Skeleton-based perpendicularly scanning: a new scanning strategy for additive manufacturing, modeling and optimization

Abstract

Actually, additive manufacturing (AM) is considered as a major class of complex parts manufacturing technologies. Including a wide range of materials, a huge set of physico-chemical phenomena are involved and adapted to control and master the variety of materials processing. On the other side, AM still knows a low productivity rate, due to different reasons that are mainly related to the material-process interaction control. In this paper, the authors propose a novel 2D scanning strategy that could be adapted to AM processes such as Laser Beam Powder Bed Fusion of Metals (PBF-LB/M) and polymers (PBF-LB/P), Electron Beam Powder Bed Fusion for metals (PBF-EB/M), and Material Extrusion-based (MEx) (ISO/ASTM 52,900 standards). The novelty presented corresponds to a Skeleton Based Perpendicularly (SBP) scanning strategy that aims to reduce the scanning lengths, and thus the production time and processing energy. The competitiveness of the new technique is mainly discussed according to the hatch space distance and the dimensions of a rectangular shape that was selected for the proof of concept of this new scanning strategy. In other words, it is proposed to investigate the competitiveness of the new scanning technique compared to four classical scanning strategies that are widely used in AM in term of process productivity. A detailed benchmark analysis has been applied to the following strategies: chess, stripe, spiral, and contour scanning. An analytical mathematical modeling was developed leading to the evaluation of the performance of SBP scanning compared to the scanning benchmark strategies by means of two proposed geometrical indices: the “gain of length” and the “specific gain of length per surface unit”. These were exploited in two separate study cases. The simulation showed that the SBP scanning length exhibits an increasing quadratic dependence on rectangle dimensions and a decreasing hyperbolic behavior according to hatch space distance. The lengths of the benchmark scanning strategies also presented a hyperbolic decreasing behavior according to hatch space distance. After that, it was proved that the SBP strategy is absolutely competitive compared to chess and stripe scanning; the competitiveness fluctuates around 95%, but it is highly concentrated around 100%. Otherwise, for the contour and stripe strategies, it has been shown that the competitiveness is strongly affected by the hatch space distance and by the dimensions of the shape being scanned. A particular behavior of the feasibility percentage of decision variable combinations, was detected as power laws or polynomials of the hatch space distance in the case of SBP/spiral comparison. In this case, the competitiveness (feasibility) ranged from 20 to 98% while it ranged from 0 to 75% in the case of SBP/contour comparison. The results of this study could also constitute a major contribution to related scientific and technical fields concerned with optimal area or volume control.

Appendices

Appendix I: SBP scanning modeling

1.1 A. Study of the area (A)

1.1.1 a. Bisectors parametrization

According to Spain 1963 [77], the coordinates of the point H which is the projection of the point C onto the bisector (B) can be calculated from the point C coordinates and the carrier vector $\overrightarrow{v}$ of the line (B), as shown in Fig. 7.

Since

$$\left\{\begin{array}{c}H\in \left(B\right)\\ \left(B\right): f\left(x,y\right)=ax-y+b=0 .\end{array}\right.$$

(51)

Then

$$a{x}_{H}-{y}_{H}+b=0.$$

(52)

1.1.2 b. Expression of the distance ${{d}}_{\overrightarrow{{C}{H}}/{A}}$

This section is developed according to the results of Sect. 3.1.1.1.

For the calculation of the distance ${d}_{\overrightarrow{CH}/A}=||\overrightarrow{CH}||$, the Cartesian distance is utilized according to the coordinates of the point $C\left({x}_{C}, {y}_{C}\right)$ and $H\left({x}_{H}, {y}_{H}\right)$, where:

$${d}_{\overrightarrow{CH}/A}=||\overrightarrow{CH}||=\sqrt{{\left({x}_{H}-{x}_{C}\right)}^{2}+{\left({y}_{H}-{y}_{C}\right)}^{2}}.$$

Since

$$\left\{C\right\}=\left(CH\right)\cap \left(D\right),$$

and

$$\left(D\right):y={L}_{2}/2,$$

thus

$${y}_{C}={L}_{2}/2.$$

(53)

The vector $\overrightarrow{n}$ which is normal to the bisector (B) is expressed by the following coordinates [54]:

$$\overrightarrow{n}=\overrightarrow{\nabla }f\left(x,y\right)=\left\{\begin{array}{c}{n}_{x}=\frac{a}{\sqrt{1+{a}^{2}}}\\ {n}_{y}=\frac{-1}{\sqrt{1+{a}^{2}}}\end{array}\right..$$

(54)

where f is the implicit function that defines the bisector (B) as described in Eq. (51).

To express the ${x}_{c}$ coordinate, we need to determine the equation of the line (CH).

Since the line (CH) is carried by the vector $\overrightarrow{n}$:

$$\left(CH\right):\;\frac{-1}{\sqrt{1+{a}^{2}}}x-\frac{a}{\sqrt{1+{a}^{2}}}y+\mathrm{C}\mathrm{s}\mathrm{t}=0.$$

(55)

The constant “Cst” could be calculated according to the point H which belongs to the line (CH).

Replacing the coordinates of point H in Eq. (51), we obtain:

$$\frac{-1}{\sqrt{1+{a}^{2}}}{x}_{H}-\frac{a}{\sqrt{1+{a}^{2}}}{y}_{H}+\mathrm{C}\mathrm{s}\mathrm{t}=0.$$

where

$$\begin{aligned} & \left(13\right)\;\;\iff \;\;{y}_{H}=a\;{x}_{H}+b\\ & \quad \Rightarrow \;\;\frac{-1}{\sqrt{1+{a}^{2}}}{x}_{H}-\frac{a}{\sqrt{1+{a}^{2}}}\left(a\;{x}_{H}+b\right)+\mathrm{C}\mathrm{s}\mathrm{t}=0 \\ & \quad \Rightarrow \;\mathrm{C}\mathrm{s}\mathrm{t}=\frac{{x}_{H}\left(1+{a}^{2}\right)+ab}{\sqrt{1+a^2}}. \end{aligned}$$

So the equation of the line (CH) becomes:

$$\left(CH\right):\;\frac{-1}{\sqrt{1+{a}^{2}}}x-\frac{a}{\sqrt{1+{a}^{2}}}y+\frac{{x}_{H}\left(1+{a}^{2}\right)+ab}{\sqrt{1+{a}^{2}}}=0.$$

