Update index.rst

Author: claudioperez

source/user/manual/analysis/transient/index.rst

@@ -15,88 +15,88 @@ Transient Analysis
    TRBDF2
 
 
+..
+   Theory
+   ------
 
-Theory
-------
+   In a nonlinear transient finite element analysis we seek a solution
+   (:math:`\boldsymbol{u}`, :math:`\dot{\boldsymbol{u}}`,
+   :math:`\ddot{\boldsymbol{u}}`) to the nonlinear residual equation
 
-In a nonlinear transient finite element analysis we seek a solution
-(:math:`\boldsymbol{u}`, :math:`\dot{\boldsymbol{u}}`,
-:math:`\ddot{\boldsymbol{u}}`) to the nonlinear residual equation
+   .. math:: 
+      
+      \boldsymbol{r}({\boldsymbol{u}},\dot{\boldsymbol{u}}, \ddot{\boldsymbol{u}}) = \boldsymbol{p}_f(t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}, \dot{\boldsymbol{u}}) = \boldsymbol{0}
 
-.. math:: 
-   
-   \boldsymbol{r}({\boldsymbol{u}},\dot{\boldsymbol{u}}, \ddot{\boldsymbol{u}}) = \boldsymbol{p}_f(t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}, \dot{\boldsymbol{u}}) = \boldsymbol{0}
+   The most widely used technique for solving the transient non-linear
+   finite element equation is to use an incremental direct integration scheme. 
+   In the incremental formulation, a solution to the equation is sought at successive time
+   steps :math:`\Delta t` apart.
 
-The most widely used technique for solving the transient non-linear
-finite element equation is to use an incremental direct integration scheme. 
-In the incremental formulation, a solution to the equation is sought at successive time
-steps :math:`\Delta t` apart.
+   .. math::
 
-.. math::
+      \boldsymbol{r}({\boldsymbol{u}}_{n \Delta t},\dot{\boldsymbol{u}}_{n \Delta t}, \ddot{\boldsymbol{u}}_{n \Delta t}) = \boldsymbol{p}_f(n \Delta t) -
+      \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}_{n \Delta t}) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}_{n \Delta t}, \dot{\boldsymbol{u}}_{n \Delta t})
 
-   \boldsymbol{r}({\boldsymbol{u}}_{n \Delta t},\dot{\boldsymbol{u}}_{n \Delta t}, \ddot{\boldsymbol{u}}_{n \Delta t}) = \boldsymbol{p}_f(n \Delta t) -
-   \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}_{n \Delta t}) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}_{n \Delta t}, \dot{\boldsymbol{u}}_{n \Delta t})
 
+   For each time step, :math:`t`, the integration schemes provide two
+   operators, :math:`\operatorname{I}_1` and :math:`\operatorname{I}_2`, to
+   relate the velocity and accelerations at the time step as a function of
+   the displacement at the time step and the response at previous time
+   steps:
 
-For each time step, :math:`t`, the integration schemes provide two
-operators, :math:`\operatorname{I}_1` and :math:`\operatorname{I}_2`, to
-relate the velocity and accelerations at the time step as a function of
-the displacement at the time step and the response at previous time
-steps:
+   .. math::
 
-.. math::
+      \dot {\boldsymbol{u}}_{t} = {\mathrm{I}}_1 ({\boldsymbol{u}}_t, {\boldsymbol{u}}_{t-\Delta t}, \dot {\boldsymbol{u}}_{t-\Delta t},
+      \ddot {\boldsymbol{u}}_{t - \Delta t}, {\boldsymbol{u}}_{t - 2\Delta t}, \dot {\boldsymbol{u}}_{t - 2 \Delta t}. ..., )
+      %\label{I1}
 
-   \dot {\boldsymbol{u}}_{t} = {\mathrm{I}}_1 ({\boldsymbol{u}}_t, {\boldsymbol{u}}_{t-\Delta t}, \dot {\boldsymbol{u}}_{t-\Delta t},
-   \ddot {\boldsymbol{u}}_{t - \Delta t}, {\boldsymbol{u}}_{t - 2\Delta t}, \dot {\boldsymbol{u}}_{t - 2 \Delta t}. ..., )
-   %\label{I1}
+   .. math::
 
-.. math::
+      \ddot {\boldsymbol{u}}_{t} = {\mathrm{I}}_2 ({\boldsymbol{u}}_t, {\boldsymbol{u}}_{t-\Delta t}, \dot{\boldsymbol{u}}_{t-\Delta t},
+      \ddot{\boldsymbol{u}}_{t - \Delta t}, {\boldsymbol{u}}_{t - 2\Delta t}, \dot{\boldsymbol{u}}_{t - 2 \Delta t}. ..., )
+      %\label{I2}
 
-   \ddot {\boldsymbol{u}}_{t} = {\mathrm{I}}_2 ({\boldsymbol{u}}_t, {\boldsymbol{u}}_{t-\Delta t}, \dot{\boldsymbol{u}}_{t-\Delta t},
-   \ddot{\boldsymbol{u}}_{t - \Delta t}, {\boldsymbol{u}}_{t - 2\Delta t}, \dot{\boldsymbol{u}}_{t - 2 \Delta t}. ..., )
-   %\label{I2}
+   These allow us to rewrite equation `fullTimeForm <#fullTimeForm>`__, in
+   terms of a single response quantity, typically the displacement:
 
-These allow us to rewrite equation `fullTimeForm <#fullTimeForm>`__, in
-terms of a single response quantity, typically the displacement:
+   .. math::
 
-.. math::
 
+      \boldsymbol{r}({\boldsymbol{u}}_t) = \boldsymbol{p}_f(t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}_t) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}_t, \dot{\boldsymbol{u}}_t)
+      %\label{genForm}
 
