## Abstract

This paper introduces heuristics based upon statistical mechanics to assist in additive manufacturability analysis of multiscale aperiodic structures. The heuristics associate structural properties at a statistical level with manufacturability. They are derived from four topological properties of the complex network representations of multiscale aperiodic structures. The validity of these heuristics is assessed in two ways. First, cross-model validation compares the manufacturability determined by these heuristics when applied to computationally designed crumpled structures and a microCT scan of the same structures when additively manufactured. Second, external validity assesses the correctness of the heuristics given design parameters that increase the potential for manufacturing errors. The results show the significance of statistical mechanics in providing insight into the additive manufacturability of multiscale aperiodic structures. The paper concludes by discussing the generality of this approach for alternative geometries and provides designers with a framework for interpreting manufacturability from a statistical mechanics perspective.

## 1 Introduction

Aperiodic structures occur in numerous applications ranging from porous bone for cranial defect repair [1] to graphene nanostructures for energy storage [2]. These aperiodic crumpled structures are usually produced by applying an external load to generate plastic deformations of the material within a confined geometry. Drawbacks include a high level of input energy and low reliability and consistency. Additive manufacturing (AM) emerges as a promising technology to fabricate multiscale aperiodic structures [3].

Analyzing the additive manufacturability of multiscale aperiodic structures poses a challenge due to a large number of geometric features at each length scale and their inter-scale relationships. Discrete topological analysis and heuristic-based design for additive manufacturing (DfAM) [3,4] using homogenization techniques are sufficient for periodic multiscale structures with a representative volume element. In contrast, aperiodic multiscale objects lack a representative volume element. This paper addresses this challenge by introducing a heuristic-based approach to DfAM using statistical mechanics. Network approximation techniques developed in statistical mechanics are correlated with complex network properties of multiscale geometric features to assess additive manufacturability.

The paper proceeds in the following order. Section 2 summarizes applications of statistical mechanics for the study of complex systems in engineering design. The summary reviews existing approaches to DfAM and discusses a gap in the scope of manufacturability analysis that can be addressed using a statistical mechanics approach. Section 3 describes the development of four network-based heuristics and how they address additive manufacturability. Section 4 describes case studies on the implementation and validation of these heuristics. Section 5 concludes the paper with a discussion on the generality, limitations, and outlook of a statistical mechanics-based approach for the additive manufacturability analysis of multiscale aperiodic structures.

## 2 Background

Statistical mechanics is the study of bulk patterns of individual, interconnected components from a systems-level perspective to discover structure–function relationships. Modeling and analysis of interconnected components as networks is one popular class of methods in statistical mechanics. Existing applications of network-based approaches to examine structure–function relationships in engineering design include the discovery of statistical properties of large informational networks formed during new product development [5] and the description of the physical architecture of complex products as modular or hierarchically modular [6]. Aperiodic multiscale objects are amenable to analysis through statistical mechanics approaches due to the sheer number of complex interactions between geometric features at multiple scales. In this research, we will exploit the experimentally-observed statistical mechanics properties of aperiodic multiscale objects to inform their design for additive manufacture.

Design for additive manufacturing (DfAM) is a specific approach to concurrent design and engineering. DfAM assesses the influence of designed geometric features and available manufacturing processes on outcomes such as part reliability, manufacturing cost, manufacturing time, and feature accuracy [4,7]. For this research, the scope of manufacturability is narrowed to focus on geometric fidelity. Fidelity is a measure of the accurate physical placement of material. Topological deviation between the as-printed and as-designed computer-aided design (CAD) model defines the degree of fidelity.

DfAM produces rules or heuristics to increase fidelity. Rule-based DfAM assesses designs against experimentally derived models of material behavior under process-specific manufacturing constraints and tool paths to generate geometric constraints such as feature size, overhangs, clearance, or enclosed voids [4,8]. These rules affect the fidelity of the printed shapes and enable designers to adjust the design and process parameters to account for anticipated printing problems. Design heuristics for additive manufacturing (DHAM) [9] propose general design principles for geometric features derived from the practice of additive manufacturing. DHAM has been applied to hierarchical geometries that can be simplified to a representative volume element, unit cell, or standard geometric element [10,11]. The morphology of multiscale aperiodic structures such as crumple-formed thin sheets exhibits geometric properties that were not encountered in the objects from which DHAM was developed. The hierarchical geometric relationships across length scales in multiscale aperiodic structures call for new approaches. We propose that global measures of multiscale aperiodic structures could provide insight into the relationship between geometric features and the additive manufacturability of the structure. In this paper, we develop and evaluate heuristics that exploit global measures of network properties of multiscale aperiodic structures to assess their additive manufacturability. This research expands our prior research [12] in two significant ways. First, the research performs cross-model validation of the additive manufacturing heuristics using two types of designed and printed multiscale aperiodic structures, spherical and cylindrical crumple-formed thin sheets. Second, the paper reports on compression testing of printed objects to assess manufacturability based on deviation in macroscale properties. Section 3 will explain the theory underpinning the heuristics.

