adapt - mesh adaptation
|typedef std::vector< int >|
adapt - mesh adaptation
geo adapt (const field& phi); geo adapt (const field& phi, const adapt_option_type& opts);
adapt implements the mesh adaptation procedure, based on the
gmsh (isotropic) or
bamg (anisotropic) mesh generators. The
bamg mesh generator is the default in two dimension. For dimension one or three,
gmsh is the only generator supported yet. In the two dimensional case, the
gmsh correspond to the
The strategy based on a metric determined from the Hessian of a scalar governing field, denoted as
phi, and that is supplied by the user. Let us denote by
H=Hessian(phi) the Hessian tensor of the field
|H| denote the tensor that has the same eigenvector as
H, but with absolute value of its eigenvalues:
M is determined from
|H|. Recall that an isotropic metric is such that
hloc(x) is the element size field and
Id is the identity
d*d matrix, and
d=1,2,3 is the physical space dimension.
GMSH ISOTROPIC METRIC
Notice that the denominator involves a global (absolute) normalization
sup_y(phi(y))-inf_y(phi(y)) of the governing field
phi and the two parameters
opts.err, the target error, and
opts.hcoef, a secondary normalization parameter (defaults to 1).
BAMG ANISOTROPIC METRIC
There are two approach for the normalization of the metric. The first one involves a global (absolute) normalization:
The first one involves a local (relative) normalization:
Notice that the denominator involves a local value
phi(x). The parameter is provided by the optional variable
opts.cutoff; its default value is
1e-7. The default strategy is the local normalization. The global normalization can be enforced by setting
When choosing global or local normalization ?
When the governing field
phi is bounded, i.e. when
err*hcoef^2*(sup_y(phi(y))-inf_y(phi(y))) will converge versus mesh refinement to a bounded value, the global normalization defines a metric that is mesh-independent and thus the adaptation loop will converge.
phi presents singularities, with unbounded values (such as corner singularity, i.e. presents peacks when represented in elevation view), then the mesh adaptation procedure is more difficult. The global normalization divides by quantities that can be very large and the mesh adaptation can diverges when focusing on the singularities. In that case, the local normalization is preferable. Moreover, the focus on singularities can also be controled by setting
opts.hmin not too small.
The local normalization has been choosen as the default since it is more robust. When your field
phi does not present singularities, then you can swith to the global numbering that leads to a best equirepartition of the error over the domain.