Creating good meshes, and in particular making them fit the geometry you
want, is a complex topic in itself. You can find much more on this in
step-49, step-53, and step-54, among other tutorial programs that cover
-the issue. Information on curved domains can also be found in the
+the issue. step-65 shows another, less manual way to achieve a mesh
+well fit to the problem here.
+Information on curved domains can also be found in the
documentation module on @ref manifold "Manifold descriptions".
Why does it make sense to choose a mesh that tracks the internal
At this point, one might assume that curved volume descriptions are the way to
go. This is generally not wrong, though it is sometimes not so easy to
-describe how exactly this should work. Here are couple examples:
+describe how exactly this should work. Here are a couple of examples:
- Imagine that the mesh above had actually been a disk, not just a ring.
In that case the polar manifold degenerates at the origin and
solution, one calls that a super- or sub-parametric geometry representation,
respectively. In deal.II, the standard class for polynomial representation is
MappingQGeneric. If, for example, this class is used with polynomial degree $4$ in 3D, a
-total of 125 ($=(4+1)^3$) points are needed for the tri-cubic
+total of 125 ($=(4+1)^3$) points are needed for the
interpolation. Among these points, 8 are the mesh vertices and already
available from the mesh, but the other 117 need to be provided by the
manifold. In case the transfinite interpolation manifold is used, we can
* which is used in all operations of MappingQGeneric. The information of the
* mapping is pre-computed by the MappingQCache::initialize() function.
*
+ * The use of this class is discussed extensively in step-65.
+ *
* @author Martin Kronbichler, 2019
*/
template <int dim, int spacedim = dim>
* Manifold to the cell containing the center. It is advisable to combine this
* class with TransfiniteInterpolationManifold to ensure a smooth transition
* from a curved shape to the straight coordinate system in the center of the
- * ball.
+ * ball. (See also the extensive discussion in step-65.)
*
* @ingroup manifold
*
* assumed to be given by another manifold (e.g. a polar manifold on a circle).
* The mechanism to extend the boundary information is a so-called transfinite
* interpolation.
+ * The use of this class is discussed extensively in step-65.
*
* The formula for extending such a description in 2D is, for example,
* described on