Commit 8afa89e8 by Turnhout, M.C. van

### finished iso(?) (now, lets try some FibLab :-)) )

parent 16876bac
 \begin{align} \Deltaup R(\sigma_x = 0.025\,\text{MPa}) = 1.03 & + 1.04\cdot 10^{-3} L + 4.04\cdot 10^{-3} w \notag \\ R_\varepsilon(\sigma_x = 0.025\,\text{MPa}) = 1.03 & + 1.04\cdot 10^{-3} L + 4.04\cdot 10^{-3} w \notag \\ & -1.50\cdot 10^{-2} E_\text{film} + 8.40\cdot 10^{-2} E_\text{cells} \notag \\ & + 1.66\cdot 10^{-2} \nu_\text{film} + 7.25\cdot 10^{-3} \nu_\text{cells} \notag \\ & -5.17 t_\text{film} + 1.10\cdot 10^{1} t_\text{cells} + \text{error} \label{isospace_dR_eq} ... ...
 \begin{tabular}{l c c r r r} \hline\noalign{\smallskip} $\Deltaup R(\sigma_x = 0.025\,\text{MPa}) =$ &95\,\% CI & SS/SS$_\text{tot}$ & $p$ & range & fraction\\ $R_\varepsilon(\sigma_x = 0.025\,\text{MPa}) =$ &95\,\% CI & SS/SS$_\text{tot}$ & $p$ & range & fraction\\ \noalign{\smallskip}\hline\noalign{\bigskip} $1.03$ & $\left[9.62\cdot 10^{-1} \quad 1.11\right]$ & & $0.000$&&\\ $+1.04\cdot 10^{-3} L$ & $\left[-9.25\cdot 10^{-3} \quad 1.13\cdot 10^{-2}\right]$ & $0.000$ & $0.841$ & $4.02\cdot 10^{-3}$ & $0.007$\\ ... ...
 \begin{align} \Deltaup \kappa(\sigma_x = 0.025\,\text{MPa}) = 1.73\cdot 10^{-1} & -1.33\cdot 10^{-3} L -1.12\cdot 10^{-3} w \notag \\ \kappa_\varepsilon(\sigma_x = 0.025\,\text{MPa}) = 1.73\cdot 10^{-1} & -1.33\cdot 10^{-3} L -1.12\cdot 10^{-3} w \notag \\ & -1.09\cdot 10^{-2} E_\text{film} + 3.03\cdot 10^{-1} E_\text{cells} \notag \\ & -3.87\cdot 10^{-2} \nu_\text{film} -2.38\cdot 10^{-2} \nu_\text{cells} \notag \\ & -6.45 t_\text{film} + 6.11 t_\text{cells} + \text{error} \label{isospace_dk_eq} ... ...
 \begin{tabular}{l c c r r r} \hline\noalign{\smallskip} $\Deltaup \kappa(\sigma_x = 0.025\,\text{MPa}) =$ &95\,\% CI & SS/SS$_\text{tot}$ & $p$ & range & fraction\\ $\kappa_\varepsilon(\sigma_x = 0.025\,\text{MPa}) =$ &95\,\% CI & SS/SS$_\text{tot}$ & $p$ & range & fraction\\ \noalign{\smallskip}\hline\noalign{\bigskip} $1.73\cdot 10^{-1}$ & $\left[1.03\cdot 10^{-1} \quad 2.43\cdot 10^{-1}\right]$ & & $0.000$&&\\ $-1.33\cdot 10^{-3} L$ & $\left[-1.13\cdot 10^{-2} \quad 8.59\cdot 10^{-3}\right]$ & $0.001$ & $0.791$ & $-5.11\cdot 10^{-3}$ & $-0.011$\\ ... ...
