# Assessing the Accuracy of Surface-Based Fibre Orientation Interpolation for Predicting Mechanical Stiffness in Douglas Fir Timber: A FEM Study using Serial Sectioning Determined Fibre Orientation and DIC as benchmarks

Hélène Penvern, Guillaume Pot, Joffrey Viguier

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Hélène Penvern, Guillaume Pot, Joffrey Viguier. Assessing the Accuracy of Surface-Based Fibre Orientation Interpolation for Predicting Mechanical Stiffness in Douglas Fir Timber: A FEM Study using Serial Sectioning Determined Fibre Orientation and DIC as benchmarks. 2025. hal-05292977

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Distributed under a Creative Commons Attribution 4.0 International License
Assessing the Accuracy of Surface-Based Fibre
Orientation Interpolation for Predicting
Mechanical Stiffness in Douglas Fir Timber:
A FEM Study using Serial Sectioning Determined
Fibre Orientation and DIC as benchmarks

Helene Penvern¹, Guillaume Pot¹, Joffrey Viguier¹

¹Arts et Metiers Institute of Technology, LaBoMaP, 13 rue Porte de Paris, Cluny, F-71250, France.

Contributing authors: [helene.penvern@ensam.eu](mailto:helene.penvern@ensam.eu);

**Abstract**

Accurate prediction of local stiffness in sawn timber is essential for improving
strength grading systems and optimising material use. This study systematically
compares several internal fibre orientation interpolation (IFOI) strategies used in
strength grading, by integrating them into finite element models (FEM) simula-
tions and benchmarking the related modulus of elasticity (MoE) profiles against
digital image correlation (DIC) measurements and serial-sectioning determined
internal fibre orientation (IFOD GST,DSS) FEM simulations corresponding to
10 Douglas-fir boards sawn from one log.

Results show that, though all methods provide stiffness profiles resembling those
obtained from the DIC measurements, the determined fibre orientation lead to
the most stable and accurate correlations overall, whereas interpolation-based
schemes are more prone to irregularities. These findings indicate that refining fibre
orientation interpolation schemes, improving local material characterisation from
determined internal fibre orientation, is key to strengthening predictive models.
Moreover, curvature-based MoE computations revealed greater variability and
closer correspondence with DIC data than strain-based approaches, effectively
capturing internal heterogeneity. This highlights the promising potential of
curvature-based MoE profiles for industrial grading, warranting systematic
evaluation at larger scales.

**Keywords:** grain angle, GST, MoE profiles, curvature, wood, local direction
# 1 Introduction

## 1.1 Background

Accurately predicting the mechanical behaviour of sawn timber is a critical challenge in wood engineering, with direct implications for structural design and material optimisation. Wood has gained renewed popularity as a sustainable construction material, praised for its aesthetic qualities and carbon-negative footprint. Structurally, its strength-to-weight ratio can rival that of steel. However, unlike steel, wood exhibits significant variability in mechanical properties due to its heterogeneous and anisotropic nature. This variability limits the precision of current grading systems and leads to conservative safety factors, increasing material use and construction costs. A better understanding and characterisation of local mechanical properties could improve non-destructive grading and promote more efficient use of timber resources.

Due to its anatomical structure, wood is a strongly orthotropic material, with mechanical properties that differ substantially depending on its longitudinal (fibre), radial (normal to the growth rings), and tangential (along the growth rings) directions. In defect-free, straight-grained specimens, the relationship between mechanical stiffness and fibre orientation relative to loading can be accurately described using rotated stiffness matrices or empirical models like the Hankinson formula (Hankinson, 1921; Olsson et al, 2013; Pot et al, 2024a). Values for wood anisotropic stiffness components and their dependency on variables such as moisture content and density can be found in Kretschmann (2010); Guitard (1987).

Due to the severe local fibre deviation they introduce, knots significantly reduce sawn timber strength and stiffness (Bodig and Jayne, 1982; Guindos and Guaita, 2012). Some recent strength grading machines are equipped with laser-based sensors that exploit the so-called tracheid effect to measure surface fibre orientation of timber boards. This optical phenomenon causes laser dots projected onto the wood surface to scatter into elliptical shapes, where the major axis of the ellipse aligns with the projected fibre direction, provided the ellipse aspect ratio is sufficiently high (Nyström, 2003; Simonaho et al, 2004; Besseau et al, 2020). The ability to capture surface fibre orientation in this way has enabled the development of automated strength grading systems. These systems incorporate surface fibre orientation data into predictive models of mechanical performance, either through direct statistical regression methods (Brännström et al, 2008; Viguier, 2015; Rais et al, 2021; Jehl et al, 2011) or through physics-based approaches such as beam theory and finite element modelling (FEM) (Viguier et al, 2015; Olsson et al, 2013, 2018; Briggert et al, 2020; Olsson et al, 2022; Pot et al, 2024b).

Several internal fibre orientation interpolation (IFOI) techniques have been proposed, as summarised in Table 1. Some methods are purely geometric, such as nearest-neighbour interpolations (Olsson et al, 2013) or linear interpolation from opposite surfaces (Besseau, 2021; Pot et al, 2024b). Hu et al (2018) introduced pith-informed interpolation, which showed marginal improvement in bending strength prediction as later confirmed in Olsson et al (2022). Other advanced approaches rely on Foley's equations to describe fibre deviations near knots, though these typically
require manual tuning and remain limited in generalisation (Foley, 2003; Lukacevic et al, 2019; Habite and Olsson, 2022; Huber et al, 2022).

<table><caption>**Table 1:** Summary of angular interpolation methods with associated references</caption><thead><tr><th>Denomination</th><th>1A,flat</th><th>1A,roof</th><th>1A,radial</th><th>2A</th></tr></thead><tbody><tr><th>Section wise IFOI scheme</th><td>Single angle two-side nearest IFOI</td><td>Single angle four-side nearest IFOI</td><td>Single angle with radial IFOI</td><td>Mix direction with linear IFOI</td></tr><tr><th>Logic</th><td>The in-plane fibre angle measured on the broad surfaces of the board is assigned uniformly to the entire underlying volume</td><td>The in-plane fibre angle measured on both the broad and edge surfaces of the board is assigned uniformly to the entire underlying volume</td><td>A mesh grid is defined based on the spacing of the laser dot measurements. For each sub-volume, an angle is assigned as the distance-weighted average of the angles measured on the surfaces intersected by the radial line connecting the sub-volume to the pith.</td><td>After creating a regular mesh grid of defined size and nearest-neighbour interpolation of the in-plane fibre angles on the four side surfaces, each sub-volume is assigned a global direction computed as a composition of the distance-weighted edge-wise and flat-wise average fibre in-plane angles.</td></tr><tr><th>Citation</th><td>(Olsson et al, 2018)</td><td>(Olsson et al, 2013, 2018; Hu et al, 2018; Olsson et al, 2022)</td><td>(Hu et al, 2018; Olsson et al, 2022)</td><td>(Besseau, 2021; Pot et al, 2024b)</td></tr></tbody></table>

Attempts to infer out-of-plane fibre angles directly from laser dot ellipse ratios have also been made (Briggert et al, 2018), but were shown to be unreliable due to confounding influences like moisture content and earlywood-latewood contrast (Purba et al, 2020; Besseau et al, 2020; Boivin, 2024).

Since direct measurement of internal fibre orientation is difficult, model validation must rely on indirect observations. FEM simulations using interpolated fibre orientation fields can be compared to experimentally derived stiffness distributions from digital image correlation (DIC). This approach has been adopted in several previous studies (Hu et al, 2018; Lukacevic et al, 2019; Huber et al, 2022; Penvern et al, 2025).
More recently, methods have been developed to reconstruct internal fibre ori-
entation fields from either X-ray computed tomography (CT) or destructive serial
sectioning (DSS) combined with high-resolution imaging. These approaches—referred
to as internal fibre orientation determination (IFOD) techniques—offer a more objec-
tive reference for evaluating fibre orientation interpolation (IFOI) models. While
differing in implementation, all methods are based on the principle that internal fibre
orientation follows the curvature of growth layers, whose geometry can be determined
by manual segmentation (Penvern et al, 2024), automatic segmentation from CT data
(Hu et al, 2025b), or Gradient Structure Tensor (GST) analysis applied to grayscale
DSS stacks (Penvern et al, 2025).

If growth layer geometry alone can provide an estimate of internal fibre orienta-
tion, combining it with surface fibre orientation derived from laser dot patterns yields
more accurate predictions than growth layer analysis by itself (Penvern et al, 2025).
Although such coupling currently requires destructive sectioning, recent studies sug-
gest that artificial intelligence models trained on combined CT and laser dot data may
be able to predict full internal fibre fields non-destructively in the future (Abdeljaber
et al, 2025).

The reconstructed fibre orientation fields can then be incorporated into FEM sim-
ulations to generate MoE profiles, which are compared against DIC measurements for
validation (Hu et al, 2018; Lukacevic et al, 2019; Hu et al, 2025a)s. Such comparisons
have consistently confirmed that local fibre orientation is a key driver of mechani-
cal behaviour and that accurate modelling of these fields is essential for predictive
reliability.

1.2 Aim and Scope

The aim of this study is to critically evaluate the impact of different IFOI and IFOD
strategies on local stiffness predictions in sawn timber, using multiple boards with
diverse orientation patterns. Fibre orientation fields—either interpolated or directly
determined—are incorporated into finite element models, from which local MoE pro-
files are derived using strain- and curvature-based approaches. These profiles are
compared with the relative MoE profiles produced from ground-truth measurements
from DIC, enabling a spatially resolved validation of interpolation robustness. In
this way, the evaluation of the previously developed fibre orientation reconstruction
method based on destructive serial sectioning of the board is extended beyond a sin-
gle case study, providing a more comprehensive assessment of its robustness. This
comparison also quantifies the discrepancies between interpolation-based and ground-
truth-informed models, and highlights how closely IFOI and IFOD strategies can
reproduce the reality observed in DIC measurements. In parallel, we introduce a novel
optical flow-based interpolation method and benchmark it against existing geometric
and pith-based schemes.
# 2 Materials and experimental procedure

## 2.1 Material

In the context of the *EffiQuAss* project (ANR-21-CE10-002), 16 Douglas Fir (*Pseudotsuga menziesii* (Mirb.) Franco) ~2.8 m long logs of various diameters were harvested by the *Bongard-Bazot et Fils* sawmill company, photographed and cut into boards or planks according to sawmill practice with the aim, among others, of later reconstructing the fibre orientation throughout the log.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Boards localisation at the log small end</figcaption></figure>

From this batch, a log was selected for its prominent central whorl and relatively small diameter (≈ 40 cm), which limited the number of resulting boards while still offering variation in cutting patterns and radial positions relative to the pith, as visible in Figure 1.

Board numbers were written on their broad side on the tree crown side, allowing each board orientation remained identified and associated to a reference coordinate system during the process as shown Figure 2.

## 2.2 Determination of the log pith localisation in relation to each board reference frame

After being air-dried, the boards were four-sided planed and their dimensions were measured using a tape measure and caliper. Their weight was recorded using an I-20W
<figure><img src="image_3.png" /><figcaption>Fig. 2: Board reference frame according to board numbering</figcaption></figure>

<table><caption>Table 2: Numbering, mass and dimensions of the selected air dried log boards.</caption><thead><tr><td>board number</td><td>mass (kg)</td><td>dimensions (mm)</td></tr></thead><tbody><tr><td>351</td><td>8.03</td><td>42 x 130 x 2775</td></tr><tr><td>352</td><td>8.15</td><td>42 x 130 x 2775</td></tr><tr><td>361</td><td>7.63</td><td>43 x 122 x 2775</td></tr><tr><td>362</td><td>7.66</td><td>42 x 126 x 2775</td></tr><tr><td>39</td><td>8.5</td><td>41 x 146 x 2775</td></tr><tr><td>40</td><td>7.41</td><td>41 x 142 x 2773</td></tr><tr><td>41</td><td>6.12</td><td>35 x 142 x 2774</td></tr><tr><td>42</td><td>8.29</td><td>41 x 142 x 2775</td></tr><tr><td>43</td><td>8.68</td><td>40 x 142 x 2775</td></tr><tr><td>44</td><td>7.66</td><td>36 x 142 x 2775</td></tr></tbody></table>

Ohaus industrial scale (see Table 2). Based on photographs of both the board ends and the corresponding log ends, the pith location was manually determined relative to the squared board coordinate system. This process was adapted from the method proposed by Li et al (2023).

