# A multiscale method bio-inspired by bone architecture to enhance stiffness to weight ratio of mechanical parts

Asma Jellad, Valentin Peroni, Nicolas Cadoret, Mathieu Bilhere-Dieuzeide, Julien Chaves-Jacob

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Asma Jellad, Valentin Peroni, Nicolas Cadoret, Mathieu Bilhere-Dieuzeide, Julien Chaves-Jacob. A multiscale method bio-inspired by bone architecture to enhance stiffness to weight ratio of mechanical parts. Mechanics Research Communications, 2026, 152, pp.104614. 10.1016/j.mechrescom.2026.104614. hal-05486198

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Submitted on 30 Jan 2026

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# A multiscale method bio-inspired by bone architecture to enhance stiffness to weight ratio of mechanical parts

Asma JELLADª, Valentin PERONIª, Nicolas CADORETª,ᵇ, Mathieu BILHERE-DIEUZEIDEª,ᶜ, Julien CHAVES-JACOBª*

## Abstract

A current challenge in mechanical engineering is reducing the mass of mechanisms while maintaining performance, in order to decrease energy and resource consumption. This challenge also exists in Nature, and the architecture of long bones provides an excellent example of how Evolution has solved this problem. Bone architecture is complex, highly adapted to mechanical stresses, and yet extremely resource efficient. The use of such complex structures is now feasible in industry thanks to the advent of additive manufacturing. In this paper, a bone bio-inspired, multiscale mass reduction method for mechanical parts is presented, aiming to enhance the stiffness to weight ratio. A two-scale approach is developed: the macroscale (bone shape, topological level) and the mesoscale (trabecular bone structure, structural level). This multiscale method is then compared with each method applied separately. For this comparison, the three-point bending test (representative of industrial demands) is used. Tests were performed both experimentally and through finite element analysis. The experiments show the efficiency of the multiscale method, with improvements of 24% compared to the topological level method and 7% compared to the structural level method. The finite element analysis results support and explain this trend, indicating that the multiscale structure achieves a better distribution of internal stresses. Thus, this study demonstrates that combining two mass reduction methods operating at different scales yields better results than either method alone. The results obtained for this multiscale method are highly encouraging and suggest potential applications for industrial parts in the automotive and aerospace sectors.

**Key Words** Mass reduction, multiscale, bio-inspiration, bone architecture, mechanical performance, stiffness.

<footnote>ªAix Marseille Université, CNRS, ISM, Inst Mouvement Sci, UMR 7287, 13009 Marseille, France</footnote>

<footnote>ᵇAirbus Helicopters, Aéroport de Marseille Provence, 13700 Marignane, France</footnote>

<footnote>ᶜThales LAS (Land Air System), France OME (Optronics and Missile Electronics), 78990 Elancourt, France</footnote>

<footnote>*Corresponding author: julien.chaves-jacob@univ-amu.fr, 413 Avenue Gaston Berger, 13625 Aix-en-Provence, +33 4 13 94 62 77 (J. Chaves-Jacob)</footnote>
# 1. Introduction

Mass reduction methods have been developed to reduce part weight while maintaining or enhancing their mechanical performance [1-10]. Weight reduction can be achieved either by selecting lightweight materials or by redesigning the workpiece while keeping the same material. The use of advanced materials, such as lightweight metals, composites, and polymers, can provide high stiffness to weight ratio, despite higher cost [10]. However, these materials may introduce significant challenges, including chemical incompatibilities and bonding issues, which can degrade mechanical properties and reduce the part's lifespan [11-12].

Selecting advanced materials can improve stiffness to weight ratio, but the mechanical response of components at small scales often deviates from predictions of classical elasticity due to size-dependent effects. This limitation highlights the need for advanced continuum theories capable of accurately capturing nanoscale and microscale mechanical behavior, including nonlocal and gradient effects. Recent advances in generalized continuum mechanics highlight the importance of capturing size-dependent behavior in small scale and lightweight structures, where classical elasticity often fails [13-14]. Gradient and nonlocal elasticity formulations combine strain and stress gradients with classical theory within variational frameworks, allowing accurate modeling of static and dynamic responses of nanobars, rods, and beam like components [15]. These size-dependent and microscale approaches collectively demonstrate the relevance of advanced continuum theories for designing lightweight and small scale engineering structures.

Even when the same material is used topology optimization strategies can exploit the interplay between material behavior and geometry to enhance performance. Among these, topology optimization (TO) is widely used to allocate material according to stress distributions determined via Finite Element Analysis (FEA), optimizing both geometry and topology [16-21]. Multiscale TO, inspired by natural structures such as bone and bamboo, enables the creation of lightweight, robust, and multifunctional components. Recent reviews have categorized existing multiscale TO approaches, analyzed their principles and applicability, and highlighted open research questions to guide future development [22].

However, the part's weight can be reduced by modifying its structure using other design optimization strategies. Among these structural methods, lattice structures have proven effective in enhancing both stiffness and strength [1-8]. Nevertheless, using lattice structures for mass reduction can introduce challenges, such as localized buckling of beams under certain constraints, potentially leading to structural failure [1-2]. Existing works have proposed limiting strut buckling by constructing a series of isostatic lines in relation to the local principal stresses to generate additional strut-based lattice structures within
the internal structure [23-24]. In addition, to create anisotropic porous structures and limit the periodicity
of the strut-based lattices, optimized material using Voronoi patterns was found to show improved strength
compared to isotropic structures [24-25].

