Merge remote-tracking branch 'origin/feature/documentation_improvemen…

…ts' into feature/documentation_improvements

docs/code/reliability/form/plot_FORM_structural_reliability.py

@@ -3,7 +3,7 @@
 3. FORM - Structural Reliability
 ==============================================
 
-The benchmark problem is a simple structural reliability problem
+The benchmark problem is a simple structural reliability problem (example 7.1 in :cite:`FORM_XDu`)
 defined in a two-dimensional parameter space consisting of a resistance :math:`R` and a stress :math:`S`. The failure
 happens when the stress is higher than the resistance, leading to the following limit-state function:
 
@@ -61,7 +61,7 @@
 # %% md
 #
 # With everything defined we are ready to run the first-order reliability method and print the results.
-# The analytic solution to this problem is :math:`\bm{u}^*=(-2, 1)` with a reliability index of
+# The analytic solution to this problem is :math:`\textbf{u}^*=(-2, 1)` with a reliability index of
 # :math:`\beta_{HL}=2.2361` and a probability of failure :math:`P_{f, \text{form}} = \Phi(-\beta_{HL}) = 0.0127`
 
 # %%
@@ -75,8 +75,8 @@
 
 # %% md
 #
-# This problem can be visualized in the following plots that show the FORM results in both :math:`\bm{X}` and
-# :math:`\bm{U}` space.
+# This problem can be visualized in the following plots that show the FORM results in both :math:`\textbf{X}` and
+# :math:`\textbf{U}` space.
 
 # %%
 

docs/source/bibliography.bib

@@ -879,3 +879,11 @@ @article{Kle2D
 author = {Zhibao Zheng and Hongzhe Dai},
 keywords = {Multi-dimensional random field, Karhunen–Loève expansion, Random field simulation, Fredholm integral equation},
 }
+
+@misc{FORM_XDu,
+  title     = "Probabilistic Engineering Design, Chapter 7, First Order and Second Reliability Methods",
+  author    = "Xiaoping Du",
+  year      = 2005,
+  publisher   = "University of Missouri – Rolla",
+  howpublished = "\url{https://pdesign.sitehost.iu.edu/me360/ch7.pdf}"
+}

docs/source/reliability/form.rst

@@ -2,7 +2,7 @@ FORM
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
 
-In FORM, the performance function is linearized according to
+In FORM :cite:`FORM_XDu`, the performance function is linearized according to
 
 .. math:: G(\textbf{U})  \approx  G(\textbf{U}^\star) + \nabla G(\textbf{U}^\star)(\textbf{U}-\textbf{U}^\star)^\intercal
 
@@ -17,9 +17,9 @@ is the norm of the design point known as the Hasofer-Lind reliability index calc
 Hasofer-Lind-Rackwitz-Fiessler (HLRF) algorithm.  The convergence criteria used for HLRF algorithm are:
 
 
-.. math:: \text{tolerance}_U:\ ||\textbf{U}^{k} - \textbf{U}^{k-1}||_2 \leq 10^{-3}
+.. math:: \text{tolerance}_{\textbf{U}}:\ ||\textbf{U}^{k} - \textbf{U}^{k-1}||_2 \leq 10^{-3}
 .. math:: \text{tolerance}_\beta:\ ||\beta_{HL}^{k} - \beta_{HL}^{k-1}||_2 \leq 10^{-3}
-.. math:: \text{tolerance}_{\nabla G(U)}:\ ||\nabla G(\textbf{U}^{k})- \nabla G(\textbf{U}^{k-1})||_2 \leq 10^{-3}
+.. math:: \text{tolerance}_{\nabla G(\textbf{U})}:\ ||\nabla G(\textbf{U}^{k})- \nabla G(\textbf{U}^{k-1})||_2 \leq 10^{-3}
 
 
 

docs/source/reliability/sorm.rst

@@ -1,10 +1,10 @@
 SORM
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
-In SORM the performance function is approximated by a second-order Taylor series around the design point according to
+In SORM :cite:`FORM_XDu` the performance function is approximated by a second-order Taylor series around the design point according to
 
 
-.. math:: G(\textbf{U}) = G(\textbf{U}^\star) +  \nabla G(\textbf{U}^\star)(\textbf{U}-\textbf{U}^\star)^\intercal + \frac{1}{2}(\textbf{U}-\textbf{U}^\star)\textbf{H}(\textbf{U}-\textbf{U}^\star)
+.. math:: G(\textbf{U}) = G(\textbf{U}^\star) +  \nabla G(\textbf{U}^\star)(\textbf{U}-\textbf{U}^\star)^\intercal + \frac{1}{2}(\textbf{U}-\textbf{U}^\star)\textbf{H}(\textbf{U}-\textbf{U}^\star)(\textbf{U}-\textbf{U}^\star)^T
 
 where :math:`\textbf{H}` is the Hessian matrix of the second derivatives of :math:`G(\textbf{U})` evaluated at
 :math:`\textbf{U}^*`. After the design point :math:`\textbf{U}^*` is identified and the probability of failure

src/UQpy/reliability/taylor_series/FORM.py

@@ -115,7 +115,7 @@ def __init__(
         """Design point in the parameter space X."""
         self.error_record: list = []
         """Record of the error defined by 
-         criteria :math:`\\text{tolerance}_U, \\text{tolerance}_\\beta, \\text{tolerance}_{\\nabla G(U)}`."""
+         criteria :math:`\\text{tolerance}_\\textbf{U}, \\text{tolerance}_\\beta, \\text{tolerance}_{\\nabla G(\\textbf{U})}`."""
         self.failure_probability: list = []
         """FORM probability of failure :math:`\\Phi(-\\beta)`."""
         self.g0 = None