finished (?) iso curling

parent b65aed8f
 ... ... @@ -106,7 +106,7 @@ Note that is due to the python step in building the model: the hard-coded method If we treat the model with $\frac{t_\text{film}}{L} = 1.08\cdot 10^{-3}$, as the result of an numerical rounding error, and the model with $\frac{t_\text{film}}{L} = 2.4\cdot 10^{-3}$, as an outlier', we might conclude that python FEM building does not work properly for films with a thickness/length ratio $\frac{t_\text{film}}{L}\leq 0.98\cdot 10^{-3}$. \noindent In general, less models converge for larger $\sigma_x$ (figure \ref{pspaceiso_n}), as was to be expected. The oscillations in the number of observations as a function of $\sigma_x$ stem from the fact that convergence sometimes skips' a value of $\sigma_x$ (see e.g.\ also figure \ref{lRfsigma00}). \noindent In general, less models converge for larger $\sigma_x$ (figure \ref{pspaceiso_n}), as was to be expected. The oscillations in the number of observations as a function of $\sigma_x$ is explained by the observation that convergence sometimes skips' a value of $\sigma_x$ (see e.g.\ also figure \ref{lRfsigma00}). \subsubsection{Maximum $\sigma_x$ in the models} ... ... @@ -150,7 +150,7 @@ We calculated the (normalised) projected film length $\frac{l}{L}$ for each of t \caption{Example of an isotropic film with $\frac{l}{L} = 0.102 < \frac{1}{2\piup}$. The (deformed) film is in blue, the cell layer in red. The undeformed mesh is in grey and extends beyond the image. \label{fullcircle}} \end{figure} The deformed mesh of the simulation with the smallest value of $\frac{l}{L}$ at $\sigma_x = 0.025$\,MPa is shown in figure \ref{fullcircle}. The film completes more than a circle and the deformed mesh intersects with itself over approximately half its length. On the top of the deformed mesh, at the end of the film, you see a stripe of the red film layer. This means that the cell layer has penetrated the complete mesh, film plus cells, at the tip of the film. The deformed mesh of the simulation with the smallest value of $\frac{l}{L}$ at $\sigma_x = 0.025$\,MPa is shown in figure \ref{fullcircle}. The curled film completes more than $1\frac{1}{2}$ circle and the deformed mesh intersects with itself over approximately half its length. On the top of the deformed mesh, at the end of the film, you see a stripe of the red film layer. This means that the cell layer has penetrated the complete mesh, film plus cells, at the tip of the film. This is possible in the simulations because no contact has been defined: the simulation, the math, doesn't know about self-touching or intersecting meshes. This behaviour is of course not possible in the physical world (hence the parameter \param{curlcheck}, page \pageref{p:curlcheck}). Yet, we decided not to exclude these simulations with $\frac{l}{L} < \frac{1}{2\piup}$ from further analysis. In the simulations no physics is violated because that part of physics is simply not defined, and the behaviour and analysis of projected edge length and radii of curvature etc., remain (mathematically) consistent and valid. In fact, one might argue that figure \ref{fullcircle} is a strong argument in favour of the assumption of constant curvature\footnote{Laziness reaps fruitful arguments\dots} (which \textsl{is} of course still violated in the physical world when the films self-touch).