Achieving both lightness and stiffness in mechanical structures is a recurring challenge for natural
organisms, as observed in numerous plant and animal species. Examples of these natural, lightweight yet
sturdy structures include sea sponges, bamboo, and endoskeletons [26-28]. Among these natural
structures, the endoskeleton, with its intricate internal bone architecture, is largely studied [27-28]. As a
result, bone has become one of the most emulated natural materials in engineering studies [16-18], valued
for its high stiffness to weight ratio [29]. Extensive research has explored how bone architecture can
inform the design of mechanical structures to enhance predicted stiffness and strength, often through the
lens of the bone remodeling process [30-32]. These studies, based on Wolff's law, have investigated
optimized relationships between bone density and stress orientation [30-34].

Drawing from this model, recent bio-inspired mass reduction methods have used the bone's internal
architecture for design optimization and structural adaptation [8, 35-38]. These methods were stress-
driven using FEA on initial workpiece and bio-inspired laws of the relationship between stress and bone
internal structure. Audibert et al. (2018) introduced a 2.5D (2D geometry extruded in width) stress driven
mass reduction method based on the trabecular bone architecture characterized by elliptic pores aligned
preferentially along the principal stress directions [39]. Their method employs a structural optimization
algorithm that uses elliptic cavities with major axes oriented along the stress tensor's principal directions
and distributed along a Face-Centered Cubic (FCC) structure of pattern size *p* [8]. The initial workpiece
is then discretized using this grid and at each node of the grid the bio-inspired elliptical cavity is computed
based on the local stress field [8]. Therefore, an iterative method inspired by the medullary cavity
appearance in long bones was developed to optimize a 2.5D architecture design through iterations [35].
Applied to a torsion disc study case, this method, iteratively remove material from the workpiece by
applying a cavity design process in low-stress areas determined with threshold varying at each iteration
[35]. Moreover, Cadoret et al. (2023) extended the approach to 3-dimensional, creating pores in the form
of stress-driven ellipsoids arranged in Face-Centered Cubic (FCC) to obtain 3D optimized lightweight
structures.

In the present study, a mass reduction method inspired by the multiscale bone's architecture is proposed.
By mimicking bone's multiscale aspects, we aim to increase the stiffness to weight ratio of the part, by
using two different scales (macroscale and mesoscale) to further improve mechanical performance of the
workpiece. For this purpose, the parts to be lightened are successively optimized at two levels: firstly,
topological level at the macroscale and secondly, structural level at the mesoscale, following the stress-
driven design approach presented in [36-37]. The mechanical performance of the multiscale method is
compared to the two single-scale methods alone.

To evaluate the effectiveness of the proposed approaches, the remainder of this paper is structured as
follows. Section 2 describes the three mass reduction methods ( macroscale, mesoscale and multiscale ). In
Section 3, an experimental three-point bending test was conducted to evaluate the mechanical
performance of the three proposed methods. For this purpose, three beams were designed: one with
topological level method, one with structural level method and the last with the multiscale method.
Thereafter, the three beams were fabricated using 3D additive manufacturing and tested. A relative
stiffness indicator is introduced to effectively assess mechanical performance comparison of the beams.
In Section 4, finite element simulations of three-point bending tests are conducted to clarify the
experimental results. Mechanical performance of the proposed multiscale method is discussed in
Section 5.

## 2. Mass reduction method bio-inspired by multiscale bone's architecture

The proposed method is inspired by the multiscale architecture of bone [40]. Indeed, the hierarchical
structural of bone, includes five main scales: the nanoscale (1-500 nm), the sub-microscale (0.5-10 µm),
the microscale (10-500 µm), the mesoscale (0.5-10 mm), and the macroscale (>10 mm) [40].

As shown in bone tomography (Fig. 1), bone adaptation occurs in both external shape and internal shape,
such as the medullary cavity. The tomography also reveals two bone structures related to trabecular
(spongy) and cortical (compact) bones, which exhibit variations in local volume fraction ρf. The volume
fraction ranges from solid cortical bone to low-density trabecular bone. Here, the macroscale presents the
general bone shape (external and internal) whereas the mesoscale illustrates the bone’s microstructure
(cortical and trabecular). In this work, these two bone’s architecture scales are mimicked (macroscale and
mesoscale). Thus, the workpiece can be lightweighted through bio-inspiration of these two levels. To
achieve this, the proposed multiscale method employs two different levels: a topological level at the
macroscale and a structural level at the mesoscale. All the studied mass reduction methods allow
lightweighting the part by only removing material from the initial geometry. First, as in bone, material is
removed to redesign both external and internal shape of the part using topological level method. Second,
material is removed by creating a mesoscale structure composed of pores, inspired by trabecular bone
architecture.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Bone's scales bio-inspiration to optimize structure and topology. The bone tomography has been obtained within partnership between FUNEVOL MECADEV MNHN and SBI ISM AMU teams.</figcaption></figure>

## 2.1. Macroscale: topological level

As described by Wolff's law, bone adapts to mechanical stress it undergoes [33-34, 47-48]. Hence, in bone, a macroscale stress-driven mechanism is considered for material distribution. Indeed, bone material is distributed in regions with significant mechanical stresses. However, at regions with low stresses, bone material is removed, as observed in medullary cavity (Fig. 1). The topological level method mimics this macroscale stress-driven mechanism using topological optimization techniques [17, 21]. The topology optimization process relies on finite element analysis to determine stress distribution within the initial part geometry under specified loading and boundary conditions. Through iterative steps, unnecessary material is progressively removed from low-stress regions, while material is retained in high-stress areas. After each iteration, the stresses are redistributed, and unnecessary material is removed to optimize distribution and generate a lightweighted optimized geometry. This approach optimizes the topology by redesigning the part's external shape and introducing cavities at this scale.