Initially, the board and log end images were aligned in their green (undried) configuration using the automatic image registration algorithm of Li et al (2023). Manual adjustments were then applied to the board images—typically small rotations and uniform rescaling—to optimize their alignment with the corresponding log images. For each board, a binary mask of its outline within the log image was generated. From this mask, the minimum-area enclosing rectangle was computed using OpenCV's `cv2.findContours` and `cv2.minAreaRect` functions, with the largest external contour extracted using `cv2.RETR_EXTERNAL` and `cv2.CHAINAPPROXSIMPLE`.

By comparing the dimensions of these rectangles to the actual measured board dimensions, an average image scale factor was determined. The log and board end images were then reloaded and rescaled according to this average scale, preserving their previous relative positions. The alignment process was repeated, this time permitting only minor rotational and translational adjustments. Following this refinement step,
new minimum-area rectangles were computed at both ends of the log, which were assumed to define the final (potentially twisted) bounding parallelepiped of the board within the log.

Using the board scan metadata, the fibre orientation was also established—specifically, whether the pith was located toward the positive or negative z-direction of the board. This information allowed for a consistent definition of the board's local coordinate frame (0, $\vec{x}$, $\vec{y}$, $\vec{z}$), anchored to the oriented rectangles. Pith position in relation to the board masks bounding rectangles and local coordinate system are illustrated Figure 3. The pith location was then recorded in this local reference frame at both the small-end ($x = 0$) and large-end ($x = L_x$) of each board, as reported in Table 3.

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** Bounding rectangles (dark red) of the board end masks, positioned over the log small end cross-section (blue), used to determine the pith location relative to the board's local coordinate system (indicated by black axes and red directional quivers).</figcaption></figure>

## 2.3 First Optical Scanning

Directly after begin squared, the boards were scanned using an industrial timber grading scanner equipped with near-infrared laser dot and line devices (830 nm) and matrix cameras on each side of the board. As the boards passed through the scanner at a speed of 62 m/min, the matrix cameras captured images every 1 mm along the length, synchronized via a roller encoder. From the laser line illumination, grayscale images were reconstructed, while the fibre in-plane angle map was extracted by analyzing the
<table><caption>**Table 3:** Pith location at the log ends ($x = 0$ and $x = L_x$) relative to the boards reference frames in mm. Greyed boards 37 and 38 were not subjected to bending tests due to their inadequate cross-section.</caption><thead><tr><th>Board</th><th>y<sub>pith</sub>(x = 0)</th><th>z<sub>pith</sub>(x = 0)</th><th>y<sub>pith</sub>(x = L<sub>x</sub>)</th><th>z<sub>pith</sub>(x = L<sub>x</sub>)</th></tr></thead><tbody><tr><td><b>351</b></td><td>11.71</td><td>110.79</td><td>-5.94</td><td>118.97</td></tr><tr><td><b>352</b></td><td>140.69</td><td>111.47</td><td>132.69</td><td>117.35</td></tr><tr><td><b>361</b></td><td>-19.32</td><td>120.52</td><td>-16.61</td><td>121.15</td></tr><tr><td><b>362</b></td><td>121.52</td><td>124.45</td><td>117.05</td><td>118.38</td></tr><tr><td>37</td><td>83.69</td><td>-115.59</td><td>108.01</td><td>-125.01</td></tr><tr><td>38</td><td>63.33</td><td>148.40</td><td>76.78</td><td>148.18</td></tr><tr><td><b>39</b></td><td>74.38</td><td>-91.98</td><td>62.71</td><td>-67.89</td></tr><tr><td><b>40</b></td><td>77.67</td><td>-44.78</td><td>67.73</td><td>-14.88</td></tr><tr><td><b>41</b></td><td>68.47</td><td>10.02</td><td>67.91</td><td>23.05</td></tr><tr><td><b>42</b></td><td>77.53</td><td>106.84</td><td>71.10</td><td>115.78</td></tr><tr><td><b>43</b></td><td>78.58</td><td>156.02</td><td>72.00</td><td>166.62</td></tr><tr><td><b>44</b></td><td>67.90</td><td>54.48</td><td>71.74</td><td>65.79</td></tr></tbody></table>

shape of the laser dot ellipses. The grayscale images and fibre angle measurements are given within a common coordinate system by means of an internal scanner calibration that compensates for physical offsets.

The longitudinal resolution was fixed at 1 px/mm for all data. A pressure roller ensured consistent contact between the board and conveyor belt, preventing slippage. The transverse resolution was influenced by the camera focus, which varied slightly as a result of manual adjustments to the camera positions within the scanner to accommodate differences in board thickness. The average resolutions for the bottom, top, left, and right sides were computed based on manual board measurements.

The elliptical contours of the detected laser dots were fitted using the cv2.fitEllipse function from the Python OpenCV library, yielding each ellipse's center, orientation relative to the board axis, and major and minor axis lengths. The average dimensions and spacing of the fitted ellipses, along with the camera resolutions, are summarized in Table 4. Across all boards, the global mean ellipse dimensions were found to be (3.43 ± 0.10) mm × (2.01 ± 0.05) mm, with a laser dot projected approximately every (4.84 ± 0.02) mm along the board.

<table><caption>**Table 4:** Summary of laser dot spacing, ellipse dimensions, and spatial resolution per surface side averaged across boards (mean ± std).</caption><thead><tr><th>Side</th><th>Laser dot<br>spacing (px)</th><th>Ellipse<br>major axis (px)</th><th>Ellipse<br>minor axis (px)</th><th>Resolution<br>[mm/px]</th></tr></thead><tbody><tr><td>Left</td><td>43.24 ± 0.53</td><td>29.73 ± 1.03</td><td>16.82 ± 0.78</td><td>0.1149 ± 0.0006</td></tr><tr><td>Right</td><td>43.15 ± 0.10</td><td>29.27 ± 1.07</td><td>15.66 ± 0.61</td><td>0.1158 ± 0.0009</td></tr><tr><td>Top</td><td>20.51 ± 0.03</td><td>15.36 ± 0.37</td><td>9.34 ± 0.13</td><td>0.2255 ± 0.0030</td></tr><tr><td>Bottom</td><td>20.28 ± 0.01</td><td>14.69 ± 0.37</td><td>9.30 ± 0.10</td><td>0.2367 ± 0.0015</td></tr></tbody></table>

Results of these first scanning process are visible in Appendix A showing acquired four sides gray images and 1 mm × 1 mm nearest-neighbour interpolated in-plane fibre
angle of the investigated ten boards. For more details on this scanner and possible
data processing, the reader can refer to (Longuetaud et al, 2022).

## 2.4 Bending Test and Digital Image Correlation monitoring

<figure><img src="image_5.png" /><figcaption>Fig. 4: Bending bench initial configuration</figcaption></figure>

To measure the actual stiffness of the boards, stereo digital image correlation (DIC) was employed to monitor displacements during 4-point bending tests carried out on multiple boards. Each test was conducted under symmetrical loading conditions with a loading span of 625 mm and a support span of 1875 mm, as shown in Figure 4. Before testing, a speckle pattern was applied to the surface of each board, which was then placed on a 3R RP100 ELM bending bench equipped with a LaVision StrainMaster Portable system. This system comprised a laptop, a device control unit with an integrated AD converter, two 9 Mpx cameras, two continuous LED illumination units, and a tripod. Correlated images were captured and processed using the LaVision software DaVis 10.2.

The orientation and spacing of the cameras (36 cm) were selected as a compromise between stereo correlation accuracy and the size of the monitoring window within the constant moment zone. As a result, about 20 cm only of each board could be monitored at a time. After calibration, the cameras achieved a resolution of 12.184 px/mm. For displacement and strain calculations, a typical subset size of 31 px and a step size of 10 px were used, providing a resolution of 1.096 px/mm. Edge-affected regions, corresponding to an approximate 5 mm border along the board, were excluded from the displacement and strain maps.

Because of the limited field of view, multiple bending tests were conducted along the length of each board at regular intervals. For each board, testing began with one end aligned with the support of the bending bench, the reference number oriented toward the cameras. The board was loaded at a rate of 5 mm/min until reaching a maximum bending stress of approximately 15 MPa, within the elastic range, before being unloaded and repositioned for the next test, spaced approximately 14 cm apart. The initial test configuration is illustrated in Figure 4, which also shows in red the cumulative area covered by the DIC system during successive tests.
Seven bending tests were performed per board, covering a total monitored length of approximately 1100 mm. During each test, images were recorded at intervals corresponding to a ~3 MPa increase in bending stress, starting from the unloaded state, and the corresponding forces were documented.

Translation monitoring across the successive tests was also performed using image correlation. Images were taken at approximately 3 cm displacement intervals, allowing the repositioning of the bending test results within each board's original reference frame. Consistency in the coordinate system was maintained across all tests using the initial DIC calibration, which ensured alignment of the *x* and *y* axes with the board's global frame. Positional offsets between distinctive patterns—typically near knots or cracks—were used to realign the DIC reference system with the global reference system of the corresponding board.

## 2.5 All through board planing and scanning operations: destructive serial sectioning

In order to reconstruct the internal fibre orientation fields of the boards, a complete process of serial sectioning using successive planing and scanning was applied to each one. For every board, a 10 mm thick plywood panel and a 16 mm thick MDF panel—both cut to match the board's exact width and length—were glued onto its top face, forming a plywood-MDF-board assembly. MDF was selected as the immediate interface with the timber to prevent laser light diffusion through the wood during the final scanning operations, while the plywood provided added rigidity to ensure stability throughout the planing and scanning steps.

Planing was performed in 1 mm increments using an *SCM S520* planer. Before the initial planing and after each subsequent increment, the boards were scanned using an industrial timber grading scanner, with the same settings as described in paragraph 2.3. For image and laser dot stacking, only the data corresponding to the board's bottom face—i.e., the exposed timber surface—were retained. This ensured consistent focus and scanning parameters throughout the serial sectioning process, thereby simplifying the alignment and integration of the successive datasets.

# 3 Methods

## 3.1 IFOD

For all boards, the internal fibre orientation was determined using the method developed by Penvern et al (2025), referred to in this study as **GST,DSS**. This denomination will be used consistently throughout the remainder of the paper.

From the serial scanning, a full map of the in-plane fibre angle and optical grayscale image stack of the boards was built and interpolated to an intermediate regular grid of 1 mm × 1 mm × 1 mm. The regularised in-plane fibre angle field was obtained by linear interpolation using the *scipy.griddata* function of the laser dot measurement scattered data, while the optical images were obtained by linear interpolation of the corrected greyscale images of the surfaces revealed every 1 mm planing step, using the *cv2_resize* function with the *cv2 INTER_AREA* interpolation scheme.
To estimate the internal fibre orientation within the board, the assumption that fibres generally follow the curvature of growth layers is leveraged. The projected fibre angle $\alpha$ obtained from laser dot measurements, defining an in-plane fibre direction :

$$ \vec{u} = \begin{pmatrix} \sin \alpha \\ \cos \alpha \\ 0 \end{pmatrix}_{(\vec{x}, \vec{y}, \vec{z})} , \qquad (1) $$

is used in combination with the local orientation of the growth layers. This local orientation is determined by performing Gradient Structure Tensor (GST) analysis on the DSS image stack. By identifying the local normal direction to the growth layers, noted $\vec{n}$, which is the local radial direction of the wood, as the direction of highest gray intensity variations, the internal fibre direction, noted $\vec{v}$, can be reconstructed by somewhat rotating the in-plane projected fibre direction so that it lies tangent to the growth layers :

$$ \vec{v} = \vec{u} - \frac{\vec{n} \cdot \vec{u}}{\vec{n} \cdot \vec{z}} \vec{z}. \qquad (2) $$

As in (Penvern et al, 2025), the GST is applied to the extrapolated image stack using the `cv2.calcopticalFlowFarneback` function in order to reduce the GST analysis edge effects.

## 3.2 IFOI

### 3.2.1 1A IFOI methods

These methods output one single interpolated fibre angle value from board surfaces laser dots measurement. This angle alone is assumed to represent the overall inclination of the fibre with respect to the longitudinal axis, without accounting for the plane in which this angle lies.

In the original interpolation methods presented in Olsson et al (2013, 2018) or Hu et al (2018), the angle between fibres and the longitudinal direction of the board were assigned for a grid corresponding to the scanning resolution on each of the four sides of the board. On the opposite, in that developed by Besseau (2021) or in the IFOD method, the surface measurements are firstly interpolated by nearest neighbour to a same 1 mm × 1 mm grid. For more consistency in this study, it was chosen to work for each interpolation or determination method with the same grid of size 1 mm × 1 mm × 1 mm. It should be noticed that when using this nearest interpolation on the board surfaces, the results are unchanged compared to original methods, except for the grid size. Thus, surface fibre orientation measurements on the four accessible faces (top, bottom, left, right) were first interpolated onto this chosen 1 mm grid using a nearest-neighbour scheme using the `scipy.interpolate NearestNDInterpolator` function.