## 3 Theory and Methodology

This section presents the theory and methodology for analyzing the additive manufacturability of multiscale aperiodic objects from a statistical mechanics perspective.

### 3.1 From Crumpled Structures to Complex Networks.

Crumpled structures form under an external load acting on a thin sheet confined within a geometry. The external load generates plastic fractures. Ridges form when the thin sheet starts to fracture. As these ridges recursively fracture within a confined geometry, vertices form at the juncture between existing and new ridges. Experimental research focusing on describing the network of ridges and vertices that form during crumpling conclude that the network is not random [13]. Direct observation of the formation of vertices and ridges confirms that the statistical characteristics of the number and lengths of ridges follow theoretical predictions about the transition of the thin sheet from a deterministic system into a complex state that can be characterized only by its statistical properties [14]. Theoretical [15], experimental [16–19], and computational simulation [20] show that the evolving network of ridges and vertices that form during crumpling within a confined geometry is well defined in a statistical sense [21]. Taken together, these studies establish the potential to exploit the characteristics of the complex network representations of crumpled structures to assess their additive manufacturability.

We design crumpled objects by confinement of a two-dimensional triangulated mesh sheet within a spherical (Fig. 2) or cylindrical (Fig. 9) shell using Rhinoceros 3D CAD software. The in-built Grasshopper visual programming language and Kangaroo live physics engine are used to place self-avoidant collision spheres at each vertex in the simulated mesh sheet. A collision sphere diameter provides a three-dimensional thickness aspect to the sheet. Self-avoidant behavior of the collision spheres minimizes self-intersection of the sheet during deformation. Dynamic crumpling deformation is initiated as the confinement shell progressively contracts around the mesh sheet, forcing the sheet to buckle and fold as self-avoidant spheres at each vertex come in contact with the outer confinement shell and each other. The crumpling simulation is complete when the confinement shell contracts to a user-defined volume. The final crumpled object is represented geometrically as a mesh structure.

The mesh structure is the basis for the complex network representation. Nodes are defined by the coordinate positions of each vertex in the crumpled mesh. Edges are generated based on spatial proximity of the node coordinates. A list of edges is generated by measuring the radial distance between each node and connecting nodes within a specified radius. In this project, 0.3 mm is chosen because this length includes the 0.25-mm sheet thickness and nodes from adjacent layers and is the minimum resolution of the chosen 3D printing technology. This dimension may be modified based on the minimum feature size capabilities of the 3D printer and the material of interest. A higher minimum may be chosen, but ridges and vertices will be missed. This method of spatial edge generation is employed rather than extracting edges directly from the CAD model because triangular mesh edges in the crumpled geometry are identical to the node connections in the flat mesh sheet prior to spherical confinement. Therefore, the triangular mesh edges alone do not provide sufficient information to approximate the crumpled geometry. This limitation is resolved in the spatial and radial distance approach to edge generation because it simultaneously connects nodes along the triangular mesh sheet and across narrow voids or ridges in the structure, providing additional information about the crumpled mesoscale geometry. To illustrate the outcome of the extraction of the vertices and ridge network from the mesh structure, a complex network representation of a crumpled sphere is plotted in Fig. 1. It is important to emphasize that the complex network represents the ridges and vertices generated from crumpling. The network is not a two-dimensional triangulated mesh sheet.

Densely populated regions appear as highly connected nodes in the network. The unweighted, undirected network of *n* nodes is represented as an *n* × *n* adjacency matrix, **A**. For connected node pairs, *A _{i,j}* = 1 and zero otherwise, resulting in a symmetric square adjacency matrix. Having generated the complex network representation, we can now relate global measures of the network’s topology to the additive manufacturability of the structure through various heuristics.