## 2.2. Mesoscale: structural level
At mesoscale, the Bio-Inspired Mass Reduction (BIMR) method developed by Cadoret et al., [36] mimics the stress-driven architecture of trabecular bone, as described by Wolff's law. The algorithm adapts the local volume fraction and anisotropy in response to local mechanical stress. To achieve this adaptation, ellipsoidal pores are locally introduced within the material.

The process begins by meshing the initial geometry and performing finite element analysis (FEA) to compute the stress tensor under a specified loading condition. Simultaneously, a Face-Centered Cubic (FCC) pattern is applied throughout the geometry's volume. The pattern size, $p$, of the FCC unit cell, (Fig. 2), is scaled relative to the initial geometry size to replicate trabecular scaling. At each FCC node, the local stress value is extracted from the FEA result. Based on the local Von Mises stress, $\sigma_{VM}$, an ellipsoidal pore is generated at the node, and the local volume fraction, $\rho f$, defined as the ratio of solid material to the total volume within an FCC cell, is adjusted using Eq. (1) [8]. It increases in regions of high-stress concentration, mimicking cortical bone, and decreases in low-stress areas, consistent with the adaptive architecture of trabecular bone [39, 41].

$$ \rho f = 3.07 \left( \frac{\sigma_{VM}}{Re} \right)^{0.59} \qquad (1) $$

With $Re$ the material's elastic limit.

The dimensions of each ellipsoid, defined by its principal semi-axes $a$, $b$ and $c$ (Fig. 2), are computed based on the volume fraction and the fabric tensor eigenvalues. Additionally, the ellipsoid is oriented along the local principal stress directions, which are determined by the fabric tensor eigenvectors. The algorithm controlling this process is implemented as a Python macro within the FreeCAD simulation environment, automating stress mapping, node assignment, and pore generation. As a result, the BIMR method achieves significant mass reduction by emulating the bone's natural stress-driven strategy, using ellipsoids shaped according to local stress constraints.
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** The structural level porous pattern model showing pores’ parameters.</figcaption></figure>

## 2.3. Multiscale method

The proposed multiscale method combines the two above scales by applying both topological optimization and BIMR methods sequentially (Fig. 3). A target mass reduction is defined as the difference between the initial part and final part masses. This total reduction value is allocated between the two stages based on a predefined proportion.

First, the initial geometry is created, and boundary conditions are applied to perform the topological level method. In this study, the topological level method is applied before the structural level method to reduce the part's mass by removing material from the macroscale to the mesoscale. To maximize material reduction, the topological level method is implemented on the entire initial geometry. During this process, material is removed from the low-stress regions, redesigning the part's overall shape and introducing cavities. As a result, mechanical stresses are redistributed in the remaining material, producing an intermediate topology optimized (TO) geometry. The structural level method is then applied to generate a mesoscale structure by introducing pores based on the redistributed stresses. By following this sequence, the computational time of the second step (mesoscale level) is reduced due to lower remained material.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Multiscale method algorithm</figcaption></figure>

**3. Experimental validation**

**3.1. *Application case: three-point bending test***

The three-point bending test was selected as the application case for this study due to its industrial relevance and simplicity in evaluating mechanical properties. For that reason, this test serves as a reliable validation framework in numerous research studies [8, 35-38]. The initial tested beam dimensions are 150 mm in length, and 30 mm in both height and width. For each method, finite element analysis (FEA) was performed under the following boundary conditions: a clamp and a sliding support were positioned 10 mm from each lateral face (Fig. 4), and a central force *F* was applied. A 0.65 target mass reduction was chosen, consistent with Cadoret *et al.*'s experimental conditions to facilitate comparison [36].
<figure><img src="image_5.png" /><figcaption>Fig. 4. The initial beam without mass reduction showing the used boundary conditions.</figcaption></figure>

## 3.2. Mass reduction methods and application on three-point bending test

### 3.2.1. Topological level method (Macro. beam)

The topology level method was applied to optimize a 3-dimensional beam geometry called Macro. beam. The objective of this optimization was to reduce the beam's mass while maintaining its mechanical stiffness under the applied loading conditions. The initial beam geometry was modeled, and the boundary conditions were defined according to Fig. 4, with a vertical load of 5 kN applied to the center. The beam was discretized using 2 mm quadratic tetrahedral elements to ensure accurate stress and deformation predictions. The topology optimization was performed using Tosca Generative Design, a module integrated within the 3DEXPERIENCE platform by Dassault Systemes. The optimization targeted a mass fraction of 0.65, corresponding to a 0.35 mass reduction, while minimizing the overall compliance (i.e., maximizing stiffness). The iterative process converged after 33 optimization cycles, resulting in a lightweight structure that effectively redistributed material along the primary load paths.