-   \boldsymbol{r}({\boldsymbol{u}}_t) = \boldsymbol{p}_f(t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}_t) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}_t, \dot{\boldsymbol{u}}_t)
-   %\label{genForm}
+   The solution of this equation is typically obtained using an iterative
+   procedure, i.e. making an initial prediction for
+   :math:`{\boldsymbol{u}}_{t}`, denoted :math:`{\boldsymbol{u}}_{t}^{(0)}`
+   a sequence of approximations :math:`{\boldsymbol{u}}_{t}^{(i)}`,
+   :math:`i=1,2, ..` is obtained which converges (we hope) to the solution
+   :math:`{\boldsymbol{u}}_{t}`. The most frequently used iterative
+   schemes, such as Newton-Raphson, modified Newton, and quasi Newton
+   schemes, are based on a Taylor expansion of
+   equation `genForm <#genForm>`__ about :math:`{\boldsymbol{u}}_{t}`:
 
-The solution of this equation is typically obtained using an iterative
-procedure, i.e. making an initial prediction for
-:math:`{\boldsymbol{u}}_{t}`, denoted :math:`{\boldsymbol{u}}_{t}^{(0)}`
-a sequence of approximations :math:`{\boldsymbol{u}}_{t}^{(i)}`,
-:math:`i=1,2, ..` is obtained which converges (we hope) to the solution
-:math:`{\boldsymbol{u}}_{t}`. The most frequently used iterative
-schemes, such as Newton-Raphson, modified Newton, and quasi Newton
-schemes, are based on a Taylor expansion of
-equation `genForm <#genForm>`__ about :math:`{\boldsymbol{u}}_{t}`:
+   .. math::
 
-.. math::
+      \boldsymbol{r}({\boldsymbol{u}}_{t}) = 
+      \boldsymbol{r}({\boldsymbol{u}}_{t}^{(i)}) +
+      \left[ {\frac{\partial \boldsymbol{r}}{\partial {\boldsymbol{u}}_t} \vert}_{{\boldsymbol{u}}_{t}^{(i)}}\right]
+      \left( {\boldsymbol{u}}_{t} - {\boldsymbol{u}}_{t}^{(i)} \right)
 
-   \boldsymbol{r}({\boldsymbol{u}}_{t}) = 
-   \boldsymbol{r}({\boldsymbol{u}}_{t}^{(i)}) +
-   \left[ {\frac{\partial \boldsymbol{r}}{\partial {\boldsymbol{u}}_t} \vert}_{{\boldsymbol{u}}_{t}^{(i)}}\right]
-   \left( {\boldsymbol{u}}_{t} - {\boldsymbol{u}}_{t}^{(i)} \right)
+   .. math::
 
-.. math::
+      \boldsymbol{r}({\boldsymbol{u}}_{t}) = \boldsymbol{p}_f (t) - \boldsymbol{p}_{\mathrm{i}} \left( \ddot {\boldsymbol{u}}_{t}^{(i)} \right) - \boldsymbol{p}_{\sigma} \left( \dot {\boldsymbol{u}}_{t}^{(i)}, {\boldsymbol{u}}_{t}^{(i)} \right)- \left[
+      \boldsymbol{M}^{(i)} {\mathrm{I}}_2'
+      + \boldsymbol{C}^{(i)} {\mathrm{I}}_1'
+      + \boldsymbol{K}^{(i)}  \right]
+      \left( {\boldsymbol{u}}_{t} - {\boldsymbol{u}}_{t}^{(i)} \right)
+      %\label{femGenFormTaylor}
 
-   \boldsymbol{r}({\boldsymbol{u}}_{t}) = \boldsymbol{p}_f (t) - \boldsymbol{p}_{\mathrm{i}} \left( \ddot {\boldsymbol{u}}_{t}^{(i)} \right) - \boldsymbol{p}_{\sigma} \left( \dot {\boldsymbol{u}}_{t}^{(i)}, {\boldsymbol{u}}_{t}^{(i)} \right)- \left[
-     \boldsymbol{M}^{(i)} {\mathrm{I}}_2'
-   + \boldsymbol{C}^{(i)} {\mathrm{I}}_1'
-   + \boldsymbol{K}^{(i)}  \right]
-    \left( {\boldsymbol{u}}_{t} - {\boldsymbol{u}}_{t}^{(i)} \right)
-   %\label{femGenFormTaylor}
-
-To start the iteration scheme, trial values for
-:math:`{\boldsymbol{u}}_{t}`, :math:`\dot
-{\boldsymbol{u}}_{t}` and :math:`\ddot {\boldsymbol{u}}_{t}` are
-required. These are obtained by assuming
-:math:`{\boldsymbol{u}}_{t}^{(0)} = {\boldsymbol{u}}_{t-\Delta t}`. The
-:math:`\dot {\boldsymbol{u}}_{t}^{(0)}` and
-:math:`\ddot {\boldsymbol{u}}_{t}^{(0)}` can then be obtained from the
-operators for the integration scheme.
+   To start the iteration scheme, trial values for
+   :math:`{\boldsymbol{u}}_{t}`, :math:`\dot
+   {\boldsymbol{u}}_{t}` and :math:`\ddot {\boldsymbol{u}}_{t}` are
+   required. These are obtained by assuming
+   :math:`{\boldsymbol{u}}_{t}^{(0)} = {\boldsymbol{u}}_{t-\Delta t}`. The
+   :math:`\dot {\boldsymbol{u}}_{t}^{(0)}` and
+   :math:`\ddot {\boldsymbol{u}}_{t}^{(0)}` can then be obtained from the
+   operators for the integration scheme.