[Getfem-commits] r5283 - in /trunk/getfem: doc/sphinx/source/userdoc/ interface/src/

[Yves . Renard]

Author: renard
Date: Mon Apr  4 10:28:03 2016
New Revision: 5283

URL: http://svn.gna.org/viewcvs/getfem?rev=5283&view=rev
Log:
minor fixes

Modified:
    trunk/getfem/doc/sphinx/source/userdoc/model_ALE_rotating.rst
    trunk/getfem/interface/src/gf_model_get.cc
    trunk/getfem/interface/src/gf_model_set.cc

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=5283&r1=5282&r2=5283&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 10:28:03 2016
@@ -1,3 +1,162 @@
+.. $Id: model.rst 3655 2010-07-17 20:42:08Z renard $
+
+.. include:: ../replaces.txt
+
+.. highlightlang:: c++
+
+.. index:: models, model bricks
+
+.. _ud-model-ALE_rotating:
+
+ALE terms for rotating objects
+------------------------------
+
+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).
+
+Theoretical background
+++++++++++++++++++++++
+
+This strategy consists in adopting an intermediary description between an 
Eulerian and a Lagrangian ones for a rotating body having a rotational 
symmetry. This intermediary description consist in a rotating axes with respect 
to the reference configuration. See for instance [Dr-La-Ek2014]_ and 
[Nackenhorst2004]_.
+
+It is supposed that the considered body is submitted approximately to a rigid 
body motion
+
+.. math::
+  \tau(X) = R(t)X + z(t)
+
+and may have additonal deformation (exptected smaller) with respect to this 
rigid motion, where :math:`R(t)` is a rotation matrix
+
+.. math::
+  R(t) = \left(\begin{array}{ccc}
+  \cos(\theta(t)) & \sin(\theta(t)) & 0 \\
+  -\sin(\theta(t)) & \cos(\theta(t)) & 0 \\
+  0 & 0 & 1
+  \end{array} \right),
+
+and :math:`z(t)` is a translation. Note that, in order to be consistent with a 
positive translation for a positive angle, the rotation is **clockwise**. This 
illustrated in the following figure:
+
+.. _ud-fig-rotating_cylinder:
+
+.. figure:: images/ALE_rotating_body.png
+    :height: 200pt
+    :align: center
+    :alt: alternate text
+    :figclass: align-center
+
+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 }
+   A = \left(\begin{array}{ccc}
+   0 & 1 & 0 \\
+   -1 & 0 & 0 \\
+   0 & 0 & 0
+   \end{array} \right).
+
+We have then
+
+.. math::
+ \dot{r}(t,X) = \dot{\theta}AR(t)X
+
+If :math:`\varphi(t, X)` is the deformation of the body which maps the 
reference configuration :math:`\Omega^0` to the deformed configuration 
:math:`\Omega_t` at time :math:`t`, the ALE description consists in the 
decomposition of the deformation of the cylinder in 
+
+.. math::
+   \varphi(t, X) = (\tau(t) \circ \bar{\varphi}(t) \circ r(t))(X) = 
\bar{\varphi}(t, r(t, X)) + z(t)
+
+With :math:`\bar{X} = R(t)X` the new considered deformation is
+
+.. math::
+  \bar{\varphi}(t,\bar{X}) = \varphi(X) - z(t)
+
+
+Thanks to the rotation symmetry of the reference configuration 
:math:`\Omega^0:`, we note that :math:`\bar{\Omega}^0 = r(t, \Omega^0)` is 
independant of :math:`t` and will serve as the new reference configuration. 
This is illustrated in the following figure: 
+
+.. _ud-fig-rotating_cylinder_conf:
+
+.. figure:: images/ALE_rotating_conf.png
