8_discussion.tex

%\section{Discussion}


\subsection{Comparison of beading schemes}
We can see from \cref{TEST_naive_accuracy}(top) and~\ref{over_underfill} that the uniform technique causes a lot of overfills and underfills: on average \revise{approximately}{} \revise{\SI{1}{\percent}}{\SI{1.6}{\percent}} of the total target area is covered by underfill and likewise for overfill.
To our knowledge, the uniform beading scheme, as well as the outer beading scheme, is of little use to FDM printers.

The constant bead count scheme effectively deals with underfills, but generates orders of magnitude more overfills compared to the other schemes. 
Also, the scheme comes at the cost of greatly varying bead widths and an average bead width that is not close to the preferred bead width.
Note that most overfill areas occur near regions of alternating bead width. 
\revise{}{While the scheme results in short toolpaths, as indicated by the idealized print time, it also results in a wide range of bead widths, which cause the back pressure compensation print time to be very large.
See \cref{statisticsfig}.}
For an input outline shape which contains both very small and very large features, the constant bead count scheme produces bead widths which can fall outside of the range of manufacturable bead widths.
Moreover the centrality marking is not robust against small perturbations in the outline; adding a small chamfer in a corner causes the unmarked ST to be very small at that location, which results in tiny bead widths.
\revise{}{See top right of \cref{TEST_Constant_accuracy}.}

In \cref{TEST_Center_accuracy} we can see that
the centered beading scheme effectively deals with \revise{both }{}overfill \revise{and underfill }{}and produces desired bead widths in all locations, except for the extrusion paths in the center, where the bead widths \revise{are within a factor 2 off from the desired bead width.}{range between $0.25 w^*$ and $1.8w^*$.}
\revise{}{However, it does produce some narrow underfill regions.}
\revise{%gs
According to \cref{over_underfill} the overfill and underfill for the centered, the evenly distributed and the inward distributed scheme are all approximately \SI{0.2}{\percent}, which is a considerable improvement over the uniform technique.
}{%sg
Compared to the uniform technique the centered technique increases the (open) path count, but considerably reduces over- and underfill and decimates the number of toolpath angles below \SI{45}{\degree}.
See \cref{statisticsfig}.%afs
}%zgds

However, according to \cref{widthHistogram} the centered scheme exhibits a wider range of bead widths than the distributed schemes:
the standard deviation of the bead widths in the centered scheme is approximately \revise{\SI{39}{\micro\meter}}{\SI{53}{\micro\meter}}, while that of the distributed schemes is approximately \revise{\SI{14}{\micro\meter}}{\SI{23}{\micro\meter}}.%_
\revise{}{\footnote{\revise{}{Although the standard deviation $\sigma$ of the inward distributed scheme is slightly higher than that of the evenly distributed scheme, the mean absolute deviation is lower (i.e. \SI{9}{\micro\meter} versus \SI{11}{\micro\meter}), because its distribution is more peaked.}}}
\revise{}{Moreover, because the quantization operator rounds to the nearest number of beads, in the worst case where we switch from a single to two beads the widths switch from $0.75w^*$ to $1.5w^*$, which is a considerably smaller range than in the centered scheme.}
We therefore conclude that the distributed schemes \revise{result in bead widths closer to the preferred widths}{exhibit a lower bead width variation and lower (open) path count} compared to the centered scheme.
%This is desirable for the manufacturability of the beads and can therefore have a positive effect on the mechanical properties and surface quality of the 3D prints. 

\revise{It is hard to visually identify the difference between the evenly and the inward distributed scheme in \cref{visualized_accuracy}, because that particular example shape does not have wide features.}
{\Cref{distributed_comparison,TEST_Distributed_accuracy,TEST_InwardDistributed_accuracy} show that in the inward distributed scheme the outer toolpaths have the preferred width more often than in the evenly distributed scheme, which means that the outline accuracy of the inward distributed beading is less affected by inaccuracies in the adaptive width control system.}
%{\Cref{TEST_InwardDistributed_accuracy} shows that in the inward distributed scheme the outer beads are more often equal to the preferred width compared to the evenly distributed scheme in \cref{TEST_Distributed_accuracy}, but that effect is more pronounced for wider geometry such as in \cref{distributed_comparison}.}
%While the difference between the evenly and inward distributed scheme in \cref{visualized_accuracy} results only in the outer bead being the preferred width in some locations, the effect of the inward distributed scheme is more pronounced in larger geometry, as can be seen in \cref{distributed_comparison}.}
\revise{However, \cref{distributed_comparison} and  \cref{widthIndexedHistogram} confirm that the outer toolpaths have the preferred bead width more often.}{}% _
Furthermore, \revise{from \cref{smoothness} }{}we find that compared to evenly distributed, the inward distributed scheme produces \revise{smoother toolpaths overall}{less corners with angles above \SI{130}{\degree} and less overfill, because the area of influence that bead count transitions have is limited in the inward distributed scheme}.
Thus the inward distributed scheme prevents over- and underfill, generates smooth toolpaths with more homogeneous width and affects smaller more centered parts of the print than the other schemes\revise{}{, while incurring little to no extra print time}. 