 ... ... @@ -255,7 +255,7 @@ The coefficient for $E_\text{cells}$ has a minimum and maximum (figure \ref{pspa \subsection{Assumption of constant curvature} Because one edge of the film is not completely free to curve, it is attached to the world', we can expect that curvature can not keep up with with the rest of the film at that edge. An extended analysis of local curvature along the long edge(s) of our default' isotropic film (table \ref{tbl:parameter_space}) shows that indeed local curvature $\vec{\kappa}$ at the fixed side of the edge(s) ($L_d = 0$\,mm) is limited in the simulations (figure \ref{lkexam}). \begin{figure} \begin{figure}[h!] \tiny \subfloat[\label{lkexama}]{% \def\svgwidth{0.3\linewidth}\raisebox{3em}{\includesvg{pics/lkexama}}}\hfill ... ... @@ -279,26 +279,26 @@ This film curved hardly over its width (figure \ref{lkexamc}, $\kappa_x \ll 1$), \item that the assumption of constant curvature \textsl{is} violated; and \item that there is a discrepancy between the actual average curvature over the length of the film and the global' radius of curvature as calculated from the projected edge length. \end{itemize} \noindent For each simulation, we calculate the mean and standard deviation of the local curvatures $\kappa$ along each long edge to calculate $\Deltaup \kappa$ (figure \ref{isospace_dk_fig}). \Deltaup \kappa = \frac{\frac{\mean\left(\kappa(\text{left bottom edge})\right) + \mean\left(\kappa(\text{right bottom edge})\right)}{2}}{ \frac{\std\left(\kappa(\text{left bottom edge})\right) + \std\left(\kappa(\text{right bottom edge})\right)}{2}} \label{eq:lk} I.e.\: we average the means and standard deviations of the two edges before analysis (similar to equation \ref{eq:lambdal}). Then, again with $\sigma_x = 0.025$\,MPa as a reference, we find: \input{isospace_dk_eq.tex} for the model of equation \ref{mregress} with $y = \Deltaup \kappa$ (table \ref{isospace_dk_tab}). \begin{table}[p] \center\small \caption{Multiple linear regression model fitted to $\Deltaup \kappa$ for $\sigma_x = 0.025$\,MPa for the 132 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 $\kappa_\varepsilon$ for $\sigma_x = 0.025$\,MPa for the 132 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}} \input{isospace_dk_tab.tex} \end{table} \begin{figure}[p] \center \def\svgwidth{0.95\linewidth}\includesvg{pics/isospace_dk_fig} \caption{Distribution of $\Deltaup \kappa$ for the 132 successfully build models. \label{isospace_dk_fig}} \caption{Distribution of $\kappa_\varepsilon$ for the 132 successfully build models. \label{isospace_dk_fig}} \end{figure} \begin{figure}[p] \noindent For each simulation, we calculate the mean and standard deviation of the local curvatures $\kappa$ along each long edge to calculate $\kappa_\varepsilon$ (figure \ref{isospace_dk_fig}). \kappa_\varepsilon = \frac{\frac{\mean\left(\kappa(\text{left bottom edge})\right) + \mean\left(\kappa(\text{right bottom edge})\right)}{2}}{ \frac{\std\left(\kappa(\text{left bottom edge})\right) + \std\left(\kappa(\text{right bottom edge})\right)}{2}} \label{eq:lk} I.e.