\\ ... ... @@ -176,25 +176,25 @@ This is possible in the simulations because no contact has been defined: the sim \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_lL_a7}}\hfill \subfloat[\label{pspaceiso_lL_a8}]{% \def\svgwidth{0.32\linewidth}\includesvg{pics/pspaceiso_lL_a8}}\\ \caption{Parameters of equation \ref{isospace_lL_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_lL_as}} \caption{Coefficients of equation \ref{isospace_lL_eq} ($\frac{l}{L}$, 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_lL_as}} \end{figure} The films curl well. The median of $\frac{l}{L} = 0.25$ is close to $\frac{1}{2\piup}$, 'full circle curling'. There are four parameters that together explain about 80\,\% of the variation in the model: $L$ (SS/SS$_\text{tot} = 0.20$, $p < 0.001$), $E_\text{film}$ (SS/SS$_\text{tot} = 0.21$, $p < 0.001$), $t_\text{film}$ (SS/SS$_\text{tot} = 0.18$, $p < 0.001$), and $t_\text{cells}$ (SS/SS$_\text{tot} = 0.23$, $p < 0.001$). The other parameters contribute no more than 0.01 each to SS$_\text{tot}$ (table \ref{isospace_lL_tab}). The films curl well. The median of $\frac{l}{L} = 0.25$ is close to $\frac{1}{2\piup}$, 'full circle curling'. There are four parameters that together explain about 80\,\% of the variation in the model: $L$ (SS/SS$_\text{tot} = 0.20$, $p < 0.001$), $E_\text{film}$ (SS/SS$_\text{tot} = 0.21$, $p < 0.001$), $t_\text{film}$ (SS/SS$_\text{tot} = 0.18$, $p < 0.001$), and $t_\text{cells}$ (SS/SS$_\text{tot} = 0.23$, $p < 0.001$). The other parameters contribute no more than 1\,\% each to SS$_\text{tot}$ (table \ref{isospace_lL_tab}). \noindent We investigated the coefficients in equation \ref{isospace_lL_eq} over the entire range for $\sigma_x$ for the simulations. Since $\frac{l}{L}$ decreases (non-linearly) with $\sigma_x$ (e.g.\ figure \ref{lRfsigma00}), we expect the effects (magnitude of the coefficients) to increase (non-linearly) with increasing $\sigma_x$. This is indeed the case, most coefficients show a clear (non-linear) increase in magnitude over the range of cell traction stresses that was investigated (figure \ref{isospace_lL_tab}). \noindent We investigated the coefficients in equation \ref{isospace_lL_eq} over the entire range for $\sigma_x$ for the simulations. Since $\frac{l}{L}$ decreases (non-linearly) with $\sigma_x$ (e.g.\ figure \ref{lRfsigma00}), we expect the effects (magnitude of the coefficients) to increase (non-linearly) with increasing $\sigma_x$. This is indeed the case, most coefficients show a clear (non-linear) increase in magnitude over the range of cell traction stresses that was investigated (figure \ref{isospace_lL_tab}). But the analysis in table table \ref{isospace_lL_tab} turns out to be representative for the entire range of $\sigma_x$ in terms of the (approximate magnitudes of the) 95\,\% confidence intervals, sums-of-squares and $p$-values. Most coefficients show a fast, initial, response for smaller $\sigma_x$. S The intercept, shows a fast initial response for smaller $\sigma_x$ and then seems to stabilise (figure \ref{pspaceiso_lL_a0}), similar to the example in figure \ref{lRfsigma00}. A fast initial response is also obvious in the four parameters that together explain about 80\,\% of the variation in the model for $\sigma_x = 0.025$\,MPa: $L$, (figure \ref{pspaceiso_lL_a1}), $E_\text{film}$ (figure \ref{pspaceiso_lL_a3}), $t_\text{film}$ (figure \ref{pspaceiso_lL_a7}), and $t_\text{cells}$ (figure \ref{pspaceiso_lL_a8}). The 95\,\% confidence intervals at $\sigma_x = 0.001$\,MPa and at $\sigma_x = 0.025$\,MPa are markedly different for these parameters, suggesting real' effects with real' responses to $\sigma_x$. The analysis in table table \ref{isospace_lL_tab} turns out to be representative for the entire range of $\sigma_x$ in terms of the (approximate sizes of the) sums-of-squares and $p$-values. \\ Confidence intervals at the beginning and the end of the range of $\sigma_x$ overlap for the other parameters. The coefficient for $w$ seems to show a downward trend (figure \ref{pspaceiso_lL_a2}), but not enough to become relevant at $\sigma_x = 0.025$\,MPa. The coefficient for $E_\text{cells}$ shows both a minimum and a maximum in the response to $\sigma_x$ (figure \ref{pspaceiso_lL_a4}), but includes zero over most of the range. There appears to be a fast initial response with stabilisation for $\nu_\text{film}$ (figure \ref{pspaceiso_lL_a4}) and cell layer Poisson ratio $\nu_\text{cells}$ (figure \ref{pspaceiso_lL_a6}), but the size of the initial response almost drowns in the 95\,\% confidence intervals, and that interval includes zero for almost the entire range of $\sigma_x$ for $\nu_\text{cells}$ (figure \ref{pspaceiso_lL_a6}). \\ \noindent Some coefficients seem to show a minimum or maximum after their (fast) initial response to $\sigma_x$ (e.g.