From d42c562a8ba6dfc731a0e43bf95adc9a8b42a366 Mon Sep 17 00:00:00 2001 From: Martin Kronbichler Date: Mon, 4 Dec 2017 17:16:39 +0100 Subject: [PATCH] Significantly extend module on the manifold. The new section describes the mesh smoothing algorithm in deal.II --- doc/doxygen/headers/manifold.h | 159 +++++++++++++++--- .../hyperball-mesh-smoothing-interpolate.png | Bin 0 -> 82710 bytes .../hyperball-mesh-smoothing-laplace.png | Bin 0 -> 82387 bytes doc/doxygen/images/hypershell-all-3.png | Bin 15618 -> 7895 bytes .../images/hypershell-boundary-only-3-old.png | Bin 0 -> 17431 bytes .../images/hypershell-boundary-only-3.png | Bin 17431 -> 11356 bytes .../images/hypershell-boundary-only.png | Bin 24795 -> 16709 bytes .../images/hypershell-boundary-thin-3.png | Bin 0 -> 10422 bytes 8 files changed, 138 insertions(+), 21 deletions(-) create mode 100644 doc/doxygen/images/hyperball-mesh-smoothing-interpolate.png create mode 100644 doc/doxygen/images/hyperball-mesh-smoothing-laplace.png create mode 100644 doc/doxygen/images/hypershell-boundary-only-3-old.png create mode 100644 doc/doxygen/images/hypershell-boundary-thin-3.png diff --git a/doc/doxygen/headers/manifold.h b/doc/doxygen/headers/manifold.h index ec0970240a..6bcd2b827c 100644 --- a/doc/doxygen/headers/manifold.h +++ b/doc/doxygen/headers/manifold.h @@ -133,7 +133,8 @@ * GridGenerator::hyper_shell (triangulation, * center, inner_radius, outer_radius, * 10); - * const HyperShellBoundary<2> boundary_description(center); + * const SphericalManifold<2> boundary_description(center); + * triangulation.set_all_manifold_ids_on_boundary(0); * triangulation.set_boundary (0, boundary_description); * * triangulation.refine_global (3); @@ -144,12 +145,15 @@ * * The mesh looks better in that it faithfully reproduces the circular inner * and outer boundaries of the domain. However, it is still possible to - * identify the original 10 cells by the kinks in the tangential lines. They - * result from the fact that every time a cell is refined, new vertices on - * interior lines are just placed into the middle of the existing line (the - * boundary lines are handled differently because we have attached boundary - * objects). In other words, they end up in places that may be in the geometric - * middle of a straight line, but not on a circle around the center. + * identify 20 kinks in the tangential lines. They result from the fact that + * every time a cell is refined, new vertices on interior lines are just + * placed into the middle of the existing line (the boundary lines are handled + * differently because we have attached a manifold object). In the first + * refinement with 10 cells, we got improved points because both outer + * boundaries have provided a curved description according to the description + * on blending different manifolds below. In other words, the new points after + * the first refinement end up in places that may be in the geometric middle + * of a straight line, but not on a circle around the center. * * This can be remedied by assigning a manifold description not only to * the lines along the boundary, but also to the radial lines and cells (which, @@ -165,6 +169,7 @@ * center, inner_radius, outer_radius, * 10); * const SphericalManifold<2> boundary_description(center); + * triangulation.set_all_manifold_ids_on_boundary(0); * triangulation.set_manifold (0, boundary_description); * * Triangulation<2>::active_cell_iterator @@ -202,26 +207,50 @@ * outer_radius = 1.0; * GridGenerator::hyper_shell (triangulation, * center, inner_radius, outer_radius, - * 4); // four circumferential cells + * 3); // three circumferential cells * const HyperShellBoundary<2> boundary_description(center); - * triangulation.set_boundary (0, boundary_description); + * triangulation.set_all_manifold_ids_on_boundary(0); + * triangulation.set_manifold (0, boundary_description); * * triangulation.refine_global (3); * @endcode * - * @image html hypershell-boundary-only-4.png "" + * @image html hypershell-boundary-only-3.png "" * - * Here, we create only four circumferential cells in the beginning, - * and refining them leads to the mesh shown. Clearly, here we have - * cells with bad aspect ratios. + * Here, we create only three circumferential cells in the beginning, and + * refining them leads to the mesh shown. Clearly, here we have cells with bad + * aspect ratios, despite the first refinement that puts the new point into + * the middle. * - * If we drive this further and start with a coarse mesh of only - * three cells (which may be inappropriate here, since we know that it - * is not sufficient, but may also be impossible to avoid for complex - * geometries generated in mesh generators), then we obtain the following - * mesh: + * If we drive this further and start with a coarse mesh of a much thinner rim + * between the radii 0.8 and 1.0 and still start with only three cells (which + * may be inappropriate here, since we know that it is not sufficient, but may + * also be impossible to avoid for complex geometries generated in mesh + * generators), we observe the following: * - * @image html hypershell-boundary-only-3.png "" + * @code + * Triangulation<2> triangulation; + * + * const Point<2> center (1,0); + * const double inner_radius = 0.8, + * outer_radius = 1.0; + * GridGenerator::hyper_shell (triangulation, + * center, inner_radius, outer_radius, + * 3); // three circumferential cells + * const SphericalManifold<2> boundary_description(center); + * triangulation.set_all_manifold_ids(0); + * triangulation.set_manifold (0, boundary_description); + * + * Triangulation<2>::active_cell_iterator + * cell = triangulation.begin_active(), + * endc = triangulation.end(); + * for (; cell!=endc; ++cell) + * cell->set_all_manifold_ids (0); + * + * triangulation.refine_global (3); + * @endcode + * + * @image html hypershell-boundary-thin-3.png "" * * This mesh neither has the correct geometry after refinement, nor do * all cells have positive area as is necessary for the finite element @@ -233,12 +262,13 @@ * Triangulation<2> triangulation; * * const Point<2> center (1,0); - * const double inner_radius = 0.5, + * const double inner_radius = 0.8, * outer_radius = 1.0; * GridGenerator::hyper_shell (triangulation, * center, inner_radius, outer_radius, * 3); // three circumferential cells * const SphericalManifold<2> boundary_description(center); + * triangulation.set_all_manifold_ids(0); * triangulation.set_manifold (0, boundary_description); * * Triangulation<2>::active_cell_iterator @@ -269,6 +299,93 @@ * * @see @ref GlossManifoldIndicator "Glossary entry on manifold indicators" * + *

