**NAME**

adapt- mesh adaptation

**SYNOPSYS**

geo adapt (const field& phi); geo adapt (const field& phi, const adapt_option_type& opts);

**DESCRIPTION**

The functionadaptimplements the mesh adaptation procedure, based on thegmsh(isotropic) orbamg(anisotropic) mesh generators. Thebamgmesh generator is the default in two dimension. For dimension one or three,gmshis the only generator supported yet. In the two dimensional case, thegmshcorrespond to theopts.generator="gmsh". The strategy based on a metric determined from the Hessian of a scalar governing field, denoted asphi, and that is supplied by the user. Let us denote byH=Hessian(phi)the Hessian tensor of the fieldphi. Then,|H|denote the tensor that has the same eigenvector asH, but with absolute value of its eigenvalues: |H| = Q*diag(|lambda_i|)*Qt The metricMis determined from|H|. Recall that an isotropic metric is such thatM(x)=hloc(x)^(-2)*Idwherehloc(x)is the element size field andIdis the identityd*dmatrix, andd=1,2,3is the physical space dimension.

**GMSH** **ISOTROPIC** **METRIC**

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) normalizationsup_y(phi(y))-inf_y(phi(y))of the governing fieldphiand the two parametersopts.err, the target error, andopts.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: |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 valuephi(x). The parameter is provided by the optional variableopts.cutoff; its default value is1e-7. The default strategy is the local normalization. The global normalization can be enforced by settingopts.additional="-AbsError". When choosing global or local normalization ? When the governing fieldphiis bounded, i.e. whenerr*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, whenphipresents 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 settingopts.hminnot too small. The local normalization has been choosen as the default since it is more robust. When your fieldphidoes not present singularities, then you can swith to the global numbering that leads to a best equirepartition of the error over the domain.

**IMPLEMENTATION**

struct adapt_option_type { typedef std::vector<int>::size_type size_type; std::string generator; bool isotropic; Float err; Float errg; Float hcoef; Float hmin; Float hmax; Float ratio; Float cutoff; size_type n_vertices_max; size_type n_smooth_metric; bool splitpbedge; Float thetaquad; Float anisomax; bool clean; std::string additional; bool double_precision; Float anglecorner; // angle below which bamg considers 2 consecutive edge to be part of // the same spline adapt_option_type() : generator(""), isotropic(true), err(1e-2), errg(1e-1), hcoef(1), hmin(0.0001), hmax(0.3), ratio(0), cutoff(1e-7), n_vertices_max(50000), n_smooth_metric(1), splitpbedge(true), thetaquad(std::numeric_limits<Float>::max()), anisomax(1e6), clean(false), additional("-RelError"), double_precision(false), anglecorner(0) {} }; template <class T, class M> geo_basic<T,M> adapt ( const field_basic<T,M>& phi, const adapt_option_type& options = adapt_option_type());