1 Introduction

Recently the uses of additive manufacture have spread at many fields containing industries, where the weight of product is one of the main issues.

The motivation of this article is to find that specific arrangement which results the lightest support structure with identical mechanical properties. For this, first of all the existing structures should be analysed.

With the help of additive manufacturing (AM), which represents a promising and innovative technology [1], it is also possible to design complex geometrics, that is why, irregular structures can be used. On one hand irregular structures could provide more aesthetical customized products, such as arm chest and dresses [2]. On the other hand, irregular geometries improve some mechanical properties, for example anisotropic stiffness against specific directed loads [3]. The internal structure of an object can be homogeneous and heterogeneous [4]. In this research Voronoi tessellation—which is an irregular structure—was used, “since it can generate realistic homogeneous and heterogeneous structures” [4].

“Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbour rule: each point is associated with the region of the plane closest to it.” [5]

Voronoi structure provides amount of possibilities to control the material properties, for instance with this approach it is possible to create products with extreme elasticity [6], or with varying the topology perturbation, we can control compressive properties as plateau stress [7].

Typical application of Voronoi structure is Controllable Irregular Porous Scaffolds, which constructs artificial bones, and this is reaching the elastic modulus and the compressive strength of a natural ones [8].

In this area the most advanced researches [9, 10] are dealing with bone tissues and uses Voronoi structure for that. In the first step they create 3D point cloud from micro CT slide-images, after that based on these points - in Rhino3D software and Grasshopper plugin—3D Voronoi cells were created. The end of their method they got rounded shaped beam scaffold. Geometric properties, for example Porosity and Normalized Young’s modulus almost identical to the natural bone structure.

In this research we designed Voronoi structure, which is topologically similar to the previously mentioned method, however it is cell-based and it is a novel recent solution in the field of CAD technology. This Voronoi scaffold was compared to a regular beam lattice [11, 12] in the view of stress distribution, that is usually applied in AM [13].

1.1 Nomenclature

Table 1 presents the Nomenclature used in this article.

Table 1 Nomenclature
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1.2 Customization of cellular structure

In this project the spatial lines of the Voronoi scaffold base geometry were created with a cellular structure to reinforce the shell boundary. In general, the lattice frame consists of beams, which geometrically connected in spatial nodes. However, these cellular structures connect in straight edges (Fig. 1). The new method called Lightweight Voronoi Scaffold, or shortly LVS can be reviewed on Fig. 1.

Fig. 1
figure 1

Geometry comparison of Voronoi scaffold base geometry and regular beam lattice. This picture represents the geometric differences well (on one side, cellular-based walls, on the other side, connected beam structure)

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Since the properties of this cellular structure are unique, it has not fit well to the correctly published classifications (Fig. 2) [14, 15].

Fig. 2
figure 2

Cellular material classifications [14, 15]. There is no category for cellular-based Lightweight Voronoi Scaffold

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Figure 2 presents classifications of unit cell shape. In the first classification, the method presented in this article is hard to categorize. However, considering the structure shape, it is a random Voronoi shape, regarding the cell topology it is open. The cell geometry is not Cubic, Octet truss or TPMS (Triply Periodic Minimal Surface), that is why only the Other category fits to this LVS [15]. The second classification presents the categories in more detail [14]. Our method consists of these parameters:

  • Tessellation \(\rightarrow \) Stochastic \(\rightarrow \) Voronoi

  • Elements \(\rightarrow \) Surface \(\rightarrow \) Internal

  • Connectivity \(\rightarrow \) Edge

According to reviewed articles there is no solution for the combination of these categories, that is why LVS consider to be a novel method.

Beside of cell shape, beam lattice can be also an appropriate structure for inside reinforcement. With same load case and boundary conditions (in the case of the first end surface, all degrees of freedom were taken away, furthermore the second end surface was loaded multiply direction on the whole surface (Fig. 8), in regular and irregular beam lattice (Table 2A and B) the load path passes over on beams, and causes high local stresses. Nonetheless, in cellular structure, the load path passes along the edges as well, that results better load distribution and lower local stresses. That is why nowadays there are many studies published dealing with regular and irregular cellular structures (Table 2C and D). Summarising the regular beam structures are easier to generate by mathematical algorithms, and the resulting geometry is easy to control with parameters as well. In contrast, the irregular cellular structure is hard to handle concerning to the generation and parameter control.

Table 2 Beam and cell solutions for regular and irregular structures
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Moreover, researchers had shown an optimization process for the strength-to-weight ratio (with Table 2, D) [16], however the method of this study brings a new type of structure by further minimalizing material usage.