\revise{}
{
\subsection{Limitations}
% design considerations
Because the performance of the various toolpathing techniques depends on the geometry of a model, they have ramifications for the practice of design for additive manufacturing.
Because the naive method produces under- or overfill for parts of an outline with a constant diameter $d \neq 2 i w^*$ it is best practice to design a model such that horizontal cross-sections have a feature diameter of an even integer multiple $i$ of the bead width.
For the center\revise{ed} scheme and for the current state of the art one should only avoid parts for which $(2i + 1.8) w^* < d < (2i + 0.25) w^*$ in order to avoid underfill.
For the distributed schemes however, there is no diameter at which the framework produces under- or overfill for a part with a constant diameter $d$.
The design consideration therefore reduces to limiting the diameter of your parts to be within the range $[w_\text{min}, \infty)$,
where $w_\text{min}$ is a configurable parameter when using the widening meta-scheme.


% single line segments & continuity
The default limit bisector angle $\alpha_\text{max} = \SI{135}{\degree}$ ensures that we don't employ transitioning in shallow wedge regions, which would result in a lot of short odd single bead polylines, which would break up the semi-continuous nature of polygonal extrusion paths;
$\alpha_\text{max} = \SI{135}{\degree}$ corresponds to $w^* / \cos \nicefrac12 \alpha_\text{max} \approx \SI{0.4}{\milli\meter}$ long segments
and under-/overfill areas of $\nicefrac14 (w^*)^2 \left( \tan ( \alpha / 2) - \alpha / 2 \right) \approx \SI{0.05}{\milli\meter\squared}$.
However, future work might be aimed at reducing under-/overfill in regions with a low bisector angle without the introduction of short single polyline extrusion segments.
If the over-/underfill problem is also solved for non-significant regions we might be able to increase $\alpha_\text{max}$ and reduce the discontinuity introduced by short extrusion segments.

% single beads
Another limitation of our method is that in a location $v$ with locally maximal $R(v) \approx (i + \nicefrac12) w^*$ the odd bead count will result in a single polyline extrusion segment consisting of only a single point.
This can be viewed in the bottom right of \cref{TEST_InwardDistributed_accuracy} for example.
In order to print such a dot, we make it into a \SI{10}{\micro\meter} long extrusion segment, with an altered width such that the total volume remains correct.
A more principled way of dealing with such a situation remains future work.

% emulation troubles
Finally it should be noted that although our framework can accurately emulate the constant bead count approach by \citeauthor{Ding2016a}, its emulation of the centered approach by \citeauthor{Jin2017JMS} is imperfect.
The transitions resulting from out framework introduce sharper corners and there is more width variation in those corners.
Whereas the width of the connecting segment in the approach by \citeauthor{Jin2017JMS} is the preferred width $w^*$, the bead widths closer to the center resulting from our framework will be twice the local radius, which is larger than $w^*$.
However, this inflated bead width variation is expected to have an insignificant impact on the measured bead width variation.
}



\input{10_applications}






\revise{
\subsection{Discussion on implications}
Note that the current industry standard in FDM printing employs little to no bead width variation.
Properly performing bead width variation calls for adaptations and developments in printers and firmware.
In the beading schemes we set a transition length of $t(n) = w^*$.
That will demand changes in cross-sectional area of the bead up to \SI{200}{\percent} over a small distance that is comparable to the nozzle size, which is challenging for some hardware systems.
Varying the movement speed can be utilized to change the cross-sectional area, but this approach is limited, since the movement speed is constrained by acceleration considerations near bends in the toolpath~\cite{Ertay2018,Kuipers2018}.
Our schemes require a more accurate control of the volumetric flow rate in \si{\milli\meter\cubed\per\second}.
Using a filament feeder directly mounted on the print head (a.k.a. direct drive) can control the flow more dynamicaly then FDM printers where the material is fed through a Bowden tube from a feeder mounted on the frame.
Still direct drive printers require some control system in order to accurately change the volumetric flow rate such as pressure advance algorithms~\cite{tronvoll2019investigating}.
Yet inaccuracies in direct drive systems employing advance algorithms might arise due to the changes in back-pressure required by changing bead size.
We expect that developments in printing hardware and firmware will address these challenges in the future.

Another limiting factor for adopting adaptive bead width is the format of G-code which stores machine instructions.
G-code does not support moves with varying cross-sectional area.
A typical extrusion move \lstinline!G1 X$x$ Y$y$ E$v$! only specifies the total amount of volume $v$ to be extruded in the move, not how that total amount should be distributed along the extrusion move.
A workaround is to approximate a variable width extrusion segment by smaller segments with constant width.
However, this introduces errors nevertheless.
Ideally the G-code language would be expanded in some way to allow for extrusion segments with varying cross-sectional area.
}{}

% Taking a broader perspective, we note that our proposed inward distributed scheme is a pragmatic solution.
% Rather than deriving some optimal beading scheme from a clear specification of the objective, we propose some arbitrary inward distributed beading scheme and show that it is better than the other beading schemes.
% An optimal beading scheme can be derived if the objective is formalized terms of a unambiguous fitness function, but that would depend on the specific hardware setup and application for which toolpaths are generated.
% This manuscript is therefore limited to showing the flexibility and versatility of the framework, rather than deriving an optimal beading scheme.