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Article

A Biomimetic Design Method for 3D-Printed Lightweight Structures Using L-Systems and Parametric Optimization

by
Monzer Al Khalil
,
Hadrien Belkebir
,
Nadhir Lebaal
,
Frédéric Demoly
* and
Sébastien Roth
ICB UMR 6303 CNRS, Univ. Bourgogne Franche-Comté, Belfort-Montbéliard University of Technology, 90010 Belfort, France
*
Author to whom correspondence should be addressed.
Appl. Sci. 2022, 12(11), 5530; https://doi.org/10.3390/app12115530
Submission received: 30 April 2022 / Revised: 24 May 2022 / Accepted: 27 May 2022 / Published: 29 May 2022
Browse Figures
Versions Notes

Abstract

:
Biological structures and organisms are determined and optimized to adapt to changes and constraints imposed by the environment. The multiple functionalities and properties exhibited by such structures are currently a source of inspiration for designers and engineers. Thus, biomimetic design has been increasingly used in recent years with the intensive development of additive manufacturing to deliver innovative solutions. Due to their multifunctional properties combining softness, high stiffness, and light weight, many potential applications can be seen in the medical, aerospace, and automotive sectors. This paper introduces a biomimetic design and geometric modeling method of 3D-printed lightweight structures based on L-systems generated and distributed along their principal stress lines. Numerical simulations and parametric optimization were conducted with three case studies to demonstrate the relevance and applicability of this method in adapting mechanical structures to various load cases as well as ensuring a proper stiffness-to-weight ratio.