Computing the weights for combining different manifold descriptions

+ * + * In a realistic application, it happens regularly that different manifold + * descriptions need to be combined. The simplest case is when a curved + * description is only available for the boundary but not for the interior of + * the computational domain. The manifold description for a ball also falls + * into this category, as it needs to combine a spherical manifold at the + * circular part with a straight-sided description in the center of the domain + * where the spherical manifold is not valid. + * + * In general, the process of blending in deal.II is achieved by the so-called + * transfinite interpolation. Its formula 2D is, for example, described on + * + * Wikipedia. Given a point $(u,v)$ on the chart, the image of this point + * in real space is given by + * @f{align*}{ + * \mathbf S(u,v) &= (1-v)\mathbf c_0(u)+v \mathbf c_1(u) + (1-u)\mathbf c_2(v) + u \mathbf c_3(v) \\ + * &\quad - \left[(1-u)(1-v) \mathbf x_0 + u(1-v) \mathbf x_1 + (1-u)v \mathbf x_2 + uv \mathbf x_3 \right] + * @f} + * where $\bf x_0, \bf x_1, \bf x_2, \bf x_3$ denote the four bounding vertices + * bounding the image space and $\bf c_0, \bf c_1, \bf c_2, \bf c_3$ are the + * four curves describing the lines of the cell. + * + * If we want to find the center of the cell according to the manifold (that + * is also used when the grid is refined), we want to evaluate this formula in + * the point $(u,v) = (0.5, 0.5)$. In that case, $\mathbf c_2(0.5)$ is the + * position of the midpoint of the lower face (indexed by 2 in deal.II's + * ordering) that is derived from its own manifold, $\mathbf c_1(0.5)$ is the + * position of the midpoint of the upper face (indexed by 3 in deal.II), + * $\mathbf c_2(0.5)$ is the midpoint of the face on the left (indexed by 0), + * and $\mathbf c_3(0.5)$ is the midpoint of the right face. In this formula, + * the weights equate to $\frac{\displaystyle 1}{\displaystyle 2}$ for the + * four midpoints in the faces and to $-\frac{\displaystyle 1}{\displaystyle + * 4}$ for the four vertices. These weights look weird at first sight because + * the vertices enter with negative weight but the mechanism does what we + * want: In case of a cell with curved description on two opposite faces but + * straight lines on the other two faces, the negative weights of -1/4 in the + * vertices balance with the center of the two straight lines in radial + * direction that get weight 1/2. Thus, the average is taken over the two + * center points in curved direction, exactly placing the new point in the + * middle. + * + * In three spatial dimensions, the weights are +1/2 for the face midpoints, + * -1/4 for the line mid points, and +1/8 for the vertices, again balancing + * the different entities. In case all the surrounding of a cell is straight, + * the formula again reduces to weight 1/8 on the eight vertices. + * + * In the MappingQGeneric class, a generalization of this concept to the + * support points of the polynomial grid representation, the nodes of the + * Gauss-Lobatto quadrature, is implemented by evaluating the boundary curves + * in the respective points $(u_i,v_i)$ of the Gauss-Lobatto points. The + * weights have been verified to yield optimal convergence rates $\mathcal + * O(h^{k+1})$ also for very high polynomial degrees, say $k=10$. + * + * In literature, also other boundary descriptions are used. Indeed, before + * version 9.0 deal.II used something called Laplace smoothing where the + * weights that are applied to the nodes on the circumference to get the + * position of the interior nodes are determined by solving a Laplace equation + * on the unit element. However, this did lead to boundary layers close to the + * curved description, i.e., singularities in the higher derivatives of the + * mapping from unit to real cell. + * + * For example, the above case with only 3 circumferential cells leads to the + * following mesh with Laplace smoothing rather than the interpolation from + * the boundary (which may be inappropriate here, since we know that it is not + * sufficient, but may also be impossible to avoid for complex geometries + * generated in mesh generators): + * + * @image html hypershell-boundary-only-3-old.png "" + * + * To use a more practical example, consider the refinement of a ball with a + * SphericalManifold attached to the spherical surface. The Laplace smoothing + * gives the following rather poor mesh: + * + * @image html hyperball-mesh-smoothing-laplace.png "" + * + * If we, instead, use the weights derived from transfinite interpolation, the + * situation is considerably improved: + * + * @image html hyperball-mesh-smoothing-interpolate.png "" + * + * Of course, one could get even better meshes by applying the + * TransfiniteInterpolationManifold to the whole domain except the boundary + * where SphericalManifold is attached, as shown by the figures in that class, + * but in principle, the mesh smoothing implemented in deal.II is as good as + * it can get from a boundary description alone. + * * @ingroup grid - * @author Luca Heltai, 2013 + * @author Luca Heltai, 2013, Martin Kronbichler, 2017 */ diff --git a/doc/doxygen/images/hyperball-mesh-smoothing-interpolate.png b/doc/doxygen/images/hyperball-mesh-smoothing-interpolate.png new file mode 100644 index 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