### 3.2 Additive Manufacturability Heuristics.

Based upon experimental results on the morphology of crumple-formed structures [22,16], we theorize an inverse curvilinear relationship (i.e., a “U”) between the additive manufacturability of multiscale aperiodic structures and the degree of topological heterogeneity produced by crumpling. At no crumpling, a thin sheet can be additively manufactured with zero defects—it is just a flat sheet. Similarly, if the sheet were perfectly folded along an axis, producing an “accordion” with a width equal to the minimum thickness producible by the additive manufacturing device, the structure can be additively manufactured with zero defects. At very low levels of crumpling, the thin sheet will not contain layers, and its morphology will be regular. As the thin sheet is crumpled, voids or pores will begin to appear between ridges. The crumpled structure becomes less spatially isotropic. All of these features can cause additive manufacturing problems. At higher levels of compaction, the geometric features become more isotropic [22] and tend to follow power laws [16]. Hence, additive manufacturability should increase. The following four heuristics and associated topological network properties predict additive manufacturability given this theorized curvilinear relationship.

#### 3.2.1 Cross-Layer Giant Component Size.

In complex network representations, a connected component is a group of linked nodes. The giant component, or the largest connected component, is measured as the greatest quantity of connected nodes. For this property, we split the network representation into a series of cross-sectional slices mimicking layer-by-layer AM processes. In simple solid geometries, each cross-sectional layer contains one fully connected component. In heterogeneous geometries like crumpled thin sheets, the cross-sectional network representation may be disconnected. The giant component in a disconnected network layer represents the largest connected region of the material. Abrupt increases or decreases in the size of giant components provide an indication of material heterogeneity along the depth of the object. Heterogeneous layers may lead to enclosed voids or overhangs, which are both attributed to common AM defects.

Heuristic 1. *A more homogeneous distribution of cross-sectional material, or giant components, in the build direction is associated with lower probability of manufacturing defects.*

#### 3.2.2 Node Degree Distribution.

Node degree is the sum of edges connected to a node. In network representations of crumpled thin sheets, node degree provides an indication of surface area and material density. Node degree is low in regions of low material density, where the thin sheet is loosely crumpled or along the outer boundary surface of the crumpled macrostructure. Node degree is higher in regions where the simulated sheet self-intersects or where the sheet has been deformed into tightly packed layers. While regions with low curvature (low node degree) may result in overhangs, complex and fine features (high node degree) may limit manufacturing fidelity due to technology-specific clearance and feature size constraints. If the distribution is homogeneous (more convex), then most nodes have a similar node degree and material density is more uniform. If the distribution is non-homogeneous and the node degree is high, there are many pockets of fine geometric features that may limit manufacturability.

Heuristic 2. *A non-homogeneous distribution of nodes with a high degree will decrease manufacturability.*

#### 3.2.3 Average Shortest Path Length.

*l*in Eq. (1), is measured as the average distance

*d(i, j)*across the giant connected component in the network. Distance is measured as the number of edges along the shortest path between each node pair, averaged over the total nodes

*n*in the network [5]

In terms of DfAM, the shorter path length of the network representation indicates one of two structures: a low-density object with an open pore structure, or a compact object with few internal voids. Both structures may be suitable for AM technologies that require excess material removal during post-processing. The interconnected, open void structure of the former enables the removal of loose powder, resin, or other support material. The more homogeneous, high-density structure of the latter may be less influenced by trapped material because it contains fewer, smaller voids.

Heuristic 3. *A decreasing ASPL of the giant component correlates with increasing manufacturability by AM technologies that require post-processing for the removal of internal support material.*

#### 3.2.4 Network Robustness.

Robustness is a topological property that describes the error tolerance of a complex network undergoing random or targeted node removal [23]. Removing a node results in the removal of its edges. Network edges may represent anything from the flow of information to physical material connections, such as those in the multiscale aperiodic objects analyzed in this research. Depending on node degree and redundancy in the network connections, the removal of a node may lead to a disconnected network. A network topology is robust if the largest connected component contains the majority of nodes in the network during progressive node removal [23]. In this study, error tolerance is measured as a change in the size of the giant component with respect to the fraction of nodes removed by the random attack.

With respect to DfAM, a sudden decrease in the giant component size due to random node removal indicates geometric sensitivity to structural defects in the manufacturing process. Lack of material in narrow, load-bearing regions in the structure may result in total build failure, whereas material removal from a densely populated area is mitigated by surrounding reinforcement. Furthermore, material removal near the object's surface may only result in cosmetic defects. Manufacturing defects and severity are also AM technology and material specific. Node removal may simulate material gaps due to insufficient sintering, melting, material deposition, or photo-polymerization.