The first methods **1A,flat** and **1A,roof**, inspired by (Olsson et al, 2013, 2018), fills the internal volume by symmetrically propagating values from opposite faces, either only from broad bottom and top faces in **1A,flat** or from all four faces in **1A,roof**, towards the centre (see Table 1).
The second method, referred to as **1A,radial**, inspired by (Hu et al, 2018; Olsson et al, 2022), requires knowledge of the location of the pith of the tree from which the board was sawn. It accounts for the fact that fibre deviation is approximately symmetrical in relation to the radial knots centred planes. Knowing the location of the pith relative to the boards using the log small end and large end, the pith position can be linearly interpolated along the board. Then for each internal sub-volume, a radial line is traced from the local pith position to sub-volume position. This line intersects with the board edges. The angle at the sub-volume is then defined as the distance-weighted average of the closest in-plane fibre angle values to the surfaces intersections points.

### 3.2.2 2A IFOI methods

These methods output two interpolated fibre angle values in two orthogonal planes from board surfaces laser dots measurement. Thus, both edge and flat planes are considered to define the overall fibre angle relative to the longitudinal axis.

Relative to Besseau (2021)'s work, two **2A,linear** and **2A,nearest** interpolation schemes are observed. Both methods rely on the simple linear interpolation of the edge and broad sides laser dots measurements to obtain a flatwise angle $\alpha_{yx}$ and an edgewise angle $\alpha_{zx}$ with the only difference being the initial surface fibre in-plane angle interpolation.

Because the **GST,DTT** determined fibre orientation field uses linearly interpolated in-plane fibre angle to the regularised 1 mm grid, it was chosen to observe the impact of using a linear interpolation method rather than the nearest neighbour interpolation method for the surface in-plane fibre angle assignment in this simple interpolation scheme, thus resulting in two fibre angle grid denominated **2A,nearest** and **2A,linear**. For this last IFOI method, the surface fibre orientation measurements on the four accessible faces (top, bottom, left, right) are thus interpolated using the `scipy.interpolate LinearNDInterpolator` function instead of the more rapid `NearestNDInterpolator`.

As illustrated Figure 5:

* assuming a fibre direction $\vec{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}_{(\vec{x}, \vec{y}, \vec{z})}$, $\vec{v}$ being unitary;

* estimating $\alpha_{yx}$ the internal local $xy$-plane fibre angle to the board longitudinal $x$-axis by width-wise linear interpolation from the top and bottom in-plane fibre angle measurements;

* estimating $\alpha_{zx}$ the internal local $xz$-plane fibre angle to the board longitudinal $x$-axis by edge-wise linear interpolation from left and right in-plane fibre angle measurements;
<figure><img src="image_6.png" /><figcaption>Fig. 5: Schematic of mix fibre direction estimation (orange line) using surface in-plane fibre angle edge-wise (green plane) and width-wise (pink plane) interpolation</figcaption></figure>

the following equations apply :

$$ \begin{gathered} \frac{v_y}{v_x} = \tan \alpha_{yx} \\ \frac{v_z}{v_x} = \tan \alpha_{zx} \end{gathered} \quad \left| \begin{array}{l} v_x^2 + v_y^2 + v_z^2 = 1 \end{array} \right. $$

It follows that the fibre direction unit vector can be estimated as :

$$ \vec{v} = \frac{1}{\sqrt{1 + \tan^2 \alpha_{yx} + \tan^2 \alpha_{zx}}} \cdot \begin{pmatrix} 1 \\ \tan \alpha_{yx} \\ \tan \alpha_{zx} \end{pmatrix}_{(\vec{x}, \vec{y}, \vec{z})} \quad (3) $$

### 3.2.3 Optical Flow based IFOI Scheme

In addition to those first five interpolation schemes from the literature, two new ones were explored, based on optical flow computation.

To guide the interpolation of fibre orientations into the volume, dense 2D optical flows were computed between boards opposing surfaces using the Farnebäck algorithm Farnebäck (2003), a variational method that estimates image per-pixel displacements by modelling local image neighbourhoods as quadratic polynomial expansions.

Unlike sparse techniques that track discrete keypoints, the Farnebäck method computes a dense displacement field by approximating the local structure of an image $f(x)$ as a second-order polynomial:

$$ f(x) \approx x^T A x + b^T x + c $$
By estimating these polynomial coefficients in both images and observing how they
transform under motion, the algorithm infers pixel-wise displacements. A multi-scale
image pyramid is used to handle large displacements, with iterative refinement at each
scale through warping and filtering.

<figure><img src="image_7.png" /><figcaption>**Fig. 6:** Optical flow fields estimated between the bottom and top grayscale surface image sections $x \in [1000, 1500]$ mm of board #40, represented by red quivers.</figcaption></figure>
In our case, the method was applied to the board gray image pairs from:

* the **bottom to top** surface, yielding a forward flow field flow_z, and

* the **top to bottom** surface, producing the inverse flow field inverse_flow_z.

An analogous process was performed edge-wise, in the $xz$-plane, between the left and right faces. Computed optical flows are visible Figure 6. In this figure it is interesting to note:

* the computed shift motion of the knots;

* the difference between the two motion fields.

While one might expect the inverse flow to be the negation of the forward flow, in practice this symmetry does not hold due to differences in texture and illumination between opposing surfaces, image noise, scanning artifacts, or inherent approximations of the *Farnebäck algorithm*. Consequently, the flow from bottom to top differs from that of top to bottom.

To make use of both flows without biasing the interpolation toward either face, a linear blending strategy was employed. To interpolate the in-plane fibre orientation angle $\alpha_{yx}$ from the top and bottom surfaces into the interior of the board, a weighted mapping guided by the average optical flow between the two greyscale images of the board's surfaces was used.

Let the vertical direction $z$ range from the bottom surface ($z = 0$) to the top surface ($z = L_z$). For each interior slice at index $z$, the relative vertical position is defined as:

$$p = \frac{z}{n_z - 1}, \quad \text{with } p \in (0, 1)$$

Two interpolated angle maps were computed at this height by mapping pixels from the surface images to the slice using the previously computed optical flow fields.

* **From the bottom surface** (image indexed as bottom), the forward optical flow $f(z) = (f_x, f_y)$ was used to estimate where each surface point would move at relative depth $p$. The displacement field is scaled proportionally:

$$x_{\text{map}}^{\text{bottom}} = x + p \cdot f_x(x, y), \quad y_{\text{map}}^{\text{bottom}} = y + p \cdot f_y(x, y)$$

The fibre angle field $\alpha_{yx}^{\text{bottom}}(x, y)$ is then interpolated to the mapped coordinates using bilinear interpolation:

$$\hat{\alpha}_{yx}^{\text{bottom}}(x, y, z) = \text{Interp}(\alpha_{yx}^{\text{bottom}}, x_{\text{map}}^{\text{bottom}}, y_{\text{map}}^{\text{bottom}})$$

* **From the top surface** (top), we similarly use the inverse optical flow $f_z^{\text{inv}}$, scaled by $(1 - \alpha)$, to map surface pixels downward:

$$x_{\text{map}}^{\text{top}} = x + (1-p) \cdot f_x^{\text{inv}}(x, y), \quad y_{\text{map}}^{\text{top}} = y + (1-p) \cdot f_y^{\text{inv}}(x, y)$$

$$\hat{\alpha}_{yx}^{\text{top}}(x, y, z) = \text{Interp}(\alpha_{yx}^{\text{top}}, x_{\text{map}}^{\text{top}}, y_{\text{map}}^{\text{top}})$$
The final interpolated value for each voxel in the interior is then obtained by a linear blend of both contributions, as :

$$ \alpha_{yx}(x, y, z) = (1-p) \cdot \hat{\alpha}_{yx}^{\text{bottom}}(x, y, z) + p \cdot \hat{\alpha}_{yx}^{\text{top}}(x, y, z) $$

Similarly, we have :

$$ \alpha_{zx}(x, y, z) = (1-p) \cdot \hat{\alpha}_{zx}^{\text{left}}(x, y, z) + p \cdot \hat{\alpha}_{zx}^{\text{right}}(x, y, z) $$

This formulation ensures a smooth and consistent interpolation between the two surface measurements, where the influence of each surface diminishes linearly with distance into the volume. This deformed and interpolated angle map at depth z thus accounts for the estimated pixel trajectories (optical flow paths), in particular the estimated knot path and extension or reduction, rather than assuming straight-line interpolation, enabling a more physically consistent mapping of fibre angles through the board thickness.

First additional interpolation scheme, noted **1A,flow** accounts for the single $\alpha = \alpha_{yx}$ angle computed from the broad surfaces of the boards only. Second additional interpolation scheme, noted **2A,flow** accounts for both $\alpha_{yx}$ and $\alpha_{zx}$ angles to define the fibre direction as in **2A** previously described methods.

## 3.3 Finite Element Simulations

The FEM-based bending profiles were generated in *Abaqus* using a three-dimensional orthotropic model, which incorporated the determined or interpolated local fibre orientation fields, as described in paragraphs 3.1 and 3.2. A structured hexahedral mesh with element size of 2 mm was generated based on the board's measured dimensions. Default *C3D8R* element type (linear brick element, with a reduced single integration point) was kept for its low memory cost, considering the great number of elements (over 2 millions).

Local orthotropic axes, denoted hereafter as (L, R, T), following the conventional nomenclature for wood orthotropy, were defined at the element level by aligning the material direction with the determined or interpolated fibre orientation.

The **GST,DSS IFOD** method defines both the local radial direction—normal to the growth layers—and the longitudinal direction—aligned with the fibres. In contrast, the **IFOI** methods only define one or two fibre angles relative to the longitudinal axis of the board. Therefore, a default radial direction $R_0$ was introduced, depending on the pith-to-board position. Based on the local longitudinal pith position, linearly interpolated between the two log-end pith-to-board positions (see Table 3), a corresponding "clear wood" radial field was computed.

In the **1A** approach, a single angle $\alpha$ is computed as a linear combination of fibre angle measurements, either from the broad faces only or from all four faces regardless of the measurement axis (orthogonal to the scanned surface). The longitudinal direction is then defined as :

$$ \mathbf{L} = \begin{pmatrix} \cos \alpha \\ \sin \alpha \\ 0 \end{pmatrix}_{(\vec{x}, \vec{y}, \vec{z})} $$
in the board reference frame, i.e. a rotation of the longitudinal x-axis by angle α around the thickness-wise z-axis.

The FE software Abaqus requires only two non-collinear vectors to define the orthotropic material directions. The third axis is generated via a cross product, after which the second axis is recalculated, ensuring strict orthogonality. Finally, each vector is normalized, yielding a local orthonormal basis. Accordingly, the tangential direction is defined as $T = R_0 \wedge L$) and the radial direction as $R = L \wedge (R_0 \wedge L)$

This implementation of the pith position to define the radial direction in the finite element model is novel and differs from the original 1A-flat, 1A-roof, and 1A-radial schemes proposed by Hu et al (2018); Olsson et al (2013, 2018), where the pith position was either undefined or not explicitly considered in the FEM.

In the 2A approach, two angles $\alpha_{yx}$ and $\alpha_{zx}$ are defined relative to the axes of rotation associated with the fibre angle measurements from the edge and broad faces, respectively. The longitudinal direction is then given by

$$L = \frac{1}{\sqrt{1 + \tan^2 \alpha_{yx} + \tan^2 \alpha_{zx}}} \begin{pmatrix} 1 \\ \tan \alpha_{yx} \\ \tan \alpha_{zx} \end{pmatrix}_{(\vec{x}, \vec{y}, \vec{z})}$$

in the board reference frame, following the geometric principle illustrated in Figure 5. Similarly, the radial direction is defined as $R = L \wedge (R_0 \wedge L)$.

The resulting local orthotropic definitions, depending on the method, are summarized in Table 5.

