[Getfem-commits] r5282 - /trunk/getfem/doc/sphinx/source/userdoc/model_ALE_rotating.rst

[Yves . Renard]

Author: renard
Date: Mon Apr  4 07:57:53 2016
New Revision: 5282

URL: http://svn.gna.org/viewcvs/getfem?rev=5282&view=rev
Log:
work in progress

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

Modified: trunk/getfem/doc/sphinx/source/userdoc/model_ALE_rotating.rst
URL: 
http://svn.gna.org/viewcvs/getfem/trunk/getfem/doc/sphinx/source/userdoc/model_ALE_rotating.rst?rev=5282&r1=5281&r2=5282&view=diff
==============================================================================
--- trunk/getfem/doc/sphinx/source/userdoc/model_ALE_rotating.rst       
(original)
+++ trunk/getfem/doc/sphinx/source/userdoc/model_ALE_rotating.rst       Mon Apr 
 4 07:57:53 2016
@@ -12,8 +12,6 @@
 ------------------------------
 
 This section present a set of bricks facilitating the use of an ALE 
formulation for rotating bodies having a rotational symmetry (typically a train 
wheel).
-
-Work in progress ...
 
 Theoretical background
 ++++++++++++++++++++++
@@ -47,7 +45,7 @@
 Note that the description is given for a three-dimensional body. For 
two-dimensional bodies, the third axes is neglected so that :math:`R(t)` is a 
:math:`2\times 2` rotation matrix. Let us denote :math:`r(t)` the rotation:
 
 .. math::
-   r(t,X) = R(t)X, ~~~~~~~~~ mbox{ and }
+   r(t,X) = R(t)X, ~~~~~~~~~ \mbox{ and }
    A = \left(\begin{array}{ccc}
    0 & -1 & 0 \\
    1 & 0 & 0 \\
@@ -114,43 +112,47 @@
 
 
 
-Main invariants and derivatives
-###############################
+Weak formulation of the transient terms
+#######################################
+
+Asuming :math:`\rho^0` the density in the reference configuration having a 
rotation symmetry, the term corresponding to acceleration in the weak 
formulation reads (with :math:`v(X) = \bar{v}(\bar{X})` a test function):
+
+.. math::
+   \int_{\Omega^0} \rho^0 \Frac{\partial^2 u}{\partial t^2}\cdot vdX =
+
+   \int_{\bar{\Omega}^0} \rho^0 \left[\Frac{\partial^2 \bar{u}}{\partial t^2} 
+ 2\dot{\theta} \nabla\Frac{\partial \bar{u}}{\partial t}A \bar{X} +  
\dot{\theta}^2\mbox{div}(((I_d + \nabla\bar{u})A \bar{X}) \otimes (A \bar{X}) ) 
 + \ddot{\theta} (I_d + \nabla\bar{u}) A \bar{X}  + \ddot{z}(t) \right] \cdot 
\bar{v} d\bar{X}.
+   
+The third term in the right hand side can be integrated by part as follows:
+
+.. math::
+   \begin{array}{rcl}
+   \ds \int_{\bar{\Omega}^0} \rho^0 \dot{\theta}^2\mbox{div}(((I_d + 
\nabla\bar{u})A \bar{X}) \otimes (A \bar{X}) ) \cdot \bar{v} d\bar{X} &=& - \ds 
\int_{\bar{\Omega}^0} (\dot{\theta}^2 (I_d + \nabla\bar{u})A \bar{X})) \cdot 
(\nabla (\rho^0 \bar{v}) A \bar{X}) d\bar{X} \\
+   &&\ds + \int_{\partial \bar{\Omega}^0} \rho^0 \dot{\theta}^2 (((I_d + 
\nabla\bar{u})A \bar{X}) \otimes (A \bar{X}) ) \bar{N} \cdot \bar{v} 
d\bar{\Gamma}.
+  \end{array}
+
+Since :math:`\bar{N}` the outward unit normal vector on :math:`\partial 
\bar{\Omega}^0` is orthogonal to :math:`A \bar{X}` the boundary term is zero 
and :math:`\nabla (\rho^0 \bar{v}) = \bar{v} \otimes \nabla \rho^0   + \rho^0 
\nabla \bar{v}` and since :math:`\nabla \rho^0.(A\bar{X}) = 0` because of the 
asumption on :math:`\rho^0` to have a rotation symmetry, we have
+
+.. math::
+   \int_{\bar{\Omega}^0} \rho^0 \dot{\theta}^2\mbox{div}(((I_d + 
\nabla\bar{u})A \bar{X}) \otimes (A \bar{X}) ) \cdot \bar{v} d\bar{X} = - 
\int_{\bar{\Omega}^0} \rho^0 \dot{\theta}^2(\nabla\bar{u}A \bar{X}) \cdot 
(\nabla \bar{v} A \bar{X}) d\bar{X} - \int_{\bar{\Omega}^0} \rho^0 
\dot{\theta}^2 (A^2 \bar{X})\cdot \bar{v} d\bar{X}.
+
+Thus globally
+
+.. math::
+   \begin{array}{rcl}
+   \ds \int_{\Omega^0} \rho^0 \Frac{\partial^2 u}{\partial t^2}\cdot vdX &=&
+   \ds \int_{\bar{\Omega}^0} \rho^0 \left[\Frac{\partial^2 \bar{u}}{\partial 
t^2} + 2\dot{\theta} \nabla\Frac{\partial \bar{u}}{\partial t}A \bar{X} + 
\ddot{\theta} \nabla\bar{u} A \bar{X}   \right] \cdot \bar{v} d\bar{X}\\
+   &&\ds - \int_{\bar{\Omega}^0} \rho^0 \dot{\theta}^2(\nabla\bar{u}A \bar{X}) 
\cdot (\nabla \bar{v} A \bar{X}) d\bar{X} - \int_{\bar{\Omega}^0} \rho^0 
(\dot{\theta}^2 A^2 \bar{X} + \ddot{\theta} A\bar{X} + \ddot{z}(t))\cdot 
\bar{v} d\bar{X}.
+   \end{array}
+
+Note that two terms can deteriorate the coercivity of the problem and thus its 
well posedness and the stability of time integration schemes: the second one 
(convection term) and the fifth one. This may oblige to use additional 
stabilization techniques for large values of the angular velocity 
:math:`\dot{\theta}`.
 