1. Introduction

Nature exhibits many optimized structures in terms of properties and functions (i.e., light weight, flexibility, high stiffness, etc.) as survivors of natural stimuli and constraints over time. Learning from these biological structures allows their effective use in applications and provides new solutions for engineering problems in a sustainable manner. The latter statement falls under the broader paradigm of biomimicry. According to Benyus [1], biomimicry is an “innovation inspired by nature” or “the conscious emulation of nature’s genius”. To mimic complex structures and properties of biological materials, biomimetic design has been investigated in multiple application sectors such as aerospace [2], automotive [3,4], and medicine [5], to name a few. This research field and the inherent complexities of these structures have been deeply addressed over the last decade with the intensive development of additive manufacturing (AM) technologies. Since its invention in 1984 [6], multiple AM processes and techniques have been proposed to increase both design and fabrication freedoms in terms of shape complexity, hierarchical complexity, material complexity, and functional complexity [7], leading, for instance, to physical objects with the desired properties and functionality [8,9,10,11,12,13,14,15,16,17]. AM is also suitable for lattice structure design, exhibiting light weight and strength behavior, similarly to porous materials in nature like honeycombs and trabecular bone [8]. These structures are categorized as homogeneous periodic, homogeneous conformal, heterogeneous periodic, and heterogeneous conformal depending on the distribution of the unit cells [18,19,20,21,22,23,24,25]. Their mechanical behavior is determined by cell topology, geometry, orientation, and size design elements. There, to achieve performance objectives such as part stiffness or strength, these structures may be spatially tuned by altering the aforementioned variables. Although the homogeneous and periodic lattice structures allowed gains in reducing part weight, good mechanical performance can only be achieved via the consideration of coordinated load [26,27]. These kinds of material distribution for parts under realistic loads are inconsistent with the heterogeneous stress distribution. In contrast, natural organisms exhibiting lightweight structures, such as bones and wood, are non-uniform heterogeneous cellular, with cellular topology, geometry, orientation, and size tailored to the magnitude and orientation of the applied loads [28,29]. Stava et al. [30] used cyclic internal hollowing, local thickening, and support to optimize the structure for given loads and restrictions. Alzahrani et al. [31] developed a heuristic technique that totalizes the relative density information collected to automatically predict the diameter of the strut in the structure, under a variety of stress scenarios. As a result, replacing the solid structure with a non-uniform load-adapted heterogeneous lattice structure and fulfilling the design requirements of the part’s mechanical performance is a scientific challenge [32,33].
Developing biomimetic lattice structures becomes, therefore, promising to achieve lightweight stiffness performance. The varying spongy trabecular structures of bone and local tissue variation in seashells (pearl oyster) are illustrative examples of this aim [34]. In hierarchical structures, there is a gradient of increasing density in the radial direction from the interior spongy (trabecular) bone to the exterior compact (cortical) bone. This property results from a natural remodeling process and adaptation to mechanical loads described by Wolff’s law [35]. Several studies were based on the architecture of the bone, and some bone mimicking mechanical structures were implemented. Robles-Linares et al. [36] focused on the modeling of the cortical bone microstructure for AM and its characteristics in the body depending on several mechanical loads and constraints [37]. Banijamali et al. [38] described the effects of different loadings on the morphology of the trabecular bone. Design, simulation, characterization, and manufacturing of bone implants and prostheses, and femoral stems were also a large center of interest [38,39,40,41]. Daynes et al. [42] and Audibert et al. [43] also worked on the bone’s inner pores’ distribution that follows the principal stress lines’ (PSLs) trajectories and demonstrated the effectiveness of their biomimetic method in enhancing the properties and the weight of the structure. Tam et al. [44] investigated the fused filament fabrication (FFF) technique along PSLs to overcome the inherent anisotropy (direction-dependent properties) of AM. Teufelhart [45] evaluated the performance of a periodic structure with the flux of force-adapted structure with straightened struts, demonstrating that it can only handle axial force and has greater stiffness and strength.
Indeed, PSLs are orthogonal curves in which the tangent, at any point of any curve, has the direction of one of the principal stresses at that point [46]. They represent an appropriate description of the behavior of the structure, as they provide a wider visualization of the applied load effect on the object/structure. It particularly shows the lines of the material continuity within a design space, and they may derive from the classical Michell structural optimization of truss structures [47,48].
One of the main issues which can be faced is the appropriateness and the fabricability of PSLs in AM. Prior research works have successfully addressed the topology design method based on PSLs for AM of two-dimensional structures [43,49]. According to Li and Chen [50], PSLs do not depend on the scaling of the material stiffness and the applied load but rather on the design boundaries, the location and the degree of the force, and the boundary conditions. Moreover, PSLs represent a continuous field in which they can either cross multiple design boundaries or be fully periodic. These mechanical and geometric properties prove the usefulness of PSLs as AM printing paths [51]. Based on the aforementioned statements and the literature, PSLs can serve as guidelines or paths for material growth and distribution inside a structure. A similar strategy was developed by Kirk et al. [52] and Ulu et al. [53] to enhance structural performance of additively manufactured objects through build orientation and gradient paths.
Plant growth algorithms and theories developed by Lindenmayer and Prusinkiewicz are widely known as L-systems [54]. From a geometric point of view, L-systems are a formalism to model the development of growing linear and branching structures, from basal filamentous organisms to trees to entire plant ecosystems [55,56]. Efforts in this research area are mainly focused on their graphical representation and computation [57,58]. However, interesting studies showed root system models using L-systems and proved their effectiveness in estimating the shear strength of root–soil composites [59,60]. Such systems have provided promising results when coupled with topology optimization algorithms. For instance, Bielefeldt et al. [61] proposed a genetic algorithm that encodes design variables and governs the development of the structure. An interpreter is required to translate genomic information into structural topologies, therefore leading to adaptive structures capable of achieving multiple design objectives in light weight and increasing stiffness. L-systems have also been combined with cellular automaton to generate mechanical structures for multiple mechanical conditions [62]. Tree-like systems or fractals have also inspired engineers to develop heat exchangers and cooling systems. A review of fractal heat exchangers has demonstrated their big advantages in lowering the pressure drop and in maintaining a uniform temperature [63]. Then, the generalization of L-systems within a structure for objectives of light weight is a challenge. The utility of integrating L-systems inside mechanical structures can be validated via numerical simulations using different FEA software such as ANSYS Workbench and COMSOL Multiphysics. Indeed, numerical simulations of bioinspired mechanical structures and composites have proven the effectiveness of emulating nature in its aspects and shapes [64,65,66].
Despite all the research works based on L-systems in the aforementioned domains, there is still a lack in the application for biomimetic lightweight structures with high stiffness in AM. To address this challenge, this paper proposes a novel computational design method based on L-systems generated and distributed along the PSLs’ directions. By considering a rough design space, multiple load cases and boundary conditions are investigated. The resulting tree-like structures are studied through numerical simulations followed by a design of experiments (DOE). The latter scheme of parametric optimization utilized the branches’ thicknesses as two design variables for the models using the first and second PSLs directions separately. This method aims to find the optimal structure after conducting a series of simulations and sensitivity studies [67]. Thus, the effectiveness of using L-systems and PSLs in finding the best compromise between light weight and high strength can be confirmed.

2. Material and Methods

The research objective is to obtain optimized lightweight and stiff structures by a novel biomimetic method. Figure 1 describes an overview of the computational design method for AM of lightweight structures, which leverages a full control of the geometric definition. The latter is built upon the L-systems generated and distributed along PSL’s directions in a way to emulate the material growth and remodeling inside biological structures [51,68,69]. Although this method can be applied to all types of loading conditions such as flexion, torsion, and impact, this paper studies only the compression and shear cases. Three load cases were investigated in order to study different material growth behavior. The final steps of this method consisted of parametric optimization to determine the best material distribution. The resultant optimal 2D structure was then thickened for AM, leading to a 2.5D structure. To fabricate the structure, the material jetting process was utilized, especially the PolyJet technique using VeroWhite (Stratasys, Rehovot, Israel) material on a Stratasys Objet260 Connex3 machine [70]. The photosensitive VeroWhite resin has a Young’s Modulus of 2495 MPa, a yield strength of 49.9 MPa, a Poisson ratio of 0.38, and a density of 1174 kg/m3. The simulations conducted in this paper follow Hooke’s elastic law. Each step of the proposed method is further detailed in the next subsections.

