Controlling the Material Width of Equation-Based Lattices for Large-Scale Additive Manufacturing

Abstract

Additive manufacturing (AM) developments have been strongly driven by the ability of AM to improve the strength-to-weight ratios of structures, in contrast to traditional manufacturing methods, heavily supported by lattice structures. These motivations have persisted with the development of large-scale additively manufactured structures, which can offer more flexibility in manufacturing location and can often be faster than traditional manufacturing. However, current large-scale AM methods are often limited by their precision in order to maintain speed, constraining the method to manufacturing simple structures and often avoiding lattices altogether. This work proposes a mathematical framework for defining an equation-based lattice that splits the lattice into (1) build direction and (2) planar components such that their design can be altered to address AM methods restricted to three degrees of freedom. The framework is applied against a class of lattices called triply periodic minimal surfaces, which are represented using implicit equations, and it is shown that this approach allows for their use in large-scale AM technologies and enables further design control for small-scale AM design.

1. Introduction

Additive manufacturing (AM) has shaped every field of engineering, not only due to the weight reduction it offers for full-scale products [1,2], but also the time it saves in prototyping design phases [3]. These benefits are further realized by incorporating lattices into a structure, which have been shown to improve heat dissipation [4], and increase structural ductility [5,6] and energy absorption [7,8], in addition to many other applications. Triply periodic minimal surfaces (TPMSs), equation-based lattices that are a particularly advantageous subset of lattices [9,10], use combinations of periodic functions to define smoothly repeating 3D surfaces that avoid stress concentration points while exhibiting excellent stiffness and strength properties as a function of volume fraction [11,12]. The equation defines an infinitely thin isosurface; offsetting the isosurface by some value on either side, the space between the offset surfaces can be filled in to realize a solid structure. When the offsets are equal in magnitude, the resulting solid roughly follows the shape of the initial isosurface and we call it the triply periodic surface (TPSf) lattice for that equation [11]. Because of the repeating nature of lattice structures, we can define a volume that contains exactly one period of the lattice in each coordinate direction. This volume is known as a unit cell of that lattice. Unit cells are very useful for visualizing, understanding, and comparing lattices. Equation-based lattices have found application in many areas of engineering, including, but not limited to, heat exchangers, latent heat thermal energy storage, adsorption cooling systems, hydrogen storage, thermal management, and membrane distillation [13]. Despite the advantages of lattices, there is a lack of research applying them in large-scale additive manufacturing.

Current methods of manufacturing equation-based lattices rely on high-precision placement of material, often using micro-scale regions of solidification to create highly detailed parts [2,13,14]. Material requirements often dictate the AM method necessary for fabrication to meet functional requirements, where the most common materials explored in AM consist of polymers, ceramics, and metals [13]. Once a material is selected, there are several main types of additive manufacturing to select from as per the ASTM/ISO 52900 standard [15]: powder bed fusion (PBF), vat photopolymerization, material jetting (MJ), and material extrusion (MEX) additive manufacturing (AM) [13,16]. Lattice structures are often manufactured using PBF [14,17,18], photopolymerization [13], or MJ [16] due to the small region of solidification relative to the smallest features of the lattices being produced. However, the main limitation of these methods is the build volume. Liu et al. show that many metal powder bed fusion methods are typically limited to a build volume of 0.5 m × 0.5 m × 0.5 m [17]. For polymers, there are many AM methods that can generate lattice structures, such as stereolithography (SLA), selective laser sintering (SLS), and digital light processing (DLP) [14]. The resolutions in these methods are roughly 5 μm, 100 μm, and 5 μm for SLA, SLS, and DLP, respectively [16]. In general, all of these AM methods have a small region of solidification relative to the smallest features of the lattice structures being produced, allowing the manufacture of these complex geometries. In addition, all of these additive manufacturing methods have limited build volumes that cannot accommodate large-scale manufacturing. In order to achieve higher build volumes, most large-scale additive manufacturing uses material extrusion.