### 3.2.2. Structural level method (Meso. beam)

The structural level method was applied to optimize a 3-dimensional beam geometry called Meso. beam. In this work, this beam was generated under the same conditions as those reported by Cadoret *et al.* [36]. The optimization process was carried out in two steps. First, the initial beam was designed, and the boundary conditions were applied according to Fig. 4 and the target weight reduction was chosen. Then, the BIMR algorithm described in Section 2.2 was executed. A finite element analysis was conducted on the initial 3-point bending beam with a first-order tetrahedral element mesh size of 2.5 mm. The stress
distribution was obtained throughout the initial beam as illustrated in Fig. 5(a). This method generated a porous part architecture constituted of ellipsoids inspired by the bone’s mesoscale structure, as illustrated in Fig. 5(b).

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Meso. beam optimized architecture considering the local stress distribution.</figcaption></figure>

### 3.2.3. Multiscale method (Multi. beam)

The multiscale method was applied to optimize a 3-dimensional beam geometry called Multi. beam. The optimization process was carried out in three steps.

(a) First, the initial beam was designed, and the boundary conditions were applied according to Fig. 4. In this step, the target volume fraction of 0.65 was distributed between the second and third steps described below.

(b) Then, in the second step a volume fraction of 0.75 was achieved using the topological level method, as detailed in Section 3.2.1. This choice of proportion was made to allow a significant volume reduction before the third step. The beam was meshed with 2 mm quadratic tetrahedral elements and a, 3-dimensional topology optimization was conducted using Tosca Generative Design, a module within the 3DEXPERIENCE platform of Dassault Systèmes. The result of this step was the intermediate topology optimized beam as shown in Fig. 6 (b).

(c) Finally, structural level method was applied from the intermediate TO beam. To reach the target volume fraction of 0.65 the residual volume fraction was reduced using the mesoscale method. A finite element analysis was conducted on the intermediate topology optimized beam with a first-order
tetrahedral element mesh size of 2.5 mm. This method generated a porous part architecture composed of ellipsoids inspired by the bone's mesoscale structure as illustrated in Fig. 6(c). The 0.65 volume fraction multiscale beam was finally optimized by successively applying the topological and structural levels methods, successively.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Algorithm of optimized Multi. beam.</figcaption></figure>

### 3.3. Optimized beams

The three optimized beams are shown in Fig. 7. Compared to the initial beam (Fig. 4), it can be observed that all the three beams were redesigned only by removing material.

By applying topological level method, a novel beam was generated by removing material from both lateral sides of the beam, as shown in Fig. 7(a). Notably, a large cavity was created in the center of the beam, as observed in xz mid-section of this beam (Fig. 7(a)).

The Meso. beam was presented in Fig. 7(b). We can notice from this figure that the general shape has been preserved, while local pores appear depending on the variation of compactness and anisotropy within the beam. According to Fig. 7(b), the corners remained with very low volume fraction, with large pores present on both lateral sides. Moreover, no pore was generated in areas of high stress concentration, such as under load application points or supports, as illustrated in Fig. 5. The difference in compactness is
associated with the size distribution of the pores as a function of the internal stresses in the beam, as determined from FEA.

As can be seen in the Multi. beam (Fig. 7(c)), the material was removed from the corners, which are areas presenting low stresses (Fig. 5). Compared to the Macro. beam and despite a higher volume fraction in the first step (0.75), the external shape of the Multi. shows similarity on both lateral sides (Fig. 7(c)). However, the internal shape of the Multi. beam presents a smaller cavity volume, distributed into two separate cavities on each side of the beam (xz mid-section). In addition, compared to the Meso. beam, similarities in compactness distribution and pore orientation were observed for the remaining volume of the Multi. beam, which exhibits smaller pores and denser regions without pores in the zones under load application. The Multi. beam seems to allow more consistent distribution of material compared to each scale alone. Indeed, the interior of the Multi. beam integrates both trabecular bone (small pores) and cortical bone structures, as seen in Fig. 7(c).

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>
**Fig. 7.** (a-b) Optimized beams using Topology level method (c) Structural level method (d) Multiscale method.

It is notable from the three optimized beams above that they all feature dense regions without pores in areas of load application and high stress concentration (Fig. 7). This is consistent with Fig. 5, which highlights both low and high-stress areas in the beam. In high-stress areas, such as load application points or contact zones, a denser material distribution with a volume fraction of 1 was observed, resembling the architecture of cortical bone (Fig. 1). Furthermore, low load zones (the two upper corners in Fig. 5) were removed by all methods except the Meso. beam, as shown in Fig. 7. Indeed, the structural level method produces geometries with a very low volume fraction in these zones, allowing large pores that intersect each other while still retaining some material, making it possible to see through the part.

## 3.4. *Physical three-point bending test*

### 3.4.1. *Beam samples manufacturing*

In this study, the optimized beams exhibit different geometries with various pores distributions. Due to the complex shape of these structures, an additive manufacturing process was used to ensure accurate beam production while accounting for the beam geometry. For this purpose, the Material Extrusion Plastic (MEX-TRB/P/PLA) process (polylactic acid material extrusion with thermal reaction bonding), as defined in the ISO/ASTM-52900:2021 standard and noted for its efficiency in material reduction, was chosen. In this work, all the beams were produced using MEX-TRB/P/PLA process on a Volumic® Stream30 Dual Mk.2 printer, operating at a print speed of 100 mm/s with a nozzle diameter of 0.6 mm. During the printing process, the layer thickness was set to 0.15 mm, while the nozzle and build plate temperatures were set to 235°C and 65°C, respectively.

