Assessing Tetrahedral Lattice Parameters for Engineering Applications Through Finite Element Analysis

Abstract

Minimizing weight while maintaining strength in components is a continuous struggle within manufacturing industries, especially in aerospace. This study explores how controlling the dimensions of the geometric parameters of a lattice yields ideal mechanical properties for aerospace-related applications. A previously developed Bubble-mesh based computational method was used to generate a novel type of tetrahedral lattice that allows for the manipulation of three geometric parameters: cell size/density, strut diameter, and strut intersection rounding. Topology optimization and lattice generation within components are typical methods used to decrease weight while maintaining strength. Although these are robust optimization methods, each have their faults. Highly topology-optimized components may fail under unexpected loads, and lattice generation within commercial software is often limited in its ability to create ideal lattices with controlled geometric parameters, resulting in lattices with repeating unit cells. In this study, we used finite element methods (FEM)-based compression tests on latticed cubes with various parameter combinations to determine the best balance of lattice parameters. The results showed that strut diameter and strut intersection rounding were the best parameters to control to maintain strength and reduce weight. This understanding of the lattice structures was then applied to two aerospace components: a jet engine bracket and an airplane bearing bracket. By applying tetrahedral lattices with specified strut diameters and strut intersection rounding, the weight of the jet engine bracket was reduced by 51.8%, and the airplane bearing bracket was reduced by 20.5%.

Introduction

Three-dimensional (3D) lattices are micro-structures consisting of a network of nodes, struts, or thin walls. The effectiveness of lightweight lattice structures that retain strength is apparent from their prevalence in naturally occurring structures, such as bone structures¹ (Fig. 1a) and beehive honeycomb structures² (Fig. 1b). The goal is to maximize the structural strength with minimum material.

FIG. 1.

Lattice structures in nature: (a) a human bone,¹ and (b) honeycomb of a beehive.² Color images are available online.

Additive manufacturing (AM) enables the use of these complex, optimized shapes in engineering applications. When considering lattices within metal components, their high surface area, stiffness, and lightweight properties allow for multiple applications seen in biomedical implants, energy absorption, thermal control, and aerospace components.^3–5 Powder Bed Fusion AM processes, such as electron beam melting, are common methods to create these parts. Raw material in powder form, usually the titanium alloy Ti-6Al-4V (Ti64), is placed under a vacuum and fused together from the heat by an electron beam.^6,7 Ti64 is a popular metal used in AM, especially within the aerospace industry, and allows for strong lightweight components to be additively manufactured.

The need for lightweight components that can withstand high mechanical loads is necessary for a gamut of engineering applications. Multiple methods are used to achieve this and the process tends to begin with using finite element analysis (FEA) to determine where to remove excess material based on resulting stresses and deformations.^8,9 Topology optimization is a common tool within many commercial FEA software packages, as it iteratively optimizes a geometry based on the loading and boundary conditions to maximize its performance and generate new designs with less material.

Lattice optimization or generation is similar to topology optimization, but instead of directly removing material, lattice structures are implanted in regions of the geometry to reduce the weight. There have been many studies that examine the strength of different lattices, both computationally and experimentally. However, most studies examine the mechanical properties of lattices that are generated from CAD packages with a repeated pattern of an individual unit cell.^5,10–13

In this study, tetrahedral lattices are generated through a computational method that manipulates geometric parameters such as (1) cell size/lattice density, (2) strut diameter, and (3) rounding the diameter of strut intersections. These parameters were examined, because a continuous patterned lattice decreases the weight of a given component, whereas a controlled lattice allows for increased strength and additional weight loss by understanding how varying the geometric parameters of the lattice affects its strength. The lattices are generated within cubes of various geometry parameters and undergo compressive tests. The study was performed in the elastic region of the material because when considering aerospace industry applications, most components operate below the yield point, meaning within the elastic region.¹⁴ From the results of the study, we gained holistic insights on which parameters are ideal for balancing lightweight and stiffness. This insight is then applied to reduce the weight of a jet engine and airplane-bearing bracket.

Related Work

In the design for additive manufacturing (DFAM) field, optimization and lattice generation are common tools to generate high strength-to-weight ratio parts. In general, there are two typical types that have been widely developed in DFAM areas: topology optimization and lattice optimization.