<table><caption>**Table 5:** Definition of the local orthotropic axes (L, R, T) depending on the IFOI and IFOD methods</caption><thead><tr><th>Method</th><th>1A</th><th>2A</th><th>GST, DSS</th></tr></thead><tbody><tr><td>L</td><td>$\begin{pmatrix} \cos \alpha \\ -\sin \alpha \\ 0 \end{pmatrix}$</td><td>$\frac{1}{\sqrt{1+\tan^2 \alpha_{yx}+\tan^2 \alpha_{zx}}} \begin{pmatrix} 1 \\ \tan \alpha_{yx} \\ \tan \alpha_{zx} \end{pmatrix}$</td><td>$\vec{v}$</td></tr><tr><td>R</td><td colspan="2">L &wedge; (R<sub>0</sub> &wedge; L)</td><td>$\vec{n}$</td></tr><tr><td>T</td><td colspan="3">Defined as orthogonal to both L and R</td></tr></tbody></table>

In regions where laser dot measurements were unavailable—typically near the edges of the boards due to the limited spatial resolution of the scanning system (with laser dots spaced at intervals of ≈ 4.8 mm—the local orthotropic directions (L, R, T) were assigned default values. Specifically, the longitudinal direction was set as $L_0 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, the radial direction as $R_0$, and the tangential direction as $T_0 = L_0 \wedge R_0$ in accordance.

Material properties were defined using the density and moisture-adjusted estimates for generic coniferous species from Guitard (1987), following Equation 4 and
parameters given in Table 6 :

$$
S_{ij}^{-1}(\rho, H) = (S_{ij,\rho_0,12\%}^{-1} + B(\rho - \rho_0)) [1 - C_{Hij}(H - 12)] \quad \text{with} \quad \rho_0 = 0.45 \, \text{g cm}^{-3} \tag{4}
$$

Resulting parameters are listed in Table 7. A pure bending scenario was simulated by constraining edge displacements and applying a symmetric distributed pressure designed to reproduce a constant bending moment equivalent in terms of value to the one induced by a 1 kN load in the experimental setup.

<table><caption>**Table 6:** Density and moisture content regression parameters for generic softwoods according to (Guitard, 1987)</caption>
  <thead>
    <tr>
      <th>Modules z</th>
      <th>S<sup>-1</sup><sub>ij</sub>(ρ<sub>0</sub>, 12%) (MPa)</th>
      <th>B (MPa.cm<sup>3</sup>.g<sup>-1</sup>)</th>
      <th>C<sub>Hij</sub></th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>S<sup>-1</sup><sub>11</sub> = E<sub>R</sub></td>
      <td>1000</td>
      <td>2370</td>
      <td>0.03</td>
    </tr>
    <tr>
      <td>S<sup>-1</sup><sub>22</sub> = E<sub>T</sub></td>
      <td>636</td>
      <td>1910</td>
      <td>0.03</td>
    </tr>
    <tr>
      <td>S<sup>-1</sup><sub>33</sub> = E<sub>L</sub></td>
      <td>13100</td>
      <td>41700</td>
      <td>0.15</td>
    </tr>
    <tr>
      <td>S<sup>-1</sup><sub>44</sub> = G<sub>TL</sub></td>
      <td>745</td>
      <td>989</td>
      <td>0.03</td>
    </tr>
    <tr>
      <td>S<sup>-1</sup><sub>55</sub> = G<sub>LR</sub></td>
      <td>861</td>
      <td>2080</td>
      <td>0.03</td>
    </tr>
    <tr>
      <td>S<sup>-1</sup><sub>66</sub> = G<sub>RT</sub></td>
      <td>83.6</td>
      <td>228</td>
      <td>0.03</td>
    </tr>
    <tr>
      <td>-S<sup>-1</sup><sub>12</sub> = E<sub>R</sub> / ν<sub>RT</sub> = E<sub>T</sub> / ν<sub>TR</sub></td>
      <td>2050</td>
      <td>5280</td>
      <td>0.546</td>
    </tr>
    <tr>
      <td>-S<sup>-1</sup><sub>23</sub> = E<sub>L</sub> / ν<sub>LT</sub> = E<sub>T</sub> / ν<sub>TL</sub></td>
      <td>30800</td>
      <td>101000</td>
      <td>0.817</td>
    </tr>
    <tr>
      <td>-S<sup>-1</sup><sub>31</sub> = E<sub>R</sub> / ν<sub>RL</sub> = E<sub>L</sub> / ν<sub>LR</sub></td>
      <td>34200</td>
      <td>117000</td>
      <td>0.762</td>
    </tr>
  </tbody>
</table>

<table><caption>**Table 7:** Board material parameters according to Guitard (1987).</caption>
  <thead>
    <tr>
      <th>Board</th>
      <th>Density (g/cm<sup>3</sup>)</th>
      <th>MC (%)</th>
      <th>E<sub>L</sub></th>
      <th>E<sub>R</sub></th>
      <th>E<sub>T</sub></th>
      <th>G<sub>LR</sub></th>
      <th>G<sub>LT</sub></th>
      <th>G<sub>RT</sub></th>
      <th>ν<sub>LR</sub></th>
      <th>ν<sub>LT</sub></th>
      <th>ν<sub>RT</sub></th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>351</td>
      <td>529.98</td>
      <td>9.5</td>
      <td>17051</td>
      <td>1279</td>
      <td>848</td>
      <td>1104</td>
      <td>886</td>
      <td>109</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.52</td>
    </tr>
    <tr>
      <td>352</td>
      <td>537.9</td>
      <td>9.5</td>
      <td>17394</td>
      <td>1299</td>
      <td>864</td>
      <td>1122</td>
      <td>894</td>
      <td>111</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.52</td>
    </tr>
    <tr>
      <td>361</td>
      <td>524.12</td>
      <td>9.4</td>
      <td>16822</td>
      <td>1267</td>
      <td>838</td>
      <td>1094</td>
      <td>882</td>
      <td>108</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.52</td>
    </tr>
    <tr>
      <td>362</td>
      <td>521.61</td>
      <td>9.4</td>
      <td>16713</td>
      <td>1261</td>
      <td>833</td>
      <td>1089</td>
      <td>879</td>
      <td>108</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.52</td>
    </tr>
    <tr>
      <td>39</td>
      <td>511.7</td>
      <td>9.8</td>
      <td>16190</td>
      <td>1222</td>
      <td>804</td>
      <td>1055</td>
      <td>859</td>
      <td>104</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.51</td>
    </tr>
    <tr>
      <td>40</td>
      <td>458.98</td>
      <td>10.3</td>
      <td>13818</td>
      <td>1073</td>
      <td>686</td>
      <td>925</td>
      <td>792</td>
      <td>90</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.51</td>
    </tr>
    <tr>
      <td>41</td>
      <td>443.9</td>
      <td>10.7</td>
      <td>13096</td>
      <td>1024</td>
      <td>649</td>
      <td>881</td>
      <td>768</td>
      <td>85</td>
      <td>0.39</td>
      <td>0.43</td>
      <td>0.51</td>
    </tr>
    <tr>
      <td>42</td>
      <td>513.12</td>
      <td>10.4</td>
      <td>16110</td>
      <td>1205</td>
      <td>793</td>
      <td>1040</td>
      <td>846</td>
      <td>103</td>
      <td>0.39</td>
      <td>0.43</td>
      <td>0.51</td>
    </tr>
    <tr>
      <td>43</td>
      <td>550.89</td>
      <td>9.0</td>
      <td>18086</td>
      <td>1351</td>
      <td>903</td>
      <td>1167</td>
      <td>921</td>
      <td>116</td>
      <td>0.39</td>
      <td>0.44</td>
      <td>0.52</td>
    </tr>
    <tr>
      <td>44</td>
      <td>465.96</td>
      <td>8.5</td>
      <td>14488</td>
      <td>1147</td>
      <td>736</td>
      <td>988</td>
      <td>841</td>
      <td>96</td>
      <td>0.40</td>
      <td>0.45</td>
      <td>0.54</td>
    </tr>
  </tbody>
</table>

After solving the static bending step, nodal displacements and strain fields were extracted at the top surface of the board (z = l<sub>z</sub>), corresponding to the DIC-monitored face.
## 3.4 Localised Bending MoE Profiles from DIC and FEM computation

### 3.4.1 Beam theory derived MoE profiles computation

Localised bending MoE were computed from both Digital Image Correlation (DIC) measurements and Finite Element Method (FEM) simulations. In both cases, beam theory relationships for homogeneous isotropic beams were applied locally in the region of pure bending, based either on the slope of the longitudinal strain or the curvature, over a sliding window of length $w_x = 90$ mm, chosen in accordance to the literature (Lukacevic et al, 2019; Olsson et al, 2022; Penvern et al, 2025).

Beam theory provides the following analytical expressions for a prismatic, homogeneous, and isotropic beam subjected to pure bending in the xy-plane, under a bending moment $M(x)$ applied about the z-axis:

$$ \kappa(x) = \frac{M(x)}{E_b(x)I_z} \qquad (5) $$

$$ \varepsilon_{xx}(x, y) = \frac{M(x)}{E_b(x)I_z} \bar{y} \qquad (6) $$

where:

* $\kappa(x)$ is the curvature of the beam at position $x$ along its length,

* $M(x)$ is the internal bending moment at location $x$,

* $E_b(x)$ is the localised bending MoE (along the beam axis),

* $I_z$ is the second moment of area of the cross-section with respect to the z-axis (perpendicular to the bending plane),

* $\varepsilon_{xx}(x, y)$ is the corresponding normal strain,

* $\bar{y}$ is the vertical coordinate within the cross-section, measured from the neutral axis (which corresponds to zero strain).

For each FEM and DIC results, and each window position $x$, two localised MoEs were thus defined:

$$ E_{b,\varepsilon,w_x}(x) = \frac{M}{I_z \cdot \left\langle \frac{\varepsilon_{xx}(x,y)}{\bar{y}} \right\rangle_{w_x}} \qquad (7) $$

$$ E_{b,\kappa,w_x}(x) = \frac{M}{I_z \cdot \langle \kappa(x,y) \rangle_{w_x}} \qquad (8) $$

with $w_x$ the averaging window width over which the mean strain slope $\langle \frac{\varepsilon_{xx}(x,y)}{y} \rangle_{w_x}$ and the mean curvature $\langle \kappa_{DIC}(x,y) \rangle_{w_x}$ are computed from the strain field or the displacement field. The mean strain slope is obtained via linear regression using `numpy.polyfit`, while the mean curvature is obtained by performing a circle fit using the algebraic algorithm of Klear (2023), implemented via `circle-fit.TaubinSVD`, on the y-wise mean transverse displacement. The chosen averaging window width was 90 mm windows in accordance to the literature (Olsson
and Oscarsson, 2017; Olsson et al, 2018; Hu et al, 2018; Olsson et al, 2022). Such circle and linear fits over the curvature or strain slope respectively are visible in Figures 13 and 11.

### 3.4.2 DIC specificities

It should be noticed that for the DIC MoE profiles, the averaged strain slope coefficients $\langle \frac{\epsilon_{xx,s,DIC}(x,y)}{y} \rangle_{w_x}$ ($x$) were computed not on the raw strain field $\epsilon_{xx,DIC}(x, y)$, but on a smoothed strain fields $\epsilon_{xx,s,DIC}(x, y)$. A post-processing uniform filter of 20.8 mm × 20.8 mm (equivalent to 19 × 19 DIC pixels at 1.096 px/mm) was applied to the strain fields to reduce the random fluctuations inherent to the correlation noise, while preserving the real spatial gradients.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Board #35 overlaid strain-based and curvature-based MoE profiles computed over the seven bending steps, in relation to the bending bench local deflection computed MoE.</figcaption></figure>

Furthermore, because the digital correlation monitoring was achieved in several steps, the curvature and MoE profiles were computed separately for each bending operation and then concatenated to build the board's full profile of the combined monitored area. Figure 7 overlays the bending MoE profiles derived from both DIC-based strain slopes ($E_{b,ε,DIC}$, dashed lines) and DIC-based curvatures ($E_{b,κ,DIC}$, solid lines) of board #351 across the multiple loading phases. The cyan blue dots correspond to reference values of local stiffness $E_{b,bench}$ obtain by processing the bench-recorded local deflection $w_l$ as a function of the applied force during the bending tests (see Figure 4):

$$E_{b,bench} = \frac{l_1^2}{16I_z} \left\langle \frac{\Delta w_l}{\Delta M} \right\rangle \quad (9)$$
where $l_1 = \frac{5}{6} \cdot$ loading span = 521 mm is the base length for determining the local deflection, and $\langle \frac{\Delta w_l}{\Delta M} \rangle$ is the mean local deflection-to-moment slope computed by ` numpy.polyfit` over the increments of successive bending tests.

