Commit ecfda613 authored by Turnhout, M.C. van's avatar Turnhout, M.C. van
Browse files

ani delta kappa

parent 153d3393
......@@ -57,6 +57,12 @@
\usepackage{svgimport}
\usepackage{framed}
% \usepackage{coffee4}
%% tables
\usepackage{tabularx}
\usepackage{colortbl}
\usepackage{multicol}
%% floats
\usepackage{placeins}
% indexing
\usepackage{makeidx}
......@@ -68,16 +74,10 @@
\newcommand\paraminc{\texttt{param.inc}\index{(file)~param.inc@(file)~\texttt{param.inc}}}
\makeindex
%% tables
\usepackage{tabularx}
\usepackage{colortbl}
\usepackage{multicol}
% language
\usepackage[UKenglish]{babel}
\selectlanguage{UKenglish}
% document specific conventions
\newcommand\warning{\marginpar{\center\includegraphics[width=1cm]{Exclamation.png}\\}}
\newcommand\curver{\href{http://mategit.wfw.wtb.tue.nl/STEM/TFlab/tags}{v0.2}}
......
......@@ -523,13 +523,15 @@ The 95\,\% confidence intervals of the other two coefficients, $a_2$ for $\sigma
\subsection{Assumption of no contraction}
We can do the maths (table \ref{anispace_dL_tab}):
\input{anispace_dLl_eq.tex}
\input{anispace_dLr_eq.tex}
And we can plot the deformations on a scale that makes it look like something is going on (figure \ref{anispace_dL_fig}), but any discussion on `effects' would be a waste of time.
\begin{align}
\lambda_L^\text{l}(\sigma_x = 0.015\,\text{MPa}) &= 1.00 -1.12\cdot 10^{-3} \mu -9.52\cdot 10^{-4} \sigma + 1.71\cdot 10^{-3} P + \text{error} \label{anispace_dLl_eq}\\
\lambda_L^\text{r}(\sigma_x = 0.015\,\text{MPa}) &= 1.00 -9.63\cdot 10^{-4} \mu -9.24\cdot 10^{-4} \sigma + 1.51\cdot 10^{-3} P + \text{error} \label{anispace_dLr_eq}
\end{align}
And we can plot the deformations on a scale that makes it look like something is going on (figure \ref{anispace_dL_fig}), but any discussion on `effects' would be a waste of time. Not just at $\sigma_x = 0.015$\,MPa, but over the entire investigated range (figure \ref{pspaceani_dL_as}).
Nothing happens here.
Nothing happens here.\\
Anisotropic sideways curling seems even more a purely geometrical affair (no lengthening/shortening of the constructs) than isotropic curling.
\noindent Anisotropic sideways curling seems even more a purely geometrical affair (no lengthening/shortening of the constructs) than isotropic curling.
\begin{table}[h!]
\caption{Multiple linear regression model fitted to $\lambda_L$ for $\sigma_x = 0.015$\,MPa for the 100 random models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_dL_tab}}
......@@ -539,6 +541,7 @@ Anisotropic sideways curling seems even more a purely geometrical affair (no len
\subfloat[\label{anispace_dLr_tab}]{%
\input{anispace_dLr_tab.tex}}
\end{table}
\FloatBarrier
\begin{figure}[hp!]
\subfloat[\label{anispace_dLl_fig}]{%
\def\svgwidth{0.85\linewidth}\includesvg{pics/anispace_dLl_fig}}\\
......@@ -546,41 +549,46 @@ Anisotropic sideways curling seems even more a purely geometrical affair (no len
\def\svgwidth{0.85\linewidth}\includesvg{pics/anispace_dLr_fig}}
\caption{Distribution of $\lambda_L$ (equation \ref{eq:lambdal}) for the 100 anisotropic models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge.\label{anispace_dL_fig}}
\end{figure}
\begin{figure}[hp!]
