From 9f2758130243e46eaa8cc6b25a31abfccda651de Mon Sep 17 00:00:00 2001 From: moe-ad Date: Tue, 14 Jan 2025 11:16:45 +0100 Subject: [PATCH] feat: more content --- doc/source/conf.py | 2 +- .../extended_examples/sfem/realizations.png | Bin 0 -> 131959 bytes .../extended_examples/sfem/stochastic_fem.rst | 30 +++++++++++++++--- 3 files changed, 26 insertions(+), 6 deletions(-) create mode 100644 doc/source/examples/extended_examples/sfem/realizations.png diff --git a/doc/source/conf.py b/doc/source/conf.py index c7fc60490f..13506bc8e4 100755 --- a/doc/source/conf.py +++ b/doc/source/conf.py @@ -99,7 +99,7 @@ "sphinx_design", "sphinx_jinja", "sphinx_copybutton", - # "sphinx_gallery.gen_gallery", + "sphinx_gallery.gen_gallery", "sphinxemoji.sphinxemoji", "sphinx.ext.graphviz", "ansys_sphinx_theme.extension.linkcode", diff --git a/doc/source/examples/extended_examples/sfem/realizations.png b/doc/source/examples/extended_examples/sfem/realizations.png new file mode 100644 index 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zvji=SlJ?fop+l^de)a{FbOQN6iGwmmJ}B?VG%gO#u2>1&FKYQ|d_3@6`sJDj`)zv( zVK2_4^vyVeOw6G=GASiRU+OL>`@L*zo@bA{k_9`C1vrR&GH8`w>{Wu`oZCE6UjP@^ zWB)SfN`l*6ur4jiVZUtu<4BaTxw-id7@Tl=FIb+-Zvyt90yU7MTwPlCpHLOygOY`W k{PX{<0LXX#FE?H13?t67JBaGLP{5D6ijHy?_Trs?0h`V=a{vGU literal 0 HcmV?d00001 diff --git a/doc/source/examples/extended_examples/sfem/stochastic_fem.rst b/doc/source/examples/extended_examples/sfem/stochastic_fem.rst index 9d90a85f1d..28c21bf89a 100644 --- a/doc/source/examples/extended_examples/sfem/stochastic_fem.rst +++ b/doc/source/examples/extended_examples/sfem/stochastic_fem.rst @@ -84,13 +84,13 @@ Here: For example, :math:`E(x)` could be a Gaussian random field, in which case it has the stationarity property, making its statistics completely defined by its mean (:math:`\mu_E`), standard deviation -(:math:`\sigma_E`) and covariance function :math:`C_E(x_1,x_2)`. This 'stationarity' simply means +(:math:`\sigma_E`) and covariance function :math:`C_E(x_i,x_j)`. This 'stationarity' simply means that the mean and standard deviation of every random variable :math:`E(x)` is constant and equal to -:math:`\mu_E` and :math:`\sigma_E` respectively. :math:`C_E(x_1,x_2)` describes how random variables -:math:`E(x_1)` and :math:`E(x_2)` are related. +:math:`\mu_E` and :math:`\sigma_E` respectively. :math:`C_E(x_i,x_j)` describes how random variables +:math:`E(x_i)` and :math:`E(x_j)` are related. For a zero-mean Gaussian random field, the covariance function is given by: -.. math:: C_E(x_1,x_2) = \sigma_E^2e^{-\frac{\lvert x_1-x_2 \rvert}{\ell}} +.. math:: C_E(x_i,x_j) = \sigma_E^2e^{-\frac{\lvert x_i-x_j \rvert}{\ell}} where :math:`\sigma_E^2` is the variance, and :math:`\ell` is the correlation length parameter. @@ -100,4 +100,24 @@ realization/sample function assigned to each outcome of an experiment. .. figure:: realizations.png - A random field as a collection of random variables or realizations \ No newline at end of file + A random field as a collection of random variables or realizations + +.. note:: + The concepts above generalize to more dimensions, for example, a random vector instead of a random + variable, or an :math:`\mathbb{R}^d`-valued stochastic process. The presentation above is however + sufficient for this example. + +Series expansion of stochastic processes +---------------------------------------- +Since a stochastic processes involves an infinite number of random variables, most engineering applications +involving stochastic processes will be mathematically and computationally intractable if there isn't a way of +approximating them with a series of a finite number of random variables. A series expansion method which will +be used in this example is explained next. + +The Karhunen-Loève (K-L) series expansion +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +For a zero-mean stationary gaussian process, :math:`X(t)`, with covariance function +:math:`C_X(t_i,t_j)=\sigma_X^2e^{-\frac{\lvert t_i-t_j \rvert}{b}}` defined on a domain :math:`\mathbb{D}=[-a,a]`, +the K-L series expansion is given by: + +.. math:: X(t) = \sum_{n=1}^\infty \sqrt{\lambda_{n,c}}\cdot\phi_{n,c}(t)\cdot\xi_{n,c} + \sum_{n=1}^\infty \sqrt{\lambda_{n,s}}\cdot\phi_{n,s}(t)\cdot\xi_{n,s},\quad t\in\mathbb{D} \ No newline at end of file