### 3.4.2. *Three-point bending test*

A quasi-static three-point bending test on a Deckel Maho® DMU 50 eVolution machine was conducted to compare mechanical performance of the optimized beams (Fig. 7). This test was controlled in both position and speed. The tested beam was supported using two metallic cylinders spaced 130 mm apart and positioned 10 mm from the edges, as indicated in Fig. 8(b). Two 3D printed supports, used to ensure accurate positioning, were installed on a multicomponent dynamometer (Fig. 8), which measured the vertical force F. During the test, a gradual displacement δ was applied to the upper beam surface using a
cylindrical metal pusher (the point of contact), imposed in steps every 8s, as shown in Fig. 9(a). Thes steps, and the corresponding pauses of 8s, were applied to avoid viscous material effect on the measured beam stiffness, ensuring quasi-static conditions. Reaction forces were then recorded at a frequency of 30 Hz for 300 seconds using a KISTLER® 9257B force plate supporting the three-points bending setup. Corrections for drift and zero offsets were applied to avoid uncertainties arising from measurement instruments.

<figure><img src="image_11.png" /><figcaption>Fig. 8. (a) Three-point bending test apparatus (b) three points of contact and supports.</figcaption></figure>

3.4.3. *Experimental Stiffness*

An incremental normal displacement $\delta$ was applied to the pusher as illustrated in Fig. 9(a). The displacement was increased in increments of 0.05 mm up to a total of 1 mm to ensure that the test remained quasi-static, thereby avoiding any viscous material behavior. Throughout the three-point bending test, the normal reaction force *F* was recorded as a function of time, as shown in Fig. 9(b). At each displacement step, various picks were observed in the loading curve, attributable to the material's viscous behavior. To focus only on the elastic response of the material and maintain the quasi-static conditions of the test, it was necessary to allow the force to stabilize. Consequently, the force was measured once it reached a stable state. To minimize the viscous effects and ensure that only the static response was considered, an average force was computed for each displacement step $\delta_i$ over an 8 second interval. The resulting average force, $F_i$, within the stabilized zone (elastic domain), was calculated for each displacement step, as shown in Fig. 9(c). A best fit line was then conducted using the least-squares method to fit a linear function of the form, where *K* denotes the slope of the loading curve and $\delta_0$ represents the intercept of the force-displacement curve, corresponding to the experimental contact uncertainty (Fig. 9(c)). The stiffness for
each step, $K_i$, was calculated with Eq. (1) and illustrated in Fig. 9(d), following the approach described by Baumgart, E. (2000) [42].

$$K_i = \frac{F_i}{(\delta_i - \delta_0)} \qquad (1)$$

Finally, the mean stiffness, $K_{exp}$, was computed using all stiffness values according to Eq. (2), where $n$ is the number of displacement steps. The standard deviation, $\sigma_{exp}$, of $K_i$ was also computed (Eq. (3)) to quantify the variability of the measured stiffness values to provide an indication of the experimental uncertainty.

$$K_{exp} = \frac{1}{n} \sum_i K_i \qquad (2)$$

$$\sigma_{exp} = \sqrt{\frac{\sum (K_i - K_{exp})^2}{n}} \qquad (3)$$

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** (a) The applied normal displacement (b) the normal load as function of time (c-d) the computed average normal load as function of normal displacement.</figcaption></figure>

### 3.4.4. Mass measurement
Before mass reduction, all beams shared the same initial geometry, with a mass value $m_{initial}$ of 176.58 g, calculated by multiplying the CAD volume of the initial beam by the PLA material density $ρ_{PLA}$ of 1.308 $g/cm^3$ [38].

After mass reduction, the mass of the CAD-optimized beam is defined as the theoretical mass, $m_{theo}$. It is calculated by multiplying the reduced CAD estimated beam volume $V$ by the material density $ρ_{PLA}$ ($m_{theo}=ρ_{PLA} V$). Furthermore, the mass of the printed beam, $m_{print}$, is measured after 3D printing using a Precisa® 321LX scale with an accuracy of 0.1 mg. Consequently, the theoretical volume fraction, $ρ_{f theo}$, is defined as the theoretical mass of the beam divided by its initial mass (Eq. (4)) whereas the real volume fraction, $ρ_{f print}$, is defined as the mass of the printed beam relative to its initial mass (Eq. (5)).

$$ρ_{f theo} = m_{theo} / m_{initial} \qquad (4)$$

$$ρ_{f print} = m_{print} / m_{initial} \qquad (5)$$

Both theoretical and printed masses and volume fractions were calculated for all designed beams and are presented in Table 1. The printed beams show lower mass values, $m_{print}$, than the theoretical values, $m_{theo}$, as observed in Table 1. This mass difference, $Δm$, (Eq. (6)) reached -4% for the bio-inspired Meso. beam and -5.4% for the Macro. beam, as described in Table 1. This difference could be attributed to variations in the optimized beam geometry and internal porosity distribution.

$$Δm = \frac{(m_{print} - m_{theo})}{m_{theo}} \qquad (6)$$

<caption>**Table 1**</caption>

Theoretical and printed mass with beam's associated volume fraction.