Though it would be advantageous to incorporate equation-based lattices into large-scale AM methods, it is not as simple as scaling existing designs, as drastic modifications would be needed to account for the limitations of current AM methods. As parts need to be scaled in geometric size, there are limitations as to what AM methods can be applied that will effectively maintain speed and address material requirements. Even though lattice structures are a major appeal of AM, they are often difficult or impossible to produce with large-scale AM methods at sufficient resolution to realize their geometry-dependent property gains. For large-scale AM, researchers often use directed energy deposition (DED) methods, such as wire arc additive manufacturing (WAAM), laser engineered net shaping (LENS), and laser-based metal wire deposition (LMWD), as they can produce large parts efficiently [17]. However, these methods have a large region of solidification, which prevents the manufacture of highly detailed parts with relatively small minimum feature sizes. In WAAM the nozzle size is about 1–5 mm; LENS has a laser thickness of about 0.5–2 mm; and LMWD has a laser thickness of about 1–5 mm [17]. There is extensive discussion in the literature comparing different lattice designs [13,14,17], but DED is excluded from this discussion altogether, likely due to the large solidification diameter of large-scale AM. However, equation-based lattices could prove to be effective for manufacturing with DED, as many of the lattices have uniform wall thicknesses which could translate well if those thicknesses could be defined intuitively.

Current equation-based lattice definitions have no concept of thickness, which requires that they be generated based on their volume fraction relative to the level set. Unfortunately, this is not a tunable metric field, with meaningful unit values, but rather a scalar field [11] that is unitless. As the level set increases linearly, the thickness of the equation-based lattices increases with an indeterminate relationship, caused by the complexity of the functional definition. This has resulted in the development of high-dimensional lookup tables to facilitate the development of structures to meet mass requirements based on their level set and unit cell size [11,19]. The current literature exploring alternative equation-based lattice definitions is highly focused on improving the mechanical performance of unit cells [20]. Ma et al. developed an optimization scheme that enforces uniform wall thickness; however, it operates on a B-spline model which does not allow integration into typical lattice design tools [21]. Zhu et al. developed a method that specifically tuned the wall thickness to improve permeability and mechanical properties [22], but yet again it cannot specifically define a thickness discretely. The lack of precision in defining the wall thickness of smooth topologies, such as TPMS easily generated using implicit fields, coupled with the limited literature addressing the improvement of manufacturability for large-scale equation-based lattices from a design perspective, motivates this work.

Large-scale equation-based lattices could be beneficial in many fields of engineering given the value provided by small-scale equation-based lattices [23]. The current literature has demonstrated a clear need to design intricate geometries by developing custom topology optimization and lattice manufacturing design methods to address large-scale manufacturing limitations. Fernández et al. have explored the development of a topology optimization scheme that addresses the manufacturing limitations of large-scale AM, which highlights the need for large-scale manufacturing of intricate geometries to address complex loading conditions [24]. Lattice structures have also been manufactured using WAAM, which have demonstrated the feasibility of their general incorporation into other large-scale AM methods. In order to effectively manufacture strut-based lattices, Li et al. formed struts using WAAM droplets to build the intricate structures [25]. Although droplet-based deposition enabled the manufacture of strut-based lattices, the authors did not take advantage of the continuous bead deposition enabled by WAAM. As such, Campocasso et al. argue that equation-based lattices are more appropriate for DED since they are thin-walled structures that can utilize continuous bead deposition [26]. In their work, they develop single-width toolpaths for WAAM based on the underlying equation for Schwarz-P lattices and manufacture them using WAAM. However, the path optimization scheme is not generalizable to other lattice types, so it does not integrate well into the traditional design for additive manufacturing framework, as it requires designers to relinquish control over the design and slicing methodology.

This work proposes a mathematical framework to modify the implicit equation-based lattice definition to address the manufacturing constraints of large-scale print methodologies, which are often limited by large solidification diameters (manufacturing feature sizes). Spatially modifying the TPMS with local modifications encourages designers to adjust their models in order to achieve effective slicing and toolpaths for their specific AM process. The new definition can be applied to any implicit equation-based lattice to generate lattices with meaningful units to enable the design of more uniform structures to adapt to large regions of solidification. To facilitate the design of parts for AM methods limited to three degrees of freedom, the definition is split into build direction and planar components, so that they can be defined independently. To accommodate AM methods that have adaptive build directions, the build direction can be rotated to change the lattice thickness locally. Additionally, the definition is field-driven, so that it can incorporate physics-based analysis to thicken and rotate regions as needed. This definition has been implemented in the implicit modeling tool nTop v5.5.2 [27], which is made available on GitHub (https://github.com/mebaldwi/Equation-Based-Lattice-Thickness-Control (accessed on 1 August 2025)).