The curves between bending tests do not overlay each other perfectly. The reasons for these discrepancies remain unclear. Stereo-correlation error was low ($\approx 0.02$ px to 0.03 px), and boundary condition effects were likely limited thanks to the small observation window located in the constant moment zone of the bending bench. Damage is unlikely, as stiffness did not decrease during the bending tests. Another, perhaps more plausible explanation, is a slight reconfiguration of cracks and microcracks during the loading/unloading cycles. In any case, it is difficult to define a “wrong” versus a “right” profile; both may be valid. Therefore, both sets of curves were retained in the profile plots, visible as dark overlap zones in the profiles shown in Appendix A.

These profile computations are similar to those employed by Hu et al (2018) and Olsson et al (2022), and identical to those in (Penvern et al, 2025).

### 3.4.3 strain-based vs. curvature-based MoE profiles

Equations 5 and 6 arise from the standard beam theory relations (see Equations), which equate both strain slope and curvature to $M/EI_z$ under the assumptions of isotropy and homogeneity. However, since wood is neither isotropic nor homogeneous, these two quantities —although physically related— diverge when applied to either experimental or simulated data.

Pot et al (2024a) compared these two different localised MoEs on artificially generated fibre orientations around knots. The localised bending MoE computed from curvature obtained in FEM was found to behave very differently from the one computed from the strains in FEM or both curvature and strain-based MoEs obtained by analytical beam theory. In particular, curvature-based MoE profiles showed peaks of infinite stiffness. This was found to be due to localized curvature inversions implied by the grain deviation around knots, which does not stand out from the analysis of the strains nor from analytical beam theory modelling.

The curvature and strain-based MoEs can be understood as different perspectives on local stiffness, both informative yet sensitive to different aspects of the material structure. Thus, in this paper, both were computed from FEM and retained, as they offer complementary insights into the assessment of the mechanical behaviour of the board, particularly in the presence of knots and fibre orientation heterogeneity.

## 3.5 Interpretation and Comparison : Correlation Framework

To quantify the agreement between FEM and DIC strain-based MoE profiles, the Pearson correlation coefficient $R$ was computed and squared to obtain the coefficient of determination $R^2$. This metric evaluates the linear relationship between two signals and is more robust to noise and material parameter setting than RMSE or MAPE, when the signals share similar shapes but differ in scale or offset.

For the curvature-based MoE profiles, the correlation analysis was performed directly on the curvature fields due to the presence of curvature inversions, which would otherwise lead to singularities (division by zero) in the MoE calculation. Therefore,
and for the sake of clarity, the curvature-based profiles reading being made difficult by the infinite peaks, the raw curvature profiles are also presented (see Appendix A).

For these FEM-DIC comparisons, the analysis was restricted to the DIC-monitored regions of the boards, using the same longitudinal resolution of 1 mm for both FEM and DIC profiles. In the DIC overlap zones, a mean signal of the combined bending test results was obtained by smoothing the raw data through two successive convolutions with a uniform 3 px kernel, corresponding to a physical length of 1.096 * 3 = 3.288 mm at the given DIC resolution. Then, the DIC data was interpolated to the FEM sampling points along the x-axis to enable direct comparison.

As shown later, profile correlations remained sufficiently high to enable meaningful comparison between experiment and simulation, an outcome intrinsically linked to the accuracy of the assigned fibre orientations.

# 4 Results and Discussion

## 4.1 General overview

<table><caption>**Table 8:** Color-coded $R^2$ values for curvature ($\kappa$) agreement between FEM and DIC across interpolation methods. Rainbow scale from 0 (red) to 1 (blue violet).</caption><thead><tr><th>Board</th><th>1A,flat</th><th>1A,flow</th><th>1A,radial</th><th>1A,roof</th><th>2A,flow</th><th>2A,linear</th><th>2A,nearest</th><th>GST,DSS</th></tr></thead><tbody><tr><td>351</td><td>0.67</td><td>0.47</td><td>0.76</td><td>0.73</td><td>0.47</td><td>0.52</td><td>0.49</td><td>0.69</td></tr><tr><td>352</td><td>0.49</td><td>0.52</td><td>0.54</td><td>0.54</td><td>0.65</td><td>0.63</td><td>0.64</td><td>0.53</td></tr><tr><td>361</td><td>0.56</td><td>0.57</td><td>0.57</td><td>0.39</td><td>0.16</td><td>0.21</td><td>0.17</td><td>0.92</td></tr><tr><td>362</td><td>0.78</td><td>0.74</td><td>0.82</td><td>0.91</td><td>0.57</td><td>0.55</td><td>0.55</td><td>0.85</td></tr><tr><td>39</td><td>0.86</td><td>0.89</td><td>0.89</td><td>0.63</td><td>0.83</td><td>0.81</td><td>0.82</td><td>0.84</td></tr><tr><td>40</td><td>0.83</td><td>0.82</td><td>0.46</td><td>0.90</td><td>0.77</td><td>0.81</td><td>0.76</td><td>0.72</td></tr><tr><td>41</td><td>0.47</td><td>0.44</td><td>0.75</td><td>0.65</td><td>0.87</td><td>0.80</td><td>0.87</td><td>0.70</td></tr><tr><td>42</td><td>0.62</td><td>0.76</td><td>0.74</td><td>0.80</td><td>0.92</td><td>0.91</td><td>0.91</td><td>0.93</td></tr><tr><td>43</td><td>0.68</td><td>0.72</td><td>0.60</td><td>0.54</td><td>0.67</td><td>0.65</td><td>0.63</td><td>0.69</td></tr><tr><td>44</td><td>0.69</td><td>0.54</td><td>0.55</td><td>0.83</td><td>0.91</td><td>0.92</td><td>0.92</td><td>0.82</td></tr><tr><td>Average</td><td>0.66</td><td>0.65</td><td>0.67</td><td>0.69</td><td>0.68</td><td>0.68</td><td>0.68</td><td>0.77</td></tr></tbody></table>

Globally, all interpolation schemes produced realistic and coherent MoE profiles (cf. Appendix A), with a good agreement between FEM predictions and DIC measurements. This demonstrates the effectiveness of propagating fibre orientation information from surface measurements into the volume. Tables 8 and 9 present the squared Pearson correlation coefficients between FEM and DIC results, calculated on a board-by-board basis for each interpolation scheme.

The best average results were obtained for profiles simulated from the determined fibre orientations by the **GST,DSS** method. This holds for both strain-based MoE and curvature profiles. For curvature profiles, the average correlation coefficient reaches $R^2 = 0.77$ for the **GST,DSS** method, compared with values ranging from 0.65 to 0.69 for the interpolated orientations. For strain-based MoE profiles, the average correlation coefficient reaches $R^2 = 0.69$ for **GST,DSS**, compared to 0.61 to 0.66 for interpolated orientations. These results confirm the quality of fibre orientations obtained by IFOD, although they remain imperfect.
<table><caption>**Table 9:** Color-coded $R^2$ values for strain ($\epsilon$) agreement between FEM and DIC across interpolation methods. Rainbow scale from 0 (red) to 1 (blue violet).</caption><thead><tr><th>Board</th><th>1A,flat</th><th>1A,flow</th><th>1A,radial</th><th>1A,roof</th><th>2A,flow</th><th>2A,linear</th><th>2A,nearest</th><th>GST,DSS</th></tr></thead><tbody><tr><td>351</td><td>0.84</td><td>0.84</td><td>0.87</td><td>0.85</td><td>0.54</td><td>0.56</td><td>0.56</td><td>0.85</td></tr><tr><td>352</td><td>0.54</td><td>0.53</td><td>0.53</td><td>0.54</td><td>0.32</td><td>0.35</td><td>0.34</td><td>0.46</td></tr><tr><td>361</td><td>0.04</td><td>0.04</td><td>0.29</td><td>0.41</td><td>0.56</td><td>0.57</td><td>0.56</td><td>0.66</td></tr><tr><td>362</td><td>0.84</td><td>0.83</td><td>0.89</td><td>0.87</td><td>0.83</td><td>0.81</td><td>0.83</td><td>0.91</td></tr><tr><td>39</td><td>0.87</td><td>0.87</td><td>0.91</td><td>0.87</td><td>0.88</td><td>0.86</td><td>0.89</td><td>0.89</td></tr><tr><td>40</td><td>0.74</td><td>0.75</td><td>0.22</td><td>0.77</td><td>0.76</td><td>0.74</td><td>0.75</td><td>0.76</td></tr><tr><td>41</td><td>0.42</td><td>0.49</td><td>0.80</td><td>0.65</td><td>0.76</td><td>0.68</td><td>0.76</td><td>0.60</td></tr><tr><td>42</td><td>0.87</td><td>0.88</td><td>0.82</td><td>0.83</td><td>0.78</td><td>0.78</td><td>0.77</td><td>0.79</td></tr><tr><td>43</td><td>0.80</td><td>0.76</td><td>0.75</td><td>0.72</td><td>0.73</td><td>0.73</td><td>0.73</td><td>0.85</td></tr><tr><td>44</td><td>0.15</td><td>0.11</td><td>0.20</td><td>0.11</td><td>0.09</td><td>0.10</td><td>0.09</td><td>0.17</td></tr><tr><td>Average</td><td>0.61</td><td>0.61</td><td>0.63</td><td>0.66</td><td>0.62</td><td>0.62</td><td>0.63</td><td>0.69</td></tr></tbody></table>

The slightly lower correlation coefficients observed in the case of strain-based MoE may be explained by the lower variability of strain-based MoE compared to curvature (see Table 10), the relative deviation being about 2.8 times smaller.

<table><caption>**Table 10:** Board-wise comparison of apparent ($E_κ$) and average ($E_ε$) moduli of elasticity from DIC and GST (GPa). Values are given as mean ± standard deviation. The pith is located inside board #41.</caption><thead><tr><th rowspan="2">Board</th><th colspan="2">E<sub>κ</sub> (GPa)</th><th colspan="2">E<sub>ε</sub> (GPa)</th><th rowspan="2">distance to the pith (mm)</th></tr><tr><th>DIC</th><th>GST,DSS</th><th>DIC</th><th>GST,DSS</th></tr></thead><tbody><tr><td>351</td><td>11.88 ± 9.10</td><td>13.41 ± 5.25</td><td>12.25 ± 2.19</td><td>13.81 ± 2.22</td><td>121</td></tr><tr><td>352</td><td>12.54 ± 3.00</td><td>14.60 ± 3.11</td><td>12.62 ± 0.96</td><td>14.54 ± 1.52</td><td>124</td></tr><tr><td>361</td><td>14.01 ± 4.25</td><td>14.13 ± 3.32</td><td>14.13 ± 1.50</td><td>14.06 ± 0.69</td><td>126</td></tr><tr><td>362</td><td>10.81 ± 17.11</td><td>11.84 ± 5.53</td><td>13.14 ± 3.67</td><td>12.37 ± 2.31</td><td>112</td></tr><tr><td>39</td><td>12.57 ± 9.02</td><td>12.45 ± 5.93</td><td>13.02 ± 2.62</td><td>12.70 ± 2.10</td><td>90</td></tr><tr><td>40</td><td>11.28 ± 9.67</td><td>12.14 ± 4.52</td><td>12.48 ± 2.75</td><td>12.28 ± 1.47</td><td>35</td></tr><tr><td>41</td><td>10.21 ± 8.17</td><td>6.86 ± 1.94</td><td>10.66 ± 2.01</td><td>7.02 ± 1.16</td><td>5</td></tr><tr><td>42</td><td>12.05 ± 6.79</td><td>12.14 ± 6.11</td><td>11.99 ± 1.24</td><td>12.33 ± 1.73</td><td>96</td></tr><tr><td>43</td><td>10.46 ± 7.78</td><td>12.45 ± 5.36</td><td>11.65 ± 2.99</td><td>13.22 ± 2.98</td><td>147</td></tr><tr><td>44</td><td>11.89 ± 5.03</td><td>10.85 ± 3.11</td><td>12.03 ± 0.93</td><td>10.78 ± 0.71</td><td>48</td></tr><tr><td>Mean ± SD</td><td>11.77 ± 8.88</td><td>12.09 ± 5.05</td><td>12.40 ± 2.43</td><td>12.31 ± 2.74</td><td>-</td></tr></tbody></table>

Table 10 reports global statistics of the DIC and GST,DSS, curvature-, and strain-based MoE profiles. The mean and standard deviation of $E_κ$ (apparent MoE) were computed from the curvature-based DIC profiles using:

$$ \langle E_{\kappa} \rangle = \frac{M}{\langle \kappa_{\text{DIC}} \rangle I_{Gz}}, \quad \sigma_{E_{\kappa}} = \frac{M \sigma_{\kappa, \text{DIC}}}{I_{Gz} \langle \kappa_{\text{DIC}} \rangle^2} $$

where $\langle \kappa_{\text{DIC}} \rangle$ and $\sigma_{\kappa, \text{DIC}}$ are the mean and standard deviation of the curvature over the monitored region.
The global standard deviation was computed as:

$$ \langle \sigma_E \rangle = \sqrt{\frac{1}{n} \sum_n (\sigma_i^2 + (E_i - \langle E \rangle)^2)} $$

where n is the number of boards, σᵢ is the standard deviation of the MoE for board i, Eᵢ is the mean MoE of board i, and ⟨E⟩ is the overall mean MoE across all boards.

