refinement criteria
and two values between zero and one denoting the fractions of cells to be refined and coarsened. It flags cells for further processing by Triangulation::execute_coarsening_and_refinement() according to the following greedy algorithm:
criteria
.
top_fraction_of_cells
times Triangulation::n_active_cells().
bottom_fraction_of_cells
times Triangulation::n_active_cells() from the end of the sorted list.
As an example, with no coarsening, setting top_fraction_of_cells
to 1/3 will result in approximately doubling the number of cells in two dimensions. The same effect in three dimensions is achieved by refining 1/7th of the cells. These values are good initial guesses, but should be adjusted depending on the singularity of approximated function.
The sorting of criteria is not done actually, since we only need the threshold values in order to call refine() and coarsen(). The order of cells with higher and of those with lower criteria is irrelevant. Getting this value is accomplished by the nth_element
function of the C++
standard library, which takes only linear time in the number of elements, rather than N log N
for sorting all values.
criteria:
the refinement criterion computed on each mesh cell. Entries may not be negative.top_fraction_of_cells
is the fraction of cells to be refined. If this number is zero, no cells will be refined. If it equals one, the result will be flagging for global refinement.bottom_fraction_of_cells
is the fraction of cells to be coarsened. If this number is zero, no cells will be coarsened.max_n_cells
can be used to specify a maximal number of cells. If this number is going to be exceeded upon refinement, then refinement and coarsening fractions are going to be adjusted in an attempt to reach the maximum number of cells. Be aware though that through proliferation of refinement due to Triangulation::MeshSmoothing, this number is only an indicator. The default value of this argument is to impose no limit on the number of cells. void GridRefinement::refine_and_coarsen_fixed_fraction | ( | Triangulation< dim, spacedim > & | tria, | |
const Vector & | criteria, | |||
const double | top_fraction, | |||
const double | bottom_fraction, | |||
const unsigned int | max_n_cells = std::numeric_limits< unsigned int >::max() | |||
) | [inline] |
This function provides a refinement strategy controlling the reduction of the error estimate.
Also known as the bulk criterion, this function computes the thresholds for refinement and coarsening such that the criteria
of cells getting flagged for refinement make up for a certain fraction of the total error. We explain its operation for refinement, coarsening works analogously.
Let cK be the criterion of cell K. Then the total error estimate is computed by the formula
If 0 < a < 1 is top_fraction
, then we refine the smallest subset of the Triangulation
such that
The algorithm is performed by the greedy algorithm described in refine_and_coarsen_fixed_number().
criteria
.
criteria:
the refinement criterion computed on each mesh cell. Entries may not be negative.top_fraction
is the fraction of the total estimate which should be refined. If this number is zero, no cells will be refined. If it equals one, the result will be flagging for global refinement.bottom_fraction
is the fraction of the estimate coarsened. If this number is zero, no cells will be coarsened.max_n_cells
can be used to specify a maximal number of cells. If this number is going to be exceeded upon refinement, then refinement and coarsening fractions are going to be adjusted in an attempt to reach the maximum number of cells. Be aware though that through proliferation of refinement due to Triangulation::MeshSmoothing, this number is only an indicator. The default value of this argument is to impose no limit on the number of cells. void GridRefinement::refine_and_coarsen_optimize | ( | Triangulation< dim, spacedim > & | tria, | |
const Vector & | criteria | |||
) | [inline] |
Refine the triangulation by flagging certain cells to reach an optimal grid: We try to minimize the error multiplied with the number of cells in the new grid. All cells with large error indicator are refined to generate an optimal grid in the above sense. We assume that the error in one cell is reduced to a quarter after refinement. The new triangulation has three new cells for every flagged cell.
Refer to the general doc of this class for more information.
void GridRefinement::refine | ( | Triangulation< dim, spacedim > & | tria, | |
const Vector & | criteria, | |||
const double | threshold | |||
) | [inline] |
Flag all mesh cells for which the value in criteria
exceeds threshold
for refinement.
The vector criteria
contains a nonnegative value for each active cell, ordered in the canonical order of of Triangulation::active_cell_iterator.
The cells are only flagged for refinement, they are not actually refined. To do so, you have to call Triangulation::execute_coarsening_and_refinement().
This function does not implement a refinement strategy, it is more a helper function for the actual strategies.
void GridRefinement::coarsen | ( | Triangulation< dim, spacedim > & | tria, | |
const Vector & | criteria, | |||
const double | threshold | |||
) | [inline] |
Flag all mesh cells for which the value in criteria
is less than threshold
for coarsening.
The vector criteria
contains a nonnegative value for each active cell, ordered in the canonical order of of Triangulation::active_cell_iterator.
The cells are only flagged for coarsening, they are not actually coarsened. To do so, you have to call Triangulation::execute_coarsening_and_refinement().
This function does not implement a refinement strategy, it is more a helper function for the actual strategies.