2. Methods

To develop equation-based lattice structures, the most effective method is to implement an implicit design to approximate the original surface using a scalar field equation [11]. Scalar fields are the foundation of implicit modeling tools like nTop, an implicit design tool often used for lattice structural design, which makes them the ideal representation for designers. This work modifies the original field definition of any equation-based lattice such that users can generate meaningful and uniform offsets, regardless of the magnitude of the field and its gradient, while adapting to build direction (see Figure 1).

To effectively modify existing equation-based lattice definitions, the key manufacturing constraints of large-scale AM were considered in order to determine what to incorporate into the new functional definition. Within large-scale AM there is one notable method that uses material extrusion, namely, wire arc additive manufacturing [17]. The main trait of WAAM is the deposition of material in the form of a continuous bead. This bead placement serves as a major limitation, as it requires that parts can only be manufactured using discrete bead widths to maintain process stability, geometric accuracy, and avoid metallurgical defects [24]. Continuous deposition of material means that there are only six possible configurations for planar deposition to create a solid object and maintain a continuous bead on the boundary (see Figure 2). Current methods of ensuring discrete bead placement incorporate a minimum feature restriction to adhere to manufacturing constraints [24]. However, this can be further enforced by ensuring uniform wall thickness. Therefore, due to this design limitation, (1) it is desirable to design lattices such that they are meaningfully and uniformly generated such that they can maintain a boundary wall for accurate manufacturing. By discretely defining the wall thickness of the lattices, we can achieve finer features using n-paths to encourage the use of continuous beads (see Figure 2).

Notably, the bead is not a uniform sphere, but most often represented using a short rectangle [28]. This means that material placement is limited by the layer height, the height between layers of the part in the build direction, in addition to the wall thickness, which is controlled by the amount of material placed in the plane normal to the build direction. Therefore, (2) there is value in separately defining the thickness of the lattice based on the build direction, such that layer height and wall thickness can be defined independently.

The final consideration was the dynamic build directions offered by different AM methods, which suggests that the layer height and wall thickness elements need to be orientation-dependent. Additionally, the layer height and wall thickness need to adapt to various lattice orientations in order to apply them to the respective dimension when rotated. Therefore, (3) there is benefit to incorporating a rotation term to rotate the effects of the magnitudes of the wall thickness, because wall thickness changes as the build direction changes.

In summary, this work aims to modify the evaluation of equation-based lattices such that the following is true:

1.

They provide meaningful (actionable) thickness values that are uniformly generated such that they can maintain a boundary wall for accurate manufacturing;

2.

They define separately the thickness of the lattice based on the build direction, such that layer height and wall thickness can be defined independently;

3.

They incorporate a rotation term to rotate the effects of the wall thickness magnitudes.

2.1. Meaningful Surface Offset

To understand the final functional representation, this section reviews each of the components of the new definition and their effects on the final shifted field. The first item to address is incorporating meaningful offsets to the field such that uniform walls can be generated. A function can be shifted by adding or subtracting a scalar value; however, this does not work for functions with concave curvature in the offset direction without intersection. Therefore, the next option is to shift from the field using first-order gradients to provide a direction and specify a magnitude with meaningful distance, as in Equation (1):

$$\mathcal{F}\left(\mathbf{x}\pm m\widehat{\nabla }f\right)$$

(1)

where x is the input feature space; $\widehat{\nabla }f$ is the unit gradient of f, which is the lattice equation with no offset; and m is the magnitude of the vector f representing the desired offset. This form of the field shift equation uniformly offsets an implicit surface equation by a known magnitude, m, which can be visualized in Figure 3. This figure shows a single shift; however, there are actually two shifts of the original surface, a positive and a negative, which define the boundaries of the desired TPSf lattice body.

2.2. Splitting Magnitude Component

The next item to consider is separating the components of the magnitude based on their relation to the build direction. If this is to be implemented in 3D space while considering the layer height and build height, then this offset will need to shift the magnitudes relative to an ellipsoid. Consider the z-dimension as the build direction, where the desired layer height is different from the desired wall thickness. Imagine projecting an ellipsoid along the surface, such that the unit gradients are intersecting that ellipsoid at a point that determines their length (see Figure 4). This is the desired result in order to uniformly and smoothly offset the surface.