The differences observed in the mean and standard deviation of the IFOD-derived MoE values compared to the DIC-derived MoE values suggest that the material parameters used in the FEM simulations may not be optimally defined.

However, the error is most likely dominated by a constant offset and uniform scaling. Indeed, the ratios between orthotropic elastic parameters employed here follow the generic values proposed by Guitard for softwoods. While the true ratios for Douglas fir are certainly not identical, they are also unlikely to deviate strongly from these reference values. This type of discrepancy has negligible influence on Pearson correlation coefficients, but would more strongly affect error metrics such as RMSE or MAPE. For this reason, the comparisons in this study focus primarily on correlation coefficients.

Among the interpolation schemes, the best overall performance is achieved by the 1A,roof method of Olsson et al (2013). However, its results show high variability across boards. For example, its correlation with the DIC strain-based MoE profile of board #39 is markedly lower than other IFOI schemes. Interestingly, on average, this scheme outperforms the 1A,radial scheme proposed by Hu et al (2018), which was expected to yield higher correlations according to (Hu et al, 2018; Olsson et al, 2022). At the board level, however, 1A,radial achieves better correlations for about half of the boards. This variability highlights the importance of studying a sufficiently large number of boards.

Given the limited dataset and the natural variability of timber, the small differences in average correlation coefficients between schemes cannot be considered conclusive. Nonetheless, because the boards were taken from different positions within the same log, thereby providing a patchwork coverage of its cross-section, and were cut according to a typical sawmill pattern, the sample is still reasonably representative.

The large-scale study of Olsson et al (2022), involving 241 Douglas fir boards (24–36 boards per log) provides context for the present results. In that work, correlation coefficients of 0.70 were obtained for both the 1A,radial and 1A,roof schemes in predicting the local static MoE in bending (measured following (EN 408, 2010), adjusted to 12% MC). These values are consistent with those observed here.

# 4.2 Detailed board-by-board DIC–FEM comparison

## 4.2.1 Board #41: a limitation of the GST,DSS method

Board #41 perfectly illustrates the insensitivity of Pearson correlation to profile uniform scaling. While the GST,DSS method yields relatively high correlation coefficients (R² = 0.70 for curvature and R² = 0.58 for strain-based MoE), the mean FEM moduli are much lower than those obtained by DIC (Table 10: Eκ,IFOD = 6.86 GPa vs. Eκ,DIC = 10.21 GPa; Eε,IFOD = 7.02 GPa vs. Eε,DIC = 10.66 GPa).
In this case, the FEM based on **GST,DSS** accurately reproduced the DIC profiles local variations but failed to capture the mean stiffness, as is visually evident in the profile plots of Appendix A. This discrepancy arises from an artifact in the GST analysis of the radial direction: GST requires a consistent outward normal (here enforced as $+\vec{z}$). Since board #41 contains the pith, abrupt inversions of the radial vectors occur across the pith plane. When fibre directions are projected tangentially to growth rings, this results in artificial $+\vec{z}/-\vec{z}$ orientations, rather than alignment with $+\vec{x}$, as shown in Figure 8. These misoriented fibres are then propagated throughout the board, lowering the predicted stiffness overall.

<figure><img src="image_9.png" /><figcaption>**Fig. 8:** (a) Orthogonal cross-section at $x = 1000$ mm from the DSS stack with the GST radial normal field, showing quadrant segmentation induced by the enforced $+Z$ normal (green and purple arrows). (b1) Grayscale image, (b2) radial field, and (b3) longitudinal field at $z = 11$ mm near the pith plane (blue line), highlighting local $+\vec{z}/-\vec{z}$ inversions (lavender: $-\vec{z}$, olive: $+\vec{z}$).</figcaption></figure>

All IFOI methods performed well for this board except **1A,flat** and **1A,flow**. This was expected: these interpolation schemes rely only on fibre angle measurements from the top and bottom faces. For this board containing the pith, the knots were oriented laterally with their cores visible only on the side faces. Consequently, in such schemes the knots are largely disregarded, despite having a strong influence on stiffness. By contrast, the **2A** methods and **1A,radial** perform best, as they incorporate side-face fibre measurements.

### 4.2.2 Boards #44, #352, and #361: effect of low variability

Board #352 shows the poorest FEM–DIC correlations for the **GST,DSS** method. This may results from the very low stiffness variability in the board, which limits the correlation across all IFOD and IFOI methods, especially for curvature. As seen in Table 10, this board had the lowest standard deviation in curvature-based MoE for both DIC and **GST,DSS** FEM, and the second-lowest strain-based variability for DIC. However, this table also shows a significant discrepancy in terms of the average MoE value between the DIC and **GST,DSS** FEM profiles, suggesting that the material parameters for this board may be incorrect.
For strain-based MoE, the lowest standard deviations are found for boards #44 and #361 in the case of **GST,DSS** FEM profiles and boards #44 and #352 in the case of the DIC profiles. This could be the reason for the very low strain-based correlation results across all methods for board #44. The flat FEM MoE profiles contrast with the one spurious peak interrupted DIC MoE profile (spurious peak. A similar artifact (MoE peak appearing in only one of two successive bending tests) occurs for board #361, leading similarly to low strain-based correlations, especially for the **1A** methods which tend to yield flatter profiles.

This board #44 notably highlights the impact -and interest- of using curvature-
based versus strain-based MoE profiles: while its strain-based MoE profile is almost
flat, the curvature-based MoE profile exhibits strong variations, including an apparent
infinite peak caused by local curvature inversion, in line with the DIC observations.
In particular, curvature-based correlations are generally high, especially for **2A IFOI**
methods, reflecting the strong influence of interior side knots.

4.2.3 Boards #351, #352, #361, #362 and #43: limitations of the
**2A methods**

Boards #351, #352, #361, show significantly lower strain-based MoE correlations for the **2A** methods compared to other boards. The same trend is observed for curvature in boards #351, #361, #362 and #43.

Overall, except for these few boards (and board #44 for which strain-based MoE profiles correlation yields poor results altogether), the **2A** IFOI methods tend to produce comparable curvature and strain correlations than **1A** IFOI methods.

***Strain-based profiles***

The poor performance of the **2A** methods in strain-based correlations for these boards
can be explained by a closer inspection of the strain profiles. As shown in Figure 9,
these methods tend to over-interpret side fibre deviation from spike (non-traversing
splay) knots (ISO2300, 1973), producing spurious notch-shaped stiffness drops.

This notch shape originates from strong local divergences
in $\epsilon_{xx}$. As long as such local strain outliers fall within the
linear regression window $w_x = 90$ mm, they induce abrupt
stiffness drops (see the strain profiles at $x = 1240$ mm from
the **GST,DSS** results in Figure 11).

The **2A** interpolation schemes propagate the edge fibre
deviation across the entire board width by linear interpola-
tion. Consequently, as visible Figure 9, in the presence of
extrapolated spike knots, these schemes lead to erroneous
stiffness drops in the strain-based MoE profiles; in the case of
traversing splay knots, notch-shaped drops are visible in all
FEM and DIC MoE profiles, but are exaggerated when using
the **2A** schemes, hence the low determination coefficients.

It should be emphasized that the poor correlation results
of the **2A** methods are not related to the cutting pattern
of the boards. In the present dataset, the cutting pattern

<figure><img src="image_10.png" /><figcaption>**Fig. 10:** Board #43 erroneous stiffness drop in **2A** methods due to over-interpretation of side fibre deviations near a spike knot.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 9:** Spurious stiffness drops in strain-based profiles produced by **2A** methods due to over-interpretation of side fibre deviations near spike knots.</figcaption></figure>

visible in Figure 1, combined with the initial log selection to obtain boards with one large central enclosed knot (#39 in (Penvern et al, 2025)), resulted in the boards with spike knots being mostly rift-sawn (#351, #352, #361, and #362). Nevertheless, a spike knot is also present in board #43, which similarly produces an erroneous stiffness drop and low determination coefficients for **2A** IFOI methods (see Figure 10).

## Curvature-based profiles

For curvature profiles, the explanation is less straightforward. While over-interpretation of side fibre deviations likely contributes, the resulting effect is not a simple local drop beneath the knot. Instead, knots tend to produce an "M"-shaped curvature response with flanking peaks. Erroneous "M"-shape curvature thus may explain the curvature inversions observed in **2A** profiles for boards #43, #351, #361, and #362 near spike knots (Figure 12) which also result in lower determination coefficients.

This characteristic "M"-shaped curvature originates from a "W"-like displacement field beneath knots, even when the knot is located on the top surface. This is visible in the deflection profiles at $x = 1580$ mm and $x = 1700$ mm in Figure 13, where blue markers indicate displacements at the bottom edge of the board ($y = 0$ mm).
<figure><img src="image_12.png" /><figcaption>**Fig. 11:** Board #351 strain plots at various longitudinal positions obtained from FEM results, using fibre orientations determined by **GST,DSS** and interpolated by **1A,radial** and **2A,nearest**. These profiles were used in the linear regression for the computation of the strain-based local MoE.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 12:** Erroneous curvature peaks in 2A methods due to over-interpretation of side fibre deviations near spike knots.</figcaption></figure>

At *x* = 1700 mm, the **2A,nearest** interpolation produces a pronounced "blue bump" that locally reduces curvature, inverting the peak compared to the **GST,DSS** and **1A,radial** results. This bump logically arises from displacements influenced by the spike knot, whose fibre deviation is only captured on the side surface. Because the **2A,nearest** interpolation extrapolates this deviation into the interior, it induces this spurious curvature response.

## 4.3 Profile Correlation Matrix Across Interpolation Methods

To complement the board-wise Pearson correlation analysis, the overall agreement between interpolation methods across all boards was assessed through the computation of correlation matrices. These matrices were obtained by averaging, over the available boards, the Pearson determination coefficients calculated between each pair of methods. For FEM-FEM comparisons, the coefficients were evaluated over the full board length, whereas for comparisons involving DIC, the calculation was restricted to the monitored window. The resulting matrices thus capture the global similarity in fibre-informed mechanical property estimation achieved by the different interpolation strategies. The correlation matrices are reported in Tables 12 and 11 for the strain-based MoE and curvature profiles, respectively.

