<br />
<b>Warning</b>:  Undefined array key 1 in <b>/home/np29546/public_html/elmerice/wiki/inc/auth.php</b> on line <b>78</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/auth.php</b> on line <b>431</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/Action/Export.php</b> on line <b>104</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/Action/Export.php</b> on line <b>104</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/Action/Export.php</b> on line <b>104</b><br />
=====  =====
==== General Information ====

  * **Solver Fortran File:** ''MMG2D_MetricAniso.F90''
  * **Solver Name:** ''ElmerIce_MeshAdapt2D(MMG2D_MetricAniso)''
  * **Required Output Variable(s):** 
    * (1) ''Metric'' (dofs = 3)
    * (2) ''hessian'' (dofs = 3)
  * **Required Input Variable(s):** 
    * (1) ''Nodal gradient'' (dofs = 2)
  * **Optional Output Variable(s):** None  
  * **Optional Input Variable(s):** None

==== General Description ==== 

This solver is used for the mesh adaptation ([[mesh:meshadaptation|Mesh Adaptation]]) to compute the anisotropic metric **M**.


The metric //**M**// , used to define the element size,  derives from a geometric error estimate based on an upper  bound for the interpolation error of a continuous field to piecewise linear elements (Frey and Alauzet, 2005). 

For a variable //v//, //**M**// depends on the eigenvalues <m>lambda_i</m> and eigenvector  matrix //**R**// of the hessian matrix of //v//, //**H**// (i.e. small elements are required where the curvature is the highest):

<m>
M=R.Lambda.R^{-1}
</m> 
with
<m>
Lambda=(matrix{2}{2}{lambda_1 0  0 lambda_2})
</m>
and
<m>
\lambda_i=min ( max ( {{c|\lambda_i|}/{epsilon_v}},{{1}/{l^2_{max}}} ), {{1}/{l^2_min}} ),    (1)
</m>

where
  * <m>c</m> is a geometric constant equal to 2/9  in 2D
  * <m>l_min</m> (resp. <m>l_max</m>) is a prescribed minimal (resp. maximal) edge size
  * <m>epsilon_v</m>  is the prescribed maximum error

First this solver compute the hessian matrix //**H**//; As computing second derivatives in linear elements in not straightforward this is done by solving the diffusive equation <m>H_ij+ K ∇(H_ij) = 1/2 (dq_i/dx_j+dq_j/dx_i)</m>, where <m>K=k A</m> is a diffusivity proportionnal to the local element size (<m>A</m>) and <m>g_i={{\partial v}/{\partial x_i}}</m>  are the nodal gradients of the variable //v// (This can be computed using using the [[solvers:2Dnodalgradient|Compute2DNodalGradient Solver]]).

Finally, the metric //**M**// is then computed from Eq. (1)


==== SIF contents ====
<code>

Solver 5
   Equation = "Metric2"
   Variable = -nooutput dumy

   Procedure = "ElmerIce_MeshAdapt2D" "MMG2D_MetricAniso"

   Metric Variable Name = String "M2"

   Hessian Variable Name = String "ddx2"
   Gradient Name = String "Gradient2"
   Diffusivity = Real 0.5 !! the diffusivity k; the total diffusivity is kA

   Linear System Solver = Direct
   Linear System Direct Method = umfpack

  Exported Variable 1 = -dofs 3 "M2"
  Exported Variable 2 = -dofs 3 "ddx2"
End


</code>

<code>
Body Force 1
!! Parameters in Eq. 1
  M2 Hmin = Real 1.0e-3
  M2 Hmax = Real 1.0
  M2 err =  Real 0.0033
End
</code>

==== Example ====
Examples for anisotropic mesh adaptation can be found under ''[ELMER_TRUNK]/elmerice/Tests/MMG2D_Aniso1'' and ''[ELMER_TRUNK]/elmerice/Tests/MMG2D_Aniso2'', where the mesh size is adapted using 1 or 2 variables (i.e. combining metric informations), respectively.