There were two methods considered for implementing this: a vector-based projection and a scalar-based projection. The vector-based projection uses the equation of an ellipsoid defined as

$${\displaystyle \frac{x^2}{d^2}}+{\displaystyle \frac{y^2}{d^2}}+{\displaystyle \frac{z^2}{{\ell }^2}}=1$$

(2)

where ℓ is the desired layer height; d is the desired wall thickness; and x, y, and z are the feature space dimensions. Then, the magnitude can be determined by the intersection of the unit gradient vectors and the ellipsoids:

$${\displaystyle \frac{{\left(m\widehat{\nabla }{f}_x\right)}^2}{d^2}}+{\displaystyle \frac{{\left(m\widehat{\nabla }{f}_y\right)}^2}{d^2}}+{\displaystyle \frac{{\left(m\widehat{\nabla }{f}_z\right)}^2}{{\ell }^2}}=1$$

(3)

where $\widehat{\nabla }{f}_x$, $\widehat{\nabla }{f}_y$, and $\widehat{\nabla }{f}_z$ are the x, y, and z components of unit vector gradient $\widehat{\nabla }f$ and m is the magnitude at which the unit gradient vectors intersect the ellipsoids. Solving for m the following is found:

$$m_v={\displaystyle \frac{1}{\sqrt{\frac{{\left(\widehat{\nabla }{f}_x\right)}^2}{d^2}+\frac{{\left(\widehat{\nabla }{f}_y\right)}^2}{d^2}+\frac{{\left(\widehat{\nabla }{f}_z\right)}^2}{{\ell }^2}}}}$$

(4)

$$\mathcal{F}\left(\mathbf{x}\pm {\displaystyle \frac{1}{\sqrt{\frac{{\left(\widehat{\nabla }{f}_x\right)}^2}{d^2}+\frac{{\left(\widehat{\nabla }{f}_y\right)}^2}{d^2}+\frac{{\left(\widehat{\nabla }{f}_z\right)}^2}{{\ell }^2}}}}\widehat{\nabla }f\right)$$

(5)

This $m_v$ can be substituted into Equation (1), and can produce some adequate surfaces in special cases. However, this representation alone fails as the unit vector component $\widehat{\nabla }{f}_z$ approaches 1 or 0, because the vector-based definition sharply curves between ℓ and d, causing discontinuities in the surface. An alternative method is to apply a scalar-based projection, which can be implemented by applying the magnitudes directly to each desired dimension using $m_s$ in the form  

$$m_s=\left[\begin{array}{c}d\\ d\\ \ell \end{array}\right]$$

(6)

$$\mathcal{F}\left(\mathbf{x}\pm m_s\odot \widehat{\nabla }f\right)$$

(7)

where ⊙ denotes element-wise multiplication. This method holds for two special cases that we present here in limit form. First, that as the direction of the gradient approaches perpendicular to the build direction, the magnitude of the offset approaches the prescribed wall thickness $\left(\mathrm{i}.\mathrm{e}.,\left|\right|m_s\left|\right|\to d\mathrm{as}{\left(\widehat{\nabla }{f}_z\right)}^2\to 0\right)$. Secondly, that as the direction of the gradient approaches parallel to the build direction, the magnitude of the offset approaches the prescribed layer thickness $\left(\mathrm{i}.\mathrm{e}.,\left|\right|m_s\left|\right|\to \ell \mathrm{as}{\left(\widehat{\nabla }{f}_z\right)}^2\to 1\right)$. Both cases hold for Equation (7) because $\widehat{\nabla }f$ has a magnitude of one by definition.