<table><thead><tr><th rowspan="2">Mass reduction method</th><th rowspan="2">Beam</th><th colspan="2">Theoretical</th><th colspan="2">Printed</th><th rowspan="2">Δm (%)</th></tr><tr><th>m<sub>theo</sub> (g)</th><th>ρ<sub>f theo</sub> (%)</th><th>m<sub>print</sub> (g)</th><th>ρ<sub>f print</sub> (%)</th></tr></thead><tbody><tr><td>Topological level</td><td>Macro. beam</td><td>119.4</td><td>67.6</td><td>112.9</td><td>63.9</td><td>-5.4</td></tr><tr><td>Structural level</td><td>Meso. beam</td><td>110.7</td><td>62.7</td><td>106.3</td><td>60.2</td><td>-4.0</td></tr><tr><td>Multiscale</td><td>Multi. beam</td><td>116.6</td><td>66.0</td><td>116.5</td><td>66.0</td><td>-0.1</td></tr></tbody></table>
### 3.5. Stiffness to weight ratio: Relative stiffness

The stiffness to weight ratio has been widely used in the literature as a mechanical performance indicator to compare workpiece with different masses [43-44]. Indeed, in this study, each optimized beam exhibits different geometries and internal porosity distributions depending on the method used (Fig. 7), which could lead to small variations in the mass of the printed beams. Consequently, these variations in beam geometry and mass could affect properties such as structural resistance and stiffness. Therefore, to ensure an accurate comparison of the bending behavior of the optimized beams, the mass difference must be accounted for using a proportional indicator based on the stiffness to weight ratio. Thus, the relative stiffness, $K_{exp relative}$, was introduced in this study to allow performance comparison of the beams while maintaining the same stiffness unit (N/mm). The relative stiffness, or stiffness to weight ratio, is defined as the stiffness, $K_{exp}$, normalized by its volume fraction, $\rho_{f print}$, as given in Eq. (7) and following the approach described by Alshaer & Harland (2020) [45]. In the subsequent sections of this paper, the relative stiffness is used as the mechanical performance indicator to effectively assess stiffness of the beams.

$$K_{exp relative} = \frac{K_{exp}}{\rho_{f print}} \qquad (7)$$

In this study, topological level method was selected as a reference method to evaluate mechanical performance of the structural level method and the proposed multiscale method. Both experimental and relative stiffness values of the beams are presented in Table 2. The experimental stiffness, $K_{exp}$, is determined through three-point bending test, as described in Section 3.4.3 while the relative stiffness, $K_{exp relative}$, is computed following Eq. (7).

The relative stiffness of the printed beams as a function of the mass reduction method is illustrated in Fig. 10 with an error bar representing one standard deviation. The Multi. beam, which achieved the highest volume fraction ($\rho_{f print} = 0.66$), outperformed the other beams, with a relative stiffness of 5,795 N/mm ($\pm$ 85 N/mm). Compared to the Macro. beam, which has relative stiffness of 4,667 N/mm ($\pm$ 145 N/mm), the Multi. beam demonstrated a significant increase of 1,128 N/mm (24%), as seen in Fig. 10. Additionally, the Meso. beam's relative stiffness of 5,412 N/mm ($\pm$ 229 N/mm) exceeds the Macro. beam reference by 745 N/mm (16%).
<figure><img src="image_13.png" /><figcaption>Fig. 10. Experimental relative stiffness of the optimized beams.</figcaption></figure>

<caption>**Table 2**</caption>

Experimental relative stiffness values of the printed beams.

<table><thead><tr><th>Mass reduction method</th><th>Printed beam</th><th>K<sub>exp</sub> (N/mm)</th><th>K<sub>exp relative</sub> (N/mm)</th><th>Standard deviation (N/mm)</th></tr></thead><tbody><tr><td>Topological level</td><td>Macro. beam</td><td>2,982</td><td>4,667</td><td>145</td></tr><tr><td>Structural level</td><td>Meso. beam</td><td>3,258</td><td>5,412</td><td>229</td></tr><tr><td>Multiscale</td><td>Multi. beam</td><td>3,825</td><td>5,795</td><td>85</td></tr></tbody></table>

## 4. Finite Element Analysis (FEA)

Finite element simulations of three-point bending tests were conducted using CATIA V5 software to estimate the stiffness of the beams. Before the simulations, mesh convergence was verified on the Macro. beam reference by varying the mesh size and the number of elements using linear tetrahedral elements (Fig. 11). The beams were then meshed with elements of 1.5 mm size. The input values of beam's Young's modulus, density and Poisson's ratio were 2,750 MPa, 1.308 g/cm³, and 0.34, respectively.

A static linear load, *F*, of 5 kN, was applied on the center upper beam surface. The normal displacement distribution was determined under imposed stress and boundary conditions. The boundary conditions
applied consisted of a clamp at the left end of the beam and a surface sliding constraint at the right end, as illustrated in Fig. 4. The beam's stiffness, $K_{FEA}$ (in N/mm), was computed as the ratio of the applied force, $F$, to the maximum displacement, $\delta_{max}$, according to Hooke's law (Eq. (8)). The FEA relative stiffness, $K_{FEA relative}$, is defined as the ratio of the FEA beam stiffness, $K_{FEA}$, to its volume fraction, $\rho_{f theo}$, as given in Eq. (9).

$$K_{FEA} = F / \delta_{max} \qquad (8)$$

$$K_{FEA relative} = K_{FEA} / \rho_{f theo} \qquad (9)$$

As seen in Section 3.5, the experimental results highlight the bending stiffness performance of the multiscale method compared to the topological level. Finite element simulations of three-point bending tests were performed to better understand these results and to estimate the effect beam's internal structure on stiffness performance. To ensure the accuracy of the numerical predictions, a mesh convergence study was conducted on the Macro. beam reference (Table 3). It was found that a mesh size of 1.5 mm (131,635 elements) provides sufficiently accurate displacement predictions (Fig. 11(a)), while further refinement significantly increases computational cost (Fig. 11(b)) due to the geometric complexity of the beam.