Since $\forall a\in \mathbb{R}$

$$\sqrt{\left(1+{a}^{2}\right)}\ne 0.$$

We find

$$\left(CH\right):\;x+ay-\left({x}_{H}\left(1+{a}^{2}\right)+ab\right)=0.$$

(56)

Since $C\in \left(CH\right):$

$${x}_{C}+a{y}_{C}-\left({x}_{H}\left(1+{a}^{2}\right)+ab\right)=0.$$

According to the expression (53):

$$\Rightarrow {x}_{C}+\frac{a{L}_{2}}{2}-\left({x}_{H}\left(1+{a}^{2}\right)+ab\right)=0.$$

Finally

$${x}_{C}-{x}_{H}=a\left(a\;{x}_{H}+b-\frac{{L}_{2}}{2}\right),$$

(57)

$${y}_{C}-{y}_{H}=-\left(a{x}_{H}+b-\frac{{L}_{2}}{2}\right).$$

(58)

In addition to the Eqs. (57) and (58), we remark that:

$${x}_{C}-{x}_{H}=-a\left({y}_{C}-{y}_{H}\right).$$

(59)

Or

$${y}_{C}-{y}_{H}=-\frac{1}{a}\left({x}_{C}-{x}_{H}\right).$$

(60)

Thus:

$$\begin{aligned} &{d}_{\overrightarrow{CH}/A}=||\overrightarrow{CH}||=\sqrt{{a}^{2}{\left({y}_{H}-{y}_{C}\right)}^{2}+{\left({y}_{H}-{y}_{C}\right)}^{2}} \\ & \quad \Rightarrow {d}_{\overrightarrow{CH}/A}=\left|{y}_{H}-{y}_{C}\right|\sqrt{1+{a}^{2}}.\end{aligned}$$

Since

$${y}_{H}\le {y}_{C}\Rightarrow \left|{y}_{H}-{y}_{C}\right|={y}_{C}-{y}_{H}.$$

Hence

$${d}_{\overrightarrow{CH}/A}=||\overrightarrow{CH}||=-a{x}_{H}\sqrt{1+{a}^{2}}\;.$$

(61)

1.1.3 c. Expression of the command parameter ${{x}}_{{H}}$

Let x_H be the command scanning variable. To apply the jumps of the scanning, x_H must be considered as a discrete parameter. Thus x_H must be expressed according to the projection jump step on the $\overrightarrow{x}$ axis.

The jump along the bisector (B) is denoted ${e}_{s}$, its projection, perpendicularly to the skeleton, on the horizontal axis is denoted ${e}_{s}^{{'}}$ (Fig. 8).

So, the formulation of x_H is given by in the expression (62).

$${x}_{H}=k {e}_{s}^{{'}} .$$

(62)

The geometrical link between ${e}_{s}$ and ${e}_{s}^{{'}}$ is described by the Fig. 22.

Fig. 22

Relation between ${e}_{s}$ and $e{\mathrm{'}}_{s}$

Thus

$$\begin{array}{c}(\text{fig. } 9)\Rightarrow \mathrm{cos}\left(\theta \right)={e}_{s}/{e}_{s}^{{{'}}}\\ (\text{fig. } 6)\Rightarrow \left\{\begin{array}{c}\cos\left(\theta \right)={\alpha }_{1}{L}_{1}/ \Lambda\\ \Lambda =\sqrt{{\left({\alpha }_{1}{L}_{1}\right)}^{2}+{\left(\frac{{L}_{2}}{2}\right)}^{2}}\end{array}\right.\end{array}.$$

(63)

It implies that:

$${e}_{s}^{{{'}}}=\frac{{e}_{s}\Lambda }{{\alpha }_{1}{L}_{1}}=\frac{{e}_{s}}{{\alpha }_{1}{L}_{1}}\sqrt{{\left({\alpha }_{1}{L}_{1}\right)}^{2}+{\left(\frac{{L}_{2}}{2}\right)}^{2}}.$$

Finally

$${e}_{s}^{{{'}}}={e}_{s}\sqrt{1+{\left(\frac{{L}_{2}}{2{\alpha }_{1}{L}_{1}}\right)}^{2}}.$$

(64)

Let us denote

$$\beta =\sqrt{1+{\left(\frac{{L}_{2}}{2{\alpha }_{1}{L}_{1}}\right)}^{2}}.$$

(65)

And

$${e}_{s}^{{'}}=\beta {e}_{s}.$$

(66)

In the case of this study, the angle $\theta$ is equal to $\frac{\pi }{4}$, so the link between the jumps ${e}_{s}$ and ${e}_{s}^{{'}}$ could be directly expressed by the expression (67):

$$\mathrm{cos}\left(\theta \right)=\frac{\sqrt{2}}{2}={e}_{s}/{e}_{s}^{{'}}.$$

(67)

For more genericity of the modeling approach, the author wanted to express the final expression of the total scanning length according to all geometrical parameters as dummy variables. Subsequently, in the simulation, the numerical values will appropriately replace the problem parameters.

Thus, according to (62), the expression of the command x_H becomes ${x}_{{H}_{A}}$ such that:

$$\left\{\begin{array}{c}{x}_{H}\left(k\right)={x}_{H/A}(k)\\ {x}_{{H}_{A}}\left(k\right)=k{e}_{s}^{{{'}}}=k\beta {e}_{s}\\ 1\le k\le {k}_{A}\end{array},\right.$$

(68)

where ${x}_{H/A}(k)$ is the restriction of ${x}_{H}$ on the area A, function of the index k.

The index k is the increment of the command, it begins by ${k}_{1}=1$ till a limit value ${k}_{\mathrm{A}}$ that limits the area (A) as depicted in Fig. 7. From the same figure, the k index takes ${k}_{\mathrm{A}}$ value when the point C reaches the right limits of the area (A). This condition is expressed by:

$${x}_{C}\left({k}_{A}\right)={\alpha }_{1}{L}_{1},$$

(69)

$${x}_{H}\left({k}_{A}\right)={k}_{A}\beta {e}_{s}.$$

(70)

Replacing (69) and (70) in the Eq. (57):

$${\alpha }_{1}{L}_{1}=\left({a}^{2}+1\right){k}_{A}\beta {e}_{s}+a\left(b-\frac{{L}_{2}}{2}\right).$$

Thus, ${k}_{A}$ is computed using the expression (71).

$${k}_{A}=\frac{{\alpha }_{1}{L}_{1}}{\left({a}^{2}+1\right)\;{e}_{s}\beta }.$$

(71)