2.1. Rough Design Space Definition

The initial rough design space—illustrated in Step 1 of Figure 1—consists of a 2D rectangular cantilever geometry (100 × 200 mm) as already chosen as classical examples in the literature [71,72]. This design space is clamped at its lower boundary and is subjected to three load cases as illustrated in Figure 2 to investigate three different behaviors of material growth: a compressive load (CL) of 60 N (Figure 2a), a shear load (SL) of 60 N (Figure 2b), and a mixed compressive/shear loads (CSL) of 60 N, respectively (Figure 2c).

2.2. PSLs Extraction

The next step of the bioinspired design method (Step 2 of Figure 1) aims to determine and extract PSLs exhibiting load-case-dependent trajectories [73,74]. In the 2D design space, only two PSL directions (denoted as first and second PSL directions) are significant. According to Daynes et al. [42], PSLs are free of shear stress since they are aligned with the principal stress trajectories. The numerical construction of the PSLs can be performed via the use of mathematical models [27,41,71]. In this research work, numerical simulations, using a solid mechanics formulation, are conducted on the 2D design space via the commercial software COMSOL Multiphysics® 4.3a (COMSOL Inc., Stockholm, Sweden). Using the load case and the boundary conditions expressed in Figure 2, PSL’s trajectories are computed in streamline forms (see Figure 3). The extraction of PSLs includes their points and vectors coordinates for further guiding L-systems computation.

2.3. L-Systems’ Generation along PSLs

L-systems are rewriting systems that use several strings with different designations [55]. Specific characters are used for the graphical interpretation using turtle graphics, i.e., each character or string commands the turtle to achieve an action. As a matter of example, the string F commands the turtle to draw a line of length l and angle α while string X commands the turtle to skip a line of length l and angle α. Among the numerous shapes and possibilities that L-systems offer, Table 1 shows the list of strings constituting the grammar used in the proposed geometric growth modeling (see Figure 4a). It is inspired by the context-free OL systems (denoted as an axial tree) illustrated in Figure 4b according to Lindenmayer and Prusinkiewicz [54]. In the proposed study, the production rule has been slightly modified to reduce the geometric complexity and to adapt symmetrical tree growth. Table 2 enumerates and describes each input used in this study.
To construct the aforementioned L-systems along the PSLs within the design space, an algorithm has been proposed in Figure 5. The first step consists in recovering the PSLs’ data (points and vectors coordinates) with the design space. In a second step, the intersection points between the PSLs and the design space boundaries are determined. At each intersection point, the first generation or recursion of the L-systems is initiated. The geometric growth is governed by the production rules of the L-systems (see Table 2 and Figure 4a) according to several recursions. At each recursion, the algorithm fits the L-systems to be aligned with the PSLs’ direction. The L-systems and PSLs’ lines are discretized into a precise number of points in a way to allow the algorithm to undergo neighboring points’ research. The number of points that discretize the different lines and L-systems is specified by the user: increasing the discretization of the L-systems and PSLs increases the fitting precision. The algorithm continues until the L-systems reach the boundary of the design space. However, in the process, a few branches either slightly surpass the design space boundary or float inside it. Hence, the very final step of the algorithm consists in extending the floating branches linearly until reaching the nearest design space boundary or the nearest neighboring branch of other L-systems. This process necessitates the calculation of the branch angle and direction to ensure its extension linearly. This step also includes trimming the branches that surpass the design space boundary. The algorithm stops when all the L-systems are connected and delimited by the design space boundaries. An implementation of the algorithm has been made within a Rhinoceros/Grasshopper environment via dedicated components developed in C# language.