 
-
-
-the available bricks ...
+The available bricks ...
 ++++++++++++++++++++++++
 
-To be adapted ..
+To be adapted  ::
 
-This brick represents a large strain elasticity problem. It is defined in the 
files :file:`getfem/getfem_nonlinear_elasticity.h` and 
:file:`getfem/getfem_nonlinear_elasticity.cc`. The function adding this brick 
to a model is ::
+  ind = getfem::brick_name(parmeters);
 
-  ind = getfem::add_nonlinear_elasticity_brick
-    (md, mim, varname, AHL, dataname, region = -1);
+where ``parameters`` are the parameters ... 
 
-where ``AHL`` is an object of type ``getfem::abstract_hyperelastic_law`` which 
represents the considered hyperelastic law. It has to be chosen between: ::
-
-  getfem::SaintVenant_Kirchhoff_hyperelastic_law AHL;
-  getfem::Ciarlet_Geymonat_hyperelastic_law AHL;
-  getfem::Mooney_Rivlin_hyperelastic_law AHL(compressible, neohookean);
-  getfem::plane_strain_hyperelastic_law AHL(pAHL);
-  getfem::generalized_Blatz_Ko_hyperelastic_law AHL;
-
-The Saint-Venant Kirchhoff law is a linearized law defined with the two Lame 
coefficients, Ciarlet Geymonat law is defined with the two Lame coefficients 
and an additional coefficient (:math:`\lambda, \mu, a`).
-
-
-
-Here is the list of nonlinear operators in the language which can be useful 
for nonlinear elasticity::
-
-  Det(M)                                % determinant of the matrix M
-  Trace(M)                              % trace of the matrix M
-  Matrix_i2(M)                          % second invariant of M (in 3D): 
(sqr(Trace(m)) - Trace(m*m))/2
-  Matrix_j1(M)                          % modified first invariant of M: 
Trace(m)pow(Det(m),-1/3).
-  Matrix_j2(M)                          % modified second invariant of M: 
Matrix_I2(m)*pow(Det(m),-2/3).
-  Right_Cauchy_Green(F)                 % F' * F
-  Left_Cauchy_Green(F)                  % F * F'
-  Green_Lagrangian(F)                   % (F'F - Id(meshdim))/2
-  Cauchy_stress_from_PK2(sigma, Grad_u) % 
(Id+Grad_u)*sigma*(I+Grad_u')/det(I+Grad_u)
-