Figure 3 presents stress distribution in a specific cell. Here the load transfer realized along the edges, in this case all of the loads remain in planes of cell walls. Because of the fact that no significant stress rises on the center area of the walls, they can be removed. The result of this solution is not only a better material utilization, but it also meets the requirements of SLS and SLA technologies. In these technologies the remaining raw material must be removed from inside of part, and since here the cells are opened, therefore this solution meets this requirement. Thus it can be said that the LVS is compatible with most additive manufacturing procedures because of its open-cell structure. This open structure enables removing the additive material or support material after building the scaffold. Generally, generating the LVS is identical to other manufacturing methods. It requires a closed volume for the seeds of Voronoi cells. Currently, the generation is far from the commercial solutions because of its experimental state.

Fig. 3
figure 3

Stress distribution in case of solid and lightened cells. The centre area is loaded with negligible stress. (a) Cellular structure (b) Stress distribution of a selected solid cell (c) Similar stress distribution of a cell with removed centre area

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However, there are studies, that already use the open cell phrase, but some cases the shape of these structures are not cellular. For instance, the application of T-splines [17] results similar structures than open cells (Fig. 4). The T-splines require different calculation methods than open cells. The T-splines have usually meshed with 3D elements, while the open cells can be simplified by 2D shell elements in the preparation phase of finite element studies. The cross-sections of these structures are also different. The T-splines have an ellipse-like section. The open cells have traditional sections similar to open beams.

Fig. 4
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Comparison of T-spline (a) and open cell structure (b). The cross-sections are significantly different

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Fig. 5
figure 5

Wireframe based mesh model represents a typical T-spline

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The creation method of this kind of structures is totally different from the cellular scaffolds. Figure 5 shows the development of wireframe-based mesh model, with geometry smoothing by surface subdivision [18]. The cellular structures are based on cell 2D walls instead of 1D wire lines.

It is important to clarify what we consider an open cell structure compared to an internal structure consisting of beams (Table 3).

Table 3 Differences of beam- and open cell structures
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In the next chapters, Voronoi based cellular structures were used that were further modified by cutting out the centre of solid cell walls in order to minimalize material usage. An example of that process is presented in Fig. 8

2 Modelling and characterization

The goal of this phase is to compare the LVS and the Regular beam lattice (RBL), with the same boundary geometry. Here selected general freeform geometry is a transition between an ellipse and a circle (Fig. 6a) in order to gain complex boundary conditions and resulted stress distribution in the finite element simulation.

2.1 Pre-processing of geometry

2.1.1 Generation of LVS

In our research we applied similar method as the Boolean intersection of the patient bone geometry [10]. However, LVS is based on cellular structure not on beam elements.

Figure 6 presents the process, that has five main stages and for the realization of those, Rhinoceros 5 SR 14 and Grasshopper 0.9 0076 software versions were used.

Fig. 6
figure 6

Generation of LVS. Randomly generated cells are selected inside the originally imported geometry

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  1. 1.

    Step: Defining and importing a freeform model as a 3D file. The height of the model is 77.5 mm, and the size of the basic ellipse is 30 mm by 40 mm. (a)

  2. 2.

    Step: Placing seed points randomly in a Boundary Box. (b) Seeds were applied in a geometry from 40 seed points to 90 seed points.

  3. 3.

    Step: Dividing Voronoi cells according to the seeds. In this phase, the generated cells placed to equal distance from neighbouring seed points. (c)

  4. 4.

    Step: Surface cuts proportionally remove the center area of the resulted cell walls with NURBS (Non-Uniform Rational B-Splines) with identical approximate parameter. Identical means the same percentage of an approximate distance of the spline knot point to the control polygon on each cell wall. (d)

  5. 5.

    Step: Selection of the internal areas of the original geometry from the Boundary Box. The method selects the inside walls bounded by original surfaces out of the walls included the bounding box. (e)

The (f) picture shows the result of the pre-processing. After we imported it into the PTC Creo 4 CAD software, all the wall thickness was set at 1 mm.