Topology optimization

Topology optimization is a mathematical method that focuses on the optimal material distribution within a given domain. It is based on the loading and boundary conditions, and other specified constraints, to maximize the performance of a mechanical component. Several commercial software packages, including ANSYS, Autodesk, and Altair, provide efficient topology optimization solutions. Generally, the method used for topology optimization in commercial software packages is to iteratively optimize a design space under a given set of loads combined with certain boundary conditions. It aims at finding high-stress regions and distributing material to those areas. Therefore, the usual truss-like structures developed from the software are typically robust and effective.¹⁵ However, issues arise with a topology-optimized component when it experiences loads outside of what it was optimized for. Highly optimized components can fail in unexpected situations, and therefore, it is better to maintain the overall stiffness of a component rather than focus specifically on its loads. It is for this reason that lattices and cellular structures have an advantage over topology optimization, especially when the uncertainty of loading is high. Due to their multi-directionality and repetitive structures in the x, y, and z directions, lattices and cellular structures have the ability to withstand forces from indefinite directions.⁸

Lattice optimization and generation

Inspired by natural biological cellular structures, lattice optimization focuses on maximizing structural stiffness by generating suitable cellular patterns. Lattice optimization determines where to replace filled sub-volumes with a cellular pattern under specific loading and boundary constraints. Studies such as Beyer and Figueroa¹⁰ provide a solid foundation of experimental design and analysis to understand the behavior of different lattice structures under compression and bending loads. Wu et al.¹⁵ propose a method to conform to both the principal stress directions and the boundary of the optimized shape, which is a reliable solution in the two-dimensional (2D) planar and 3D volumetric domains. However, both studies examined lattices with a repeated unit cell pattern and did not really explore the potential of varying the geometric parameters of the lattice.

The following studies propose several similar techniques to realize structural optimization but differed by exploring geometric changes to the lattice geometry and applying them for AM. Li et al.⁸ present an example of cellular structures by generating variable-density gyroid-based structures with specific mechanical properties. Wang et al.¹⁶ propose a novel solution for natural frequency optimization by utilizing a homogenization-based topology optimization method to develop a honeycomb structure. Dong and Zhao⁹ present a joint stiffening element concept that represents the influence of the joint stiffness of the lattice structure. Finally, Zhang et al.¹⁷ propose an efficient technique for lattice optimization by creating a variable-density hexagonal cellular structure. These studies explored limited geometric changes to the lattice structure (such as grading the lattices or altering the strut diameter) and focused on repeated unit cell lattices under given loading or boundary conditions. They lacked the control of multiple geometric parameters to best fit among applications.

Most of the methods and studies for lattice optimization and generation discussed earlier offer strong but limited optimization approaches. Many of the lattice optimization studies do not explore the potential of controlling various geometric parameters to optimize the lattice. Some of the lattice mechanical property studies described earlier used commercial CAD software to generate tetrahedral lattices through a continuous pattern of the individual unit cells. Although studies show that tetrahedral lattices have superior compressive strength and effectively reduce the weight of components,³ a global parameter controlled tetrahedral lattice offers further weight reduction and overall strength, regardless of the loading conditions.

Proposed Methods

The proposed method creates a tetrahedral lattice structure with a tetrahedral mesh-generation technique called Bubble Mesh¹⁸ and an implicit surface generation technique called metaball.¹⁹ The method is able to control the cell size, strut diameter, and strut intersection rounding, to ensure that components are lightweight and maintain stiffness.

Generation of tetrahedral lattices

The computational method inputs a target geometric domain and creates a tetrahedral lattice as the output in the following steps, which is also illustrated in Figure 2:

FIG. 2.

The lattice generation method (a) imports a geometry, (b) uses the Bubble Mesh method¹⁸ to create a tetrahedral mesh and extracts the edges of the mesh to define the location of the lattice nodes and struts (elements are separated to visually show the volumetric tetrahedral mesh), and (c) once the strut diameter and strut intersection rounding is defined, exports a latticed model. Color images are available online.

• 1.Creates a tetrahedral mesh around the geometry using the Bubble Mesh method¹⁸ and extracts the edges of the mesh to define the location of the lattice nodes and struts
• 2.Define the strut diameter and strut intersection rounding

The first step is to create a tetrahedral mesh that fills the target geometric domain. The mesh is based on a tetrahedral meshing method called Bubble Mesh where the mesh density is locally controlled.¹⁸ This method generates a tetrahedral mesh by taking in a scalar function of position as part of the input, which controls the cell size locally. The scalar function defines the cell size for the mesh. The cell size is user-defined and determines the density of the lattice. Spherical bubbles are packed inside the volumetric domain, and diameters of the bubbles are controlled by the given scalar function. The mesh element nodes are then placed at the centers of the bubbles, which are connected to form a tetrahedral mesh. The edges of the triangular elements, as seen within the tetrahedral mesh in Figure 2b, become the struts of the tetrahedral lattice. The struts are connected at the nodes of the tetrahedral mesh. It is noted that the Bubble-Mesh method can generate an anisotropic tetrahedral mesh. However, since this study does not require an anisotropic mesh, the Bubble-Mesh method is used to create an isotropic mesh with controlled element size.

