feat: activate mathjax ext and add base rst

doc/source/conf.py

@@ -103,6 +103,7 @@
     "sphinxemoji.sphinxemoji",
     "sphinx.ext.graphviz",
     "ansys_sphinx_theme.extension.linkcode",
+    "sphinx.ext.mathjax",
 ]
 
 # Intersphinx mapping

doc/source/examples/extended_examples/index.rst

@@ -25,6 +25,8 @@ with other programs, libraries, and features in development.
 +------------------------------------------------------+--------------------------------------------------------------------------------------------+
 | :ref:`hpc_ml_ga_example`                             |  Demonstrates how to use PyMAPDL in a high-performance computing system managed by SLURM.  |
 +------------------------------------------------------+--------------------------------------------------------------------------------------------+
+| :ref:`stochastic_fem_example`                        |  Demonstrates using PyMAPDL for stochastic FEA via the Monte Carlo simulation.             |
++------------------------------------------------------+--------------------------------------------------------------------------------------------+
 
 
 .. toctree::

doc/source/examples/extended_examples/sfem/stochastic_fem.rst

@@ -0,0 +1,40 @@
+.. _stochastic_fem_example:
+
+Stochastic finite element method with PyMAPDL
+=============================================
+
+This example leverages PyMAPDL for stochastic finite element analysis via the Monte Carlo simulation.
+Numerous advantages / workflow possibilities that PyMAPDL affords users is demonstrated through this
+extended example. Important concepts are first explained before the example is presented.
+
+Introduction
+------------
+Often in a mechanical system, system parameters (geometry, materials, loads, etc.) and response parameters
+(displacement, strain, stress, etc) are taken to be deterministic. This simplification, while sufficient for a
+wide range of engineering applications, results in a crude approximation of actual system behaviour.
+
+To obtain a more accurate representation of a physical system, it is essential to consider the randomness
+in system parameters and loading conditions, along with the uncertainty in their estimation and their
+spatial variability. The finite element method is undoubtedly the most widely used tool for solving deterministic
+physical problems today and to account for randomness and uncertainty in the modeling of engineering systems,
+the stochastic finite element method (SFEM) was introduced.
+
+The stochastic finite element method (SFEM) extends the classical deterministic finite element approach
+to a stochastic framework, offering various techniques for calculating response variability. Among these,
+the Monte Carlo simulation (MCS) stands out as the most prominent method. Renowned for its versatility and
+ease of implementation, MCS can be applied to virtually any type of problem in stochastic analysis.
+
+Random variables vs stochastic processes
+----------------------------------------
+A distinction between random variables and stochastic processes (also called random fields) is attempted in this
+section. Explaining these concepts is important since they are used for modelling the system randomness.
+Random variables are easier to understand from elementary probability theory, the same cannot be said for stochastic
+processes. Readers are advised to consult books on SFEM if the explanation here seems to brief.
+
+Random variables
+~~~~~~~~~~~~~~~~
+Imagine a beam with a concentrated load :math:`P` applied at a specific point on the beam. The value of :math:`P`
+is uncertain — it could vary due to manufacturing tolerances, loading conditions, or measurement errors. Mathematically,
+:math:`P` is a random variable:
+
+.. math:: P : \Omega \longrightarrow \mathbb{R}