+    :height: 200pt
+    :align: center
+    :alt: alternate text
+    :figclass: align-center
+
+The denomination ALE of the method is justified by the fact that 
:math:`\bar{\Omega}^0` is an intermediate configuration which is of Euler type 
for the rigid motion and a Lagrangian one for the additional deformation of the 
solid. If we denote
+
+.. math::
+  \bar{u}(t,\bar{X}) = \bar{\varphi}(t, \bar{X}) - \bar{X}
+
+the displacement with respect to this intermediate configuration, the 
advantage is that if this additional displacement with respect to the rigid 
body motion is small, it is possible to use a small deformation model (for 
instance linearized elasticity).
+
+Due to the objectivity properties of standard consistutive laws, the 
expression od these laws in the intermediate configuration is most of the time 
identical to the expression in a standard reference configuration except for 
the expression of the time derivative which are modified because the change of 
coordinate is  nonconstant in time :
+
+.. math::
+
+  \Frac{\partial \varphi}{\partial t} = \Frac{\partial \bar{\varphi}}{\partial 
t} + \dot{\theta} \nabla \bar{\varphi} A \bar{X} + \dot{z}(t),
+
+  \Frac{\partial^2 \varphi}{\partial t^2} = \Frac{\partial^2 
\bar{\varphi}}{\partial t^2} + 2\dot{\theta} \nabla\Frac{\partial 
\bar{\varphi}}{\partial t}A \bar{X} + 
\dot{\theta}^2\mbox{div}((\nabla\bar{\varphi}A \bar{X}) \otimes (A \bar{X}) )  
+ \ddot{\theta}\nabla\bar{\varphi}A \bar{X} + \ddot{z}(t).
+
+Note that the term :math:`\dot{\theta} A \bar{X} = 
\left(\hspace*{-0.5em}\begin{array}{c} \dot{\theta}\bar{X}_2 \\ 
-\dot{\theta}\bar{X}_1 \\ 0 \end{array}\hspace*{-0.5em}\right)` is the rigid 
motion velocity vector. Now, If :math:`\Theta(t,X)` is a quantity attached to 
the material points (for instance the temperature), then, with 
:math:`\bar{\Theta}(t,\bar{X}) = \Theta(t,X)` , one simply has
+
+.. math::
+
+  \Frac{\partial \Theta}{\partial t} = \Frac{\partial \bar{\Theta}}{\partial 
t} + \dot{\theta} \nabla \bar{\Theta} A \bar{X}
+
+This should not be forgotten that a correction has to be provided for each 
evolving variable for which the time derivative intervene in the considered 
model (think for instance to platic flow for plasticity). So that certain model 
bricks canot be used directly (plastic bricks for instance).
+
+|gf| bricks for structural mecanics are mainly considering the displacement as 
the amin unknown. The expression for the displacement is the following:
+
+.. math::
+  \Frac{\partial u}{\partial t} = \Frac{\partial \bar{u}}{\partial t} + 
\dot{\theta} (I_d + \nabla \bar{u}) A \bar{X} + \dot{z}(t),
+
+  \Frac{\partial^2 u}{\partial t^2} = \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).
+  
+  
+
+
+
+Weak formulation of the transient terms
+#######################################
+
+Assuming :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 
assumption 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 ...
+++++++++++++++++++++++++
+
+To be adapted  ::
+
+  ind = getfem::brick_name(parmeters);
+
+where ``parameters`` are the parameters ... 
+
+=======
 .. $Id: model.rst 3655 2010-07-17 20:42:08Z renard $
 