2.4. Finite Element Analysis and Design Regions Decomposition

Numerical simulations are then conducted on the six L-systems structures via COMSOL Multiphysics® 4.3a software. A beam elements formulation is used for the calculations in this step of the proposed method. Similar to the 2D rough geometry, the L-systems’ structures are all clamped at their lower boundary and subjected to different loading types. For example, CL-1 and CL-2 are subjected to a compressive load of 60 N. Structures SH-1 and SH-2 are subjected to a shear load of 60 N, and structures CL-SL-1 and CL-SL-2 are subjected to a compressive load of 60 N and a shear load of 60 N (see Figure 6).
In order to reach the objectives of light weight and increasing stiffness by emulating natural aspects, it was necessary to decompose the structures into two regions (see Figure 7). The first region represents the tree’s trunk and the rectangular contour (in red), while the second region represents the tree branches (in blue). Beam elements with rectangular sections are used in these simulations. These sections present a thickness in the z-axis of 12.7 mm for both regions and are equal to the thickness used in the 2D rough design static and stationary simulation. Their thickness in the y axis is variable according to the region decomposition and presents three levels for each region. The thicknesses used for the first region vary between 1.5, 2, and 2.5 mm (denoted tFR1, tFR2, and tFR3, respectively) while those used for the second region vary between 1, 1.25, and 1.25 mm (denoted tSR1, tSR2, and tSR3, respectively).

2.5. DOE-Driven Parametric Optimization and Determination of the Optimal L-Systems Based Structure

This section consists of a study of the influence of region 1 and region 2 cross-sections, on both the weight and the von Mises stress. This study was conducted for each of the six L-systems models via a design of experiments (DOE). One can identify the individual and interaction impacts of many elements that might influence the output findings of the measurements using DOE. The latter can be also utilized to learn about a system, process, or product, and determine the ideal operating conditions. It may be used for a variety of research objectives, but it is especially useful in a screening study to assist in figuring out the most significant parameters. Therefore, DOE provides assistance in optimizing and better understanding how the most significant control parameters that affect replies or crucial quality features. In the case of the proposed study, the DOE presents two factors: the y-axis thickness of the first region and the y-axis thickness of the second region. Each factor presents three levels as described in Table 3. The effect of these two factors was then analyzed in function of the von Mises stress and the structure weight using an L9 Taguchi Orthogonal Array Design [75].

3. Results

The results of the numerical simulations via COMSOL Multiphysics® 4.3a, which were conducted on the 2D rough design space and the L-systems’ structures, are presented in this section. It is important to remind that the simulations conducted on the 2D rough design space used the solid mechanics formulation, while those carried out on the L-systems’ structures used the beam elements formulation. Table 4 shows the maximal von Mises stress and the weight results for each load case (CL, SL, and CSL, respectively), and Figure 8 illustrates the corresponding von Mises stress contours.
Figure 9 represents the implementation of the algorithm for the construction of L-systems along the PSLs directions within the Rhinoceros/Grasshopper environment. The L-systems’ components required all the parameters enumerated in Table 2 as entry inputs and had two outputs: The L-systems branches are represented by lines and the points of intersection of each line. The PSL component required one text entry: it was the file that contained all the data exported by COMSOL Multiphysics® 4.3a. It had one output, which was the assembled lines. The third component permitted gathering all the output data of the first two components as well as the design space geometry in B-Rep type. It also used two number sliders to allow the user to control the discretization of the L-systems and PSLs’ segments. The output of this definition was then the L-systems’ structures constructed and fitted along the PSLs’ directions. These structures, as described in the previous section, were used later for numerical simulations and parametric optimization using Taguchi L9 DOE.
Table 5 and Table 6 represent the L9 Taguchi Orthogonal Array Design consisting of two factors and three levels. Table 5 gathers the von Mises and weight results for the L-systems following the first PSLs’ direction (CL-1, SL-1, and CSL-1), while Table 6 is dedicated to the second PSLs’ direction (CL-2, SL-2, and CSL-2). The numbers 1, 2, and 3 correspond to the level of each factor as explained in Table 3. Each experiment in the DOE contains the corresponding von Mises stress and weight results.
Figure 10 shows the von Mises stress contour for Experiments 1, 3, and 9 in the DOE of Table 5 and Table 6 for each L-systems’ structure:
-
Exp. 1, where the thicknesses were at their lowest levels. The thickness of the first region was tFR1 = 1.5 mm and that of the second was tSR1 = 1 mm (see Figure 10 CL-1a, CL-2a, SL-1a, SL-2a, CSL-1a, CSL-2a).
-
Exp. 3, where both regions had the same thickness (level 1 for the first region and level 3 for the second), tFR1 = tSR3 = 1.5 mm (see Figure 10 CL-1b, CL-2b, SL-1b, SL-2b, CSL-1b, CSL-2b).
-
Exp. 9, where the thicknesses were at their highest levels. The thickness of the first region was tFR3 = 2.5 mm and that of the second was tSR3 = 1.5 mm (see Figure 10 CL-1c, CL-2c, SL-1c, SL-2c, CSL-1c, CSL-2c).
The main effect plot for the six structures is shown in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. The influence of each factor on the response function was analyzed by the main effect plot, respectively, for the maximum value of the von Mises stress (Figure 11a, Figure 12a, Figure 13a, Figure 14a, Figure 15a and Figure 16a) and the structure weight (Figure 11b, Figure 12b, Figure 13b, Figure 14b, Figure 15b and Figure 16b). For the main effect plot, the slope justified the effect of each parameter. Parameters having the highest inclination had a greater effect, while parameters having a horizontal plot had a minimal effect on the response function.