Heuristic 4. *Additive manufacturability correlates positively with the robustness of the network representation of the aperiodic structure.*

#### 3.2.5 Validation of Heuristics.

Two forms of validity were used to validate the heuristics: cross-model validity and external validity. Cross-model validity checks whether multiple models produce the same results. In this research, the network properties of the as-designed are checked against the properties of the as-printed crumpled-formed structures for similarity. There should be congruency between the four global network properties of the as-designed and as-printed structures as described in Sec. 3.2. If the global network properties do not align, then they would be inappropriate to assess these types of structures.

External validity determines whether the heuristics correctly assess the manufacturability of the objects under known conditions that should cause additive manufacturing errors. For this research, five printed specimens of each designed cylindrical structure were evaluated under uniaxial compression. Even relatively small imperfections could substantially affect the mechanical properties of crumple-formed structures [22]. If there are no errors in AM, then the compressive yield stress of the specimens should be similar among the structures designed, printed, and post-processed with the same set of parameters. Variation in yield stress is measured as a way to assess the external validity of the heuristics.

To support both types of validation, we additively manufacture the simulated crumpled-formed structures shown in Figs. 2 and 9 on a Figure 4^{™} Modular 3D printer, a vat photo-polymerization print platform from 3D Systems using PRO-BLK 10 resin. We chose a high-precision AM technology and durable material, designed for the manufacture of complex forms on a length scale suitable for microCT imaging. Even with considerable precision, variation may occur due to the multiscale geometric features of crumple-formed thin sheets. Random AM errors may be magnified across length scales, resulting in multiscale defects that are difficult to predict using existing DfAM techniques.

Network representations of the additively manufactured crumpled structures were generated based on images captured using micro-computed tomography (microCT) on a customized Hamamatsu L10711-19. MicroCT is a non-destructive X-ray scanning technique that enables imaging of internal topology at micrometer resolution and is used in other research to understand the morphology of crumple-formed structures [22]. Initial reconstruction of microCT scan data produces a three-dimensional stack of cross-sectional image slices, reminiscent of the layer-by-layer print process of many AM technologies. The individual image slices are converted to cross-sectional contours with a two-dimensional Marching Squares algorithm [24], isolating the surface of the crumpled object based on pixel intensity. To reconstruct a complete three-dimensional surface from the microCT data and reduce noise in the scans, we implement the Marching Cubes algorithm to sample intensity values across the full stack of cross-sectional image slices [25]. The Marching Cubes surface reconstruction process results in a triangular surface mesh that is analyzed in the same manner as the CAD crumpled object.

The microCT scans and network edge generation process may contain noise, which can obscure the underlying structure of crumpled objects. To eliminate noise and reveal statistical properties of the crumpled structure, we apply Eigenvalue Decomposition (EVD) to the geometric features [12]. In the following previous work employing EVD or singular value decomposition (SVD) [12], the network adjacency matrix **A** may be factorized into an *n* × *n* matrix of eigenvectors and a diagonal matrix of eigenvalues. Plotting eigenvalues by descending magnitude reveals characteristics of the network modularity [6]. High-magnitude eigenvalues contain significant information about network topology, while low-magnitude eigenvalues are treated as noise or redundant spatial information in the network. Dimensionality reduction is achieved by identifying the extent of low-magnitude eigenvalue terms, balancing noise reduction and information loss. For consistent crumpled object analysis, dimensionality reduction is performed by retaining the first 20% of high-magnitude eigenvalue terms.

As an external validity check of the proposed heuristics, mechanical testing of the additively manufactured crumpled cylinders is performed under uniaxial compression on an Instron^{®} 5982 electromechanical universal testing system. Compressive load is applied along the AM layer-by-layer build axis. Specimen dimensions of the designed crumpled cylinders approximate a right cylinder whose length is twice its diameter, in accordance with ASTM D695-15. The length and diameter dimensions are varied to achieve the desired crumpling compaction percentage. The uniaxial compression tests are performed with a crosshead speed of 1.3 mm/min.

## 4 Case Studies

This section presents the analysis of two crumple-formed structures, a sphere and a cylinder, to illustrate the application of the heuristics and validate their suitability for additive manufacturability analysis.