1.1.4 d. Correction of the index ${{k}}_{{A}}$

Since ${k}_{A}$ is an integer number, the corrected ${k}_{{A}_{C}}$ must is expressed by:

$${k}_{{A}_{C}}=E\left(\frac{{\alpha }_{1}{L}_{1}}{\left({a}^{2}+1\right)\;{e}_{s}\beta }\right),$$

(72)

and

$${x}_{{H}_{A}}=\frac{{\alpha }_{1}{L}_{1}}{1+{a}^{2}}.$$

(73)

1.1.5 e. Productive length of the area (A)

On the area (A), the productive length ${L}_{{P}_{A}}$ is corresponds to the following summation:

$${L}_{{P}_{A}}=\sum _{k=1}^{{k}_{{A}_{C}}}{d}_{\overrightarrow{CH}/A}+{\left({d}_{\overrightarrow{CH}/A}\right)}_{f}.$$

where ${\left({d}_{\overrightarrow{CH}/A}\right)}_{f}$ is the distance $CH$ at the limit of the area (A). According to the expression (61):

$${\left({d}_{\overrightarrow{CH}/A}\right)}_{f}=-a{x}_{{H}_{A}}\sqrt{1+{a}^{2}},$$

where ${x}_{{H}_{A}}$ is expressed by the Eq. (73).

So

$${L}_{{P}_{A}}=\sum _{k=1}^{{k}_{A}}{d}_{\overrightarrow{CH}/A}+{\left({d}_{\overrightarrow{CH}/A}\right)}_{f}=\begin{array}{c}\sqrt{1+{a}^{2}}\sum _{k=1}^{{k}_{{A}_{C}}}\left(-a{x}_{H}\right)\end{array}+\left(-a{x}_{{H}_{A}}\right)\sqrt{1+{a}^{2}}.$$

Since ${x}_{H}=k\;{e}_{s}^{{{'}}}$ (62)

$$\begin{aligned}{L}_{{P}_{A}}& =\begin{array}{c}\sqrt{1+{a}^{2}}\left[-\frac{a}{2}{k}_{{A}_{C}}\left({{k}_{A}}_{C}+1\right){e}_{s}^{{{'}}}+\left(-a{x}_{{H}_{A}}\right)\right],\end{array}\\{L}_{{P}_{A}}& =\begin{array}{c}\sqrt{1+{a}^{2}}\left[-\frac{a\beta }{2}{e}_{s}{k}_{{A}_{C}}\left({k}_{{A}_{C}}+1\right)+\left(-a{x}_{{H}_{A}}\right)\right]\end{array},\\ {L}_{{P}_{A}}& =\begin{array}{c}-a\sqrt{1+{a}^{2}}\left[\frac{\beta }{2}{e}_{s}{k}_{{A}_{C}}\left({k}_{{A}_{C}}+1\right)+{x}_{{H}_{A}}\right]\end{array}.\end{aligned}$$

(74)

1.1.6 f. Jump length of the area (A)

On the other hand, the non-productive or jump length ${L}_{{J}_{A}}$ is calculated by multiplying the jumps lengths ${e}_{s}$ and ${e}_{s}^{{'}}$ by their respective number of repetitions.

The author analyzes two cases:

${k}_{{A}_{C}}$ -is an impair number:

$$\begin{aligned}{L}_{{J}_{A}}&=\sum _{k=1}^{E\left({k}_{{A}_{C}}/2\right)}({e}_{s}+{e}_{s}^{{{'}}})=\sum _{k=1}^{E\left({k}_{{A}_{C}}/2\right)}{e}_{s}\left(1+\beta \right),\\ {L}_{{J}_{A}}&=E\left({k}_{{A}_{C}}/2\right)\left(1+\beta \right){e}_{s}.\end{aligned}$$

(75)

${k}_{{A}_{C}}$ -is pair number:

$$\begin{aligned}{L}_{{J}_{A}}&=\sum _{k=1}^{\frac{{k}_{{A}_{C}}}{2}-1}{e}_{s}+\sum _{k=1}^{\frac{{k}_{{A}_{C}}}{2}}{e}_{s}^{{{'}}},\\{L}_{{J}_{A}}& =\left(\frac{{k}_{{A}_{C}}}{2}-1\right){e}_{s}+\frac{{k}_{{A}_{C}}}{2}{e}_{s}^{{{'}}}.\end{aligned}$$

(76)

1.2 B. Study of the area (B)

Given the rectangular geometry analyzed in this paper, the region (A) and (B) are symmetrical according to the red line of Fig. 7. Then the productive and non-productive lengths formulated for the area (A) are adopted for the area (B).

1.3 C. Study of the area C

According to Fig. 4, the area (C) corresponds to a simple rectangle. The scanning schema to adopt can be composed by the classical chess policy or it is possible to create a second level of skeleton scanning on this area. In order to proceed as simple as possible for this first stage of skeleton scanning modeling, the authors decided to adopt the classical chess strategy on the area (C). This is consistent with the scanning principal adopted in this study, as the chess schema remains perpendicular and parallel to the skeleton. Thus, the constraint of the perpendicularity to the skeleton is well assured. It is possible to adopt a meander scanning perpendicular to the $\overrightarrow{X}$ axis which will correspond to a scanning perpendicularly to the skeleton of the region (C). In this case the scanning parameters of the (C) region will be $\left({n}_{{C}_{x}},{n}_{{C}_{y}}\right)=\left(\mathrm{1,1}\right)$.

The productive length ${L}_{{P}_{C}}$ and the non-productive or jump length ${L}_{{J}_{C}}$ can be similarly formulated from Eqs. (81) and (82) that are developed for the chess scanning Appendix II.A. As expressed by these equations, the scanning schema can be fixed according to the number or the dimensions of the patterns of Figs. 5 and 6.