4. Discussion

Table 4 represents the numerical simulation results of the 2D rough geometry for the three load cases. It was noticed that von Mises stress was in the range between 0.043 and 0.173 MPa, which were values lower than the yield stress of the VeroWhite material (49.9 MPa). The weight of the structure was equal to 298.2 g. As mentioned before, this study aimed to find a better compromise between weight and strength. By integrating the biomimetic method of L-systems along the PSLs of the structure, it could be possible to reduce the initial weight in a way that the von Mises stress stays lower than the yield stress of the VeroWhite material. To validate the hypothesis, numerical simulations were conducted on six L-systems’ structures. The results of von Mises stress and weight were collected in the L9 DOE (Table 5 and Table 6).
Firstly, by analyzing the von Mises values in Table 5 and Table 6, it was noticed that some values were higher than the yield stress of the VeroWhite material. This was the case of the values in experiments 1 to 7 for the CL-1 configuration (Table 5), experiments 1 to 6 for the CSL-1 configuration (Table 5), and experiments 1 to 4 for the SL-2 configuration (Table 6). The following combinations are hereafter excluded and cannot be considered as optimal configurations:
-
For the CL-1 configuration, a thickness of 1.5 mm for the first region and thicknesses of 1, 1.25, and 1.5 mm for the second region cannot be used. Moreover, a thickness of 2 mm for the first region and a thickness of 1.5 mm for the second region cannot be employed together.
-
For the CSL-1 configuration, thicknesses of 1.5 and 2 mm for the first region cannot be used with any of the thicknesses of the second region (1, 1.25, and 1.5 mm).
-
For the SL-2 configuration, a thickness of 1.5 mm for the first region cannot be used with any of the thicknesses of the second region.
Furthermore, it was noticed that all configurations, including the ones that were excluded from the study, presented a weight value lower than that of the 2D rough structure (298.2 g). The weight reduction percentage is shown in Table 7 below. The weight reduction with respect to the weight of the 2D rough structure was in a range between 74.55 and 86.82%. Logically when the thicknesses of the first and second regions were at level 1 (1.5 and 1 mm, respectively), the weight reduction was the highest. Inversely, when the thicknesses of the first and second regions were at level 3 (2.5 and 1.5 mm, respectively), the weight reduction was the lowest.
Figure 10 represents the von Mises stress results for each load case and PSLs’ direction configuration. It represents particularly the following cases:
-
The case where both regions were at their lowest level 1 (1.5 mm and 1 mm for first and second region, respectively). This case is denoted as Case 1 (a in Figure 10).
-
The case where both regions had the same thickness (level 1 for the first region and level 3 for the second region). tFR1 = tSR3 = 1.5 mm. This case is denoted as Case 2 (b in Figure 10).
-
The case where both regions were at their highest level 3 (2.5 mm and 1.5 mm for the first and second regions, respectively). This case is denoted as Case 3 (c in Figure 10).
According to Figure 10, and as expected for the three load cases, the structures that had tree growth along the PSLs and the load direction provided a lower von Mises value than those that had growth orthogonal to the load direction. Until this stage of the study, one could predict that the CL-2 configuration was more suitable for the compressive load case, the SL-1 configuration was more adequate for the shear load case, and the CSL-2 configuration was better than the CSL-1 configuration for the compressive/shear load case.
In order to determine the optimal configuration for each load case, a sensibility study was conducted in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. For von Mises stress, the effect slope was descending. Figure 11a and Figure 12a illustrate for CL-1 and CL-2 configurations the effect of the thickness of the first and second regions on the von Mises stress. The effect of the thickness of both regions was high, but it was noticed that the thickness of the first region was more important than that of the second region. Figure 13a shows the opposite behavior than that mentioned before: for the SL-1 configuration, the effect of the thickness of the second region was higher than that of the first region. For SL-2, CSL-1, and CSL-2 configurations, Figure 14a, Figure 15a and Figure 16a show that the effect of the thickness of the first region was much more important than that of the second region. As for the weight, Figure 11b to Figure 16b show that the effect of the thicknesses of the first and second regions was merely the same with more importance on the effect of the second region’s thickness.
For the compressive load case, both CL-1 and CLN-2 showed the effectiveness of the method of integrating L-systems along PSLs’ directions in a structure in order to reduce its weight while keeping a reasonable stiffness and strength. However, it was noticed that the CL-1 structures presented higher von Mises values than CL-2 did. For a load of 60 N, the CL-1 configuration presented high von Mises values while for the CL-2 configuration, the von Mises stress stayed lower than the yield stress of the VeroWhite material. Even if the CL-1 structure presented lower weight values than that of the CL-2 structure for each experience in the DOE, the CL-2 structure fulfilled the objective of this study in a better way. The latter statement can be justified by the growth direction of the L-systems in the CL-2 configuration. Indeed, the direction of the branches was colinear to the compressive load direction, thus, the material distribution was closer to the zone where the load was applied, as shown in Figure 6b. Meanwhile for the CL-1 structure, the L-systems branches’ growth direction was orthogonal to the load direction, which explained the higher von Mises values (Figure 6a). That means that the structure CL-2 was better adapted to the compressive load case.
For the shear load case for both SL-1 and SL-2 structures, the effectiveness of the method of integrating L-systems along PSLs’ directions was again validated. SL-1 and SL-2 structures presented approximatively the same weight values, but the SL-1 structure had remarkably lower von Mises stress values. This difference could be justified by the L-systems’ growth along the shear load direction (Figure 6a). SL-1 is then the structure that was better adapted to the shear loads. Hence, the effectiveness of this method was also proved for the combined compressive/shear load cases. The same interpretation done for the CL configurations could be used for the CSL configurations: CSL-2 was better adapted for the combined compressive/shear load.
To summarize the analysis of the study of the effect of the first and second region thicknesses on the von Mises stress and the weight for the six structures, the following observations can be stated:
-
A thickness of 2 mm (level 1) for the first region and a thickness of 1 mm (level 1) for the second region gave a maximal von Mises stress value and a minimal weight (Exp. 1 in Table 5 and Table 6).
-
A thickness of 3 mm (level 3) for the first region and a thickness of 2 mm (level 3) for the second region gave a minimal von Mises stress value and a maximal weight (Exp. 9 in Table 5 and Table 6).
Finally, it was concluded that integrating L-systems generated along the PSLs’ directions in a structure reduced its weight and maintained a reasonable stiffness. The latter statement validated the hypothesis raised at the beginning of the discussion section. For the three load cases, CL-2, SL-1, and CSL-2 configurations were taken into account. Considering that for all combinations these configurations have shown low von Mises stress values, one could choose the minimal thicknesses levels as the parameters used for the optimal structures: tFR1 = 1.5 mm and tSR1 = 1 mm for CL-2 and SL-1 configurations. Meanwhile, for the CSL-2 configuration, a minimal thickness for the first region equal to tFR1 = 1.5 mm combined with any of three thicknesses of the second region could be taken into account. A CSL-2 structure presenting first region thickness of 1.5 mm and a second region thickness of 1.25 mm was considered. The DOE optimization technique then proved its effectiveness in determining structures having light weight and high stiffness. Further studies using different optimization algorithms will be conducted in order to obtain optimal structures with several variations of sections and thicknesses like the Moving Least Square (MLS) and the Kriging methods and sequential quadratic programming (SQP) techniques [73,74]. These three optimal structures were then additively manufactured by the material jetting process and more particularly with the PolyJet technique using VeroWhite material on a Stratasys Objet260 Connex3 machine (see Figure 17).