In Step 2, the user specifies strut diameter and strut intersection rounding. The units of both these parameters are based on the units of the imported geometry. The strut diameter and intersection rounding are simple numeric inputs, but the strut intersection rounding is done by using implicit surface generation techniques from Blinn¹⁹ to utilize metaballs. Metaballs are spheres that can create smooth transitions between 3D objects. In this article, a sphere of a given diameter is placed at the node or center of a strut intersection, and it blends with the struts to round the intersection. The size of the metaball diameter creates different levels of curvature at the strut intersections.

The program extends the struts of a latticed region outside of the envelope of the original geometry in each direction by the distance of the strut radius. An example of this can be seen in Figure 3. It is for this reason that there is an option in the program to cut off excess geometry or end the struts at the volumetric boundary of the original geometry.

FIG. 3.

An example of how the computational method extends the lattice struts beyond the geometry envelope. Color images are available online.

This lattice generation method differs from other conventional methods due to its utilization of bubble packing to generate a locally sized tetrahedral mesh around a given geometry. The method creates ideally spaced nodes with respect to the given mesh sizing and, thus, avoids ill-shaped elements created when two nodes are too close or where the density of the nodes is less than ideal.¹⁸ More control over the mesh is possible, and the locally sized tetrahedral meshing leads to a suitable mesh throughout the geometry. The mesh is directly correlated to the tetrahedral lattice that is formed, and it is for this reason that the final lattice structure is unique.

Relative density relationship to lattice parameters

To understand the structural effects of altering the different geometric parameters of the lattice structure, we examine the relative density of our generated lattices. The relative density is defined by:

$${\rho }_r=\frac{{\rho }_l}{{\rho }_s},$$ (1)

where ${\rho }_r$ is the relative density, ${\rho }_l$ is the density of the lattice structure (ratio of the mass of the lattice structure divided by the volume the lattice structure would occupy), and ${\rho }_s$ is the density of the solid material. This definition is meant for physical components and does not properly estimate how the cell size, strut diameter, and rounding the strut intersections impact the relative density of the tetrahedral lattice.

Gibson and Ashby have developed equations relating this definition of relative density (equation 1) to mechanical properties such as the elastic modulus.^20,21 Although their work is relevant, it is outside the scope of this study as most studies that use their equations use repeating unit cell lattices (namely Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) lattice structures) and use the Gibson and Ashby equations by experimentally determining the values needed to find the relationship between relative density and mechanical properties.^5,13 This is understandable, as Ashby²¹ states that for 2D cellular structures, the equations can be used to analytically determine the relative density and mechanical property relationship. It is far more difficult to do the same relative density analysis for a 3D cellular structure, because the response is an average of cell walls in random orientation in space. It is better to rely on experiments to analyze 3D cellular structures.²¹ Therefore, a relationship between the three parameters and relative density was defined to better estimate a wider range of relative densities. Other studies reflect similar relationships for the relative density as well.^3–5,8–13

We apply lattices to multiple 30 × 30 × 30 mm cubes, vary the parameters of cell size/density, strut diameter, and the rounding of strut intersections, and finally measure the resulting relative density. To ensure the dimensions are the same for each cube, the cutoff option within our computational method is used to remove excess geometry. We focus on a range of relative density values between 0.1 and 0.9, as this range is most useful for engineering applications. Ultimately, the relationship between cell size, strut diameter, and rounding with relative density can be used to estimate the weight of the component.

To determine this relationship, we vary the diameters of the metaballs from 0 to 8 mm in increments of 2; cell sizes 5, 10, 11, 13, 15, and 17 mm; and the strut diameter from 0.5 to 3 mm. Relative density values below 0.1 and above 0.9 that do not fit the correlation were removed, because a relative density value of 1 is a solid geometry. Figure 4 displays a biharmonic surface plot of the data.

FIG. 4.

This biharmonic surface plot relates the different geometric lattice parameters to the volume ratio of latticed cubes, $V_L∕V_S$, also considered the relative density. SD/CS, strut diameter and cell size ratio. Color images are available online.

As the strut diameter and cell size ratio (SD/CS) increase, the volume ratio between the lattice and solid cube, $V_L∕V_S$, essentially the relative density, increases and is estimated with a linear fit. As the metaball diameter increases, the relative density increases slightly. This is also evident when both parameters are correlated with relative density. Table 1 displays the results of applying a Pearson correlation between the differing parameters; SD/CS and metaball diameter have 0.745 and 0.125 coefficients of correlation when graphed with relative density, meaning that SD/CS has a greater impact on the relative density. Figure 4 complements these results, as a clear correlation is seen between SD/CS and the volume ratio. However, when the metaball diameter nears impractical values of more than 10 mm, the relative density increases dramatically. Based on these results, it is inferred that decreasing the strut diameter or increasing the cell size decreases weight.