After manual development adaptation, the productive and the jump lengths of the area (C) are expressed by the Eqs. (77) and (78), respectively:

$${{L}_{P}}_{C}=\frac{{N}_{1}{\alpha }_{1}{L}_{1}}{2}\left(\frac{1}{{n}_{{C}_{x}}}\left(\frac{{L}_{2}}{2{n}_{{C}_{y}}{e}_{s}}+1\right)+\frac{1}{{n}_{{C}_{y}}}\left(\frac{{\alpha }_{2}{L}_{1}}{2{n}_{{C}_{x}}{e}_{s}}+1\right)\right).$$

Since

$$\left\{\begin{array}{c}{\alpha }_{1}{L}_{1}=\frac{{L}_{2}}{2}\\ {\alpha }_{2}{L}_{1}={L}_{1}-{L}_{2}\end{array}\right.,$$

the productive and non-productive lengths of C area become:

$${{L}_{P}}_{C}=\frac{{N}_{1C}{L}_{2}}{4}\left(\frac{1}{{n}_{{C}_{x}}}\left(\frac{{L}_{2}}{2{n}_{{C}_{x}}{n}_{{C}_{y}}{e}_{s}}+1\right)+\frac{1}{{n}_{{C}_{y}}}\left(\frac{\left({L}_{1}-{L}_{2}\right)}{2{n}_{{C}_{x}}{e}_{s}}+1\right)\right),$$

(77)

$${{L}_{J}}_{C}=\frac{{L}_{2}}{2}\left(\frac{{N}_{1C}}{{n}_{{C}_{y}}}+\frac{{N}_{2C}{L}_{2}}{2{n}_{{C}_{x}}}\right),$$

(78)

$$\begin{array}{c}\text{if }\left(\left({n}_{{C}_{x}}\;\mathrm{i}\mathrm{s}\;\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\;\right)\;\text{or}\;\left({n}_{{C}_{y}}\;\mathrm{i}\mathrm{s}\;\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\right)\right): {N}_{1C}={N}_{2C}=\frac{{n}_{{C}_{x}}{n}_{{C}_{y}}}{2}\\ \text{else }\left\{\begin{array}{c}\text{ if chess begins with pattern} 1: {N}_{1C}=E\left(\frac{{n}_{{C}_{x}}{n}_{{C}_{y}}}{2}\right)+1\\ \text{ if chess begins with pattern} 2: {N}_{2C}=E\left(\frac{{n}_{{C}_{x}}{n}_{{C}_{y}}}{2}\right)+1\\ {N}_{1}+{N}_{2}={n}_{{C}_{x}}{n}_{{C}_{y}}\end{array}\right.\end{array}.$$

(79)

where N_1C is the total number of chess pattern 1 (Fig. 8a); N_2C is the total number of pattern 2 (Fig. 8b); ${n}_{{C}_{x}}$ is the number of divisions of the dimension of the rectangle according to the pattern along the axis $\overrightarrow{x}$; ${n}_{{C}_{y}}$ is the number of divisions of the dimension of the rectangle according to the pattern along the axis $\overrightarrow{y}.$

Appendix II: Benchmark strategies lengths modeling

2.1 A. Classical chess strategy: production time formulation

2.1.1 a. Geometry parametrization

Figure 23 presents a typical example of the classical chessboard scanning strategy. According to this figure, the system (80) presents the possible count of pattern 1 and pattern 2 as displayed in Fig. 8 of Sect. 3.1.2. The system (80) expresses also the divisions of the $\left({L}_{1}\times {L}_{2}\right)$ rectangle into regular $\left({n}_{x}\times {n}_{y}\right)$ portions.

$$\left\{\begin{array}{c}{n}_{x}=E\left(\frac{{L}_{1}}{{a}_{x}}\right); {n}_{x}=\frac{{L}_{1}}{{a}_{x}}\in {\mathbb{N}}^{\mathrm{*}};{a}_{x}=\frac{{L}_{1}}{{n}_{x}}\\ {n}_{y}=E\left(\frac{{L}_{1}}{{a}_{x}}\right); {n}_{y}=\frac{{L}_{2}}{{a}_{x}}\in {\mathbb{N}}^{\mathrm{*}};{a}_{y}=\frac{{L}_{2}}{{n}_{y}}\end{array}\right.,$$

(80)

where a_x is the length according to the $\overrightarrow{x}$ axis; a_y is the length according to the $\overrightarrow{y}$ axis; e is the hatch space of the chess strategy.

Fig. 23

Classical Chess scanning policy in the additive manufacturing processes

2.1.2 b. Total scanning length

2.1.2.1 i. Productive time: active scanning

The productive time of the pattern 1 and pattern 2 could be determined directly from Fig. 8a and b, respectively.

The productive lengths of patterns 1 and 2, are respectively noted as ${L}_{{P}_{1}}$ and ${L}_{{P}_{2}}$, and are expressed by the Eqs. (81.1) and (81.1):

$$\begin{aligned}{L}_{{P}_{1}} &={a}_{x}\left({m}_{y}+1\right) \quad (81.1)\\ {L}_{{P}_{2}} & ={a}_{y}\left({m}_{x}+1\right)\quad (81.2)\\ {m}_{x} &=E\left(\frac{{a}_{x}}{e}\right) \;\;\;\qquad (81.3)\\ {m}_{y} &=E\left(\frac{{a}_{y}}{e}\right),\end{aligned}$$

(81)

s.t.

${m}_{x}$ and ${m}_{y}$ are, respectively, the number of divisions, resp. of the patterns 1 and 2 along the axis $\overrightarrow{x}$ and $\overrightarrow{y}$.

2.1.2.2 ii. Inactive/jump scanning lengths

The inactive or jump scanning could be defined as the scanning during which the laser or the electron beam is switched off in the case of PBF-LB/M, PBF-LB/P, PBF-EB/M processes. For Material Extrusion-based AM, this step is considered as productive. The jump scanning lengths, ${L}_{{J}_{1}}$ and ${L}_{{J}_{2}}$, related to pattern 1 and pattern 2 respectively are expressed by expressions (82.1) and (82.2):

$$\begin{aligned}{L}_{{J}_{1}}& ={a}_{y}\quad (82.1)\\ {L}_{{J}_{2}} & ={a}_{x} \quad (82.2)\end{aligned}$$

(82)

2.1.3 c. Total scanning length

Hence, the total lengths ${L}_{\mathrm{c}{\mathrm{h}}_{1}}$ and ${L}_{\mathrm{c}{\mathrm{h}}_{2}}$ of resp. pattern 1 and pattern 2 of the chess strategy are expressed by the system (83):

$$\left\{\begin{array}{c}{L}_{\mathrm{c}{\mathrm{h}}_{1}}={a}_{x}\left(E\left(\frac{{a}_{y}}{e}\right)+1\right)+{a}_{y} \left(83.1\right)\\ {L}_{\mathrm{c}{\mathrm{h}}_{2}}={a}_{y}\left(E\left(\frac{{a}_{x}}{e}\right)+1\right)+{a}_{x} \left(83.2\right)\end{array}\right..$$

(83)