## 4.4 1-Angle vs 2-Angle Interpolation

Tables 11 and 12 demonstrate a clear dichotomy between the 1-angle (1A) and 2-angle (2A) interpolation methods, evident in two dark violet blocks (indicating a correlation factor close to 1). This indicates that the primary difference among these interpolation techniques lies in the decision to compute a combined angle or not.
<figure><img src="image_14.png" /><figcaption>Fig. 13: Displacement fields of board #351 at various longitudinal positions from FEM simulations, based on fibre orientations determined by GST,DSS and interpolated using 1A,radial and 2A,nearest methods for curvature computation.</figcaption></figure>

Within each family, the methods **1A,flat** vs.**1A,flow** and **2A,flat** vs.**2A,flow** yield nearly identical results. The rationale behind the flow-based methods was to improve upon nearest-neighbour interpolation in the 1A case and linear interpolation in the 2A case by using optical flow to detect the orientation of knot cores and to interpolate fibre orientation along these directions. However, the results show
<table><caption>**Table 11:** Pearson correlation ($R^2$) between FEM curvature profiles from different interpolation methods. Rainbow scale from 0 (red) to 1 (blue/violet).</caption><thead><tr><th rowspan="2">Method</th><th colspan="4">1A IFOI</th><th colspan="3">2A IFOI</th><th rowspan="2">IFOD<br/>GST,DSS</th><th rowspan="2">DIC</th></tr><tr><th>flat</th><th>flow</th><th>radial</th><th>roof</th><th>nearest</th><th>linear</th><th>flow</th></tr></thead><tbody><tr><td>1A,flat</td><td>1.00</td><td>0.94</td><td>0.76</td><td>0.83</td><td>0.66</td><td>0.70</td><td>0.65</td><td>0.73</td><td>0.66</td></tr><tr><td>1A,flow</td><td>0.94</td><td>1.00</td><td>0.75</td><td>0.79</td><td>0.64</td><td>0.69</td><td>0.65</td><td>0.68</td><td>0.65</td></tr><tr><td>1A,radial</td><td>0.76</td><td>0.75</td><td>1.00</td><td>0.77</td><td>0.55</td><td>0.59</td><td>0.55</td><td>0.67</td><td>0.67</td></tr><tr><td>1A,roof</td><td>0.83</td><td>0.79</td><td>0.77</td><td>1.00</td><td>0.63</td><td>0.68</td><td>0.63</td><td>0.68</td><td>0.69</td></tr><tr><td>2A,nearest</td><td>0.66</td><td>0.64</td><td>0.55</td><td>0.63</td><td>1.00</td><td>0.95</td><td>1.00</td><td>0.64</td><td>0.68</td></tr><tr><td>2A,linear</td><td>0.70</td><td>0.69</td><td>0.59</td><td>0.68</td><td>0.95</td><td>1.00</td><td>0.96</td><td>0.67</td><td>0.68</td></tr><tr><td>2A,flow</td><td>0.65</td><td>0.65</td><td>0.55</td><td>0.63</td><td>1.00</td><td>0.96</td><td>1.00</td><td>0.63</td><td>0.68</td></tr><tr><td>GST,DSS</td><td>0.73</td><td>0.68</td><td>0.67</td><td>0.68</td><td>0.64</td><td>0.67</td><td>0.63</td><td>1.00</td><td>0.77</td></tr><tr><td>DIC</td><td>0.66</td><td>0.65</td><td>0.67</td><td>0.69</td><td>0.68</td><td>0.68</td><td>0.68</td><td>0.77</td><td>1.00</td></tr></tbody></table>

<table><caption>**Table 12:** Pearson correlation ($R^2$) between FEM MoE strain-based profiles from different interpolation methods. Rainbow scale from 0 (red) to 1 (blue/violet).</caption><thead><tr><th rowspan="2">Method</th><th colspan="4">1A IFOI</th><th colspan="3">2A IFOI</th><th rowspan="2">IFOD<br/>GST,DSS</th><th rowspan="2">DIC</th></tr><tr><th>flat</th><th>flow</th><th>radial</th><th>roof</th><th>nearest</th><th>linear</th><th>flow</th></tr></thead><tbody><tr><td>1A,flat</td><td>1.00</td><td>0.98</td><td>0.81</td><td>0.89</td><td>0.73</td><td>0.74</td><td>0.73</td><td>0.71</td><td>0.61</td></tr><tr><td>1A,flow</td><td>0.98</td><td>1.00</td><td>0.82</td><td>0.88</td><td>0.73</td><td>0.74</td><td>0.73</td><td>0.71</td><td>0.61</td></tr><tr><td>1A,radial</td><td>0.81</td><td>0.82</td><td>1.00</td><td>0.81</td><td>0.73</td><td>0.74</td><td>0.73</td><td>0.73</td><td>0.63</td></tr><tr><td>1A,roof</td><td>0.89</td><td>0.88</td><td>0.81</td><td>1.00</td><td>0.86</td><td>0.87</td><td>0.86</td><td>0.73</td><td>0.66</td></tr><tr><td>2A,nearest</td><td>0.73</td><td>0.73</td><td>0.73</td><td>0.86</td><td>1.00</td><td>0.98</td><td>1.00</td><td>0.69</td><td>0.63</td></tr><tr><td>2A,linear</td><td>0.74</td><td>0.74</td><td>0.74</td><td>0.87</td><td>0.98</td><td>1.00</td><td>0.98</td><td>0.70</td><td>0.62</td></tr><tr><td>2A,flow</td><td>0.73</td><td>0.73</td><td>0.73</td><td>0.86</td><td>1.00</td><td>0.98</td><td>1.00</td><td>0.69</td><td>0.62</td></tr><tr><td>GST,DSS</td><td>0.71</td><td>0.71</td><td>0.73</td><td>0.73</td><td>0.69</td><td>0.70</td><td>0.69</td><td>1.00</td><td>0.69</td></tr><tr><td>DIC</td><td>0.61</td><td>0.61</td><td>0.63</td><td>0.66</td><td>0.63</td><td>0.62</td><td>0.62</td><td>0.69</td><td>1.00</td></tr></tbody></table>

that, for the board thicknesses and widths investigated here, interpolation along the optical flow field produces outcomes almost indistinguishable from simpler methods: the curvature and strain-based MoE profiles correlation coefficient are 0.94 and 0.98 respectively between **1A,flat** and **1A,flow**, and twice 1.0 respectively between **2A,nearest** and **2A,flow** and, meaning close identity between the width-wise and thickness-wise nearest-neighbour interpolation and optical field directed linear interpolation. Likewise, applying linear interpolation to the laser dot angle measurements (**2A,linear**) has negligible effect on the resulting profiles ($R^2_{E,ε} = 0.95$ and $R^2 = 0.98$ respectively) and does not improve correlation with the **DIC** profiles while increasing computational cost.

The **1A,radial** scheme differs more from **1A,flat** and **1A,flow** than the **1A,roof** scheme, despite the fact that most investigated boards were being flat-sawn. The **1A,roof** scheme yields strain-based MoE profiles that are most similar to those obtained with the **2A** scheme, as it likewise systematically incorporates fibre angle measurements along the board edges.

On average, correlations with DIC profiles are similar for **1A** and **2A** schemes ($R^2 \approx 0.67$ vs. $R^2 \approx 0.68$ for curvature; $R^2 \approx 0.63$ vs. $R^2 \approx 0.62$ for strain). Relative to the **GST,DSS** reference, however, **1A** schemes consistently yield higher correlations.
Assuming that **GST,DSS** provides a reliable approximation of the true fibre orientation, this indicates that the described single-angle interpolations capture internal fibre orientation variations more effectively than dual-angle approaches, probably because of the erroneous impact of the spike knots described paragraph 4.2.3.

In this correlation matrix, the **1A,roof** scheme shows slightly higher correlations with the **GST,DSS** profiles ($R_{\kappa,1A,roof}^2 = 0.68$ vs. $R_{\kappa,1A,radial}^2 = 0.67$, $R_{E(\epsilon),1A,roof}^2 = R_{E(\epsilon),1A,radial}^2 = 0.73$) as well as with the **DIC** profiles ($R_{\kappa,1A,roof}^2 = 0.69$ vs. $R_{\kappa,1A,radial}^2 = 0.67$, $R_{E(\epsilon),1A,roof}^2 = 0.66$ vs. $R_{E(\epsilon),1A,radial}^2 = 0.63$). Given the limited number of boards and the variability across them, this ranking could well be reversed in a larger dataset or under different parameter choices. Nevertheless, within the present analysis, **1A,roof** emerges as the most balanced option.

However, despite the notable shortcomings in the treatment of spike knots discussed in paragraph 4.2.3, the correlations of the **2A** methods with the **GST,DSS**, and **DIC** profiles are ultimately close to those obtained with the **1A** methods, and in particular with the **1A,roof** scheme. Visual inspection of the profiles provided in the appendix further suggests that the **2A** methods may reproduce local curvature variations more faithfully.

<table><caption>**Table 13:** Method wise mean and standard deviation values of $E_ε$ and $E_κ$ across the boards in their monitored DIC section.</caption><thead><tr><th>Method</th><th><i>E</i><sub>ε</sub> Mean (GPa)</th><th><i>E</i><sub>ε</sub> Std (GPa)</th><th>κ Mean (km<sup>-1</sup>)</th><th>κ Std (km<sup>-1</sup>)</th></tr></thead><tbody><tr><td><strong>DIC</strong></td><td>11.77</td><td>8.88</td><td>12.40</td><td>2.43</td></tr><tr><td><strong>1A,flat</strong></td><td>13.51</td><td>5.48</td><td>13.75</td><td>2.17</td></tr><tr><td><strong>1A,flow</strong></td><td>13.73</td><td>5.54</td><td>13.92</td><td>2.09</td></tr><tr><td><strong>1A,radial</strong></td><td>14.01</td><td>5.56</td><td>14.12</td><td>2.01</td></tr><tr><td><strong>1A,roof</strong></td><td>13.82</td><td>5.07</td><td>14.01</td><td>2.09</td></tr><tr><td><strong>2A,flow</strong></td><td>12.23</td><td>5.88</td><td>12.78</td><td>2.64</td></tr><tr><td><strong>2A,linear</strong></td><td>12.20</td><td>5.66</td><td>12.67</td><td>2.52</td></tr><tr><td><strong>2A,nearest</strong></td><td>12.12</td><td>5.85</td><td>12.71</td><td>2.69</td></tr><tr><td><strong>GST,DSS</strong></td><td>12.09</td><td>5.05</td><td>12.31</td><td>2.74</td></tr></tbody></table>

From the mean and standard deviation values reported in Table 13, the proximity of the mean values obtained with the **2A** methods to those of DIC cannot be considered a genuine advantage unless the material parameters used in the simulations are assumed to be accurate. The systematically lower average moduli predicted by the **2A** methods, compared with the **1A** methods, are consistent with the way the combined fibre direction is calculated (see Figure 5), whereas in the **1A** schemes, each interpolated angle is treated separately. On the other hand, it is worth noting that the mean-to-standard-deviation ratios of the **2A** methods are closer to those of the **DIC** profiles than are those of the **1A** methods. This observation is in line with the visual impression of a more accurate reproduction of local curvature and strain-based MoE variations by the **2A** schemes. Together, these results highlight the potential of the **2A** approaches, while also underscoring the need for refinements to their angle-combination strategy in order to reduce sensitivity to spike knots.
# 5 Conclusion

This study proposes an original and comprehensive comparison between interpolation schemes for internal fibre orientation and directly determined orientations (GST,DSS), based on FEM-computed localized MoE profiles derived both from axial strain and from curvature, enabling comparison against experimental DIC measurements.

Despite its limitations on heartwood boards, where abrupt sign changes in the radial direction presently impair fibre orientation determination, the GST-DSS approach consistently delivers the most reliable outcomes. The correlation levels obtained from this method are not only the most stable across boards but also the highest among all strategies tested.

In contrast, interpolation-based IFOI schemes frequently exhibit irregularities and reduced accuracy. These findings confirm that the GST-DSS IFOD method currently provides a robust and effective means of reconstructing internal fibre orientation, and represents a promising tool for enhancing current angle-based IFOI methods, with the aim of improving predictions of board mechanics.

The IFOI schemes grouped into two main families: one-angle (1A) interpolations, which propagate a single angle field across the board thickness, and two-angle (2A) interpolations, which combine separately thickness-wise and width-wise interpolated angles. On average, the 2A schemes underperform, as they are highly sensitive to surface fibre deviations introduced by spike knots, leading to spurious stiffness drops or even curvature inversions. Nevertheless, the 2A interpolations, despite yielding lower average MoE values, capture stiffness variations more effectively—showing the best correlations for the quarter-sawn heartwood board and often reproducing local patterns closer to those observed by DIC. This makes them potentially advantageous in applications where detecting local stiffness minima is critical, such as strength grading, which encourages further research efforts to make this type of interpolation more robust.

In the meantime, among the investigated methods, 1A,roof emerged as the most reliable and economical compromise, consistently providing robust profiles without introducing artefacts.

The novel optical flow interpolation did not significantly improve correlations, but remains attractive for knot tracking issues without prior identification of the pith location.

Another key result of this study is that curvature-based MoE computations reflects the internal variability of the boards more effectively, therefore correlating better with the DIC data than strain-based MoE profiles, even revealing heterogeneity in cases where MoE profiles derived from strain measurements indicate none. Such profiles may therefore be of particular relevance—especially when combined with curvature-based strain indicators—to develop new indicative properties for board grading. However, as curvature-based MoE profiles have not yet been systematically evaluated for grading purposes, further investigation is required to establish their practical applicability.
## Limitations

The present study is restricted to a limited dataset, as all boards were cut from a single log. This narrow sampling limits the generality of the results with respect to growth conditions, sawing patterns, and natural variability between trees. Furthermore, the FEM simulations relied on material parameters linearly interpolated from global density and moisture content, which do not accurately capture local heterogeneities.

## Perspectives

Future work should first address the limitations of the **GST-DSS** IFOD method, which, despite its robustness, still encounters difficulties on heartwood boards where abrupt radial sign changes impair fibre orientation determination. Further refinements are also needed in knot and crack regions, where fibre orientation is particularly challenging to resolve.