\: we average the means and standard deviations of the two edges before analysis (similar to equation \ref{eq:lambdal}). Then, again with $\sigma_x = 0.025$\,MPa as a reference, we find: \input{isospace_dk_eq.tex} for the model of equation \ref{mregress} with $y = \kappa_\varepsilon$ (table \ref{isospace_dk_tab}). \begin{figure}[b!] \tiny \subfloat[\label{pspaceiso_dk_a0}]{% \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_dk_a0}}\hfill ... ... @@ -318,26 +318,35 @@ for the model of equation \ref{mregress} with $y = \Deltaup \kappa$ (table \ref{ \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_dk_a7}}\hfill \subfloat[\label{pspaceiso_dk_a8}]{% \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_dk_a8}}\\ \caption{Parameters of equation \ref{isospace_dk_eq} (red) with their 95\,\% confidence intervals (blue) as function of $\sigma_x$. With the coefficients for \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} film length $L$ $(a_1)$, \textbf{(c)} film width $w$ $(a_2)$, \textbf{(d)} film stiffness $E_\text{film}$ $(a_3)$, \textbf{(e)} cell layer stiffness $E_\text{cells}$ $(a_4)$, \textbf{(f)} film Poisson ratio $\nu_\text{film}$ $(a_5)$, \textbf{(g)} cell layer Poisson ratio $\nu_\text{cells}$ $(a_6)$, \textbf{(h)} film thickness $t_\text{film}$ $(a_7)$, and \textbf{(i)} cell layer thickness $t_\text{cells}$ $(a_8)$. \label{fig:pspaceiso_dk_as}} \caption{Coefficients of equation \ref{isospace_dk_eq} ($\kappa_\varepsilon$, red) with their 95\,\% confidence intervals (blue) as function of $\sigma_x$. With the coefficients for \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} film length $L$ $(a_1)$, \textbf{(c)} film width $w$ $(a_2)$, \textbf{(d)} film stiffness $E_\text{film}$ $(a_3)$, \textbf{(e)} cell layer stiffness $E_\text{cells}$ $(a_4)$, \textbf{(f)} film Poisson ratio $\nu_\text{film}$ $(a_5)$, \textbf{(g)} cell layer Poisson ratio $\nu_\text{cells}$ $(a_6)$, \textbf{(h)} film thickness $t_\text{film}$ $(a_7)$, and \textbf{(i)} cell layer thickness $t_\text{cells}$ $(a_8)$. \label{fig:pspaceiso_dk_as}} \end{figure} The parameters have little effect on $\kappa_\varepsilon$: $R_a^2 = 0.13$, SS$_\text{error}$/SS$_\text{tot} = 0.85$ and the effects are at least an order of magnitude smaller than the intercept. Of the 15\,\% of the variation left for the parameters to explain, half is due to $t_\text{cells}$ (SS/SS$_\text{tot} = 0.08$, $p = 0.003$). Since the parameters hardly affect $\kappa_\varepsilon$ at $\sigma_x = 0.025$\,MPa, it is not surprising that $\kappa_\varepsilon$ (figure \ref{isospace_dk_fig}) and the coefficients of equation \ref{isospace_dk_eq} (figure \ref{fig:pspaceiso_dk_as}) seem fairly constant over the range of $\sigma_x$ that we tested. Some coefficients hardly play any role at all (zero included in the 95\,\% confidence interval, noisy), e.g.\ $E_\text{cells}$ ($a_4$, figure \ref{pspaceiso_dk_a4}), $\nu_\text{film}$ ($a_5$, figure \ref{pspaceiso_dk_a5}), and $\nu_\text{cells}$ ($a_6$, figure \ref{pspaceiso_dk_a6}). The other coefficients only exclude zero from the 95\,\% confidence interval at small $\sigma_x$, e.g.\ $L$ ($a_1$, figure \ref{pspaceiso_dk_a1}), $w$ ($a_1$, figure \ref{pspaceiso_dk_a1}), $E_\text{film}$ ($a_3$, figure \ref{pspaceiso_dk_a3}), $t_\text{cells}$ ($a_8$, figure \ref{pspaceiso_dk_a8}) and indeed $t_\text{film}$ ($a_7$, figure \ref{pspaceiso_dk_a7}). The intercept itself increases with increasing $\sigma_x$ (figure \ref{pspaceiso_dk_a0}), but also the coefficients for most effects become less positive, or more negative, with increasing $\sigma_x$ (note that our parameters are positive, by definition): e.g.