Thus the total scanning time of the filling area related to the chess scanning strategy in the case of a rectangle ${L}_{1}\times {L}_{2}$ is given by the Eq. (84) and the system (85):

$${L}_{\mathrm{c}\mathrm{h}}={N}_{1} {L}_{\mathrm{c}{\mathrm{h}}_{1}}+{N}_{2}{L}_{\mathrm{c}{\mathrm{h}}_{2}},$$

(84)

$$\begin{array}{c}\text{if }\left(\left({n}_{x} \mathrm{i}\mathrm{s}\;\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r} \right) \mathrm{o}\mathrm{r} \left({n}_{y} \mathrm{i}\mathrm{s}\;\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\right)\right): {N}_{1}={N}_{2}=\frac{{n}_{x}{n}_{y}}{2}\qquad \;\;\;\; \;\left(85.1\right)\\ \text{else }\left\{\begin{array}{c}\text{if }\left(\begin{array}{c}\text{the scanning begins }\\ \text{with pattern }1\end{array}\right):{N}_{1}=E\left(\frac{{n}_{x}{n}_{y}}{2}\right)+1 \quad \left(85.2\right)\\ \text{if }\left(\begin{array}{c}\text{the scanning begins}\\ \text{ with pattern }2\end{array}\right):{N}_{2}=E\left(\frac{{n}_{x}{n}_{y}}{2}\right)+1 \quad\left(85.3\right)\\ {N}_{1}+{N}_{2}=1\end{array}\right.\end{array}.$$

(85)

where N₁ is the total number of pattern 1; N₂ is the total number of pattern 2; ${n}_{x}$ is the number of divisions of the dimension of the rectangle along the axis $\overrightarrow{x}$; ${n}_{y}$ is the number of divisions of the dimension of the rectangle along the axis $\overrightarrow{y}$;

2.2 B. Stripe strategy

Figure 9 od Sect. 3.1.3 presents the geometry parametrization of a stripe strategy scanning.

The distance “d” can be negative in the case of scanning overlap, positive or zero. In the case of this study, “d” is considered zero, which means that no overlap is considered. Other stripes strategies could be analyzed by the same methodology.

2.2.1 a. Stripe strategy parallel to x axis: pattern 1

The geometry analysis allows the calculation of the length of scanning according to the procedure described by the Eq. (86) and the system (87). The geometry parametrization if detailed in Fig. 9, Sect. 3.1.3:

$${L}_{\mathrm{s}\mathrm{t}1}={m}_{y}\left({m}_{x} e+{a}_{x}\right)+{L}_{x}^{r},$$

(86)

$$\begin{array}{c}{m}_{x}=\left\{\begin{array}{c}{m}_{x0}+1 \text{ if }{a}_{x}-{m}_{x0} e<e\\ {m}_{x0} \text{ if }{a}_{x}-{m}_{x0} e\ge e\end{array}\qquad (87.1)\right.\\ {m}_{y}=\left\{\begin{array}{c}{m}_{y0}+1 \text{ if }{a}_{y}-{m}_{y0} e<e\\ {m}_{y0} \text{ if }{a}_{y}-{m}_{y0} e\ge e\end{array}\right. \qquad(87.2)\\ {m}_{x0}=E\left(\frac{{a}_{x}}{e}\right) \qquad\qquad\qquad\qquad\qquad(87.3)\\ {m}_{y0}=E\left(\frac{{a}_{y}}{e}\right) \qquad\qquad\qquad\qquad\qquad(87.4)\\ \left\{\begin{array}{c}{L}_{x}^{r}={a}_{x}+{e}_{x}^{r} {m}_{x}\\ {e}_{x}^{r}=\left\{\begin{array}{c}{a}_{y}-{m}_{y} e \text{ if }{a}_{y}-{m}_{y} e \ge e\\ 0 \text{ else }\end{array}\right.\end{array}\right. \qquad(87.5)\end{array},$$

(87)

${L}_{x}^{r}$ is the length of the last raw for which the height does not necessarily correspond to the hatch spacing e but to a residual hatch e_r.

2.2.2 b. Stripe strategy parallel to y axis: pattern 2

The analytical expression for the length of pattern 2, that is denoted ${L}_{\mathrm{s}\mathrm{t}2}$, is systematically deduced from pattern 1 by permuting x and y indices (see Eq. (87)):

$${L}_{\mathrm{s}\mathrm{t}2}={m}_{x}\left({m}_{y} e+{a}_{y}\right)+{L}_{y}^{r},$$

(88)

where

$$\begin{array}{c}{L}_{y}^{r}={a}_{y}+{e}_{y}^{r} {m}_{y}\\ {e}_{y}^{r}=\left\{\begin{array}{c}{a}_{x}-{m}_{x} e \;if\; {a}_{x}-{m}_{x} e \ge e\\ 0 \;else\end{array}\right.\end{array}.$$

(89)

2.2.3 c. Total stripe strategy scanning length

The numbers N₁ and N₂ of patterns 1 and 2 respectively are calculated by the same procedure like chess scanning strategy.

Thus, the total scanning length of $\left({L}_{1}\times {L}_{2}\right)$ rectangle is calculated by the Eq. (90):

$${L}_{\mathrm{s}\mathrm{t}}={L}_{\mathrm{s}\mathrm{t}1} {N}_{1}+{L}_{\mathrm{s}\mathrm{t}2} {N}_{2}.$$

(90)

2.3 C. Spiral scanning strategy

Figure 10 of Sect. 3.1.4 presents the geometry parametrization of spiral strategy scanning.

The calculation procedure can be applied to both outer or internal spiral scanning strategy.

2.3.1 a. Spiral strategy starting by y axis: pattern 1

2.3.1.1 i. Sum of lengths along the x axis: ${\boldsymbol{L}}_{\boldsymbol{x}}$

Following the same modeling procedure and according to Fig. 10:

$$\begin{aligned} {a}_{{x}_{1}}&={a}_{x}\\ {a}_{{x}_{2}}&={a}_{{x}_{1}}-e\\ {a}_{{x}_{3}}&={a}_{{x}_{2}}-e\\ &\vdots \\ {a}_{{x}_{i}}&={a}_{{x}_{i-1}}-e.\end{aligned}$$