2.3. Rotating the Magnitude Effects

Another benefit of the scalar-based projection is the ability to apply rotation to the magnitude vector in order to dynamically change the build direction. Using Rodrigues’ rotation formula, we can develop a rotation matrix:

$$\mathbf{R}=\mathbf{I}+sin\theta \mathbf{K}+\left(1-cos\theta \right){\mathbf{K}}^2$$

(8)

$$\mathbf{K}=\left[\begin{array}{ccc}0& -k_z& k_y\\ k_z& 0& -k_x\\ -k_y& k_x& 0\end{array}\right],\mathrm{where}:k={\displaystyle \frac{\mathbf{a}\times \mathbf{b}}{∥\mathbf{a}\times \mathbf{b}∥}}$$

(9)

where $\mathbf{R}$ is Rodrigues’ rotation formula, $\mathbf{K}$ is the cross-product matrix, $\mathbf{I}$ is the identity matrix, $\mathbf{a}$ is the current assumed build direction vector [0, 0, 1], and $\mathbf{b}$ is the new desired build direction vector. Then, Rodrigues’ rotation formula can be further simplified by applying the trigonometric cross-product formula and applying the known assumptions of vectors $\mathbf{a}$ and $\mathbf{b}$. Knowing $\mathbf{a}$ is a unit vector since it defines the original build direction, and $\mathbf{b}$ can be assumed as a unit vector of the new build direction:

$$\begin{array}{cc}\hfill sin\theta & =∥\widehat{\mathbf{a}}\times \widehat{\mathbf{b}}∥\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill cos\theta & =\widehat{\mathbf{a}}·\widehat{\mathbf{b}}\hfill \end{array}$$

(11)

If $\widehat{\mathbf{a}}=[0,0,1]$, our current build direction, is to be rotated to align to $\widehat{\mathbf{b}}=[b_x,b_y,b_z]$, some field of build direction vectors, then to describe the rotation between the build directions we solve the simplified Rodrigues formula by combining Equations (8), (9), (10), and (11) to produce the rotation build direction matrix ${\mathbf{R}}_{\mathrm{BD}}$ in Equation (12).

$${\mathbf{R}}_{\mathrm{BD}}=\left[\begin{array}{ccc}{\displaystyle \frac{b_x^2(b_z-1)+b_x^2+b_y^2}{b_x^2+b_y^2}}& {\displaystyle \frac{b_x{b}_y(b_z-1)}{b_x^2+b_y^2}}& b_x\\ {\displaystyle \frac{b_x{b}_y(b_z-1)}{b_x^2+b_y^2}}& {\displaystyle \frac{b_y^2(b_z-1)+b_x^2+b_y^2}{b_x^2+b_y^2}}& b_y\\ -b_x& -b_y& {\displaystyle \frac{b_x^2(b_z-1)+b_y^2(b_z-1)+b_x^2+b_y^2}{b_x^2+b_y^2}}\end{array}\right]$$

(12)

There are two special cases that must be accounted for to address the limitations of the rotation matrix: division by zero and rotation of ${180}^{\circ }$. To avoid division by zero in the case where $\mathbf{b}=[0,0,1]$, we apply the signum function defined in Equation (13).

$$\mathrm{sgn}\left(x\right):=\left\{\begin{array}{cc}1\hfill & \mathrm{if}x>0,\hfill \\ 0\hfill & \mathrm{if}x=0,\hfill \\ -1\hfill & \mathrm{if}x<0.\hfill \end{array}\right.$$

(13)

Since ${∥\mathbf{a}\times \mathbf{b}∥}^2=b_x^2+b_y^2$ is non-negative, the signum(${∥\mathbf{a}\times \mathbf{b}∥}^2$) will be 0 or 1. Therefore, ${∥\mathbf{a}\times \mathbf{b}∥}^2$ becomes ${\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}$ to account for this special case. A second modification was made to account for the fact that the Rodrigues formula does not allow rotations of ${180}^{\circ }$, which occur when the original and new vectors are aligned, so it does not perceive the need for rotation. Therefore, a fix was developed to address the case where $\widehat{\mathbf{a}}=-\widehat{\mathbf{b}}$. If ${∥\mathbf{a}\times \mathbf{b}∥}^2=b_x^2+b_y^2$, then ${∥\mathbf{a}\times \mathbf{b}∥}^2-b_y^2=b_x^2$; applying this to the rotation matrix will allow the $b_z$ term to address the negative components necessary; since the only cases to address are those where ${∥\mathbf{a}\times \mathbf{b}∥}^2=0$, we can apply the signum function:

$${\mathbf{R}}_{\mathrm{BD}}=\left[\begin{array}{ccc}{\displaystyle \frac{\left({\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}-b_y^2\right)(b_z-1)+b_x^2+b_y^2}{{\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}}}& {\displaystyle \frac{b_x{b}_y(b_z-1)}{{\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}}}& b_x\\ {\displaystyle \frac{b_x{b}_y(b_z-1)}{{\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}}}& {\displaystyle \frac{\left({\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}-b_y^2\right)(b_z-1)+b_x^2+b_y^2}{{\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}}}& b_y\\ -b_x& -b_y& {\displaystyle \frac{b_x^2(b_z-1)+b_y^2(b_z-1)+b_x^2+b_y^2}{{\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}^{\mathrm{sgn}\left({∥\mathbf{a}\times \mathbf{b}∥}^2\right)}}}\end{array}\right]$$

(14)

The rotation formulation of Equation (14) was reduced to its algebraic form and implemented in a rotation block in nTop to perform rotations of fields; this can be found in ‘CB Rotation Transformation’. The matrix must be applied to a unit vector based on our prior assumption of $\mathbf{b}$ being a unit vector, and the magnitude can be reapplied after the rotation transformation is performed. When applied to our magnitude unit vector, ${\widehat{m}}_r$, the rotation can produce a vector with negative components due to the nature of rotating vectors. Therefore, the absolute value of the rotation is used, since the magnitude should have no effect on the direction of the gradients. This ensures that the effects of the magnitude are rotated effectively. This results in the final scalar-based projection shift formulation:

$$\mathcal{F}\left(\mathbf{x}\pm \mathrm{abs}\left({\mathbf{R}}_{\mathrm{BD}}{\widehat{m}}_s\right)∥m_s∥\odot \widehat{\nabla }f\right)$$

(15)

3. Results

The new equation-based lattice definition serves to address the need for lattices with meaningful units, as original lattice offsets were unitless, thereby requiring the use of high-dimensional lookup tables [11]. These lookup tables change as the scale of the lattice varies as well as when the lattice is stretched. This section will analyze the progression of the lattice definition and compare the original and new lattice definitions. This definition progressed through many formats to improve and expand its applicability; this section demonstrates that progression using the common definition of the gyroid [29]:

$$sinxcosy+sinycosz+sinzcosx=t$$

(16)

3.1. Meaningful Offsets

Equation (1) defines the method for generating equation-based lattices with a uniform offset of the entire body in order to give the unit cell meaningful values of thickness. If this equation were applied to the gyroid definition, the x, y, and z terms in Equation (16) would be replaced with the shifted x, y, and z terms of Equation (1):  

$$\begin{array}{cc}\hfill & sin\left(x\pm m\widehat{\nabla }{f}_x\right)cos\left(y\pm m\widehat{\nabla }{f}_y\right)\hfill \\ \hfill +& sin\left(y\pm m\widehat{\nabla }{f}_y\right)cos\left(z\pm m\widehat{\nabla }{f}_z\right)\hfill \\ \hfill +& sin\left(z\pm m\widehat{\nabla }{f}_z\right)cos\left(x\pm m\widehat{\nabla }{f}_x\right)=t\hfill \end{array}$$

(17)

This form can be tested using the nTop block ‘Uniform Shift’, which uses the block ‘Gyroid Generator v4 Level Set’ to define the original isosurface equation. The figures generated to compare these functions use a unit cell size of 1 mm cubed, with the period in the x, y, and z directions being 1 mm. The contours display the fields where z = 0 mm. Figure 5 demonstrates some of the limitations of the shifting method, as it does not maintain uniform shifting between each previous offset. Unfortunately, this means large-scale AM methods will be limited to using only 10% of the lattice size as the wall thickness if uniform wall thickness is to maintained. In the case of the gyroid, only about 20% wall thickness is achievable before becoming a solid unit cell, so the method covers the area where the lattice is most often used.