\ $w$ ($a_2$, figure \ref{pspaceiso_dk_a2}), $E_\text{film}$ ($a_3$, figure \ref{pspaceiso_dk_a3}) and $t_\text{film}$ ($a_7$, figure \ref{pspaceiso_dk_a7}). As a result, the distribution of $\kappa_\varepsilon$ shifts to a slightly larger value, and becomes slightly less sharp', for increasing values of $\sigma_x$ (figure \ref{isospace_dk_fig}). Overall, the standard deviation of the local curvature along the edges is about 10\,\% of the mean local curvature of the edges, and this value is hardly affected by the parameters or $\sigma_x$. Which suggests that the edge effects (figure \ref{lkexam}) scale linearly with geometry and curvature in isotropic films.\\ \noindent We finally quantify how much the curvature as assessed with the projected edge length deviates from the average curvature as calculated from the local $\vec{\kappa}$'s (equation \ref{eq:lk}) by calculating the fraction of the two radii of curvature $\Deltaup R$ (figure \ref{isospace_dR_fig}): \noindent We finally quantify how much the curvature as assessed with the projected edge length deviates from the average curvature as calculated from the local $\vec{\kappa}$'s (equation \ref{eq:lk}) by calculating the fraction of the two radii of curvature $R_\varepsilon$ (figure \ref{isospace_dR_fig}): \Deltaup R = \frac{R}{\frac{1}{\mean\left(\kappa\right)}} = R \mean\left(\kappa\right) R_\varepsilon = \frac{R}{\frac{1}{\mean\left(\absv{\vec{\kappa}}\right)}} = R \mean\left(\kappa\right) And with $y = R_\varepsilon$, equation \ref{mregress} then yields (table \ref{isospace_dR_tab}): \input{isospace_dR_eq.tex} The parameters have little effect on $R_\varepsilon$: $R_a^2 = 0.10$, SS$_\text{error}$/SS$_\text{tot} = 0.86$ (very similar to $\kappa_\varepsilon$, table \ref{isospace_dk_tab}). The intercept is slightly larger than 1, and the effects are all at least two magnitudes of order smaller than that intercept: they affect $R_\varepsilon$ by few percent, at most. Two of the three largest effects have a negative coefficient: $E_\text{film}$ (-0.015, SS/SS$_\text{tot} = 0.04$, $p = 0.045$) and $t_\text{film}$ (-5.17, SS/SS$_\text{tot} = 0.05$, $p = 0.020$). Together, they seem to overrule the positive coefficient of the third of the largest effects $t_\text{cells}$ (SS/SS$_\text{tot} = 0.04$, $p = 0.033$) so that the intercept overestimates the median (and certainly: modus, figure \ref{isospace_dR_fig}) of the distribution. Finally note that there is a very large (single) outlier at $R_\varepsilon = 1.53$ which is likely to remain unexplained' and adds to SS$_\text{error}$. \begin{table}[p] \center\small \caption{Multiple linear regression model fitted to $\Deltaup R$ for $\sigma_x = 0.025$\,MPa for the 132 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_dR_tab}} \caption{Multiple linear regression model fitted to $R_\varepsilon$ for $\sigma_x = 0.025$\,MPa for the 132 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_dR_tab}} \input{isospace_dR_tab.tex} \end{table} \begin{figure}[p] \center \def\svgwidth{0.95\linewidth}\includesvg{pics/isospace_dR_fig} \caption{Distribution of $\Deltaup R$ for the 132 successfully build models. \label{isospace_dR_fig}} \caption{Distribution of $R_\varepsilon$ for the 132 successfully build models. Outliers are