For interpolation-based schemes, an optimal strategy may involve a hybrid use of **2A**-type interpolations—applied selectively in regions around internal knots to better capture local stiffness variations—while avoiding excessive sensitivity to edge spike knots that lead to erroneous stiffness loss.

Broader investigations should also consider boards from multiple logs and diverse sawing patterns to strengthen the generality of the findings. Incorporating improved local material characterization (e.g. microstructural imaging, vibratory testing) would further enhance the accuracy of FEM simulations.

Finally, the practical relevance of curvature-based MoE profiles for timber grading should be investigated in larger-scale studies due to their ability to further capture internal variability.

## Declarations

This research was funded by the French National Research Agency (EffiQuAss project ANR-21-CE10-0002).

The authors have no relevant financial or non-financial interests to disclose.

The authors are deeply grateful to **Thomas Bultel**, to **Ameur Ben Chaabane** and to **Julien Brosset-Heckel** for their help in the investigations.

The authors contributed to this article as follows:

* Conceptualisation: GP, JV, HP;

* Investigation: HP, JV, GP;

* Data Curation: HP;

* Software: HP, GP, JV;

* Formal Analysis, writing – original draft and visualisation: HP;

* Funding Acquisition and project administration: GP;

* Methodology: HP;

* Supervision and writing – review & editing: GP, JV;
References

Abdeljaber O, Olsson A, Hu M (2025) Determination of 3d fiber orientations in timber boards using x-ray ct scanning and deep learning. In: ECCOMAS Thematic Conference on Computational Methods in Wood Mechanics (CompWood 2025), June 4-6, 2025, Vienna, Austria, TU Wien, p 106, f4:1

Besseau B (2021) Contribution au développement de procédés innovants pour une transformation plus efficiente du chêne. These de doctorat, Ecoles des Arts et Métiers, campus de Cluny, URL http://www.theses.fr/s216173

Besseau B, Pot G, Collet R, et al (2020) Influence of wood anatomy on fiber orientation measurement obtained by laser scanning on five European species. Journal of Wood Science 66(1):74. https://doi.org/10.1186/s10086-020-01922-y, URL https://jwoodscience.springeropen.com/articles/10.1186/s10086-020-01922-y

Bodig J, Jayne BA (1982) Mechanics of wood and wood composites. Van Nostrand Reinhold

Boivin J (2024) Analyse et modélisation des phénomènes d'interaction de la lumière et du bois : application pour le contrôle non destructif. Theses, École Nationale Supérieure des Arts et Métiers, URL https://pastel.hal.science/tel-04988386

Brännström M, Manninen J, Oja J (2008) Predicting the strength of sawn wood by tracheid laser scattering. BioResources 3(2):437-451

Briggert A, Hu M, Olsson A, et al (2018) Tracheid effect scanning and evaluation of in-plane and out-of-plane fiber direction in norway spruce timber. Wood and Fiber Science 50(04):411-429. URL https://wfs.swst.org/index.php/wfs/article/view/2728

Briggert A, Olsson A, Oscarsson J (2020) Prediction of tensile strength of sawn timber: definitions and performance of indicating properties based on surface laser scanning and dynamic excitation. Materials and Structures 53:1-20

EN 408 (2010) Timber structures – structural timber and glued laminated timber – determination of some physical and mechanical properties. Tech. Rep. EN 408:2010+A1:2012, Comité Européen de Normalisation (CEN), with Amendment A1 approved 16 June 2012

Farnebäck G (2003) Two-frame motion estimation based on polynomial expansion. In: Image Analysis, SCIA 2003, Lecture Notes in Computer Science, vol 2749. Springer, pp 363-370, https://doi.org/10.1007/3-540-45103-X_50, URL https://doi.org/10.1007/3-540-45103-X_50

Foley C (2003) Modeling the effects of knots in structural timber. PhD thesis, Lund University
Guindos P, Guaita M (2012) A three-dimensional wood material model to simulate the behavior of wood with any type of knot at the macro-scale. Wood Science and Technology 47. https://doi.org/10.1007/s00226-012-0517-4, URL https://doi.org/10.1007/s00226-012-0517-4

Guitard D (1987) Mécanique du matériau bois et composites. Cépaduès

Habite T, Olsson A (2022) Modelling of knots and 3d fibre orientation within timber boards based on information obtained from optical scanning. SSRN Electronic Journal https://doi.org/10.2139/ssrn.4110671

Hankinson (1921) Investigation of crushing strength of spruce at varying angles of grain. Air service information circular 3(259):16

Hu M, Olsson A, Johansson M, et al (2018) Modelling local bending stiffness based on fibre orientation in sawn timber. European Journal of Wood and Wood Products 76(6):1605-1621. https://doi.org/10.1007/s00107-018-1348-2, URL https://doi.org/10.1007/s00107-018-1348-2

Hu M, Abdeljaber O, Schirén W (2025a) Fe modeling of timber boards integrating 3d fiber orientation data obtained from x-ray ct and deep learning. In: ECCOMAS Thematic Conference on Computational Methods in Wood Mechanics (CompWood 2025), June 4-6, 2025, Vienna, Austria, TU Wien, p 107, f4:2

Hu M, Olsson A, Abdeljaber O, et al (2025b) Determining growth surfaces and fiber orientation in norway spruce sawn timber using x-ray computed tomography and optical scanning. Construction and Building Materials 482:141734. https://doi.org/10.1016/j.conbuildmat.2025.141734, URL https://www.sciencedirect.com/science/article/pii/S0950061825018859

Huber JA, Broman O, Ekevad M, et al (2022) A method for generating finite element models of wood boards from x-ray computed tomography scans. Computers & Structures 260:106702. https://doi.org/10.1016/j.compstruc.2021.106702, URL https://www.sciencedirect.com/science/article/pii/S0045794921002248

ISO2300 (1973) Iso 2300:1973: Sawn timber of broadleaved species — defects — terms and definitions. Tech. rep., International Organization for Standardization, URL https://www.iso.org/standard/7127.html, edition 1, confirmed 2021

Jehl A, Bleron L, Meriaudeau F, et al (2011) Apport de la mesure de pente de fil au classement mécanique des bois de structures. In: CFM 2011-20ème Congrès Français de Mécanique, AFM, Maison de la Mécanique, 39/41 rue Louis Blanc-92400 Courbevoie

Klear M (2023) circle-fit: A python package for circular fitting. URL https://pypi.org/project/circle-fit/, licence: MIT License (Copyright (c) 2023 Michael Klear
Kretschmann DE (2010) Mechanical properties of wood. Environments 5:34

Li X, Pot G, Ngo P, et al (2023) An image processing method to recognize position of sawn boards within the log. Wood Science and Technology 57(6):1401–1420. https://doi.org/10.1007/s00226-023-01495-1

Longuetaud F, Pot G, Mothe F, et al (2022) Traceability and quality assessment of Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) logs: the Tree-TraceDouglas database. Annals of Forest Science 79(46). https://doi.org/10.1186/s13595-022-01163-7, URL https://hal.science/hal-03658479

Lukacevic M, Kandler G, Hu M, et al (2019) A 3d model for knots and related fiber deviations in sawn timber for prediction of mechanical properties of boards. Materials and Design 166:107617. https://doi.org/10.1016/j.matdes.2019.107617, URL https://www.sciencedirect.com/science/article/pii/S0264127519300541

Nyström J (2003) Automatic measurement of fiber orientation in softwoods by using the tracheid effect. Computers and Electronics in Agriculture 41(1-3):91–99. https://doi.org/10.1016/S0168-1699(03)00045-0, URL https://linkinghub.elsevier.com/retrieve/pii/S0168169903000450

Olsson A, Oscarsson J (2017) Strength grading on the basis of high resolution laser scanning and dynamic excitation: a full scale investigation of performance. European Journal of Wood and Wood Products 75(1):17–31. https://doi.org/10.1007/s00107-016-1102-6, URL https://doi.org/10.1007/s00107-016-1102-6

Olsson A, Oscarsson J, Serrano E, et al (2013) Prediction of timber bending strength and in-member cross-sectional stiffness variation on the basis of local wood fibre orientation. European Journal of Wood and Wood Products 71(3):319–333. https://doi.org/10.1007/s00107-013-0684-5, URL https://doi.org/10.1007/s00107-013-0684-5

Olsson A, Pot G, Viguier J, et al (2018) Performance of strength grading methods based on fibre orientation and axial resonance frequency applied to norway spruce (picea abies l.), douglas fir (pseudotsuga menziesii (mirb.) franco) and european oak (quercus petraea (matt.) Liebl./Quercus robur l.). Annals of Forest Science 75(4):102

Olsson A, Pot G, Viguier J, et al (2022) Performance of timber board models for prediction of local bending stiffness and strength—with application on douglas fir sawn timber. Wood and Fiber Science 54(4):226–245

Penvern H, Demoulin L, Pot G, et al (2024) A laboratory method to determine 3d fibre orientation around knots in sawn timber: case study on a douglas fir
specimen. Wood Science and Technology 58(5):1735-1760. https://doi.org/10.1007/s00226-024-01583-w
URL https://doi.org/10.1007/s00226-024-01583-w

Penvern H, Pot G, Viguier J (2025) Comparison of various internal fibre orientation determination methods for predicting the mechanical properties of one douglas-fir timber board. URL https://hal.science/hal-05265237

Pot G, Duriot R, Girardon S, et al (2024a) Comparison of classical beam theory and finite element modelling of timber from fibre orientation data according to knot position and loading type. European Journal of Wood and Wood Products 82(3):597-617

Pot G, Viguier J, Besseau B, et al (2024b) Modelling tensile mechanical properties of oak timber from fibre orientation scanning for strength grading purpose. In: 11th Hardwood Conference, Sopron, Hungary, https://doi.org/10.35511/978-963-334-518-4

Purba CYC, Viguier J, Denaud L, et al (2020) Contactless moisture content measurement on green veneer based on laser light scattering patterns. Wood Science and Technology https://doi.org/10.1007/s00226-020-01187-0, URL https://doi.org/10.1007/s00226-020-01187-0, aCL

Rais A, Bacher M, Khaloian-Sarnaghi A, et al (2021) Local 3d fibre orientation for tensile strength prediction of european beech timber. Construction and Building Materials 279:122527. https://doi.org/10.1016/j.conbuildmat.2021.122527, URL https://www.sciencedirect.com/science/article/pii/S0950061821002877

Simonaho SP, Palviainen J, Tolonen Y, et al (2004) Determination of wood grain direction from laser light scattering pattern. Optics and Lasers in Engineering 41(1):95-103. https://doi.org/10.1016/S0143-8166(02)00144-6, URL https://linkinghub.elsevier.com/retrieve/pii/S0143816602001446

Viguier J (2015) Classement mécanique des bois de structure. prise en compte des singularités dans la modélisation du comportement mécanique. PhD thesis, Phd thesis, Université de Lorraine, Metz, France, URL http://docnum.univ-lorraine.fr/public/DDOC_T_2015_0142_VIGUIER.pdf

Viguier J, Jehl A, Collet R, et al (2015) Improving strength grading of timber by grain angle measurement and mechanical modeling. Wood Material Science & Engineering 10(1):145-156. https://doi.org/10.1080/17480272.2014.951071, URL http://dx.doi.org/10.1080/17480272.2014.951071, aCL
A Boards enhanced visualisation, along with the
strain-based MoE, curvature-based MoE and
curvature profiles

The enhanced board visualisations display the scanner measurements, where linearly interpolated laser dot angle data are colour-mapped from pink to green alongside the corresponding in-plane fibre orientation fields represented as streamlines. These are shown next to the original greyscale scan images of the boards for comparison.

Then, in order, for each board, the computed strain-based and curvature-based MoE profiles, followed by the computed curvature profile are shown.
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<figure><img src="image_37.png" /><figcaption>Comparison of FEM and DIC Curvature Profiles of Board 40 with w<sub>x</sub> = 90mm</figcaption></figure>

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<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>
<figure><img src="image_41.png" /><figcaption>Comparison of FEM and DIC Curvature Profiles of Board 41 with w<sub>x</sub> = 90mm</figcaption></figure>

<figure><img src="image_42.png" /></figure>
<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>
<figure><img src="image_45.png" /><figcaption>Comparison of FEM and DIC Curvature Profiles of Board 42 with w<sub>x</sub> = 90mm</figcaption></figure>

<figure><img src="image_46.png" /></figure>
<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>
<figure><img src="image_49.png" /><figcaption>Comparison of FEM and DIC Curvature Profiles of Board 43 with w<sub>x</sub> = 90mm</figcaption></figure>

<figure><img src="image_50.png" /></figure>
<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>
<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>