# Influence of geometric partitioning on the finite element analysis of a face-centered cubic (FCC) lattice cell under mechanical loading

Yann Julius Aba'a Ekoto

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Yann Julius Aba'a Ekoto. Influence of geometric partitioning on the finite element analysis of a face-centered cubic (FCC) lattice cell under mechanical loading. 2026. hal-05454496

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# Influence of geometric partitioning on the finite element analysis of a face-centered cubic (FCC) lattice cell under mechanical loading

Yann Julius Aba'a Ekoto<sup>a</sup>

<sup>a</sup> Laboratory of Materials and Mechanical Engineering, University of Reims Champagne-Ardenne, 51100 Reims, France

## Abstract

Finite element analysis of lattice structures proves very useful for predicting their behavior under mechanical stress. In this article, we propose a geometric partitioning of a face-centered cubic (FCC) lattice cell. From the resulting new geometry, which enables the hexahedral meshing of the entire cell, we perform a finite element analysis of its compression under quasi-static and dynamic regimes using the ABAQUS software. The results obtained are compared with those of the analysis of the same cell with a standard (unpartitioned) geometry and a default tetrahedral mesh. The results obtained with the two mesh types are quite similar, with only slight differences. However, it is worth noting that the hexahedral mesh (known for its higher numerical accuracy) of the cell generates nearly 8 times fewer elements and nodes than the tetrahedral mesh of the same size. Furthermore, the simulation calculation time with the hexahedral mesh is almost 7 times shorter in quasi-static regime and almost 10 times shorter in dynamic regime than with the tetrahedral mesh. The overall results obtained in this study demonstrate that the proposed geometric partitioning for the FCC lattice cell allows for a more precise and, above all, much faster finite element analysis. This geometric partitioning technique could therefore be a highly effective tool for improving finite element analysis of FCC lattice structures containing multiple cells.

**Keywords:** Lattices structures, Face-centered cubic (FCC) cell, Finite element analysis, Geometric partitioning, Hexahedral mesh, Tetrahedral mesh.
# 1. Introduction

Recent technological and industrial advances have given rise to new techniques for designing mechanical structures, notably additive manufacturing (known to the general public as 3D printing) [1-13], which allows for the layer-by-layer production of lightweight and strong parts, including lattice cellular structures [14-22]. These structures, formed by the repetition of an elementary cell in spatial directions, possess particular mechanical properties, including excellent energy absorption performance under mechanical stress [23-26]. This is why they are widely used in many fields, particularly in the automotive industry, to limit damage during impacts [27,28]. There are three main classes of lattice structures [29], the most commonly used being strut-based lattices [20-22] whose elementary cells consist of struts interconnected at junctions. To study their behavior under mechanical loads, numerous researchers have conducted experimental tests and finite element analyses on these structures [30-46]. Finite element analysis of lattice cellular structures, essential for predicting their behavior, requires prior geometric modeling. Once the geometry of the structure is designed, it must be meshed. There are two main types of mesh for three-dimensional mechanical structures: hexahedral meshing, which involves decomposing the structure into finite elements in the shape of hexahedra (with eight or twenty nodes), and tetrahedral meshing, which involves decomposing the structure into finite elements in the shape of tetrahedra (with four or ten nodes). Hexahedral meshing offers better numerical accuracy and lower computation time [47,48]; however, its generation is difficult, especially for structures with complex geometries and irregular shapes [49]. Tetrahedral meshing, while offering less numerical accuracy than hexahedral meshing and requiring more computation time, is easy to generate and is well-suited for structures with complex geometries, such as lattice structures. This is why, in most articles including finite element analysis of lattice structures, tetrahedral meshing (very often with ten nodes due to its better accuracy compared to four-node meshing) is generally used [50-52]. Therefore, finding a way to mesh lattice structures with hexahedra would represent a gain in numerical accuracy and computation time during finite element analyses.

There are various unit lattice cells, one of the best known being the face-centered cubic (FCC) cell, composed of struts forming a cube with two diagonals on each of its six faces. Several studies have been conducted on FCC lattice structures using finite element analysis with tetrahedral meshing [53-55]. In this study, we propose a geometric partitioning of a FCC lattice cell that allows the entire cell to be meshed with hexahedra. In Section 2, we describe the cell geometry and express its volume parameters as a function of its geometric parameters. In Section 3, we perform a finite element analysis of the compression of the cell, starting with
the standard geometry and then the partitioned geometry, under both quasi-static and dynamic regimes. The results obtained will be rigorously discussed. Section 4 will be dedicated to a summary conclusion of this study.