Table 1.

Results of Performing a Pearson Correlation Between the Different Parameters

	SD/CS	Metaball diameter	$V_L∕V_S$
$V_L∕V_S$	0.745	0.125	1

SD/CS, strut diameter and cell size ratio.

The results from Figure 4 could be represented as a 2D plot and as illustrated in Figure 5, due to the SD/CS ratio having a larger effect on relative density, multiple linear fits and their equations are formed by keeping the metaball diameter constant. It can be seen that as the metaball diameter values increase, the slope between SD/CS and relative density decreases. This is because as the metaball diameter increases in size, it occupies more volume. As the volume increases, so does the relative density. The SD/CS ratio still governs the relative density, but smaller values of the ratio are needed to reach higher relative densities as the metaball increases. It must also be noted that once the metaball diameter nears 8 mm, it is large enough that for almost all cell sizes and strut diameters, the relative density is near 1 and has the smallest slope. In addition, the deviation from the linear fit is also much larger compared with other metaball diameter values, particularly when both cell size and strut diameter are either very large or very small. Ultimately, using metaball diameters from 2 to 6 mm was ideal for 30 × 30 × 30 mm cubes. To generalize this, the ideal metaball size for a cube ranges from being equal to or three times the strut diameter. A metaball diameter of 8 mm was still examined for the latticed cubes, but these results show that this metaball diameter size was too large for the 30 × 30 × 30 mm cubes and reiterates that the strut diameter dictates the ideal metaball diameter size.

FIG. 5.

Displays multiple linear fits and their equations that describe the relative density as a function of SD/CS. The linear fits are based on the metaball diameter. Color images are available online.

FEA procedure

The tetrahedral lattice is applied to multiple cubes of 30 × 30 × 30 mm, each having a parameter of the lattice altered while keeping the other two parameters constant to analyze the structural integrity of the tetrahedral lattice. The different lattice parameter combinations can be seen in Figure 6 and Table 2. It must be noted that the cutoff option to remove excess geometry is not utilized to alleviate altering the stiffness results in the compression tests. In addition, the relative density for the latticed cube with a metaball diameter of 8 mm (MBD-8mm) is estimated to be a little higher based on the correlation data from Figure 5.

FIG. 6.

The examined latticed cubes with different parameters.

Table 2.

Breaking Down of the Geometric Lattice Parameter Combinations That Were Tested

Name	CS (mm)	SD (mm)	Intersection rounding: MBD (mm)	Relative density
CS-12	12	2	0	0.230
CS-16	16	2	0	0.125
CS-20	20	2	0	0.062
SD-2mm	16	2	0	0.125
SD-3mm	16	3	0	0.282
SD-4mm	16	4	0	0.438
MBD-4mm	16	2	4	0.196
MBD-6mm	16	2	6	0.328
MBD-8mma	16	2	8	0.814

^a MBD-8mm relative density is most likely a little higher.

CS, cell size; MBD, metaball diameter; SD, strut diameter.

To simulate the deformation of the tetrahedral lattice structures, we perform FEA by using the ANSYS software. The latticed cubes for the compression tests are created by using the commercial CAD software, SOLIDWORKS, to make the 30 × 30 × 30 mm cube. It is then converted to an STL file and imported to the tetrahedral lattice generating program. Once the program generates the lattice within the cube, similar to other studies,^4,5,11,12 two plates are modeled on the top and bottom of the latticed cube in the y-direction to imitate a typical Instron compression machine (Fig. 7). The material properties of the lattice structure are based on the data sheets of Arcam Ti64 powder, as shown in Table 3. It is noted that these simulations did not consider typical defects that occur during the AM process, such as internal porosity and thermal warpage. The work primarily focused on computationally analyzing the bulk properties of the lattice structures.

FIG. 7.

The finite element analysis model used in compression simulations for lattice mechanical property analysis. Color images are available online.

Table 3.