<figure><img src="image_14.png" /><figcaption>Fig. 11. (a) Mesh convergence study results of displacement (b) CPU estimation time.</figcaption></figure>

The mesh was selected based on the maximum displacement error, $\delta_{max error}$, (Eq. (10 and 11)). The goal was to achieve an error below 2%, which is considered acceptable for the purposes of this study. As shown in Table 3, an element size of 1.50 mm results in a $\delta_{max error}$ of 1.990% when compared to the finest
mesh (element size of 0.5 mm, 1,187,444 elements), thereby satisfying the convergence criterion while maintaining a reasonable computational cost. Therefore, this mesh density was chosen for all subsequent simulations. These results confirm that the chosen mesh achieves an effective compromise between computational efficiency and numerical accuracy, with consistent convergence behavior observed in the calculated stiffness values.

$$ \delta_{max ref} = 1.8639 \qquad (10) $$

$$ \delta_{max error} = | \delta_{max ID} / \delta_{max ref} | \qquad (11) $$

In this expression, $\delta_{max ID}$, denotes the maximum displacement obtained for each mesh configuration identified by its mesh ID in the convergence study. The denominator, $\delta_{max ref}$, corresponds to the maximum displacement computed using the finest mesh (ID = 8), which has an element size of 0.5 mm and serves as the reference solution. The ratio therefore quantifies the relative error of each mesh compared to this finest resolution reference.

<caption>**Table 3**</caption>

Macro. beam mesh convergence study results.

<table><thead><tr><th>ID</th><th>Element size (mm)</th><th>Number of elements</th><th>Number of nodes</th><th>δ<sub>max</sub> (mm)</th><th>CPU time (s)</th><th>δ<sub>max error</sub> (%)</th></tr></thead><tbody><tr><td>1</td><td>3.50</td><td>53,378</td><td>12,377</td><td>1.6873</td><td>&gt; 1</td><td>9.475</td></tr><tr><td>2</td><td>2.50</td><td>63,143</td><td>18,889</td><td>1.7809</td><td>1</td><td>4.453</td></tr><tr><td>3</td><td>1.50</td><td>131,635</td><td>33,122</td><td>1.8268</td><td>6</td><td><strong>1.990</strong></td></tr><tr><td>4</td><td>1.25</td><td>183,389</td><td>43,762</td><td>1.8480</td><td>9</td><td>0.853</td></tr><tr><td>5</td><td>1.00</td><td>296,689</td><td>67,021</td><td>1.8429</td><td>20</td><td>1.127</td></tr><tr><td>6</td><td>0.75</td><td>519,232</td><td>117,193</td><td>1.8644</td><td>80</td><td>0.027</td></tr><tr><td>7</td><td>0.60</td><td>860,897</td><td>188,090</td><td>1.8568</td><td>200</td><td>0.381</td></tr><tr><td>8</td><td>0.50</td><td>1,187,444</td><td>261,632</td><td>1.8639</td><td>400</td><td>Ref.</td></tr></tbody></table>

Finite element simulations were then conducted on the Multi. beam and the Macro. beam reference using linear tetrahedral elements with mesh size of 1.5 mm, corresponding to 297,141 elements and 131,635
elements, respectively. Numerical results show better stiffness performance of the multiscale method compared with the topological level method. Specifically, the Multi. beam outperforms the Macro. beam, with a relative stiffness $K_{FEA relative}$ of 4,270 N/mm compared to 4,066 N/mm, reflecting a difference of 204 N/mm (5%). Fig. 12 presents the Von Mises stress distribution across the xz mid-sections of the Multi. beam. and the Macro. beam. As shown, the Multi. beam exhibits a more uniform stress distribution throughout the section, indicating efficient load transfer and reduced stress concentrations. However, the Macro. beam shows stress localization primarily at the upper and lower portions of the section, which may lead to premature yielding or localized deformation under bending loads. These observations not only confirm the higher stiffness performance of the multiscale method but also provide insight into its improved structural behavior under applied loads.

<figure><img src="image_15.png" /><figcaption>Fig. 12. Von Mises stress field (xz mid-section) under three-point bending test (a) Multi. beam (b) Macro. beam reference.</figcaption></figure>

## 4. Discussion

As seen in Fig. 7(a), the geometry generated using the topological level method presents a central cavity in the xz mid-section of the beam. A parallel between this cavity and the medullary cavity present in the center of long bones may be drawn [46]. As shown in Fig. 5(a), the horizontal central zone of the beam is
subjected to relatively low stress (but higher than in the two upper corners) which allows material to be
removed from the center, similar to the case of long bones in animals. This means that the cortical bone
tube architecture is preserved, as described in [46].

In this work, the Multi. beam geometry was generated by achieving a volume fraction of 0.65 of the initial
weight of the part. Here, the total lightening of the Multi. beam geometry (0.35) was achieved in two steps
using both topological level and structural level methods. In the first step, an intermediate geometry was
generated using the topological level method, applied on the initial geometry of the part by optimizing
material distribution at the macroscale (Fig. 6(b)). Consequently, a volume fraction of 0.75 was achieved
by removing unnecessary material from low-stress regions. In the second step, an additional volume
fraction of 0.1 was removed by creating a porous mesostructure in the part. It's worth noting that Nature
could exhibit lower volume fractions compared to topological or structural lightweighting, as done in this
study [30-33].