Summation:

$${a}_{{x}_{i}}={a}_{x}-\sum _{j=2}^{i}e$$

So:

$$\begin{aligned}& \quad\left\{\begin{array}{c}\forall i\ge 2 : {a}_{{x}_{i}} ={a}_{x}-e\left(i-1\right)\\ {a}_{{x}_{1}}={a}_{x}\end{array}\right.\\{L}_{x}&=\sum _{i=1}^{{m}_{x}}{a}_{{x}_{i}}={a}_{{x}_{1}}+\sum _{i=2}^{{m}_{x}}{a}_{{x}_{i}}\\{L}_{x}&={a}_{x}+\sum _{i=2}^{{m}_{x}}\left({a}_{x}-e\left(i-1\right)\right)={m}_{x}{a}_{x}-\sum _{i=2}^{{m}_{x}}e\left(i-1\right)\\{L}_{x}&={m}_{x}{a}_{x}-e\sum _{i=2}^{{m}_{x}}i+\sum _{i=2}^{{m}_{x}}e={m}_{x}{a}_{x}-e\left(\sum _{i=1}^{{m}_{x}}i-1\right)+e\left({m}_{x}-1\right)\\{L}_{x}&={m}_{x}{a}_{x}-e\left(\frac{{m}_{x}}{2}\left({m}_{x}+1\right)-1\right)+e\left({m}_{x}-1\right)\\{L}_{x}& ={m}_{x}{a}_{x}-e\frac{{m}_{x}}{2}\left({m}_{x}-1\right),\end{aligned}$$

(91)

${m}_{x}$ is computed according to the system (87).

2.3.1.2 ii. Sum of lengths along the y axis: ${{L}}_{{y}}$

Following the same modeling procedure and according to Fig. 10:

$$\begin{aligned}{a}_{{y}_{1}}& ={a}_{y}\\ {a}_{{y}_{2}}& ={a}_{y}\\{a}_{{y}_{3}}&={a}_{{y}_{2}}-e\\ & \vdots\\ {a}_{{y}_{i}}& ={a}_{{y}_{i-1}}-e.\end{aligned}$$

Summation:

$${a}_{{y}_{i}}={a}_{y}-\sum _{j=3}^{i}e$$

So:

$$\begin{aligned}& \quad\left\{\begin{array}{c}\forall i\ge 3 : {a}_{{y}_{i}} ={a}_{y}-e\left(i-2\right)\\ {a}_{{y}_{1}}={a}_{{y}_{2}}={a}_{y}\end{array}\right.\\{L}_{y}&=\sum _{i=1}^{{m}_{y}}{a}_{{y}_{i}}={a}_{{y}_{1}}+{a}_{{y}_{2}}+\sum _{i=3}^{{m}_{y}}{a}_{{y}_{i}}=2 {a}_{y}+\sum _{i=3}^{{m}_{y}}\left({a}_{y}-e\left(i-2\right)\right)\\{L}_{y}&=2 {a}_{y}+\left({m}_{y}-2\right){a}_{y}-e\sum _{i=3}^{{m}_{y}}\left(i-2\right)\\{L}_{y}&={m}_{y}{a}_{y}+2e\left({m}_{y}-2\right)-e\left(-1-2+\sum _{i=1}^{{m}_{y}}i\right)\\{L}_{y}&={m}_{y}{a}_{y}+2 e {m}_{y}-4e+3e-e\frac{my}{2}\left({m}_{y}+1\right)\\{L}_{y}&={m}_{y}{a}_{y}+e\left(2{m}_{y}-1-\frac{{m}_{y}}{2}\left({m}_{y}+1\right)\right)\\{L}_{y}& ={m}_{y}{a}_{y}-\frac{e}{2}\left(-4{m}_{y}+2+{m}_{y}\left({m}_{y}+1\right)\right)\\ {L}_{y}& ={m}_{y}{a}_{y}-\frac{e}{2}\left({m}_{y}^{2}-3{m}_{y}+2\right),\end{aligned}$$

(92)

${m}_{y}$ is computed according to the system (87).

2.3.1.3 iii. Total length of pattern 1

The total length of pattern 1 is expressed by Eq. (93):

$$\begin{aligned}{L}_{1}&={L}_{x}+{L}_{y}\\{L}_{sp1}&={m}_{x}{a}_{x}+{m}_{y}{a}_{y}-\frac{e}{2}\left({m}_{x}\left({m}_{x}-1\right)+\left({m}_{y}^{2}-3{m}_{y}+2\right)\right).\end{aligned}$$

(93)

2.3.2 b. Spiral strategy starting by y axis: pattern 2

Pattern 2 of the spiral strategy corresponds to scanning starting parallel to the x axis. The total scanning length of spiral pattern 2 can be deduced from expression (93) by swapping the indices x and y resulting in the Eq. (94):

$${L}_{\mathrm{s}\mathrm{p}2}={m}_{x}{a}_{x}+{m}_{y}{a}_{y}-\frac{e}{2}\left({m}_{y}\left({m}_{y}-1\right)+\left({m}_{x}^{2}-3{m}_{x}+2\right)\right).$$

(94)

2.3.3 c. Total spiral strategy scanning length

The numbers N1 and N2 of patterns 1 and 2, respectively, are calculated by the same procedure as the chess scanning strategy.

The total spiral strategy scanning length of the ${L}_{1}\times {L}_{2}$ is calculated by the Eq. (95):

$${L}_{\mathrm{s}\mathrm{p}}={N}_{1} {L}_{\mathrm{s}\mathrm{p}1}+{N}_{2}{L}_{\mathrm{s}\mathrm{p}2}.$$

(95)

2.4 D. Contour scanning strategy

Figure 11 of Sect. 3.1.5 presents the geometry parametrization of contour strategy scanning.

2.4.1 a. Contour strategy starting by y axis: pattern 1

Pattern 1 of the contour strategy corresponds to scanning starting parallel to the y axis. The total scanning length of contour pattern 1 is modeled according to the following sections.

2.4.1.1 i. Sum of lengths along the x axis: ${{L}}_{{x}}$

Following the same modeling procedure and according to Fig. 11:

$$\begin{aligned}{a}_{{x}_{1}}&={a}_{x}\\{a}_{{x}_{2}}&={a}_{{x}_{1}}\\{a}_{{x}_{3}}&={a}_{{x}_{2}}-2e\\{a}_{{x}_{4}}&={a}_{{x}_{3}}-2e\\{a}_{{x}_{i}}&={a}_{{x}_{i-1}}-2e\end{aligned}$$

