Rheolef
7.2
an efficient C++ finite element environment
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geo adapt (const field& criterion); geo adapt (const field& criterion, const adapt_option& aopt);
The adapt
function implements an adaptive mesh procedure, based either 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 option aopt.generator="gmsh"
, where aopt
is an adap_option
variable (see adapt
).
The strategy bases on a metric determined by the Hessian of a scalar criterion field, denoted here as phi
, and that is supplied by the user as the first argument of the adapt
function.
Let us denote by H=Hessian(phi)
the Hessian tensor field of the scalar field phi
. Then, |H|
denotes the tensor that has the same eigenvector as H
, but with absolute value of its eigenvalues:
|H| = Q*diag(|lambda_i|)*Qt
The metric M
is determined from |H|
. Recall that an isotropic metric is such that M(x)=hloc(x)^(-2)*Id
where 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.
max_(i=0..d-1)(|lambda_i(x)|)*Id M(x) = ----------------------------------------- err*hcoef^2*(sup_y(phi(y))-inf_y(phi(y)))
Notice that the denominator involves a global (absolute) normalization sup_y(phi(y))-inf_y(phi(y))
of the criterion field phi
and the two parameters aopt.err
, the target error, and aopt.hcoef
, a secondary normalization parameter (defaults to 1).
There are two approach for the normalization of the metric. The first one involves a global (absolute) normalization:
|H(x))| M(x) = ----------------------------------------- err*hcoef^2*(sup_y(phi(y))-inf_y(phi(y)))
The first one involves a local (relative) normalization:
|H(x))| M(x) = ----------------------------------------- err*hcoef^2*(|phi(x)|, cutoff*max_y|phi(y)|)
Notice that the denominator involves a local value phi(x)
. The parameter is provided by the optional variable aopt.cutoff
; its default value is 1e-7
. The default strategy is the local normalization. The global normalization can be enforced by setting aopt.additional="-AbsError"
.
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.
Otherwise, when phi
presents singularities, with unbounded values (such as corner singularity, i.e. presents picks 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 controlled by setting aopt.hmin
not too small.
The local normalization has been chosen as the default since it is more robust. When your field phi
does not present singularities, then you can switch to the global numbering that leads to a best equirepartition of the error over the domain.
This documentation has been generated from file main/lib/adapt.h