The multiscale method produced a geometry with a very interesting central section (Fig. 7(c)). Indeed,
similar to the topological level method, a parallel can be drawn with the long bones of animals. As
Houssaye shows in her works [47-48], relatively light animals can have an empty central cavity (Giraffa
camelopardalis), whereas heavier animals (Ceratotherium simum) must have trabecular structure in this
central part of their bones to resist the loads they are subjected to. This parallel can be drawn between the
topological level method (which cannot make a structure similar to trabecular bone) and multiscale
method (which combines structures resembling both trabecular and cortical bones).

Furthermore, a parallel can be made between the topological level method and the bone remodelling
process. The topological level method calculates the stress field at each iteration and determines whether
material needs to be conserved or removed. The topological level method presents only two types of
material structure: cortical bone and void, whereas the proposed multiscale method has three types of
material structure: cortical bone, trabecular bone, and void, making it more efficient. As seen in Section
3.3, the structural level method preserves unnecessary material in low loading areas, which may explain
why it is a slightly less efficient than the multiscale method. However, the Meso. beam shows an
improvement in structural stiffness compared to the Macro. beam with a stiffness difference of 753 N/mm
(16%) (Table1). Consequently, the proposed multiscale method exhibits a better stiffness to weight ratio
than either the topological or structural level method alone. The Multi. beam exhibited a relative stiffness
surpassing the two other beams, as shown in Fig. 10 and Table 2. The relative stiffness difference between
the Multi. beam and the Meso. beam is 383 N/mm (7%), likely due to the similarity in their architectures, which feature small porosities that enhance stress distribution and stiffness. However, Fig. 10 also highlights a significant difference in relative stiffness between the Multi. beam and the Macro. Beam (24%).

Finite element analysis, similar to experimental tests, highlights that the proposed multiscale approach provides higher stiffness performance than the topological level reference method. Indeed, finite element simulations conducted on the proposed Multi. beam and the Macro. beam reference, outline a better stress distribution for the Multi. beam due to its central material arrangement (Fig. 12). This can be explained by its internal geometry integrating both trabecular bone (small pores) and cortical bone, as seen in the Multi. beam sections presented in Fig. 7(c). However, the Macro. beam topology shows a large central cavity, leading to low stress distribution in the center of the beam, as observed in Fig. 12 and the mid-section in Fig. 7(c). As mentioned previously, a parallel can be drawn between these two geometries and natural bones. Lighter animals do not need to reinforce the center of their bones with trabecular bone, whereas heavier animals must reinforce the interior. For this reinforcement, which is located in an area of lower mechanical stress than the surface, a less dense material (trabecular bone) must be used, and only a multiscale approach enables this. This is observed in the porous wall at the centre of the Multi. beam, Fig. 7(c). These observations underscore the effective stiffness performance achieved by the proposed multiscale method, despite its complex architecture.

Ultimately, the multiscale method has demonstrated promising performance but presents three major limitations:

- The current implementation is limited to two scales, primarily due to constraints of existing manufacturing capabilities. To extend the method to additional scales, manufacturing processes that allow precise control over material anisotropy at the local level would be required.

- The proposed method involves a dual optimization process, which increases computational resource demands. However, the mesoscale optimization is non-iterative, and thus remains computationally efficient.

- The method proceeds in sequential steps: first, topological level optimization, followed by structural optimization. It would be valuable to quantify the potential gains achieved through iterative optimization to explore the possibility of finding a globally optimal solution.

**5. Conclusion**
This work proposes a mass reduction method inspired by the multiscale architecture of bone to enhance the stiffness to weight ratio of mechanical parts. In this method, two scales are considered: macroscale (overall bone shape, topological level) and mesoscale (trabecular bone structure, structural level). Subsequently, to assess the performance of the proposed multiscale method, a comparison was conducted with each of its constituent methods applied individually. The comparison was evaluated through three-point bending tests, which are representative of industrial requirements. These tests were carried out both experimentally and via finite element analysis. For this purpose, three optimized beam geometries were designed: one using the topological level method (Macro. beam), one using the structural level method (Meso. beam) and the last one using the proposed multiscale method (Multi. beam). These three beams were fabricated using 3D additive manufacturing, and their performance was tested experimentally.

Four main contributions stand out in this work:

- A bone bio-inspired multiscale mass reduction method (considering macroscale and mesoscale) is proposed to generate lightweight part using two optimization levels (topological and structural).

- The proposed multiscale method demonstrates greater efficiency than either the topological or the structural level method alone. In the three-point bending case, the multiscale method shows an improvement in stiffness to weight ratio of 24% compared to the topological level method.

- Finite element simulations confirm this trend, showing a better internal stress distribution for the multiscale method, which may explain its stiffness to weight ratio performance compared to the topological level method.

- More generally, this study shows that optimizing workpiece using multiple scales is more effective than single scale optimization. This result therefore tends to justify the multiscale optimization in bones.

This article introduces a method for reducing part weight without compromising mechanical performance. The approach is inspired by the resource efficient material distribution observed in endoskeleton bones and employs multiscale optimization techniques. By optimizing material usage in mechanical components, this method aims to minimize raw material and energy consumption, particularly in transportation sectors such as automotive and aerospace.

## Funding sources
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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