3.2. Addressing Manufacturing Constraints

When looking to separate the layer height and wall thickness into two separate input variables, the method progressed by exploring a vector-based projection (Equation (5)) and a scalar-based projection (Equation (15)). The nTop blocks that implements these equations can be found in ‘Ellipse Intersect’ and ‘Shift Rotation’, respectively. The first priority is to maintain uniformity in the $x-y$ plane as thickness increases, as this enables large-scale manufacturing of the design. The second priority is to ensure that the layer height is effectively applied such that it is unaffected as the wall thickness varies; this can be monitored by the $y-z$ plane. Figure 6 demonstrates what these planes represent as we compare them in

Table 1. Comparison of vector- and scalar-based projection methods using their scalar field contours. The scalar field contours of the vector-based projection represent the fields produced by Equation (5), while the scalar-based projection fields are produced by Equation (7).

	$\mathit{x}-\mathit{y}$ Contours	$\mathit{z}-\mathit{y}$ Contours
Vector-based Projection
Scalar-based Projection

. The parameters of the unit cells are defined as 1 mm in size, layer height is 5% unit cell size, and wall thickness varies from 0 to 20% of unit cell size.

Note that in the $x-y$ and $y-z$ contours, the ellipsoidal definition does not maintain uniform curves in the cases where $d<\ell$, whereas the scalar-based definition actually maintains smooth curves through all different values of d. However, the curves of these plots do not demonstrate uniformity between the lines, again demonstrating that staying within 10% of the unit cell size for the wall thickness will yield the most uniformity. This motivated the use of the scalar-based projection to explore rotation of the build direction.

3.3. Dynamic Build Direction

The dynamic build direction shift promotes the use of AM methods with more degrees of freedom during manufacturing, as well as developing lattices with new build directions to explore their properties. The exploration of the rotation term usage is limited to simple demonstrations of the changing of axes as the build direction is redefined. The changes in these contours demonstrate the rotation of the gyroid as if the build direction is changing; therefore, the layer height is being reassigned to the new build direction dimension (see

Table 2. Comparison of scalar-based projection for various build directions to confirm the effects of rotation.

Build Direction	$\mathit{x}-\mathit{y}$ Contours	$\mathit{z}-\mathit{y}$ Contours
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]

). Due to the computation with the rotation, if the build direction aligns with the original z-axis, there is an error in the visualization in nTop, where it displays a cube. However, the field is still correct and the image of the implicit body shows the correct shape.

3.4. Original vs. New Definition

The benefit of the new definition is highlighted when it is compared to the original definition of the gyroid (Equation (16)), where it is shifted by setting the t-value. Displayed in

Table 3. Comparison of original gyroid definition (red) to gyroid defined using the scalar-based projection method (blue), using scalar contours to demonstrate the effects of scaling on the methods.

Unit Cell Size	Contours of Original Gyroid Equation vs. Scalar-Based Projection Gyroid Definition
1 mm
10 mm
50 mm

are the plots demonstrating the differences between the definitions with uniform shifting. Both methods use values ranging from 0 to 20% of the unit cell size to show how the scale of the unit cell affects the definitions. The original gyroid is shown in red color mapping, and the new definition is shown in a blue color mapping to emphasize the differences in the contours.

Table 3 demonstrates that the original gyroid definition must have the t-values scaled with the unit cell size in order to generate a desired volume fraction, which is a physical demonstration of how complex the lookup tables are for these structures. These plots affirm that the new method is immune to scaling effects of the unit cell size, as the contour plots are identical as the scale is increased. This enables the development of lattices with thickness with respect to meaningful units.

3.5. Other Lattices

To evaluate the applicability of the proposed method on other lattice topologies, we selected two common lattices from the literature: the D-surface and the IWP [11]. These offer some distinct differences from the gyroid in identifying any issues with the method. The parameters of the unit cells are defined as 1 mm in size, and the layer height and wall thickness consist of all combinations of 5 and 10% of the unit cell size. The orientation of the lattices in

Table 4. Comparison of shifting methods for various lattices to demonstrate the effects of the scalar-based projection method on alternative lattice definitions.

Input Parameters	IWP	D-Surface
Layer height = 0.1 mm Wall thickness = 0.1 mm
Layer height = 0.05 mm Wall thickness = 0.1 mm Build direction = [1, 0, 0]
Layer height = 0.1 mm Wall thickness = 0.05 mm Build direction = [1, 0, 0]

matches the orientation in Figure 7.