Material Properties of Arcam Ti64 Powder^23–26

Arcam Ti64 powder material properties
Yield strength	950 MPa
Ultimate tensile strength	1020 MPa
Elongation	14%
Modulus of elasticity	120 GPa
Tangent modulusa	125 GPa
Poisson's ratioa	0.31

^a Some values were not stated in the Arcam Ti64 data sheet, so values were based on nonpowder Ti64 from other sources.^24–26

The boundary conditions on the plates are meant to imitate a physical compression experiment based on ASTM standards.²² The top plate moves only in the y-direction for both tests; first, a 1000 N load is applied, and then it is lowered at a constant rate of 0.02 mm/s. The bottom plate is fixed in all directions, and no boundary conditions were applied to the side of the lattice models. In addition, the two plates are rigid, and the contact between the lattice and the two plates was set to bond with automatic surface-to-surface contact. A bonded contact was selected, as it does not allow for slipping, therefore resulting in higher stresses and deformations compared with a friction-based contact. This was done to consider a worst-case scenario for the lattices.

Defining Lattice Mechanical Properties

A 1000 N load test was used to gain an initial idea of the behavior of each lattice parameter. The model from Figure 7 displays how the load was applied from the top plate. In addition, it was used to ensure that mesh sensitivity would not alter the results and use these data to determine the final mesh size when compressing the different lattices at a constant rate.

The compression test for each latticed cube followed the configuration seen in Figure 7 and was similar to the 1000 N load test; however, the top plate was displaced downward at a constant rate of 0.02 mm/s. This test helped to better understand the mechanical properties of the different lattice parameters. Stress–strain and force–displacement plots were created to examine how varying the lattice parameters alters the elastic modulus and stiffness when compared with solid Arcam Ti64 powder. The max von mises stress and strain were used to create the stress–strain plot. The max displacement and reactionary forces due to the lowered plate were used to create the force–displacement plot. Again, the elastic region was examined in the stress–strain plot, as the focus of this article is to determine the ideal lattice parameters for aerospace–industry applications. The data seen in the following plots are when the stress reached the yield strength of Arcam Ti64 powder, 950 MPa. The relationship between stress and strain is defined by the elastic modulus, the material property that measures the stiffness of a solid material,²⁷ E, defined by:

$$E=\frac{\sigma }{\epsilon },$$ (2)

where $\sigma$ is the stress and $\epsilon$ is the strain. It is essentially the slope of a stress–strain plot. Similarly, the stiffness values were found by considering Hooke's Law and the relationship between force, P, and displacement, ΔL, is known as the stiffness, K. The stiffness is the extent to which a solid body will resist deformation from an applied force.²⁷ It is defined as follows:

$$K=\frac{P}{\Delta L}.$$ (3)

1000N load results

The displacement and stress results for each lattice configuration are shown in Figures 8 and 9. It must be noted that the smaller the mesh size, the greater the number of elements. Similar to what is seen in FEA simulations,^28,29 the mesh size did not affect the displacement results. It can be seen that increasing the lattice strut diameter best limits displacement. In addition, rounding the intersections also keeps the displacement relatively low, but it does not vary as much compared with altering the strut diameter and cell size.

FIG. 8.

1000 N Load displacement results for each lattice parameter. Color images are available online.

FIG. 9.

1000 N Load stress results for each lattice parameter. Color images are available online.

The stress results in Figure 9, however, are mesh sensitive, meaning they did not converge when changing strut diameter and cell size. The lattices that varied with cell size received the most stress, whereas the lattices with different metaball diameters for rounding intersections sustained the least, but maintained consistent stress values (less than 5%). Similar to other findings and studies,^9,28,29 this phenomenon is due to the fact that lattices without intersection rounding have stress concentrations at the strut intersections, which leads to stress increasing as the mesh element size decreases.

The 1000 N load results indicate that, to develop high strength-to-weight ratio components, the ideal parameters to change are strut diameter and intersection rounding radius. Decreasing the cell size definitely increases strength because of the higher density of lattices, but it is not worth the additional weight. Based on this study, when comparing the max stress at the mesh element sizes of 1 and 3 mm, most of the latticed cubes had max stress values within a percentage difference of 10%. Therefore, we concluded that a mesh size of 3 mm would suffice to perform the computationally expensive simulation of compressing the lattices at a constant rate.

Lattice compression at constant rate results

Based on the results from the 1000 N load test, a mesh size of 3 mm was used for all simulations in this section. The general trend seen within Figures 10 and 11 is that the larger the strut or metaball diameter, or the smaller the cell size, the stiffer the lattice, and the closer the elastic modulus of the lattice would be to the elastic modulus of Arcam Ti64 Powder (120 GPa). This agrees with past studies,^5,9,10,12,13 because the elastic modulus measures the stiffness of a solid material and stiffness is a solid body's resistance to deformation under an applied force. The elastic modulus is material dependent, where stiffness is dependent on the material and the object's structure and boundary conditions. Therefore, once the lattices grow closer to being a solid model, their elastic modulus grows closer to the elastic modulus of the material, and the lattice becomes stiffer. Inversely, when the lattice deviates from becoming a solid model, the elastic modulus and stiffness decreases. This is seen through the weaker lattices, for example, CS-16 and SD-2mm, as they begin to plateau before reaching 950 MPa. It must also be noted that on average, rounding the intersections with metaballs had higher elastic modulus and stiffness values. These results show that rounding the intersections can greatly increase the strength of the lattice. Overall, although the strut diameter and cell size are the dominant parameters that control weight, for ideal lightweight and stiffness, the cell size should remain high to best reduce weight and altering both the strut diameter and intersection rounding should be done to maintain strength.