\tiny
\subfloat[\label{pspaceani_dLl_a0}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a0}}\hfill
\subfloat[\label{pspaceani_dLr_a0}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a0}}\\
\subfloat[\label{pspaceani_dLl_a1}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a1}}\hfill
\subfloat[\label{pspaceani_dLr_a1}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a1}}\\
\subfloat[\label{pspaceani_dLl_a2}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a2}}\hfill
\subfloat[\label{pspaceani_dLr_a2}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a2}}\\
\subfloat[\label{pspaceani_dLl_a3}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a3}}\hfill
\subfloat[\label{pspaceani_dLr_a3}]{%
\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a3}}\\
\caption{Coefficients of the equations for $\lambda_L$ (red) with the left edge on the left (equation \ref{anispace_dLl_eq}) and the right edge on the right (equation \ref{anispace_dLr_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_dL_as}}
\end{figure}
%\begin{figure}
%\tiny
%\subfloat[\label{pspaceani_dLl_a0}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a0}}\hfill
%\subfloat[\label{pspaceani_dLr_a0}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a0}}\\
%\subfloat[\label{pspaceani_dLl_a1}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a1}}\hfill
%\subfloat[\label{pspaceani_dLr_a1}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a1}}\\
%\subfloat[\label{pspaceani_dLl_a2}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a2}}\hfill
%\subfloat[\label{pspaceani_dLr_a2}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a2}}\\
%\subfloat[\label{pspaceani_dLl_a3}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLl_a3}}\hfill
%\subfloat[\label{pspaceani_dLr_a3}]{%
%\def\svgwidth{0.37\linewidth}\includesvg{pics/pspaceani_dLr_a3}}\\
%\caption{Coefficients of the equations for $\lambda_L$ (red) with the left edge on the left (equation \ref{anispace_dLl_eq}) and the right edge on the right (equation \ref{anispace_dLr_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_dL_as}}
%\end{figure}
\subsection{Assumption of constant curvature}
Before we assess we look at the difference in curvature between the left and right bottom edges. We calculate the mean local curvature $\kappa$ for each edge, and normalise the difference between these values with the average curvature of the two edges (figure \ref{anispace_dk_fig}):
\begin{equation}
\Deltaup \kappa = \frac{2\left(\kappa^\text{left} - \kappa^\text{right}\right)}{\kappa^\text{left} + \kappa^\text{right}}
\end{equation}
At $\sigma_x = 0.015$\,MPa, the model (equation \ref{aregress}) predicts for this difference (table \ref{anispace_dk_tab}):
\input{anispace_dk_eq.tex}
The intercept of a difference of about 40\,\% seems to represent the modus of the highly skewed distribution of $\Deltaup \kappa$ over a wide range of $\sigma_x$ (figure \ref{anispace_dk_fig}), but the model does not fit well and fails to explain most of the variation in the data ($R_a^2 = 0.10$, SS$_\text{error}$/SS$_\text{tot} = 0.86$).
\input{anispace_dk_eq.tex}
\begin{table}
\begin{table}[hp]
\center\small
\caption{Multiple linear regression model fitted to $\Deltaup \kappa$ for $\sigma_x = 0.015$\,MPa for the 100 random models. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{isospace_dk_tab}}
\caption{Multiple linear regression model fitted to $\Deltaup \kappa$ for $\sigma_x = 0.015$\,MPa for the 100 random models. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_dk_tab}}
\input{anispace_dk_tab.tex}
\end{table}
\begin{figure}
\begin{figure}[hp]
\center
\def\svgwidth{0.95\linewidth}\includesvg{pics/anispace_dk_fig}
\caption{Distribution of $\Deltaup \kappa$ for the 100 anisotropic models. \label{anispace_dk_fig}}
......@@ -598,9 +606,14 @@ Anisotropic sideways curling seems even more a purely geometrical affair (no len
\def\svgwidth{0.47\linewidth}\includesvg{pics/pspaceani_dk_a3}}\\
\caption{Coefficients of equation \ref{anispace_dk_eq} ($\Deltaup \kappa$, red) with \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} the mean of the Gaussian distribution $\mu$ ($a_1$), \textbf{(c)} the standard deviation of the Gaussian distribution $\sigma$ ($a_2$), and \textbf{(d)} the anisotropic fraction $P$ ($a_3$). \label{pspaceani_dk_as}}
\end{figure}
\FloatBarrier
\input{anispace_kel_eq.tex}
\input{anispace_ker_eq.tex}
\begin{align}
\kappa_\varepsilon^\text{l}(\sigma_x = 0.015\,\text{MPa}) & = 6.36\cdot 10^{-1} -3.17\cdot 10^{-1} \mu -1.66\cdot 10^{-1} \sigma + 5.21\cdot 10^{-2} P + \text{error} \label{anispace_kel_eq}\\
\kappa_\varepsilon^\text{r}(\sigma_x = 0.015\,\text{MPa}) &= 6.39\cdot 10^{-1} -3.56\cdot 10^{-1} \mu -1.52\cdot 10^{-1} \sigma -3.33\cdot 10^{-2} P + \text{error} \label{anispace_ker_eq}
\end{align}
\begin{table}
\caption{Multiple linear regression model fitted to $\kappa_\varepsilon$ for $\sigma_x = 0.015$\,MPa for the 100 random models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_ke_tab}}
......@@ -639,8 +652,12 @@ Anisotropic sideways curling seems even more a purely geometrical affair (no len
\caption{Coefficients of the equations for $\kappa_\varepsilon$ (red) with the left edge on the left (equation \ref{anispace_kel_eq}) and the right edge on the right (equation \ref{anispace_ker_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_ke_as}}
\end{figure}
\input{anispace_Rel_eq.tex}
\input{anispace_Rer_eq.tex}
\begin{align}
R_\varepsilon^\text{l}(\sigma_x = 0.015\,\text{MPa}) & = 0.953 -9.04\cdot 10^{-2} \mu + 2.07\cdot 10^{-2} \sigma + 2.53\cdot 10^{-1} P + \text{error} \label{anispace_Rel_eq}\\
R_\varepsilon^\text{r}(\sigma_x = 0.015\,\text{MPa}) & = 1.47 -1.97\cdot 10^{-1} \mu -3.34\cdot 10^{-2} \sigma + 3.07\cdot 10^{-1} P + \text{error} \label{anispace_Rer_eq}
\end{align}
%\input{anispace_Rel_eq.tex}
%\input{anispace_Rer_eq.tex}
\begin{table}
\caption{Multiple linear regression model fitted to $R_\varepsilon$ for $\sigma_x = 0.015$\,MPa for the 100 random models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_Re_tab}}
......
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