These test cases demonstrated that some lattice implicit approximations can cause nonlinear sections when the gradients are computed, in this case the D-surface. In Table 4, the D-surface boundaries became nonlinear, this is much clearer in Figure 7, which uses a larger wall thickness. However, the test case of the IWP lattice did demonstrate that the method was generalizable to other lattice types. It should be noted that for the tested lattice definitions, nTop was able to produce a gradient field directly from the defined implicit surface; however, for other surfaces, the gradient field produced may not be meaningful and should be computed analytically to provide the gradient.

Overall, the scalar-based projection method supports the development of many areas of lattice prototyping. The benefit of all of these methods is the ability to incorporate physics-based analysis of implicit field results to dynamically increase the thickness of regions of the lattice where necessary. The ‘Uniform Shift’ nTop block enables the definition of meaningfully thick lattices, which could be useful for all methods of AM. The ‘Shift Rotation’ nTop block will enable designers to discretely define the layer height and wall thickness separately based on their print parameters, which is especially useful in material extrusion methods. Additionally, as the layer height term approaches 0, this inherently causes a reduction of material in the build direction, which could potentially improve the strength-to-weight ratio of the lattice. Finally, the method provides a starting point for manufacturing large-scale lattice structures, as low-precision manufacturing methods work best with uniform structures that enable simpler toolpaths.

4. Conclusions

The benefits of additive manufacturing have been exemplified by the many advantages of lattice structures: improved strength-to-weight ratios, effective heat dissipation, increased ductility, and many more. However, these benefits have yet to be realized in large-scale additive manufacturing as it prioritizes high material placement over precision, which limits the ability to print complex geometries. To support the manufacture of large-scale lattices, this work proposes a new equation-based lattice definition to directly define lattice wall thickness. This definition was motivated by the three key design features of large-scale AM:

1.

It is desirable to design lattices such that they are meaningfully and uniformly generated such that they can maintain a boundary wall for accurate manufacturing;

2.

There is value in separately defining the thickness of the lattice based on the build direction;

3.

There is benefit to rotating the effects of the wall thickness magnitudes to accommodate build direction-agnostic AM methods.

The new definition will benefit both small- and large-scale AM as it empowers designers to have more control over how their lattices are generated with respect to wall thickness rather than volume fraction. This definition was implemented using implicit fields directly in nTop to enable accessibility.

The primary benefit of the method was demonstrated in the ‘Uniform Shift’ nTop block, which develops a uniform offset from the isosurface, allowing designers to explicitly define the thickness of the lattice. This demonstrated that the field is only uniform when the offset is less than 10% of the unit cell body width for gyroid unit cells. The new definition proved immune to scaling the unit cell width, whereas with the original definition, scaling requires careful tuning to determine its effects. This ensures the development of unit cells with meaningful units. For applications where the build direction can change, such as WAAM, it is beneficial to dynamically define the build direction such that the nozzle orientation is considered. This benefit was realized in the ‘Shift Rotation’ nTop block, which can accept a field of unit build directions to dynamically define the layer height and wall thickness throughout the unit cell as it is built. This feature will enable designers to print more detailed lattice structures by taking advantage of the finer precision offered by layer height rather than nozzle size. Finally, the field definition of the function still enables the application of physics-informed analysis, where the fields can be used to thicken and rotate fields as needed.

Limitations to this method appear when the applied offset thickness is below a critical value relative to the lattice unit cell size. For example, the current definition only maintains uniformity for gyroids when the thickness is less than 10% of the unit cell width; outside of those values the holes in the lattice start to negatively affect the definition, which is a limitation of the first-order gradient shifting technique. Other lattice fields should be adjusted based on their individual fields, as it is unclear if they will maintain uniformity as the wall thickness increases to larger than 10% of the unit cell width. Another major limitation of large-scale AM is often the maximum overhang angles, which will require further investigation to incorporate into the method. Finally, the effects of rotation were not directly measured and will require further validation before implementing complex rotation fields.

Future work will explore how these functions translate to the physical space. This will include building equation-based lattice structures using DED or MEX systems, and comparing the as-built structures to the intended as-designed geometry. Further testing would include an evaluation of the effects on the equation-based lattice mechanical properties from the defined thickness method presented here relative to the conventionally defined geometry. Additional work will be required to adapt this equation-based lattice definition to multi-axis AM systems, including those based in a polar coordinate system [30].