FIG. 10.

Stress–strain plot of latticed cubes with various parameters. Color images are available online.

FIG. 11.

Force–displacement plot of latticed cubes with various parameters. Color images are available online.

When considering how the elastic modulus and stiffness each relate to the relative density, Figures 12 and 13 illustrate their relationships. It must be noted that Figure 12 examines the effective elastic modulus, $E_L∕E$, which is the ratio of the elastic modulus of the lattice to the elastic modulus of the solid material.

FIG. 12.

Effective elastic modulus ($E_L∕E$) versus relative density. As relative density increases, so does the elastic modulus. Color images are available online.

FIG. 13.

Stiffness (K) versus relative density. As relative density increases, so does the stiffness. Color images are available online.

In general, it is seen that as the relative density increased, so did the stiffness and elastic modulus. This was expected, because it reflects what was seen in Figures 8 and 9 and aligns with other studies.^5,9,10,12,13 However, it was not expected to see such a large difference in behavior for CS-16 between the elastic modulus and stiffness. CS-16 has a lower elastic modulus compared with CS-20, but then has a higher stiffness than CS-20. After further examination, it was determined that for a cube of 30 × 30 × 30 mm with a cell size of 16, the program may have produced defects within the lattice tessellation that caused nonuniform lattice distribution (Fig. 14). The top half of the lattice had 52% of the lattice volume. This suggests that although CS-16 had a higher density of struts compared with CS-20, if one area of the CS-16 has less struts compared with another, that area will most likely deform differently and at a faster rate when compared with the rest of the lattice. In addition, the metaball rounding grew closer to a relative density of 1 at a higher rate compared with the other lattice parameters. This reflects what is seen in Figure 5, where the relative density equation with a metaball diameter of 8 mm resulted in relative density values near 1. There was also a small decline in stiffness with SD-4mm, whereas its elastic modulus was greater than SD-3mm. Stiffness is based on the geometry, so it suggests that there may have been a slight geometric defect with this latticed cube.

FIG. 14.

Tessellation defect seen in CS-16. The upper portion of the lattice seems to have a slightly higher density of struts compared with the lower portion, as the top half of the lattice retained 52% of the lattice volume compared with the bottom. Color images are available online.

These collective results show that our lattice generation methods have a great potential to maintain stiffness and the mechanical properties of the given material. All of the latticed cubes were able to maintain elastic modulus values close to Arcam Ti64 (120 GPa) despite having much less material, which demonstrates the structural integrity of our computationally generated lattices, as well as their resistance to deform under given loading. Further, the ability to control the geometric parameters of the lattice improves on the conventional lattice generation process, as it allows for additional weight reductions.

Applying Lattices to Case Studies

To apply the latticed cube results and further explore the strength capabilities of altering the geometric lattice parameters, case studies were performed where lattices were generated for two aerospace components: a jet engine bracket and an airplane-bearing bracket.

Description of case studies

The components are based on the 2013 GE jet engine bracket challenge³⁰ and the 2016 Alcoa airplane-bearing bracket challenge.³¹ The goal for both challenges is to optimize each design under their respective boundary and loading conditions (seen in Fig. 15) to best satisfy performance requirements and reduce the amount of material. It must be noted that the original loads of the airplane-bearing bracket were meant for a material stronger than Arcam Ti64 powder. Therefore, proportional loads were applied to all Arcam Ti64 powder airplane-bearing brackets.

FIG. 15.

The boundary and loading conditions used for: (a) the jet engine bracket,³⁰ and (b) airplane-bearing bracket³¹ challenge. Color images are available online.

Similar to numbers 1 and 2 in Figure 15b, interfaces 2 through 5 in Figure 15a are considered fixed and fastened by high-strength bolts as well. By applying their respective loading and boundary conditions, as well as the material properties from the simulations discussed earlier, the original brackets were analyzed by using ANSYS. The max stress results, seen in Figure 16, were used to determine where to remove excess material within both components. Again, defects that occur during the AM process were not considered for the unlatticed and latticed bracket simulations, as the goal is to computationally analyze their strength capabilities.

