From ace66c78c7fd4aec256cf8a652bb364c24d1eab6 Mon Sep 17 00:00:00 2001
From: Lambert Theisen <lambert.theisen@rwth-aachen.de>
Date: Mon, 9 Dec 2019 13:15:27 +0000
Subject: [PATCH] Add more documention to assemble
- Derive the used identities
---
src/solver.py | 159 ++++++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 159 insertions(+)
diff --git a/src/solver.py b/src/solver.py
index 6a113a6..043cb86 100644
--- a/src/solver.py
+++ b/src/solver.py
@@ -277,6 +277,165 @@ class Solver:
.. [5] M. Torrilhon, N. Sarna (2017). Hierarchical Boltzmann
Simulations and Model Error Estimation
+ Identities:
+
+ - :math:`\boldsymbol{x}_1 \in \mathbb{R}^3`,
+ :math:`\boldsymbol{x}_2 \in \mathbb{R}^{3 \times 3}`,
+ ...
+ - Test function :math:`\boldsymbol{\psi}` is assumed to be
+ symmetric and trace-less:
+
+ .. math::
+
+ \boldsymbol{\psi} : \boldsymbol{I} = 0 \\
+ (\boldsymbol{\psi})_{\text{sym}} = \boldsymbol{\psi}
+
+ - Trace of vector gradient is divergence of vector:
+
+ .. math::
+
+ \langle
+ \boldsymbol{I},\boldsymbol\nabla \boldsymbol{x}_1
+ \rangle
+ =
+ \boldsymbol{I} : \boldsymbol\nabla \boldsymbol{x}_1
+ =
+ \text{div}(\boldsymbol{x}_1)
+
+ - Inner product has orthogonality property with respect to the
+ additive symmetric/skewsymmetric tensor decomposition:
+
+ .. math::
+
+ \begin{align}
+ \langle
+ (\boldsymbol{x}_2)_{\text{sym}}
+ ,
+ \boldsymbol{y}_2
+ \rangle
+ &=
+ \langle
+ (\boldsymbol{x}_2)_{\text{sym}}
+ ,
+ (\boldsymbol{y}_2)_{\text{sym}}
+ +
+ (\boldsymbol{y}_2)_{\text{skew}}
+ \rangle
+ \\
+ &=
+ \langle
+ (\boldsymbol{x}_2)_{\text{sym}}
+ ,
+ (\boldsymbol{y}_2)_{\text{sym}}
+ \rangle
+ +
+ \langle
+ (\boldsymbol{x}_2)_{\text{sym}}
+ ,
+ (\boldsymbol{y}_2)_{\text{skew}}
+ \rangle
+ \\
+ &=
+ \langle
+ (\boldsymbol{x}_2)_{\text{sym}}
+ ,
+ (\boldsymbol{y}_2)_{\text{sym}}
+ \rangle
+ \end{align}
+
+ Tricks of the trade:
+
+ .. math::
+
+ \begin{align}
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{STF}}
+ ,
+ \boldsymbol\nabla \boldsymbol{r}
+ \rangle
+ &=
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{sym}}
+ -
+ \frac13\text{tr}(\boldsymbol\nabla \boldsymbol{s})\boldsymbol{I}
+ ,
+ \boldsymbol\nabla \boldsymbol{r}
+ \rangle
+ \\
+ &=
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{sym}}
+ ,
+ \boldsymbol\nabla \boldsymbol{r}
+ \rangle
+ -\frac13\text{tr}(\boldsymbol\nabla \boldsymbol{s})
+ \langle
+ \boldsymbol{I},(\boldsymbol\nabla \boldsymbol{r})
+ \rangle
+ \\
+ &=
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{sym}}
+ ,
+ {(\boldsymbol\nabla \boldsymbol{r})}_{\text{sym}}
+ \rangle
+ -\frac13\text{tr}(\boldsymbol\nabla \boldsymbol{s})
+ \text{tr}(\boldsymbol\nabla \boldsymbol{r})
+ \\
+ &=
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{sym}}
+ ,
+ {(\boldsymbol\nabla \boldsymbol{r})}_{\text{sym}}
+ \rangle
+ -\frac13\text{div}(\boldsymbol{s})
+ \text{div}(\boldsymbol{r})
+ \end{align}
+
+ .. math::
+
+ \begin{align}
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{STF}}
+ ,
+ \boldsymbol{\psi}
+ \rangle
+ &=
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{sym}}
+ -
+ \frac13\text{tr}(\boldsymbol\nabla \boldsymbol{s})\boldsymbol{I}
+ ,
+ \boldsymbol{\psi}
+ \rangle
+ \\
+ &=
+ \langle
+ {(\boldsymbol\nabla \boldsymbol{s})}_{\text{sym}}
+ ,
+ \boldsymbol{\psi}
+ \rangle
+ -\frac13\text{tr}(\boldsymbol\nabla \boldsymbol{s})
+ \langle
+ \boldsymbol{I},\boldsymbol{\psi}
+ \rangle
+ \\
+ &=
+ \langle
+ {\boldsymbol\nabla \boldsymbol{s}}
+ ,
+ {\boldsymbol{\psi}}
+ \rangle
+ -\frac13\text{div}(\boldsymbol{s})
+ \text{tr}(\boldsymbol{\psi})
+ \\
+ &=
+ \langle
+ {\boldsymbol\nabla \boldsymbol{s}}
+ ,
+ {\boldsymbol{\psi}}
+ \rangle
+ \end{align}
"""
# Check if all mesh boundaries have bcs presibed frm input
--
GitLab