2.1.2 Generation of RBL

PTC Creo regular square (S), triangular (T), hexagonal (H) and octagonal (O) types of feature were applied (Fig. 7) with the following initial parameters:

  • Beam diameter 1 mm (as a characteristic value)

  • Centre ball diameter 3 mm (to provide manifold attachment of beams)

  • Edge length from 9 to 20 mm (to cover the whole design space)

Fig. 7
figure 7

Generation of RBL. The highlighted beams can be placed in regular (S)quare (T)riangular (H)exagonal or (O)ctagonal arrangements. The number of these elements depends on the length of the edges. The number of these components are depending on the length of edges

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2.2 Pre-processing of simulation

The finite element calculations are processed by the PTC Creo Simulation module. This module is based on a polynomial mesh. Increasing the polynomial order, the P-Mesh is highly adaptive to the geometry. The single-pass adaptive method was applied by calculating the stresses and the deformation. This method always calculates two phases. After the first phase, the mesh is automatically modified at the singular points, ensuring that the second phase of the calculation is going to be mesh independent.

ABS (Acrylonitrile butadiene styrene) type material (\(\nu \) = 0.3; E = 2.4 GPa; \(\rho \) = 1.06 g/\(cm^3\)) was given to the geometry, which was assigned only the purpose of comparison. Since the simulation is based on ABS material, therefore this common thermoplastic polymer is suggested for fabrication.

Two filled end-surfaces were placed in the model (Fig. 8). In case of the first end surface, all degrees of freedom were taken away, the second end surface was loaded with −100 [N] in the direction of x, y, z distributed on the whole surface. The LVS walls have a multidirectional orientation, therefore this complex geometry has notable resistance against the multiaxial loads.

The simulation has run solid B-REP geometry in both cases (LVS and RBL).

Fig. 8
figure 8

Pre-processing of simulation. The components of displacement at the left end-surface are fixed. The force load at the right end-surface highlighted in orange

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The method of creating LVS is presented in Fig. 9. As we reviewed the cellular material modelling methods [19], we decided to use a hybrid solution by combining the function based BRep (Boundary Representation) and CSG (Constructive Solid Geometry) methods.

Fig. 9
figure 9

The final step of LVS method is based on CGS subtraction of the generated lightened walls from the original solid geometry. (a) Original solid geometry (b) Volumes of thickened Voronoi cells (c) Finalized LVS with cut-outs

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In this method the volumes of Voronoi cells were cut-out from the initial solid geometry. These result hollow Voronoi supported structure. As a novel feature, the wall placements are based on randomly generated seed points. The walls divide the inside volume applying the Voronoi principle.

2.3 Simulation of method

2.3.1 Simulation of LVS

The seeds of LVS geometry was generated with Monte Carlo method, that is, “in essence, the generation of random objects or processes by means of a computer” [20].

As the first stage of this research, 600 geometries were generated, and from these, 178 models resulted invalid, non-manifold geometry. Although the non-manifold models have proper geometric representation, these models have improper details for manufacturing. This issue is originated from inauspicious locations of specific walls caused by random point seeds. That is why 422 simulations were considered.

In order to monitor the results, it is essential to convince that the resultant displacement (d) [mm] and the total mass (m) [g] values are between the limits of entire solid (ES) and empty shell geometries (Table 4). The resultant magnitude is the vectorized sum of the displacement-components in the default (world) coordinate system.

Table 4 Analyses of the limits of displacements (d) and total masses (m)
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After the analyses process, 4 from the simulations should be excluded, since the total mass values are almost the same than the case of ES. To find these simulations, the next inequality was used:

$$\begin{aligned} \frac{|m_{ES}-m_{LVS}|}{m_{ES}} < 10\% \end{aligned}$$
(1)

In Table 4, limits of LVS simulations can be also seen, which contains the maximum and minimum result of d and m.

2.3.2 Simulation of RBL

The same simulation was applied on four arrangements (square, triangular, hexagonal, octagonal), where the size of these geometries were varied from 9 to 20 mm. This way, all the LVS sized comparable variations of the regular beam support structures were covered. The geometry (like the wall thickness of bounding surfaces and the diameter of supporting beams) was aligned to the LVS models. The characteristic dimension was 1 mm in both cases (LVS and RBL simulations). The results of the FEA confirm this effort well. The PTC Creo software uses P method to mesh the original part’s geometry. In this case, the generated elements are larger than the H method, but these elements cover the freeform surfaces much better than the H elements. A usual H element has linear properties and planar bounding geometry. Against a P element has higher-order bounding geometry, which adapts well to the original surfaces of the meshed part. To avoid the unwanted effect of mesh size, the “Single-Pass Adaptive” method was used in simulations. With this method, after the first simulation, the mesh is automatically corrected to eliminate the local singularities in stress distribution. After this correction, the result of the second run was included in this paper.

Although the internal support geometry was regular, sometimes the connection with the outer shell was irregular and in some further cases it was non-manifold as well. These connection errors made the meshing process impossible, that is why some displacement values were interpolated from the neighbouring results.