 .. include:: ../replaces.txt

Modified: trunk/getfem/interface/src/gf_model_get.cc
URL: 
http://svn.gna.org/viewcvs/getfem/trunk/getfem/interface/src/gf_model_get.cc?rev=5283&r1=5282&r2=5283&view=diff
==============================================================================
--- trunk/getfem/interface/src/gf_model_get.cc  (original)
+++ trunk/getfem/interface/src/gf_model_get.cc  Mon Apr  4 10:28:03 2016
@@ -912,21 +912,21 @@
       `dispname` is the displacement variable.
       `multname` is the plastic multiplier.
       `pressname` is an optional pressure multiplier for a mixed
-                  displacement pressure formulation, otherwise it is an
-                  empty string.
+      displacement pressure formulation, otherwise it is an
+      empty string.
       `lawname` is the name of the plasticity model (for the moment it can
-                only be the name Simo_Miehe)
+      only be the name Simo_Miehe)
       For the Simo_Miehe model:
-        `params1` is the bulk modulus,
-        `params2` is the shear modulus,
-        `params3` is the name of a user defined function expressing isotropic
-                  hardening in terms of the Frobenius norms of the strain
-                  and stress tensors.
-        `params4` is the name of the plastic strain variable (which is 
updated).
-        `params5` is the name of the vector field for storing all non-repeated
-                  components of the inverse plastic right Cauchy-Green tensor
-                  (which is updated).
-        address@hidden/
+      `params1` is the bulk modulus,
+      `params2` is the shear modulus,
+      `params3` is the name of a user defined function expressing isotropic
+      hardening in terms of the Frobenius norms of the strain
+      and stress tensors.
+      `params4` is the name of the plastic strain variable (which is updated).
+      `params5` is the name of the vector field for storing all non-repeated
+      components of the inverse plastic right Cauchy-Green tensor
+      (which is updated).
+      @*/
     sub_command
       ("finite strain elastoplasticity next iter", 10, 10, 0, 1,
        getfem::mesh_im *mim = to_meshim_object(in.pop());

Modified: trunk/getfem/interface/src/gf_model_set.cc
URL: 
http://svn.gna.org/viewcvs/getfem/trunk/getfem/interface/src/gf_model_set.cc?rev=5283&r1=5282&r2=5283&view=diff
==============================================================================
--- trunk/getfem/interface/src/gf_model_set.cc  (original)
+++ trunk/getfem/interface/src/gf_model_set.cc  Mon Apr  4 10:28:03 2016
@@ -1792,7 +1792,7 @@
       and for a quasistatic model. `multname` is a plastic multiplier name
       previously defined by the user on a mesh_fem. `pressname` can either
       be an empty string, resulting in a pure displacements based formulation,
-       or it can contain the name of a user defined scalar fem variable to be
+      or it can contain the name of a user defined scalar fem variable to be
       used as the pressure multiplier in a mixed displacement-pressure
       formulation. The fem spaces of `dispname` and `pressname` have to
       fulfill an appropriate inf-sup condition.
@@ -1803,17 +1803,17 @@
       `param1` is an expression for the initial bulk modulus K
       `param2` is an expression for the initial shear modulus G,
       `param3` is the name of a user predefined function that decribes
-               the yield limit as a function of the hardening variable
-               (both the yield limit and the hardening variable values are
-               assumed to be Frobenius norms of appropriate stress and strain
-               tensors, respectively),
+      the yield limit as a function of the hardening variable
+      (both the yield limit and the hardening variable values are
+      assumed to be Frobenius norms of appropriate stress and strain
+      tensors, respectively),
       `param4` is the name of a (scalar) fem_data or im_data field that holds
-               the plastic strain at the previous time step, and
+      the plastic strain at the previous time step, and
       `param5` is the name of a fem_data or im_data field that holds all
-               non-repeated components of the inverse of the plastic right
-               Cauchy-Green tensor at the previous time step
-               (it has to be a 4 element vector for plane strain 2D problems
-               and a 6 element vector for 3D problems).
+      non-repeated components of the inverse of the plastic right
+      Cauchy-Green tensor at the previous time step
+      (it has to be a 4 element vector for plane strain 2D problems
+      and a 6 element vector for 3D problems).
       As usual, `region` is an optional mesh region on which the term is added.
       If it is not specified, it is added on the whole mesh.
       Return the brick index in the address@hidden/