FIG. 16.

The max stress contour plot results of: (a) the jet engine bracket, and (b) the airplane-bearing bracket. These results determine where to implant the lattices. Color images are available online.

The prime areas to remove excess material and to apply the lattice are seen in dark blue within Figure 16, as they received the minimum amount of stress for each loading condition. With this understanding, lattices were strategically generated in these areas to minimize weight and maintain strength. In addition, the base of the airplane-bearing bracket (where the part is fastened) was redesigned to have an incline for additional weight reduction as the stresses were low there. Depending on their respective constraints and geometry, the lattice parameter specifications for each bracket are shown in Table 4 and the latticed brackets are shown in Figure 17. The relative densities were found by using the equations seen in Figure 5. Figure 17b and d are the cross-sections of each latticed bracket to show that the solid looking latticed regions are, indeed, latticed. When the strut diameters are large enough in a given volume, they will merge together and create hollow regions. Knowing that the previous compression simulations demonstrated that the cell size should remain high to reduce weight, and primarily controlling the strut diameter and intersection rounding maintains strength, the selected parameter sets best display the strength capabilities of our tetrahedral lattice for these brackets.

Table 4.

Lattice Parameters for the Optimized Regions of the Jet Engine Bracket and Airplane-Bearing Bracket

	CS (mm)	SD (mm)	MBD (mm)	Relative density
Jet engine bracket	30	5	6	0.431
Airplane-bearing bracket	30	6	6	0.514

FIG. 17.

Latticed (a) jet engine bracket and (c) airplane-bearing bracket. Their respective cross-sections, (b, d) display the lattices at solid-looking regions. Color images are available online.

Case study FEA results

After applying the same loading conditions (seen in Fig. 15) to the latticed components, Tables 5 and 6 show the max stress and deformation results of the original and latticed brackets. A global mesh element size of 10 mm was used to analyze each component; however, since the latticed portions are faceted bodies, they were meshed at 1.5 mm by using Patch Independent, a meshing method within ANSYS that is efficient for meshing faceted bodies. “LC” stands for “loading conditions,” and each LC number is correlated to loads numbered in Figure 15.

Table 5.

Max Stress Results of Both the Original and Latticed Brackets

	Mass (kg)	LC1 max stress (MPa)	LC2 max stress (MPa)	LC3 max stress (MPa)	LC4 max stress (MPa)
Jet engine bracket	2.14	567.14	288.78	315.88	211.52
Latticed jet engine bracket	1.03	709.74	697.26	514.47	222.65
Airplane-bearing bracket	0.486	613.74	758.98	573.56	NA
Latticed airplane-bearing bracket	0.386	505.62	485.39	930.73	NA

NA, not applicable.

Table 6.

Max Deformation Results of Both the Original and Latticed Brackets

	Mass (kg)	LC1 max deformation (mm)	LC2 max deformation (mm)	LC3 max deformation (mm)	LC4 max deformation (mm)
Jet engine bracket	2.14	0.289	0.202	0.143	0.123
Latticed jet engine bracket	1.03	0.474	0.357	0.232	0.177
Airplane-bearing bracket	0.486	1.46	1.16	0.939	NA
Latticed airplane-bearing bracket	0.386	1.61	1.17	1.34	NA

It can be seen that the lattices reduced the mass of both brackets. The jet engine bracket had a weight reduction of 51.8%, whereas the airplane-bearing bracket was reduced by 20.5%. The difference in weight reduction between the two brackets was due to their respective geometry and loading conditions. The loads on the original airplane-bearing bracket generally produced higher stresses compared with the respective loads for the jet engine bracket. As seen in Table 4, it was for this reason that the relative density of the latticed regions of the airplane-bearing bracket was higher than the jet engine bracket. The higher relative density meant less of a weight reduction, and the latticed airplane-bearing bracket needed thicker struts to sustain the given loads.

When examining the stress results seen in Table 5, they were all under the yield strength (950 MPa). It was expected that the stresses would be higher than the original brackets, as there is less material within the latticed brackets. When examining the stress contour plots for both brackets in Figure 18, the stresses throughout both brackets are generally below the yield strength of the material. The location of the higher stresses differs for each bracket.

FIG. 18.

Stress contour plots for: (a) latticed jet engine bracket, and (b) airplane-bearing bracket, at each loading condition. Color images are available online.