### 4.1 Crumple-Formed Spheres.

Computationally designed crumpled spheres were generated starting with thin triangulated mesh sheets composed of 13,189 vertices with self-avoidant spheres (0.3 mm in diameter) positioned at each vertex to simulate material thickness and minimize self-intersection of the sheet. Crumple compaction is calculated as the total volume of self-avoidant spheres positioned at each vertex of the mesh sheet divided by the final volume of the confinement shell. Figure 2 depicts rendered CAD models of the five crumple-formed spheres with incrementally increasing compaction from 20% to 100%. A solid sphere (far right in Fig. 2) was modeled to compare crumple-formed spheres to a solid homogeneous object of similar macroscale geometry. Topological features of the additively manufactured crumpled spheres are measured using microCT, and the images are reconstructed into triangular surface meshes from which complex networks are extracted for comparison to those networks generated from the CAD models. Equatorial microCT slices in Fig. 3 show that each crumple sphere is composed of a mesostructure of ridges and folds with a non-homogeneous material distribution at lower compaction. This mesostructure exhibits an aperiodic distribution of material (shown in white in Fig. 3) and internal voids (black regions within the white circular cross section).

Complex network approximations are generated for the five designed crumpled spheres (Fig. 2) and the reconstructed microCT surface mesh of the same objects after fabrication by AM. Eigenvalue decomposition is performed on the adjacency matrices as explained in Sec. 3.2.5. The eigenvalue spectra for the complex networks generated from the microCT images are plotted in Fig. 4. The solid and crumpled spheres exhibit varying degrees of hierarchical modularity. Modular and hierarchically modular networks have an eigenvalue spectrum characterized by a stepped pattern with groups of singular values clustered together [6]. None of the eigenvalue spectra exhibit the properties of a random network. The eigenvalue spectrum of a random network always has a recognizable peak difference between the magnitudes of the first and the second singular value [6].

Figure 5 displays variation in the layer-by-layer giant component size for both the designed (Fig. 5(a)) and AM (Fig. 5(b)) crumpled spheres. In addition to demonstrating cross-layer structural homogeneity, this network property is an initial indication that the proposed methodology produces congruent results when applied to computationally designed or manufactured objects. Figure 5 reveals that the cross-sectional giant component size is more variable at low levels of compaction, while high crumpling compaction results in a more homogeneous distribution of material, as explained in Heuristic 1. The designed and additively manufactured objects at 80% and 100% compaction most closely approximate an inverted U-distribution. An inverted “U” describes the giant component distribution of a solid sphere. This distribution of the cross-layer giant component size reflects increasing cross-sectional area toward the equator of the sphere-like macrostructure. Another indication of congruent results is the peak and subsequent dip in giant component size from 0.6 to 0.8 of the fractional depth of the crumpled object at 20% compaction. This feature is reflected in both the designed and AM object at loose crumpling compaction.

The node degree distribution is the second network property analyzed for the designed and manufactured crumpled spheres. Complementary cumulative frequency distribution of node degree is plotted for the designed crumpled spheres at varying compaction percentages in Fig. 6(a), and the microCT reconstructed models of their additively manufactured counterparts in Fig. 6(b). More non-homogeneous distribution of node degrees (few well-connected nodes with very high degrees and many nodes with low degrees) is shown in the AM crumpled objects in Fig. 6(b) due to the Marching Cubes method used to reconstruct each series of cross-sectional microCT scans as a surface mesh. Triangulation of the reconstructed surface mesh was performed to maximize topological accuracy using the same number of vertices as the original CAD model, with few large triangles in regions of low curvature and many small triangles detailing regions of high curvature. In contrast, the designed crumpled objects described in Fig. 6(a) were modeled from a mesh sheet composed of uniform triangles. Figure 6(a) highlights the separation between the more non-homogeneous distribution of high-degree nodes at 20% compaction and the more homogeneous node degree distribution at higher compaction. As expected, node degree distribution becomes more homogeneous at higher levels of compaction (i.e., the cumulative distribution shifts from left to right as crumpling compaction increases). Heuristic 2 suggests that the loosely crumpled (20% compaction) sphere would be more challenging to manufacture. Greater compaction results in more highly connected nodes for both the designed and AM crumpled spheres.

Figure 7 plots the average shortest path length (ASPL) of the largest connected component in each crumpled sphere network representation with respect to compaction percentage for both the designed and AM structures. In both the designed and AM crumple spheres, ASPL is the longest at intermediate (60%) crumpling compaction, with shorter ASPL at either end of the compaction spectrum. The non-monotonic relationship between ASPL and crumpling compaction is attributed to the counteractive effects of high node density at high crumpling compaction versus small giant component size at low compaction. The giant component size of the full network (as opposed to individual cross-sectional layers described in Fig. 5) is small at low crumpling compaction because nodes are loosely spread out across the geometry. Sparse node placement results in multiple isolated components, for which ASPL is measured across the largest of these connected components. Crumpled structures at high compaction exhibit a larger giant component than the structures at low compaction, which is offset by greater node density. High-degree, well-connected nodes reduce the path length between any one node and others within the giant component. At intermediate compaction, moderate giant component size is not sufficiently offset by high node degree, resulting in long ASPL as shown in Fig. 7.