In total 50 geometries were analysed and compared to the LVS results, which is going to be demonstrated in the next chapter.

3 Results and discussion

For the comparison of RBL and LVS first of all, those should be normalized in the values of mass and displacement at the same time.

In this case of bounding geometry, the mass of empty shell (\(m_{min}\)) was the minimum (0%), while the mass of ES geometry (\(m_{max}\)) was the maximum mass (100%). All of the mass (m) values were normalized (\(m_n\)) on this range.

$$\begin{aligned} m_n=\frac{m-m_{min}}{m_{max}-m_{min}} \end{aligned}$$
(2)

Considering the displacements, the displacement of ES geometry (\(d_{min}\)) was the minimum (0%), while the displacement of empty shell (\(d_{max}\)) was the maximum displacement (100%). All of the considering resultant values (d) were normalized (\(d_n\)) on this range.

$$\begin{aligned} d_n=\frac{d-d_{min}}{d_{max}-d_{min}} \end{aligned}$$
(3)

Because of Monte Carlo Method large number of geometries were analysed (Fig. 10), however for comparison purpose the lightest ones were selected among same displacement magnitude values.

Fig. 10
figure 10

Distribution of mass and displacement results of the Monte Carlo Method

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In Fig. 11 the selected normalized displacement values can be seen. Mass of LVS was compared to the mass of RBL (triangular, square, hexagonal, octagonal types), and it turned out that LVS has the lightest structure. Concerning the results, from 3 percent up to 15 percent of reduction is going to be realized depending on the base geometry of the original lattice structure.

Fig. 11
figure 11

Normalized mass as a function of displacement shows the comparison of LVS and different RBL types. The masses of LBS based models are lower than the concurrent RBL type models in each case. The average stress error is less than ±0.6% of maximum stress

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Although for this research only Monte Carlo Method was used which resulted in random arrangement of cells, the trend of total mass clearly shows that LVS has lighter construction at least with a few percentiles compare to the others. It is great result, since nowadays the analysing of weight reduction is essential in many field, for instance in aviation or automotive industry. As an example of that passengers are going to be charged according to their weight when they fly with certain airlines, since studies have shown that each 1% of weight loss results in around 0.75% of fuel consumption reduction [21].

The analysis of stress distribution is also essential. Figure 12 shows that LVS loaded very similarly as RBL by using the same load pass (according to Fig. 8), that encourages to continue this research.

Fig. 12
figure 12

Comparison of RBL and LVS with von Mises stress distribution [MPa]

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4 Future research

Considering the future research, varying more parameters, such as wall thickness and the relative sizes of cut-outs would cause better results, since these have a major effect on the thermal and mechanical properties. As a matter of course, the beam support structure can be further improved. This paper only discusses the totally regular beam structure. By varying some parameters of the geometry, the stiffness of the support is changing as well. Figure 13 introduces an internal beam support structure with varying density of beams. Further research will be the comparison of this wall-like LVS with beam-like supports with advanced varying geometry.

Fig. 13
figure 13

Variable beam structure with variable density. The beam diameters are changing from 0.8 to 1.5 mm

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In this case, the beam diameters are changing from 0.8 mm to 1.5 mm. The result of the FEA represented with an asterisk (* mass Var RBL). The mass of varying structure differentiates well from the results of totally regular beam structures. This result is not better than the comparable mass of LVS. With further optimization of geometry, the mass might have decreased value, but the manufacturability might be a new issue as well. Optimizing the connection of Voronoi cells manufacturability would increase in additive case. Although, there is technology, for instance DLP (digital light processing) where is not need to use support structure, which enable to print our model of 3D Voronoi tessellation [22]. As the next step, manufacturing the physical model is necessary to validate the printing characteristics and prosperous mechanical properties.

5 Conclusion

This research is dealing with interior support structures in order to find the arrangement that provides the lightest solution. For this, the state of the art Voronoi tessellation has been used.

At the beginning it is important to compare different types of lattices. Our conclusion is that cellular structures are basically better than the common beam lattices. For the reason to reach the lightest structure, cut-outs were applied in the center of the cells, since no significant stress loads on that area.

For the simulation a freeform geometry was defined, and we introduced the Lightweight Voronoi Scaffold in five main steps (2.1.1. chapter), and regular beam lattice versions were used as a reference. In the comparison same boundary condition and resultant displacement magnitude was utilized, and it results that LVS has lower mass than in the cases of regular beam lattices.