\ $E_\text{film}$ and $t_\text{film}$, figures \ref{pspaceiso_lL_a2} and \ref{pspaceiso_lL_a7}; perhaps $t_\text{cells}$, figure \ref{pspaceiso_lL_a8}). Such minima and maxima were not expected. The values of $\sigma_x$ for where they occur seem similar between the parameters, about 0.007\,MPa. Something seems to happen around this value of $\sigma_x$ that reverses' these effects. Pending further investigation, we hypothesise that this is related to some geometrical effect in film curling, e.g.\ completing a quarter circle or a full circle.\\ \noindent \textbf{round up} Longer isotropic constructs and isotropic constructs with thicker cell layer curl more, isotropic constructs with stiffer films and isotropic constructs with thicker films curl less. \noindent \textbf{round up} Longer isotropic constructs and isotropic constructs with thicker cell layer curl more, isotropic constructs with stiffer films and isotropic constructs with thicker films curl less. Film stiffness has the largest coefficient (considering the orders of magnitude of the parameters, table \ref{tbl:parameter_space}). Not too far behind are construct length and cell layer thickness, at about equal footing. And of these four parameters, film thickness has the smallest effect (about a third of cell layer thickness, a quarter of film stiffness, give or take). Film stiffness has the largest effect (considering the orders of magnitude of the parameters, table \ref{tbl:parameter_space}). Not too far behind are construct length and cell layer thickness, at about equal footing. And of these four parameters, film thickness has the smallest effect (about a third of cell layer thickness, a quarter of film stiffness, give or take). \subsection{Assumption of no contraction} ... ... @@ -370,7 +370,7 @@ for the model of equation \ref{mregress} with $y = \Deltaup \kappa$ (table \ref{ \includegraphics[width=0.47\linewidth]{lfmu_def1.pdf}}\hfill \subfloat[\label{lfmu_def2}]{% \includegraphics[width=0.47\linewidth]{lfmu_def2.pdf}}\\ \caption{Stress fibre orientation and projected edge $l$. With \textbf{(a)} we simulate 10 films with different main' orientations of the stress fibres, $90 \leq \mu\,[\degree] \leq 180$ and otherwise equal fibre distributions (solid) and other properties. Note that in sampling of the analytical function into 30 bins (dots) a little asymmetry may be introduced; \textbf{(b)} results for the projected edge length for 25 values of $0 \leq \sigma_x\,\text{[MPa]}\leq 0.025$ for each of the ten films (successful simulations marked with a dot); and examples of the ten curled films in the simulations at $\sigma_x = 0.02577$\,MPa, the last value of $\sigma_x$ where all models still converged, \textbf{(c)} from the top (projected view), and \textbf{(d)} obliquely. \label{lgmu}} \caption{Stress fibre orientation and projected edge $l$. With \textbf{(a)} we simulate 10 films with different main' orientations of the stress fibres, $90 \leq \mu\,[\degree] \leq 180$ and otherwise equal fibre distributions (solid) and other properties; \textbf{(b)} results for the projected edge length for 25 values of $0 \leq \sigma_x\,\text{[MPa]}\leq 0.025$ for each of the ten films (successful simulations marked with a dot); and examples of the ten curled films in the simulations at $\sigma_x = 0.02577$\,MPa, the last value of $\sigma_x$ where all models still converged, \textbf{(c)} from the top (projected view), and \textbf{(d)} obliquely. \label{lgmu}} \end{figure} ... ...
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