According to Heuristic 3, crumpled spheres at 20% and 100% compaction are more suitable for AM technologies that require the removal of internal support material than the structures at intermediate crumpling compaction. At 20% compaction, this behavior is attributed to an open cavity structure that facilitates the removal of internal support material. At 100% compaction, the pore structure becomes closed, but overall porosity also decreases. Therefore, trapped material will have a smaller impact on the final geometry at high compaction.

The fourth and final network topology property analyzed for the designed and additively manufactured crumpled spheres is robustness as explained in Sec. 3.2.4. Figure 8 displays the size of the giant component in the network representation with respect to extent of random node removal in the designed and AM crumpled spheres.

In a perfectly robust network, this plot appears as a diagonal line, because the giant component size is only reduced by a single node at each stage of the random attack process. In an imperfect network, node removal may create a disconnected topology shown by an abrupt reduction in the size of the giant component. Robustness increases with higher compaction for both the designed and additively manufactured objects as suggested by Heuristic 4. Crumpled spheres at 20% compaction display the greatest sensitivity to random node removal.

The designed solid sphere in Fig. 8(a) and additively manufactured crumpled spheres at 80% and 100% compaction in Fig. 8(b) embody the most robust networks. These networks only become disconnected after more than 80% of nodes have been removed. Robustness similarity between the CAD solid sphere and manufactured crumpled sheets at high compaction reveals loss of topological complexity during additive manufacturing, indicating that the Figure 4^{™} Modular photo-polymerization platform may not sufficiently capture the multiscale complexity of the designed crumple spheres at high compaction.

The output from a built-in 3D-printing mesh analysis tool in an open-source computer graphics and CAD software (Blender) emphasizes the challenge of interpreting the manufacturability of multiscale aperiodic geometries using conventional surface feature metrics. The three metrics of interest for AM listed in Table 1 are the number of intersecting, thin, and overhanging faces. Table 1 provides inconclusive design advice because the number of intersecting and thin faces tends to increase with compaction, but the number of overhanging faces is fairly stable. As such, manufacturability analysis based on these results would be challenging for multiscale aperiodic objects.

### 4.2 Crumple-Formed Cylinders.

This case study examines the generality of the proposed statistical mechanic’s approach to DfAM when applied to a different type of crumpled structure: a thin sheet crumpled within a cylindrical shell.

Retaining the same sheet size and thickness parameters from the first case study, cylindrically crumpled thin sheets were generated at five levels of compaction: 20%, 40%, 60%, 80%, and 100%. Figure 9 depicts the five rendered crumpled cylinders and one solid, homogeneous cylinder. Similar to the solid sphere in Fig. 2, a solid cylinder is modeled to compare crumpled cylinders to a simple homogeneous geometry with similar macrostructure and known manufacturability success. In this case study, we additively manufacture the solid cylinder using the same AM process and material as the crumpled cylinders. Figure 10 contains cross-sectional microCT images of the AM crumpled and solid cylinders with printed material shown in white for high contrast. As-designed and as-printed geometric comparison is made by overlaying the topological robustness property of the solid and crumpled cylinders.

The same manner of EVD and dimensionality reduction is applied to the crumpled spheres and cylinders, retaining only the top 20% of terms with high-magnitude eigenvalues. Furthermore, crumpled cylinders are evaluated by the same four topological network properties and design heuristics. This case study enables relates network topology to multiscale geometry, providing designers with the building blocks to interpret manufacturability of disparate aperiodic objects.

Figure 11 describes the cross-layer homogeneity of crumpled cylinders at varying degrees of crumpling compaction for the designed and AM objects. This network property demonstrates the impact of macrostructure on crumpled network topology. For the crumpled spheres, homogeneous distribution of cross-sectional material approximates an inverted-U shape (Fig. 5). Homogeneous distribution of cross-sectional material in the cylinders should be a horizontal line in Fig. 11, describing constant giant component size along the axial depth of the cylinder. Crumpled spheres and cylinders both exhibit more homogeneous cross-layer material distribution with increasing compaction. Evaluating Heuristic 1 for different macroscale geometries reveals that the designer should understand what a perfectly homogeneous distribution would look like for the macroscale geometry of interest. AM build orientation will also impact this network property in objects like cylinders with non-isotropic macrostructure. The designer may opt to perform this analysis in the desired build orientation or test the network property in multiple orientations to determine which results in a more homogeneous distribution of cross-layer material.