## 2. Geometrical design

The geometrical designs of the FCC cell (standard and partitioned) are illustrated in Fig.

1. Fig. 2 represents the partitioned geometry of the cell with its features.

The cell is cut from a cube of edge $\ell$. It is composed of five types of solids; 14 cube-shaped solids $S_1$ (represented in red) of edge $a$, 24 rectangular parallelepiped-shaped solids $S_2$ (represented in green), 24 triangular prism-shaped solids $S_3$ (represented in blue), 12 rectangular parallelepiped-shaped solids $S_4$ (represented in brown) and 24 rectangular parallelepiped-shaped solids $S_5$ (represented in violet). The geometric features of those five solids are shown in Fig. 3. Let $V_{S_1}$, $V_{S_2}$, $V_{S_3}$, $V_{S_4}$ and $V_{S_5}$ be the respective volumes of the five solids. The volume $V_{FCC}$ of the FCC cell can then be expressed according to the following relation:

$$V_{FCC} = 14V_{S_1} + 24V_{S_2} + 24V_{S_3} + 12V_{S_4} + 24V_{S_5}$$

Since the five solids have regular shapes, their volumes can easily be expressed in terms of their geometric parameters. We have:

$$V_{S_1} = a^3; V_{S_2} = \frac{a^3\sqrt{2}}{2}; V_{S_3} = \frac{a^3}{4};$$

$$V_{S_4} = \ell a^2 - (2 + \sqrt{2})a^3; V_{S_5} = \frac{\ell a^2 \sqrt{2} - (2 + 2\sqrt{2})a^3}{2}$$

Finally, the volume of the cell can be expressed according to relation (1) below:

$$V_{FCC} = -(28 + 24\sqrt{2})a^3 + (12 + 12\sqrt{2})\ell a^2 \quad (1)$$

We can also express the relative density $\bar{\rho}_{FCC}$ of the FCC cell which is the ratio of its volume and the volume $V_{Cube}$ of the cube in which the cell is cut. We have:

$$V_{Cube} = \ell^3$$

Thus:

$$\bar{\rho}_{FCC} = \frac{V_{FCC}}{V_{Cube}} = -(28 + 24\sqrt{2})\left(\frac{a}{\ell}\right)^3 + (12 + 12\sqrt{2})\left(\frac{a}{\ell}\right)^2 \quad (2)$$

So, thanks to this proposed geometric partitioning, it becomes very simple to obtain the analytical expression of the volumetric parameters of the FCC cell as a function of its geometric parameters.
# 3. Numerical simulations

## 3.1. Simulations data

In this section, we perform numerical simulations of the compression of a FCC cell using the ABAQUS (2018) software, under two regimes: quasi-static and dynamic. In both regimes, the cell is resting on a fixed rigid bottom plate (Fig. 4).

In the quasi-static regime, the cell is compressed by a rigid top plate moving vertically with a very small and constant velocity $v = 10^{-4}$ m. s$^{-1}$. The strain rate is $\dot{\epsilon} \approx \frac{v}{l} = 10^{-2}$ s$^{-1}$. Inertia effects are neglected. For this regime, the simulations will be performed with the ABAQUS/Standard solver. In the dynamic regime, the cell is impacted by a rigid top plate moving vertically with an initial velocity $v_0 = 1$ m. s$^{-1}$. The strain rate is $\dot{\epsilon} \approx \frac{v_0}{l} = 10^2$ s$^{-1}$. The initial distance between the bottom of the top plate and the top of the cell is $d_0 = 0.1$ mm. A concentrated mass is added to the top plate, so that its total mass is $m_{TP} = 12$ kg, and its initial kinetic energy is $\epsilon_0 = 6$ J. For this regime, the simulations will be performed with the ABAQUS/Explicit solver. In both regimes, the simulations will be performed with 12 CPUs and 12 domains.