added to the count in the last bin. \label{isospace_dR_fig}} \end{figure} \begin{figure}[p] \tiny ... ... @@ -359,11 +368,16 @@ for the model of equation \ref{mregress} with $y = \Deltaup \kappa$ (table \ref{ \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_dR_a7}}\hfill \subfloat[\label{pspaceiso_dR_a8}]{% \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_dR_a8}}\\ \caption{Parameters of equation \ref{isospace_dR_eq} (red) with their 95\,\% confidence intervals (blue) as function of $\sigma_x$. With the coefficients for \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} film length $L$ $(a_1)$, \textbf{(c)} film width $w$ $(a_2)$, \textbf{(d)} film stiffness $E_\text{film}$ $(a_3)$, \textbf{(e)} cell layer stiffness $E_\text{cells}$ $(a_4)$, \textbf{(f)} film Poisson ratio $\nu_\text{film}$ $(a_5)$, \textbf{(g)} cell layer Poisson ratio $\nu_\text{cells}$ $(a_6)$, \textbf{(h)} film thickness $t_\text{film}$ $(a_7)$, and \textbf{(i)} cell layer thickness $t_\text{cells}$ $(a_8)$. \label{fig:pspaceiso_dR_as}} \caption{Coefficients of equation \ref{isospace_dR_eq} ($R_\varepsilon$, red) with their 95\,\% confidence intervals (blue) as function of $\sigma_x$. With the coefficients for \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} film length $L$ $(a_1)$, \textbf{(c)} film width $w$ $(a_2)$, \textbf{(d)} film stiffness $E_\text{film}$ $(a_3)$, \textbf{(e)} cell layer stiffness $E_\text{cells}$ $(a_4)$, \textbf{(f)} film Poisson ratio $\nu_\text{film}$ $(a_5)$, \textbf{(g)} cell layer Poisson ratio $\nu_\text{cells}$ $(a_6)$, \textbf{(h)} film thickness $t_\text{film}$ $(a_7)$, and \textbf{(i)} cell layer thickness $t_\text{cells}$ $(a_8)$. \label{fig:pspaceiso_dR_as}} \end{figure} The coefficients in equation \ref{isospace_dR_eq} table \ref{isospace_dR_tab} turn out to be rather insensitive to $\sigma_x$ (figure \ref{fig:pspaceiso_dR_as}). Although the 95\,\% confidence intervals are narrow and constant, there are no obvious trends for the responses to $\sigma_x$, except perhaps for $t_\text{cells}$ ($a_8$, figure \ref{pspaceiso_dR_a8}). Such a trend for a real' (albeit small) effect $t_\text{cells}$ would explain why in figure \ref{isospace_dR_fig}) distribution seems to skew to the left for increasing $\sigma_x$. \\ \noindent \textbf{round up} The assumption of constant curvature is violated by a small amount ($\frac{\std\kappa}{\mean\kappa}\approx 0.1$) that varies little over a wide range of parameters and $\sigma_x$'s (figure \ref{isospace_dk_fig}); and the radius of curvature $R$ as estimated from projected edge length $l$ estimates the mean local curvature over the length of the film within a few percent over a wide range of parameters and $\sigma_x$'s (figure \ref{isospace_dR_fig}). In isotropic films, the radius of curvature $R$ as estimated from projected edge length $l$ is a good and valid estimate of the actual mean local curvature over the length of the film. \clearpage \section{Anisotropic films} \begin{figure}[b!] ... ... @@ -385,15 +399,15 @@ for the model of equation \ref{mregress} with $y = \Deltaup \kappa$ (table \ref{ \tiny \subfloat[\label{anikappaxl}]{% \def\svgwidth{0.47\linewidth}\includesvg{pics/anikappaxl}}\hfill \subfloat[\label{anikappaxl}]{% \subfloat[\label{anikappaxr}]{% \def\svgwidth{0.47\linewidth}\includesvg{pics/anikappaxr}}\\ \subfloat[\label{anikappayl}]{% \def\svgwidth{0.47\linewidth}\includesvg{pics/anikappayl}}\hfill \subfloat[\label{anikappayl}]{% \subfloat[\label{anikappayr}]{% \def\svgwidth{0.47\linewidth}\includesvg{pics/anikappayr}}\\ \subfloat[\label{anikappazl}]{% \def\svgwidth{0.47\linewidth}\includesvg{pics/anikappazl}}\hfill \subfloat[\label{anikappazl}]{% \subfloat[\label{anikappazr}]{% \def\svgwidth{0.47\linewidth}\includesvg{pics/anikappazr}}\\ \caption{The three components of the local curvature vector along the deformed edges of the films in figure \ref{lgmu}. With the left edge on the left, the right edge at the right, and from top to bottom $\kappa_x$, $\kappa_y$, and $\kappa_z$. \label{fig:anikappaxyz}} \end{figure} \ No newline at end of file
 ... ... @@ -50,7 +50,7 @@ \makeatother% \begin{picture}(1,0.77832577)% \put(0,0){\includegraphics[width=\unitlength]{isospace_dR_fig.pdf}}% \put(0.4953588,0.0053223){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{$\Deltaup R$\,[-]}}}% \put(0.4953588,0.0053223){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{$R_\varepsilon$\,[-]}}}% \put(0.01944014,0.42427606){\color[rgb]{0,0,0}\rotatebox{90}{\makebox(0,0)[b]{\smash{$\sigma_x$\,[MPa]}}}}% \put(0.13484786,0.03884335){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{0.85}}}% \put(0.22548734,0.03884335){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{0.9}}}% ... ...
 ... ... @@ -50,7 +50,7 @@ \makeatother% \begin{picture}(1,0.78508084)% \put(0,0){\includegraphics[width=\unitlength]{isospace_dk_fig.pdf}}% \put(0.4953588,0.0053223){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{$\Deltaup \kappa$\,[-]}}}% \put(0.4953588,0.0053223){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{$\kappa_\varepsilon$\,[-]}}}% \put(0.01944014,0.42427606){\color[rgb]{0,0,0}\rotatebox{90}{\makebox(0,0)[b]{\smash{$\sigma_x$\,[MPa]}}}}% \put(0.13484786,0.03884335){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{0}}}% \put(0.25570049,0.03884335){\color[rgb]{0,0,0}\makebox(0,0)[b]{\smash{0.1}}}% ... ...
 ... ... @@ -238,7 +238,7 @@ for y = 2:size(l, 2) end figure imagesc(range, -sigma, htd) xlabel('$\Deltaup \kappa$\,[-]') xlabel('$\kappa_\varepsilon$\,[-]') ylabel('$\sigma_x$\,[MPa]') colormap(mjet); caxis([0 max(htd(:))]); colorbar svgprint(get(gca, 'parent'), 'pics/isospace_dk_fig') ... ... @@ -259,7 +259,7 @@ end %%% analysis for delta kappa: sigma_x = 0.01 Y = sk(gidx, end)./mk(gidx, end); pspaceiso_eqstabs(X(gidx, :), Y, '\Deltaup \kappa(\sigma_x = 0.025\,\text{MPa})', 'isospace_dk'); pspaceiso_eqstabs(X(gidx, :), Y, '\kappa_\varepsilon(\sigma_x = 0.025\,\text{MPa})', 'isospace_dk'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ... ... @@ -288,7 +288,7 @@ for y = 2:size(l, 2) end figure imagesc(range, -sigma, htd) xlabel('$\Deltaup R$\,[-]') xlabel('$R_\varepsilon$\,[-]') ylabel('$\sigma_x$\,[MPa]') colormap(mjet); caxis([0 max(htd(:))]); colorbar svgprint(get(gca, 'parent'), 'pics/isospace_dR_fig') ... ... @@ -308,4 +308,4 @@ end % analysis for R * mean(kappa) @ sigma_x = 0.025 Y = R(gidx, end).*mk(gidx, end); pspaceiso_eqstabs(X(gidx, :), Y, '\Deltaup R(\sigma_x = 0.025\,\text{MPa})', 'isospace_dR'); pspaceiso_eqstabs(X(gidx, :), Y, 'R_\varepsilon(\sigma_x = 0.025\,\text{MPa})', 'isospace_dR');
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