As seen in Figure 18a, the latticed jet engine bracket had higher stresses along the outside surfaces of the lattices and the max stresses were usually found at the intersections of the struts. This relates to Figure 3 and the cut-off option within the computational method that generates the lattices. To ensure that the lattice does not extend beyond the volume of the original geometry, the cut-off option removes any part of the lattice outside of the original volume. Although this geometric change alters the lattice and potentially the strength, the stress results show that the latticed jet engine bracket was still able to withstand the given loads. Despite the direction of the loads, each loading condition for the jet engine bracket caused the latticed regions of the bracket to experience tension. Although the latticed cube simulations focused on compression, these results provided additional confidence in the strength of the lattices.

The higher stress locations for the latticed airplane-bearing bracket were generally closer to the actual loading region. This can be seen for LC1 and LC2 in Figure 18b, and their max stresses were lower than the original bracket. This builds on what was seen in the jet engine bracket, as LC1 and LC2 for the airplane-bearing bracket caused tension within the latticed regions, but they were still able to withstand the given loads. However, the max stress results for LC3 were much higher compared with the other two. The downward vertical load for LC3 creates a moment that the latticed region needs to withstand, and the higher stress areas are where the latticed region connects to the base of the bracket. The high stress in this area was most likely due to the lack of transition from the latticed region to the solid region of the base. This suggests that the connection between the solid regions and latticed regions may be weaker areas. However, physical testing would help to give a better understanding of this, and the deformation results for both brackets provide additional confidence in the stiffness of the lattices.

The deformation results for the latticed brackets provide additional information in understanding the stiffness behavior of our lattices. The deformation of the latticed brackets seen in Table 6 was all less than double the deformation seen in the original brackets. LC1 and LC2 for the airplane-bearing bracket, particularly, were very close to deformation values seen in the original bracket, and these were also the loading cases where the stress was lower than the original bracket. Considering the weight reduction for each bracket, the deformation results seen for the brackets indicate their level of stiffness and suggest that physical testing would help to determine whether further weight reductions are possible.

The max deformation and stress results demonstrate that our lattice generation methods are capable of reducing weight and maintaining the strength of aerospace and other engineering applications. After understanding the loads needed for a given component or system, our novel tetrahedral lattices can be used to redesign the component for lightweight and stiffness. In addition, we take the conventional lattice generation process further due to the capability of manipulating the lattices for lightweight and stiffness by controlling its geometric parameters.

Conclusions

Redesigning components for high strength and low weight is important in many engineering fields, especially within the aerospace industry. Topology and lattice optimization are common methods of doing this, however they each have limitations. Highly topologically optimized components may fail under unexamined loads, which limits the overall stiffness. Although lattice optimization can better maintain the overall stiffness of a component, it often does not control the geometric parameters of the lattice to obtain the best mechanical properties. In this article, tetrahedral lattices created by a computational method were numerically analyzed to understand their effects on Ti64 Arcam powder mechanical properties and their performance. The computational method allows for the manipulation of the lattice's cell size, strut diameter, and strut intersection rounding. These different parameters affect the strength and weight of the lattice, and the study concludes the following:

• Multiple equations that estimated the relative density of the lattices were developed to better understand how the parameters affect the lattice's relative density/weight. Altering the strut diameter and cell size had the largest effects on the weight, and the relative density equations varied based on the metaball diameters (controls the strut intersection rounding).

• A metaball diameter of 8 mm or higher is expected to reach higher relative density values at quicker rates and should not be used for the examined cube size due to the weight increase. Metaball diameters between 2 and 6 mm are ideal. Essentially, the metaball diameter should range from being equal to or three times the strut diameter.

• The best parameters to maintain stiffness and reduce weight are strut diameter and intersection rounding since adding round features at the strut intersections can help remove stress concentration. A balance of these two parameters allows for high stiffness and a high elastic modulus. Although decreasing the cell size increases the strength of the lattice, it is not worth the increased weight. Therefore, it is suggested to keep the cell size as large as possible, and fine tune the strut diameter and strut intersection rounding for optimal results.

• When the lattices were applied to the jet engine bracket and airplane-bearing bracket, the jet engine bracket had a weight reduction of 51.8% and the airplane bearing bracket, 20.5%. Their stress and deformation results reflect their strength and stiffness, as they can still withstand their respective loading, regardless of their reduced weight. These results prove the effectiveness of our lattices. With additional physical testing results, our lattices can be used as a tool to redesign engineering components for lightweight and stiffness for various loading configurations, especially within the aerospace field.

As future work, we will expand this study by using AM to create the latticed cubes and components examined in this article. We will conduct physical tests to better understand the mechanical properties of our lattices. We also plan to explore anisotropic lattices and graded lattices or lattices with differing densities, and the multi-physics functionalities of our lattice by considering other important factors such as temperature shifts and heat transfer.

Author Disclosure Statement

There was no potential conflict of interest reported by the authors.

Funding Information

No funding was received for this article.