Complementary cumulative frequency distribution of node degree for the designed and AM crumpled cylinders are displayed in Fig. 12. Like the node degrees of the crumpled spheres (Fig. 6), the distribution may not be identical between as-designed and as-manufactured objects, but the overall trend is in agreement. Greater compaction of the crumpled cylinders results in a more homogeneous (convex) distribution of node degree. Per Heuristic 2, manufacturability increases in Fig. 12 from left (low compaction) to right (high compaction).

Figure 13 shows the ASPL of the giant component in each crumpled cylinder with respect to compaction. ASPL of the crumpled cylinders in Fig. 13 exhibits a step-wise behavior. The discontinuity between shorter ASPL at low compaction and longer ASPL at high compaction may be due to a jump in the total crease length of the crumpled cylinders above 40% compaction. This may result in a fully connected network that is not sufficiently offset by node density to reduce the ASPL at higher compaction. Per Heuristic 3, crumpled cylinders at 20% and 40% compaction are more suitable for manufacture by AM technologies that require an open pore structure for support material removal during post-processing. Figure 10 only depicts a single cross-sectional layer within the additively manufactured cylinders, but it effectively demonstrates that internal voids (shown in black) are more likely to be enclosed within the structure as crumpling compaction increases. At 60% compaction and greater, the pore structure becomes more closed due to the self-intersection of the thin sheet and tight crumpling near the boundary shell.

The designed crumpled cylinders exhibit fairly consistent robustness regardless of compaction level as shown in Fig. 14(a). Figure 14(b) shows disperse robustness behavior of the AM crumpled cylinders because the microCT reconstructed node locations of the manufactured objects are less precisely known than nodes in the designed crumpled structures. Even so, the same trend proposed by Heuristic 4 is observed: increasing robustness at greater crumpling compaction for the designed and AM crumpled spheres and cylinders in Figs. 8 and 14, respectively. Furthermore, each crumpled object is more sensitive to random node removal than the homogeneous solid sphere or cylinder. These homogeneous structures effectively demonstrate how objects with high manufacturing fidelity and low sensitivity to microscale AM defects would perform under Heuristic 4.

Uniaxial compression tests of the AM crumpled and solid cylinders were performed to validate whether the network heuristics correctly analyze the manufacturability of the crumpled objects. As stated previously, even minor defects in printing can substantially affect the mechanical properties of crumpled structures. Five replicates of each sample are manufactured using the same AM process and material. Each specimen is tested under uniaxial compression (Fig. 15) to evaluate compressive yield stress, *σ _{y}*, measured as compression force at the yield point, applied across the cross-sectional area of the sample. Cross-sectional area is a non-trivial measurement due to the non-homogeneous geometry of crumpled objects. In this study, the average cross-sectional area from each microCT image slice of the manufactured samples was evaluated. Figure 16 displays the uniaxial force versus crosshead displacement curve averaged across each specimen. The compressive yield point for each individual specimen is marked where the test transitions from a linear elastic regime to a non-linear plastic regime. Mean compressive yield stress of each sample is listed in Table 2. In following Heuristics 1, 2, and 4, manufacturing defects that affect mechanical strength are more pronounced at low (20%) compaction. This behavior is evidenced by the large relative standard deviation (RSD) on the average compressive yield stress of cylinders at 20% compaction in Table 2, followed by consecutively decreasing RSD with increasing crumpling compaction. As expected, solid cylinder specimen exhibit the narrowest spread of yield stress values.