In the ABAQUS/CAE meshing interface, the displayed color indicates whether a geometry, or part of a geometry, can be meshed with hexahedra. In figure **Fig. 5(a)**, the cell (with a standard geometry) is entirely colored orange, meaning it cannot be meshed with hexahedra and therefore can only be meshed with tetrahedra. Conversely, in figure **Fig. 5(b)**, the cell (with a partitioned geometry) is entirely colored green, meaning it can be entirely meshed with hexahedra. Thus, for the simulations we will perform, the standard geometry will be meshed with ten-node tetrahedral elements (C3D10 in ABAQUS/Standard solver and C3D10M in ABAQUS/Explicit solver), while the partitioned geometry will be meshed with eight-node hexahedral elements (C3D8). For both types of mesh, the overall size chosen is $\frac{a\sqrt{2}}{6} = 0.2357$ mm. **Fig. 6** illustrates the two types of mesh for the cell. The tetrahedral mesh generated 126890 elements and 201930 nodes, while the hexahedral mesh generated 15872 elements and 23890 nodes (**Fig. 7**). The two rigid plates are 20 mm x 20 mm x 0.5 mm blocks and have been meshed with hexahedral elements (C3D8) of size 1 mm.
The values of the geometric parameters of the FCC cell are $l = 10$ mm and $a = 1$ mm. The material chosen for the cell is made of a titanium alloy (Ti-6Al-4V) and obeys the Johnson-Cook law given by equation (3) [56]:

$$ \sigma = (A + B\varepsilon_p^n) \left[ 1 + C \ln \left( \frac{\dot{\varepsilon}}{\dot{\varepsilon}_0} \right) \right] \quad (3) $$

Where $\sigma$ is the flow stress, $\varepsilon_p$ is the effective plastic strain and $\dot{\varepsilon}$ is the strain rate. The values of the mechanical parameters of the material are [57]:

Density: $\rho = 4430$ kg.m$^{-3}$; Elastic Modulus: $E = 113$ GPa; Poisson's ratio: $\nu = 0.33$;

Yield stress: $A = 1098$ MPa; Plastic hardening parameter: $B = 1092$ MPa;

Plastic hardening exponent: $n = 0.93$; Strain rate dependent parameter: $C = 0.014$;

Reference strain rate: $\dot{\varepsilon}_0 = 1$ s$^{-1}$.

In quasi-static regime, the effects of velocity are neglected, and thus, the parameter C will be considered to be zero. Whether in the quasi-static regime or the dynamic regime, damage will not be taken into account. To avoid problems related to self-contact, we will ensure that the deformation along the direction of compression remains low (less than 10%). In quasi-static regime, we perform a compression of 8%. In both regimes, the contact between the cell and the two rigid plates have been modelled by a surface-to-surface contact described by a hard normal contact and a tangential contact with a friction coefficient $\mu = 0.1$.

## 3.2. Results and discussion

**Fig. 8** presents a comparison of the simulation times for the two mesh types and under the two regimes. We observe that in the quasi-static regime, the simulation time with the tetrahedral mesh is 26929 seconds, nearly 7 hours and 29 minutes, while with the hexahedral mesh of the same size, the simulation time is 3750 seconds, nearly 1 hour and 3 minutes; in this regime, the simulation time with the hexahedral mesh is therefore almost 7 times shorter than with the tetrahedral mesh. On the other hand, in the dynamic regime, we note that the simulation time with the tetrahedral mesh is 15509 seconds, nearly 4 hours and 19 minutes, while with the hexahedral mesh of the same size, the simulation time is 1448 seconds, nearly 25 minutes. In this regime, the simulation time with the hexahedral mesh is therefore nearly 10 times shorter than with the tetrahedral mesh. Thus, the proposed geometric partitioning allows for a much faster finite element analysis.
**Fig. 9, 10, 11 and 12** illustrate the distribution of the Von Mises stress and the distribution of the equivalent plastic strain in the cell in the quasi-static regime (at 8% crushing) and in the dynamic regime. In the dynamic regime, the distributions are shown at the instant when the velocity of the top plate reaches zero; that is, at approximately 7.94% crushing with the tetrahedral mesh and approximately 7.78% crushing with the hexahedral mesh. Overall, the deformation of the cell is the same in both regimes and with both types of mesh. The distributions are quite similar. The maximum stresses and plastic strains are located in the same areas, in a region in the middle of the vertical struts. In the quasi-static regime, the maximum stress is 1679 MPa with the tetrahedral mesh and 1532 MPa with the hexahedral mesh; a difference of approximately 8.75%. In this same regime, the maximum value of the equivalent plastic strain is 0.43 with the tetrahedral mesh and 0.37 with the hexahedral mesh; a difference of approximately 13.67%. In the dynamic regime, the maximum stress is 1745 MPa with the tetrahedral mesh and 1559 MPa with the hexahedral mesh; a difference of approximately 10.65%. In this same regime, the maximum value of the equivalent plastic strain is 0.415 with the tetrahedral mesh and 0.351 with the hexahedral mesh; a difference of approximately 15.42%.

