[Getfem-commits] r5074 - /trunk/getfem/doc/sphinx/source/userdoc/model_Nitsche.rst

[logari81]

Author: logari81
Date: Wed Aug 19 17:06:47 2015
New Revision: 5074

URL: http://svn.gna.org/viewcvs/getfem?rev=5074&view=rev
Log:
some further corrections in the documentation of Nitsche's method

Modified:
    trunk/getfem/doc/sphinx/source/userdoc/model_Nitsche.rst

Modified: trunk/getfem/doc/sphinx/source/userdoc/model_Nitsche.rst
URL: 
http://svn.gna.org/viewcvs/getfem/trunk/getfem/doc/sphinx/source/userdoc/model_Nitsche.rst?rev=5074&r1=5073&r2=5074&view=diff
==============================================================================
--- trunk/getfem/doc/sphinx/source/userdoc/model_Nitsche.rst    (original)
+++ trunk/getfem/doc/sphinx/source/userdoc/model_Nitsche.rst    Wed Aug 19 
17:06:47 2015
@@ -12,16 +12,16 @@
 Nitsche's method for dirichlet and contact boundary conditions
 --------------------------------------------------------------
 
-|gf| provides a generic implementation of Nitche's method which allows to take 
into account Dirichlet type boundary conditions or contact with friction 
boundary conditions in a weak sense without the use of Lagrange multipliers.
+|gf| provides a generic implementation of Nitche's method which allows to 
account for Dirichlet type or contact with friction boundary conditions in a 
weak sense without the use of Lagrange multipliers.
 The method is very attractive because it transforms a Dirichlet boundary 
condition into a weak term similar to a Neumann boundary condition.
-However, this advantage is paid by the fact that the implementation of 
Nitche's method is model dependent, because it requires an approximation of the 
corresponding Neumann term.
-In order to add a boundary condition with Nitsche's method on a variable of a 
model, the corresponding brick has to have access to an approximation of the 
Neumann term of all partial differential terms applied to this variables.
+However, this advantage is at the cost that the implementation of Nitche's 
method is model dependent, since it requires an approximation of the 
corresponding Neumann term.
+In order to add a boundary condition with Nitsche's method on a variable of a 
model, the corresponding brick needs to have access to an approximation of the 
Neumann term of all partial differential terms applied to this variable.
 In the following, considering a variable :math:`u`, we will denote by
 
 .. math::
   G
 
-the sum of all Neumann terms on this variables.
+the sum of all Neumann terms on this variable.
 Note that the Neumann term :math:`G` will often depend on the variable 
:math:`u` but it may also depend on other variables of the model.
 This is the case for instance for mixed formulations of incompressible 
elasticity.
 The Neumann terms depend also frequently on some parameters of the model 
(elasticity coefficients ...) but this is assumed to be contained in its 
expression.
@@ -47,20 +47,25 @@
 Generic Nitsche's method for a Dirichlet condition 
 ++++++++++++++++++++++++++++++++++++++++++++++++++
 
-Assume that the variable :math:`u` is considered and that on wants to 
prescribe the condition
+Assume that the variable :math:`u` is considered and that one wants to 
prescribe the condition
 
 .. math::
   Hu = g
 
-on a part :math:`\Gamma_D`  of the boundary of the considered domain. Here 
:math:`H` is considered equal to one in the scalar case or can be either the 
identity matrix in the vectorial case either a singular matrix having only 1 or 
0 as eigenvalues. This allow here to prescribe only the normal or tangent 
component of :math:`u`. For instance if one wants to prescribe only the normal 
component, :math:`H` will be chosen to be equal to :math:`nn^T` where :math:`n` 
is the outward unit normal on :math:`\Gamma_D`.
-
-Nitsche's method to prescribe this dirichlet condition consists in adding to 
the weak formulation of the problem the following term
+on a part :math:`\Gamma_D`  of the boundary of the considered domain.
+Here :math:`H` is considered equal to one in the scalar case or can be either 
the identity matrix in the vectorial case either a singular matrix having only 
1 or 0 as eigenvalues.
+This allow here to prescribe only the normal or tangent component of :math:`u`.
+For instance if one wants to prescribe only the normal component, :math:`H` 
will be chosen to be equal to :math:`nn^T` where :math:`n` is the outward unit 
normal on :math:`\Gamma_D`.
+
+Nitsche's method for prescribing this Dirichlet condition consists in adding 
the following term to the weak formulation of the problem
 
 .. math::
   \int_{\Gamma_D} \Frac{1}{\gamma}(Hu-g-\gamma HG).(Hv) - 
\theta(Hu-g).(HD_uG[v])d\Gamma,
 
-where :math:`\gamma` and :math:`\theta` are two parameters of Nitsche's method 
and :math:`v` is the test function corresponding to :math:`u`. The parameter 
:math:`\theta` can be chosen positive or negative. :math:`\theta = 1` 
corresponds to the more standard method which leads to a symmetric tangent term 
in standard situations, :math:`\theta = 0` corresponds to a non-symmetric 
method which has the advantage to have a reduced number of terms and especially 
not to need the second derivatives of :math:`G` in the nonlinear case, and 
:math:`\theta = -1` is a kind of skew-symmetric method which ensure an 
inconditonal coercivity (which means independent of :math:`\gamma`) at least in 
standard situations.
-The parameter :math:`\gamma` is a kind of penalization parameter (although the 
method is consistent) which is taken to be :math:`\gamma = \gamma_0 h_T` where 
:math:`\gamma_0` is taken uniform on the mesh and :math:`h_T` is the diameter 
of the element :math:`T`. Note that, in standard situations, except for 
:math:`\theta = -1` the parameter :math:`\gamma_0` has to be taken sufficiently 
small in order to ensure the convergence of Nitsche's method.
+where :math:`\gamma` and :math:`\theta` are two parameters of Nitsche's method 
and :math:`v` is the test function corresponding to :math:`u`.
+The parameter :math:`\theta` can be chosen positive or negative. :math:`\theta 
= 1` corresponds to the more standard method which leads to a symmetric tangent 
term in standard situations, :math:`\theta = 0` corresponds to a non-symmetric 
method which has the advantage of a reduced number of terms and not requiring 
the second derivatives of :math:`G` in the nonlinear case, and :math:`\theta = 
-1` is a kind of skew-symmetric method which ensures an inconditonal coercivity 
(which means independent of :math:`\gamma`) at least in standard situations.
+The parameter :math:`\gamma` is a kind of penalization parameter (although the 
method is consistent) which is taken to be :math:`\gamma = \gamma_0 h_T` where 
:math:`\gamma_0` is taken uniform on the mesh and :math:`h_T` is the diameter 
of the element :math:`T`.
+Note that, in standard situations, except for :math:`\theta = -1` the 
parameter :math:`\gamma_0` has to be taken sufficiently small in order to 
ensure the convergence of Nitsche's method.
 
 The bricks adding a Dirichlet condition with Nitsche's method to a model are 
the following::