Sample | σ (MPa)_{y} | RSD (%) |
---|---|---|

20% Crumpling compaction | 3.60 | 22.12 |

40% Crumpling compaction | 5.50 | 18.69 |

60% Crumpling compaction | 9.27 | 13.01 |

80% Crumpling compaction | 16.48 | 8.77 |

100% Crumpling compaction | 53.15 | 4.60 |

Solid cylinder | 80.62 | 1.13 |

Sample | σ (MPa)_{y} | RSD (%) |
---|---|---|

20% Crumpling compaction | 3.60 | 22.12 |

40% Crumpling compaction | 5.50 | 18.69 |

60% Crumpling compaction | 9.27 | 13.01 |

80% Crumpling compaction | 16.48 | 8.77 |

100% Crumpling compaction | 53.15 | 4.60 |

Solid cylinder | 80.62 | 1.13 |

## 5 Discussion and Conclusion

This paper examined the statistical mechanics properties of complex network representations of ridges and vertices resulting from crumpling a thin sheet as the basis for additive manufacturability analysis. Two types of crumpled structures were tested: a crumpled sphere and a crumpled cylinder. Additive manufacturability heuristics were developed from four global topological properties of complex networks: cross-layer giant component size, node degree distribution, average shortest path length (ASPL), and robustness. The heuristics were applied to the complex network representations of crumpled spheres and cylinders at varying compaction to understand the relationship between material distribution and additive manufacturability.

The first case study compared the network properties of designed and additively manufactured crumpled spheres. Each network property produced corresponding results between the designed and additively manufactured objects. The network properties confirmed that the crumpled spheres are increasingly homogeneous and more tolerant to additively manufactured defects at higher compaction (*>*20%). ASPL does not trend monotonically with compaction. ASPL results indicated that crumpled spheres at opposite extremes of the compaction spectrum are more suitable for fabrication by AM processes that require support structure removal than the spheres at intermediate compaction. This behavior is attributed to an open pore structure at low compaction and fewer internal voids to trap material at high compaction.

The second case study investigated crumple-formed cylinders to test the effect of changing macroscale geometry on network properties and the interpretation of the heuristics. Like the crumpled spheres, crumpled cylinders become more homogeneous at high compaction. Unlike the spheres, cylinder network robustness does not trend strongly with compaction. Furthermore, cylinder ASPL approximates a step-wise distribution, with short path lengths at low compaction and longer path lengths at 60% compaction and greater. This behavior indicates that the crumpled cylinders have an open pore structure at 20% and 40% compaction for easier removal of support material. Long ASPL at high compaction diverges from the crumpled sphere behavior and indicates that the cylinders have a different mesoscale void distribution.

In Sec. 3.2, we theorized an inverse curvilinear relationship between additive manufacturing fidelity and topological heterogeneity of multiscale aperiodic structures. The results from both case studies generally describe a monotonically increasing relationship between 20% and 100% compaction and manufacturability. While objects with very low levels of crumpling such as 1% or 5% were not produced, it is evident that such objects would not have additive manufacturing problems since this level of crumpling would produce a simple geometry. Rather than a curvilinear relationship, there could be an abrupt transition to additive manufacturing problems when the crumpling is sufficient to produce folds and networks of ridges. This research demonstrated that additive manufacturing problems will typically occur at modest levels of compaction. Therefore, if a design application calls for mechanical properties produced at low to medium levels of compaction, the results show that those mechanical properties will be the most challenging to produce reliably given the potential for additive manufacturing defects.

Heuristic identification and interpretation is only one aspect of successful DfAM. If manufacturing errors are predicted for the crumpled structure, what might the designer change to increase additively manufactured fidelity? Improving the manufacturability of multiscale aperiodic structures is not as simple as altering individual geometric features in the structure. Isolated feature changes may propagate along the structure or across length scales. In lieu of isolated changes, the designer may change parameters of the crumpling generation or consider a different AM process. For example, if design requirements mandate a low compaction ratio, the designer may vary the geometry of the confinement shell to achieve a more robust structure. Manufacturability may also improve with an alternative AM technology, depending on predictions of the ASPL property. In this paper, the crumpled structures are all designed using a confinement shell approach. However, applying alternative crumpling methods, such as deformation by growing the sheet within a set volume [20], or multiple iterations of crumpling and unfolding [16], may produce a different material distribution and different manufacturability results.

This research provided a framework for the additive manufacturability analysis of multiscale aperiodic objects. We explained how designers can approach the interpretation of network properties of the ridges and vertices produced during crumpling for different multiscale aperiodic geometries. These results, combined with prior theoretical and experimental observations on the statistical mechanics properties of the network of the ridges and vertices produced during crumpling, provide evidence to support the use of statistical mechanics as an approach to additive manufacturability analysis. The corresponding heuristics provide consistent results when applied to designed and additively manufactured structures. In the future, this framework of approaching DfAM from a statistical mechanics perspective may be extended to other fields of design for X, facilitating concurrent engineering of complex engineered structures and systems.

## Acknowledgment

This work was made possible by the use of Oregon State University’s microCT facility, a user facility developed with support from the Major Research Instrumentation Program of NSF’s Earth Sciences (EAR) directorate under award #1531316.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.