The curves from the quasi-static regime simulations are shown in **Fig. 13**. In this regime, the normal force exerted by the top plate on the cell is equal in magnitude to the normal force exerted by the bottom plate. Therefore, we only show the evolution curve of one of these curves as a function of the top plate displacement in **Fig. 13(a)**, while **Fig. 13(b)** shows the evolution curve of the cell internal energy as a function of the top plate displacement. The normal reaction force evolution curves obtained with the two mesh types coincide perfectly from the initial position of the plate up to a displacement of approximately 0.1 mm. As we approach the point where the force reaches its maximum value, the curves begin to separate slightly. The maximum value of the reaction force is 8486.84 N with the tetrahedral mesh and 8655 N with the hexahedral mesh. This represents a difference of approximately 1.98%. The internal energy evolution curves of the cell with the two meshes coincide perfectly from the initial position up to a displacement of 0.2 mm, after which they diverge slightly. After 8% compression, the internal energy absorbed by the cell with the tetrahedral mesh is 5726.59 mJ and 5875.25 mJ with the hexahedral mesh; a difference of approximately 2.59%. These different values are summarized in **Table 1**.

The curves from the dynamic simulations are shown in **Fig. 14**. **Fig. 14(a)** and **14(b)** represent the respective evolution curves of the top plate displacement and velocity as a function
of time. These curves coincide perfectly from the initial instant until approximately 1.25 ms. The curves begin to separate slightly as we approach the point at which the top plate velocity becomes zero. This zeroing of the top plate velocity occurs after 1.73 ms with the tetrahedral mesh and 1.68 ms with the hexahedral mesh; a difference of approximately 2.89%. The maximum distance traveled by the top plate is 0.894 mm with the tetrahedral mesh and 0.878 mm with the hexahedral mesh; a difference of approximately 1.78%. Fig. 14(c) and 14(d) show the respective time-dependent curves of the kinetic energy of the top plate and the internal energy absorbed by the FCC cell. Here again, the curves coincide until the point at which the velocity of the top plate (and therefore its kinetic energy) reaches zero. The maximum energy absorbed by the cell is 5963.67 mJ with the tetrahedral mesh and 5958.48 mJ with the hexahedral mesh; a very small difference of approximately 0.08%. As soon as the velocity of the top plate reaches zero, the cell releases some of the energy it absorbed during the impact, and the top plate thus acquires kinetic energy that allows it to reverse direction. Fig. 14(e) and 14(f) respectively represent the curves showing the evolution of the top plate normal force and the bottom plate normal force as a function of the top plate displacement. These forces have almost the same shape, with the sole difference being an initial peak on the curves showing the evolution of the top plate normal force. Here again, the curves for the two meshes coincide until the point at which the force reaches its maximum value. The value of the initial peak is 1569.70 N with the tetrahedral mesh and 1382.36 N with the hexahedral mesh; a slightly significant difference of approximately 11.93%. The maximum value of the top plate force is 8980.07 N with the tetrahedral mesh and 9062.72 N with the hexahedral mesh; a difference of approximately 0.92%. Finally, the maximum value of the bottom plate force is 8977.47 N with the tetrahedral mesh and 9063.05 N with the hexahedral mesh; a difference of approximately 0.95%. These different values are summarized in Table 2.

We therefore realize that in both regimes, the results obtained with the two types of mesh are quite similar, with only minor differences. Ultimately, the proposed geometric partitioning for the FCC cell allows for a complete hexahedral mesh with which we obtain results quite close to (and even more precise than) those obtained with the tetrahedral mesh of the same size on the standard geometry cell. Moreover, the numerical results with the hexahedral mesh made possible by this geometric partitioning are obtained in much less time than with the tetrahedral mesh.
**4. Conclusions**

In this study, we proposed a geometric partitioning of a face-centered cubic (FCC) lattice cell. We then performed a hexahedral finite element analysis of its compression under quasi-static and dynamic regimes. The results were compared with those of the tetrahedral finite element analysis for the same cell with a standard (unpartitioned) geometry. The observations that can be made from all the results obtained in this study are as follows:

(1) The proposed geometric partitioning for the cell facilitates the determination of the analytical expression of its volumetric parameters (volume, relative density) as a function of its geometric parameters.