5. Conclusions

In this paper, a novel biomimetic design and modeling method based on L-systems distributed along the PSLs’ directions has been proposed. Numerical simulations and parametric optimization schemes based on an L9 DOE sensibility study were conducted to prove the effectiveness of this method in adapting mechanical structures to various loading cases as well as ensuring a good stiffness-to-weight ratio. The structures studied were all fixed at their lower boundary. Although the proposed method could be applied to any load cases (flexural, torsional, and impact), only three loading cases were considered: compression (CL), shear (SL), and combined compression/shear (CSL). Each load case presented two structures derived from the two significant PSLs’ directions. Indeed, the DOE sensibility study helped us identify the optimal structure for each load case:
-
For the compressive load case (CL-2), the weight reduction was estimated between 74 and 84% with respect to the initial rough geometry, and the maximal von Mises stress was equal to 5.61 MPa, a value much lower than the yield stress of VeroWhite material (49.9 MPa).
-
For the shear load case (SL-1), the weight reduction was in the same range as the one of the compressive load case, and the maximal von Mises stress was equal to 17.04 MPa.
-
For the combined compressive/shear load case (CSL-2), the weight reduction was between 75 and 84%, and the maximal von Mises stress was equal to 32.22 MPa.
The latter results of the conducted numerical simulations have demonstrated that this method helped reinforce the structure, thus obtaining high specific strength while reducing its weight. This study also highlighted the need of producing lightweight and stiff structures by AM since it was developed in a way to respect AM constraints and build orientations. In future horizons, this generative design method will be matured in terms of designing and optimizing 3D complex models and in conducting a combined parametric and topology optimization scheme. In addition, future work will focus on different loading conditions, especially impact study cases.