Summation:

$$\begin{aligned}& \quad\left\{\begin{array}{c}\forall i\ge 3 {a}_{{x}_{i}} ={a}_{x}-2\sum _{j=3}^{i}e\\ {a}_{{x}_{1}}={a}_{{x}_{2}}={a}_{x}\end{array}\right.\\{a}_{{x}_{i}}&={a}_{x}-2e\left(i-2\right)\\{L}_{x}&=\sum _{i=1}^{{m}_{x}}{a}_{{x}_{i}}={a}_{{x}_{1}}+{a}_{{x}_{2}}+\sum _{i=3}^{{m}_{x}}\left({a}_{x}-2e\left(i-2\right)\right)\\{L}_{x}&=\sum _{i=1}^{{m}_{x}}{a}_{{x}_{i}}=2{a}_{x}+{a}_{x}\left({m}_{x}-2\right)-2e\sum _{i=3}^{{m}_{x}}\left(i-2\right)\\{L}_{x}&={m}_{x}{a}_{x}-2e\left(-1-2+\sum _{i=1}^{{m}_{x}}i\right)+4e\left({m}_{x}-2\right)\\{L}_{x}&={m}_{x}{a}_{x}+e\left(6-{m}_{x}\left({m}_{x}+1\right)+4\left({m}_{x}-2\right)\right)\\{L}_{x}&={m}_{x}{a}_{x}-e\left({m}_{x}^{2}-3{m}_{x}+2\right),\end{aligned}$$

(96)

${m}_{x}$ is computed according to the system (87).

2.4.1.2 ii. Sum of lengths along the y axis: ${{L}}_{{y}}$

Following the same modeling procedure and according to Fig. 11:

$$\begin{aligned}{a}_{{y}_{1}}&={a}_{y}\\{a}_{{y}_{2}}&={a}_{{y}_{1}}-e\\{a}_{{y}_{3}}&={a}_{{y}_{2}}-2e\\{a}_{{y}_{4}}&={a}_{{y}_{3}}+e\\{a}_{{y}_{5}}&={a}_{{y}_{4}}-2e\\{a}_{{y}_{6}}&={a}_{{y}_{5}}-e\\{a}_{{y}_{7}}&={a}_{{y}_{6}}-2e\\{a}_{{y}_{8}}& ={a}_{{y}_{7}}+e\\& \quad \vdots\\ & \left\{\begin{array}{c}\forall i\ge 3 : {a}_{{y}_{i}} ={a}_{y}-2 e \left(E\left(\frac{i-3}{2}\right)+1\right)\\ {a}_{{y}_{1}}={a}_{y}\end{array}\right.\end{aligned}$$

Summation:

$$\begin{aligned}{L}_{y}&=\sum _{i=1}^{{m}_{y}}{a}_{{y}_{i}}={a}_{{y}_{1}}+{a}_{{y}_{2}}+\sum _{i=3}^{{m}_{y}}{a}_{{y}_{i}}\\{L}_{y}&=2{a}_{y}+\sum _{i=3}^{{m}_{y}}\left({a}_{y}-2 e \left(E\left(\frac{i-3}{2}\right)+1\right)\right)\\{L}_{y}&={m}_{y}{a}_{y}-2 e\left({m}_{y}-2+\sum _{i=3}^{{m}_{y}}E\left(\frac{i-3}{2}\right)\right),\end{aligned}$$

(97)

${m}_{y}$ is computed according to the system (87).

2.4.1.3 iii. Total length of scanning along the diagonal

The total length along the diagonal is the sum of the e’ distances (Fig. 11).

Flowing the same techniques of indexing, it can be stated that:

$${e}_{i}^{{'}}=\left\{\begin{array}{c}\sqrt{2}e;\quad \text{if }i\text{ pair}\\ \text{NaN}\quad \text{else}\end{array}\right..$$

(98)

Thus, the total length along the diagonal is expressed by:

$${L}_{\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}}=\sum _{i=1}^{m}{e}_{i}^{{'}}+{e}_{r}^{{'}}.$$

(99)

The parameter m and ${e}_{r}^{{'}}$ are calculated using the procedure (100) and (101). ${e}_{r}^{{'}}$ is a residual distance to scan in order to avoid a lack of matter in the last processing step which can be caused by the integer count ${m}_{x}$ and ${m}_{y}$ of the scanning steps.

$$\begin{array}{c}\text{if }{a}_{{x}_{mx}}<e:\left\{\begin{array}{c} m={m}_{x}\\ {e}_{r}^{{'}}=0\end{array}\right.\\\text{ else} : \left\{\begin{array}{c}\text{if }{a}_{{y}_{my}}<e :\left\{\begin{array}{c}m={m}_{y}\\ {e}_{r}^{{'}}=\sqrt{{a}_{{y}_{my}}^{2}+{\left({e}_{y}^{r}\right)}^{2}}\\ {e}_{y}^{r}={a}_{{x}_{mx}}-e\end{array}\right.\\ \text{else }:\left\{\begin{array}{c}m={m}_{y}\\ {e}_{r}^{{'}}={e}^{{'}}=\sqrt{2} e\end{array}\right.\end{array}\right.\end{array} ,$$

(100)

$$\left\{\begin{array}{c}\text{if }{a}_{{x}_{my}}<e:\left\{\begin{array}{c} m={m}_{y}\\ {e}_{r}^{{'}}=0\end{array}\right.\\ \text{else }: \left\{\begin{array}{c}\text{if }{a}_{{x}_{mx}}<e :\left\{\begin{array}{c}m={m}_{x}\\ {e}_{r}^{{'}}=\sqrt{{a}_{{x}_{mx}}^{2}+{\left({e}_{x}^{r}\right)}^{2}}\\ {e}_{x}^{r}={a}_{{y}_{my}}-e\end{array}\right.\\ \text{else }:\left\{\begin{array}{c}m={m}_{x}\\ {e}_{r}^{{'}}={e}^{{'}}=\sqrt{2} e\end{array}\right.\end{array}\right.\end{array}\right. .$$

(101)

2.4.1.4 iv. Total length of contour pattern 1

The total length of contour pattern is computed according to Eq. (102):

$${L}_{\mathrm{c}\mathrm{o}\mathrm{n}{\mathrm{t}}_{1}}={L}_{x}+{L}_{y}+{L}_{\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}}.$$

(102)

2.4.2 b. Contour strategy starting by x axis: pattern 2

Pattern 2 of the contour strategy corresponds to scanning starting parallel to the x axis. The total scanning length of contour pattern 2 can be deduced from the previous paragraph by swapping the indices x and y of Eq. (102) and related equations.

2.4.3 c. Total contour strategy scanning length

The total contour strategy scanning length of the ${L}_{1}\times {L}_{2}$ is calculated by Eq. (103).