(2) This geometric partitioning allows the entire cell to be meshed with hexahedra, unlike the standard geometry which can only be meshed with tetrahedra; the numerical results obtained with this hexahedral mesh will therefore be more accurate.

(3) The numerical results obtained with the two types of meshes are quite close to each other, with very small differences.

(4) With the hexahedral mesh made possible by this proposed geometric partitioning, the simulation computation time is significantly shorter (7 times faster in quasi-static regimes and 10 times faster in dynamic regimes) than with the tetrahedral mesh obtained by default with the standard geometry.

Thus, the geometric partitioning proposed for the FCC lattice unit cell has a positive influence on the numerical accuracy and computation time in the finite element analysis of its behavior under mechanical compression. This partitioning will also allow for faster and more accurate simulations of matrix lattice structures (n × m × l) containing multiple FCC cells.
Figures

<figure><img src="image_2.png" /><figcaption>Fig. 1. Geometrical designs of FCC unit cell: (a) Standard; (b) Partitioned.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Features of the partitioned geometry.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(e)</figcaption></figure>

**Fig. 3.** Geometric features of the five types of solids that make up the FCC cell : (a) $S_1$ ; (b) $S_2$ ; (c) $S_3$ ; (d) $S_4$ ; (e) $S_5$.
<figure><img src="image_9.png" /><figcaption>**Fig. 4.** Physical configuration of the compression tests: (a) Quasi-static regime; (b) Dynamic regime.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 5.** Color indication of the possibility of a hexahedral mesh for the FCC cell: (a) Standard geometry; (b) Partitioned geometry.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 6.** Different meshes for the FCC cell: (a) Tetrahedral mesh; (b) Hexahedral mesh.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 7. Mesh features for the FCC cell.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 8. Calculation times.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 9. Contour plot of Von-Mises stress – Quasi-static regime: (a) Tetrahedral mesh; (b) Hexahedral mesh.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 10. Contour plot of Plastic equivalent strain – Quasi-static regime: (a) Tetrahedral mesh; (b) Hexahedral mesh.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 11. Contour plot of Von-Mises stress – Dynamic regime: (a) Tetrahedral mesh; (b) Hexahedral mesh.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 12.** Contour of Plastic equivalent strain – Dynamic regime: (a) Tetrahedral mesh; (b) Hexahedral mesh.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 13.** Quasi-static regime results: (a) Reaction force-top plate displacement curve; (b) Internal cell energy-top plate displacement curve.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(d)</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Fig. 14.** Dynamic regime results: (a) Top plate displacement curve; (b) Top plate velocity curve; (c) Top plate kinetic energy curve; (d) Cell internal energy curve; (e) Top plate force-top plate displacement curve; (f) Bottom plate force-top plate displacement curve.</figcaption></figure>
Tables

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

Results of quasi-static regime analyses

<table><caption>**Table 2**</caption><thead><tr><th>Mesh type</th><th>Maximum force (N)</th><th>Maximum energy absorbed (mJ)</th></tr></thead><tbody><tr><td>Tetrahedral</td><td>8486.84</td><td>5726.59</td></tr><tr><td>Hexahedral</td><td>8655.00</td><td>5875.25</td></tr><tr><td>Difference (%)</td><td>1.98</td><td>2.59</td></tr></tbody></table>

Results of dynamic regime analyses

<table><thead><tr><th>Mesh type</th><th>Maximum distance (mm)</th><th>Instant of velocity zeroing (ms)</th><th>Maximum energy absorbed (mJ)</th><th>Initial peak force (N)</th><th>Maximum top plate force (N)</th><th>Maximum bottom plate force (N)</th></tr></thead><tbody><tr><td>Tetrahedral</td><td>0.894</td><td>1.73</td><td>5963.67</td><td>1569.70</td><td>8980.07</td><td>8977.47</td></tr><tr><td>Hexahedral</td><td>0.878</td><td>1.68</td><td>5958.48</td><td>1382.36</td><td>9062.72</td><td>9063.05</td></tr><tr><td>Difference (%)</td><td>1.78</td><td>2.89</td><td>0.08</td><td>11.93</td><td>0.92</td><td>0.95</td></tr></tbody></table>
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