Author Contributions

Conceptualization, M.A.K., N.L., F.D. and S.R.; methodology, M.A.K., N.L., F.D. and S.R.; software, H.B.; writing—original draft preparation, M.A.K.; writing—review and editing, N.L., F.D. and S.R.; supervision, N.L., F.D. and S.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

This study did not report any data.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Figure 1. Overview of the proposed computational design method for AM of lightweight structures using L-systems and parametric optimization.
Figure 1. Overview of the proposed computational design method for AM of lightweight structures using L-systems and parametric optimization.
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Figure 2. The three studied load cases applied to a 2D design space: (a) compression (CL), (b) shear (SL), and (c) compression-shear (CSL).
Figure 2. The three studied load cases applied to a 2D design space: (a) compression (CL), (b) shear (SL), and (c) compression-shear (CSL).
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Figure 3. First (in blue color) and second (in red color) directions of PSLs for the three load cases: (a) CL, (b) SL, and (c) CSL.
Figure 3. First (in blue color) and second (in red color) directions of PSLs for the three load cases: (a) CL, (b) SL, and (c) CSL.
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Figure 4. (a) L-systems axioms and production rules used in this study based on (b) the context-free OL systems defined by Lindenmayer and Prusinkiewicz [54].
Figure 4. (a) L-systems axioms and production rules used in this study based on (b) the context-free OL systems defined by Lindenmayer and Prusinkiewicz [54].
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Figure 5. Flowchart describing the L-systems’ generation along PSLs’ directions within a 2D design space.
Figure 5. Flowchart describing the L-systems’ generation along PSLs’ directions within a 2D design space.
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Figure 6. L-system models studied in this paper: (a) CL-1, (b) CL-2, (c) SL-1, (d) SL-2, (e) CSL-1, and (f) CSL-2.
Figure 6. L-system models studied in this paper: (a) CL-1, (b) CL-2, (c) SL-1, (d) SL-2, (e) CSL-1, and (f) CSL-2.
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Figure 7. Design regions decomposition: region 1 (in red color) and region 2 (in blue color).
Figure 7. Design regions decomposition: region 1 (in red color) and region 2 (in blue color).
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Figure 8. von Mises contours for the three load cases: (a) CL, (b) SL, and (c) CSL.
Figure 8. von Mises contours for the three load cases: (a) CL, (b) SL, and (c) CSL.
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Figure 9. Grasshopper definition of the L-systems’ construction along the PSLs’ directions in the 2D design space.
Figure 9. Grasshopper definition of the L-systems’ construction along the PSLs’ directions in the 2D design space.
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Figure 10. von Mises contours of Experiments 1, 3, and 9 for each load case and PSLs’ direction.
Figure 10. von Mises contours of Experiments 1, 3, and 9 for each load case and PSLs’ direction.
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Figure 11. Study of the effect of the thickness of the first and second regions for the CN-1 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
Figure 11. Study of the effect of the thickness of the first and second regions for the CN-1 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
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Figure 12. Study of the effect of the thickness of the first and second regions for the CL-2 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
Figure 12. Study of the effect of the thickness of the first and second regions for the CL-2 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
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Figure 13. Study of the effect of the thickness of the first and second regions for the SL-1 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
Figure 13. Study of the effect of the thickness of the first and second regions for the SL-1 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
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Figure 14. Study of the effect of the thickness of the first and second regions for the SH-2 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
Figure 14. Study of the effect of the thickness of the first and second regions for the SH-2 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
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Figure 15. Study of the effect of the thickness of the first and second regions for the CSL-1 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
Figure 15. Study of the effect of the thickness of the first and second regions for the CSL-1 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