The numbers N₁ and N₂ of patterns 1 and 2, respectively, are calculated by the same procedure like chess scanning strategy:

$${L}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}={N}_{1} {L}_{\mathrm{c}\mathrm{o}\mathrm{n}{\mathrm{t}}_{1}}+{N}_{2}{L}_{\mathrm{c}\mathrm{o}\mathrm{n}{\mathrm{t}}_{2}}.$$

(103)

Appendix III: Scanning parameters according to the bibliography

See Table 6.

Table 6 AM filling parameters—case of PBF-LB/M processes

Appendix IV: Curves and surfaces modeling

4.1 A. Case of study 1: Modeling of benchmark strategies lengths ${L}_{i}$ according to the hatch space distance e (Fig. 14)

According to Fig. 14, the authors propose to fit hyperbolic curves to the $(e,{L}_{i})$ dispersions for the set of benchmark scanning strategies. Equation (104) present the proposed model:

$${\stackrel{\sim }{L}}_{i}=\frac{a}{{e}^{n}},$$

(104)

where $\left\{\begin{array}{c}a>0\\ n>0\end{array}\right.. {\stackrel{\sim }{L}}_{i}$ is the approximation of the L_i distribution; i is a given strategy from the set: $\left\{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{s} \left(i=1\right), \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{e} \left(i=2\right), \mathrm{s}\mathrm{p}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}\;\left(i=3\right), \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{u}\mathrm{r} \left(i=4\right)\right\}$.

A preliminary linearization of the distribution was applied by means of the log function, as follows:

$$\mathrm{log}\left({\stackrel{\sim }{L}}_{i}\right)=\mathrm{log}\left(a\right)-n \mathrm{l}\mathrm{o}\mathrm{g}\left(e\right).$$

(105)

Hence, linear regression was applied for all L_i curves. The models parameters estimation is presented in Table 7 with the correspondent R² statistics related to the linear fitting.

Table 7 Hyperbolic fitting of scanning benchmark strategies lengths—case of study 1 (Fig. 14)

4.2 B. Case of study 2: modeling of the first order slopes of ${L}_{{S}{B}{P}}$ according to the hatch space distance e (Fig. 18)

According to Fig. 18, the authors propose to fit hyperbolic curves to the first order variations or slopes of ${L}_{\text{SBP}}$ that were noted in Fig. 18 as:

$$\left\{\begin{array}{c}{b}_{1}=\partial {L}_{\text{SBP}}/ \partial {L}_{1}\\ {b}_{2}=\partial {L}_{\text{SBP}}/\partial {L}_{2}\\ {b}_{3}=\partial {L}_{\text{SBP}}/\partial {n}_{{C}_{x}}\\ {b}_{4}=\partial {L}_{\text{SBP}}/\partial {n}_{{C}_{y}}\end{array}\right..$$

The hyperbolic models that are proposed are expressed by Eq. (106):

$${b}_{i}=\frac{a}{{e}^{n}}.$$

(106)

where $\left\{\begin{array}{c}i\in \left\{1,..,4\right\}\\ a>0\\ n>0\end{array}\right..$

The same linearization and fitting procedures were applied similarly to the modeling of the previous paragraph.

The models parameters estimation is presented in Table 8 with the correspondent R² statistics related to the linear fitting.

Table 8 Hyperbolic fitting of ${L}_{\text{SBP}}$ slopes—case of study 1 (Fig. 18)

4.3 C. Case of study 2: modeling of ${L}_{\mathrm{S}\mathrm{B}\mathrm{P}}$ according to $\left({L}_{1},\;{L}_{2},\;{n}_{{C}_{x}},{n}_{{C}_{y}}\right)$

For each hatch space, linear, interactions, pure quadratic, and full quadratic models were tested in order to propose a regression modeling for ${L}_{\mathrm{S}\mathrm{B}\mathrm{P}}$ as simple as possible. Figure 24 shows the R² statistics related to each regressive model. It is interesting to see that full quadratic models produce high values of R², generally higher than 75%.

Hence, one can argue that the family of functions ${L}_{SB{P}_{e}}\left({L}_{1}, {L}_{2}, {n}_{{C}_{x}},{n}_{{C}_{y}}\right)$ could be described as a set of quadratics of $\left({L}_{1}, {L}_{2}, {n}_{{C}_{x}},{n}_{{C}_{y}}\right)$ according to the formulation (107).

For each hatch space distance e:

$${\stackrel{\sim }{L}}_{\mathrm{S}\mathrm{B}{\mathrm{P}}_{e}}\left(\overrightarrow{x}\right)=\frac{1}{2}{\overrightarrow{x}}^{T}{A}_{e}\overrightarrow{x}+{\overrightarrow{{b}_{e}}}^{T}\overrightarrow{x}+{\gamma }_{e}.$$

(107)

where $\overrightarrow{x}={\left({L}_{1}, {L}_{2}, {n}_{{C}_{x}},{n}_{{C}_{y}}\right)}^{T}$ is the model geometrical features; ${A}_{e}$ is the family of symmetric matrices associated to the quadratic terms of the models ${\stackrel{\sim }{L}}_{\mathrm{S}\mathrm{B}{\mathrm{P}}_{e}}$; $\overrightarrow{{b}_{e}}$ is the family of the multiplications of the first order terms of the function ${\stackrel{\sim }{L}}_{\mathrm{S}\mathrm{B}{\mathrm{P}}_{e}}$; ${\gamma }_{e}$ is the family of constant terms associated to the function ${\stackrel{\sim }{L}}_{\mathrm{S}\mathrm{B}{\mathrm{P}}_{e}}$ (see Fig. 24).

Fig. 24

R² statistics VS hatch space for

Appendix V: Specific gains ${S}{{G}}_{\max}$ plot

The maximal specific gains $S{G}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ were computed at minimal values of ${L}_{\text{SBP}}$ corresponding to $\left(\mathrm{m}\mathrm{i}\mathrm{n}\left({n}_{{C}_{x}}\right), \mathrm{m}\mathrm{i}\mathrm{n}\left({n}_{{C}_{y}}\right)\right)$ (see Fig. 25).

Fig. 25

Specific gains $S{G}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ according to $\left({L}_{1}, {L}_{2},e\right)$ a SBP VS Chess b SBP VS Stripe c SBP VS Spiral d SBP VS Contour. $S{G}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are computed for the higher values of ${L}_{\mathrm{S}\mathrm{B}\mathrm{P}}$ at $\left(\mathrm{m}\mathrm{a}\mathrm{x}\left({n}_{{C}_{x}}\right), \mathrm{m}\mathrm{a}\mathrm{x}\left({n}_{{C}_{y}}\right)\right)$