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Figure 16. Study of the effect of the thickness of the first and second regions for the CSL-2 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
Figure 16. Study of the effect of the thickness of the first and second regions for the CSL-2 structure. A and B refer, respectively, to the thickness of the first and second regions. (a) von Mises stress. (b) structure weight.
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Figure 17. Structures (a) CL-1, (b) SL-2, and (c) CSL-1 printed with the PolyJet technique on a Stratasys Objet260 Connex3 machine with VeroWhite material. All scale bars: 50 mm.
Figure 17. Structures (a) CL-1, (b) SL-2, and (c) CSL-1 printed with the PolyJet technique on a Stratasys Objet260 Connex3 machine with VeroWhite material. All scale bars: 50 mm.
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Table
Table 1. L-systems’ structure and related turtle commands.
Table 1. L-systems’ structure and related turtle commands.
L-Systems StructureTurtle Command
FDraw a segment of length l and angle α
XSkip a segment of length l and angle α
+Turn left
−Turn right
[Store the turtle’s current position
]Retrieve the turtle’s current position
Table
Table 2. Grammar of the L-systems used in the biomimetic design method.
Table 2. Grammar of the L-systems used in the biomimetic design method.
InputDescriptionSpecific L-Systems in the Study
AxiomStarting string for first recursionF
RuleProduction rule that generates the L-systemsF = F[+F][−F]F[+F][−F]F[+F][F]F[+F][−F]
nNumber of recursions2
lBranch lengthAdapted to the PSLs length
αBranch angleAdapted to the tilt of PSLs
Start PointSpecify the point where the L-systems’ generation startsAn extremity of the PSL on the design space boundary
DirectionSpecify the plane and the vector to indicate the L-systems’ generation directionAdapted to the direction of the PSLs
Table
Table 3. DOE levels.
Table 3. DOE levels.
LevelRegion 1 Thickness [mm]Region 2 Thickness [mm]
11.51
221.25
32.51.5
Table
Table 4. Simulation results of the 2D rough structure for each load case.
Table 4. Simulation results of the 2D rough structure for each load case.
Load CaseVon Mises [MPa]Weight [g]
CL0.043298.2
SL0.173298.2
CSL0.13298.2
Table
Table 5. L9 DOE of von Mises and weight for the CL-1, SL-1, and CSL-1 configurations.
Table 5. L9 DOE of von Mises and weight for the CL-1, SL-1, and CSL-1 configurations.
CL-1 StructureSL-1 StructureCSL-1 Structure
Exp.Region 1 Thickness LevelRegion 2 Thickness LevelVon Mises [MPa]Weight [g]Von Mises [MPa]Weight [g]Von Mises [MPa]Weight [g]
11191.4939.3017.0448.15119.8743.79
21273.3745.0212.1355.54118.3750.35
31361.4150.749.1162.93117.4856.91
42167.3044.7714.7254.3469.3649.63
52255.0150.4910.6661.7368.1856.20
62346.0256.218.0869.1267.4362.76
73151.1750.2512.9560.5345.6455.48
83243.2455.979.5367.9244.7062.04
93336.7461.687.3075.3144.0868.61
Table
Table 6. L9 DOE of von Mises and weight for the CL-2, SL-2, and CSL-2 configurations.
Table 6. L9 DOE of von Mises and weight for the CL-2, SL-2, and CSL-2 configurations.
CL-2 StructureSL-2 StructureCSL-2 Structure
Exp.Region 1 Thickness LevelRegion 2 Thickness LevelVon Mises [MPa]Weight [g]Von Mises [MPa]Weight [g]Von Mises [MPa]Weight [g]
1115.6148.8882.6648.0732.2247.37
2125.0057.2669.2556.1231.4754.75
3134.9665.6558.5164.1830.9962.14
4215.0354.0055.3053.3519.0453.31
5223.6562.3848.8661.4018.4160.70
6232.7970.7642.8969.4618.0068.08
7314.7759.1138.9558.6314.9859.26
8323.5267.4935.5766.6812.2966.64
9332.7375.8832.2974.7411.9374.02
Table
Table 7. Weight reduction percentage of the L-systems-based structures with respect to the weight of the 2D rough structure (298.2 g).
Table 7. Weight reduction percentage of the L-systems-based structures with respect to the weight of the 2D rough structure (298.2 g).
CL-1 ConfigurationSL-1 ConfigurationCSL-1 ConfigurationCL-2 ConfigurationSL-2 ConfigurationCSL-2 Configuration
86.8283.8585.3283.6183.8884.11
84.9081.3783.1280.8081.1881.64
82.9878.9080.9277.9878.4879.16
84.9981.7883.3681.8982.1182.12
83.0779.3081.1579.0879.4179.64
81.1576.8278.9576.2776.7177.17
83.1579.7081.4080.1880.3480.13
81.2377.2279.2077.3777.6477.65
79.3274.7576.9974.5574.9475.18
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Al Khalil, M.; Belkebir, H.; Lebaal, N.; Demoly, F.; Roth, S. A Biomimetic Design Method for 3D-Printed Lightweight Structures Using L-Systems and Parametric Optimization. Appl. Sci. 2022, 12, 5530. https://doi.org/10.3390/app12115530

AMA Style

Al Khalil M, Belkebir H, Lebaal N, Demoly F, Roth S. A Biomimetic Design Method for 3D-Printed Lightweight Structures Using L-Systems and Parametric Optimization. Applied Sciences. 2022; 12(11):5530. https://doi.org/10.3390/app12115530

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Al Khalil, Monzer, Hadrien Belkebir, Nadhir Lebaal, Frédéric Demoly, and Sébastien Roth. 2022. "A Biomimetic Design Method for 3D-Printed Lightweight Structures Using L-Systems and Parametric Optimization" Applied Sciences 12, no. 11: 5530. https://doi.org/10.3390/app12115530

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