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Implicit Slicing for Functionally Tailored Additive Manufacturing

Conference Paper · August 2016

DOI: 10.1115/DETC2016-59638

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Proceedings of the ASME 2016 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2016August 21-24, 2016, Charlotte, NC, USA

DETC/CIE2016-59638

IMPLICIT SLICING FOR FUNCTIONALLY TAILORED ADDITIVE MANUFACTURING

John C. SteubenAthanasios P. Iliopoulos

John G. MichopoulosComputational Multiphysics Systems Laboratory

Center of Materials Physics and TechnologyNaval Research LaboratoryWashington DC 20375, USA

ABSTRACTOne crucial component of the additive manufacturing softwaretoolchain is a class of geometric algorithms known as “slicers.”The purpose of the slicer is to compute a parametric toolpath de-fined at the mesoscale and associated g-code commands, whichdirect an additive manufacturing system to produce a physicalrealization of a three-dimensional input model. Existing slicingalgorithms operate by application of geometric transformationsupon the input geometry in order to produce the toolpath. Inthis paper we introduce an implicit slicing algorithm that com-putes mesoscale toolpaths from the level sets of heuristics-basedor physics-based fields defined over the input geometry. Thisenables computationally efficient slicing of arbitrarily complexgeometries in a straight forward fashion. The calculation ofcomponent “infill” is explored, as a process control parame-ter, due to its strong influence on the produced component’sfunctional performance. Several examples of the application ofthe proposed implicit slicer are presented. It is demonstrated–via proper experimentation– that the implicit slicer can pro-duce a mesoscale structure leading to objects of superior func-tional performance such as greatly increased stiffness and ulti-mate strength without an increase of mass. We conclude withremarks regarding the strengths of the implicit approach relativeto existing explicit approaches, and discuss future work requiredin order to extend the methodology.

INTRODUCTIONAdditive manufacturing (AM), also known as layered manufac-turing, rapid prototyping, or less formally as 3D printing, is anincreasingly important family of fabrication techniques for theproduction of a wide variety of components. These fabricationtechniques are characterized by successive additions of material

to a domain, as opposed to the repeated subtractions that areemployed by most traditional fabrication technologies [1]. Re-cent years have seen a surge of interest in additive manufacturingtechnology from a broad number of engineering and manufactur-ing disciplines. Besides the emergence of inexpensisve consumergrade systems, this interest is primarily driven by the opportunityof the relative freedom from geometric constraints provided byadditive manufacturing methods; geometries that are difficult orimpossible to produce by conventional means are often readilyachievable via AM.

At the present time, a variety of AM technologies exist.Common techniques include stereolithography [2], Fused De-position Modeling (FDM) [3], Selective Laser Sintering (SLS)[4–6], Electron Beam Melting (EBM) [7], and Direct Metal De-position (DMD) [8,9]. The mechanical details of these processesvary considerably, but they share a common software toolchain,sometimes refered to as as the “digital thread.” A block diagramof the AM-specific digital thread concept is shown in Fig. 1.It should be noted that the conventional usage of the term “digi-tal thread” in the AM community refers only to CAD modeling,STL represenation, and transmission to the 3D-printer. What ispresented in Fig. 1 represents our own extended definition of theterm “digital thread.” Because of the highly integrated nature ofthe additive manufacturing software implementations producedby AM hardware manufacturers, most of the individual com-ponents of the digital thread (usually those that are executed onthe runtime system of the 3d-printer itself) are are frequentlyinvisible and not accessible by the end-users.

As Fig. 1 shows, the AM-specific digital thread is subdi-vided into three major stages; a design environment, a pre-processing environment, and a manufacturing environment. The

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DesignEnvironment

Triangularization

Design Model

PreprocessingEnvironment

Process ControlParameters

SlicingBuild Layout

ManufacturingEnvironment

Output Object

Motion Control

Design Intent

FIGURE 1: An overview of the ”digital thread” concept.

digital thread begins in the design environment, and originatesfrom a Computer-Aided Design (CAD) model produced by a de-signer. The ultimate goal of the additive manufacturing process isto produce this model within acceptable constraints on accuracy,time, cost, and other parameters. Within the design environment,this geometry is converted to a triangular mesh form, typicalyencoded as a stereolithography (STL) or an additive manufac-turing file (AMF), or similar file. It is important to note that thisconversion preserves approximate [10] geometric information re-garding the original model only. Any other ancillary informationencoded within the original model is lost in this process. Mov-ing to the preprocessing stage, the position of the resultant meshwithin the build volume of the additive manufacturing processis determined by a layout optimization routine. In practice, acollection of many meshes is packed into the build volume ofthe machine in order to reduce per-unit production costs, using amethod such as that presented in [11]. The mesh (or collectionof meshes) is then processed by an algorithm known as “slicer.”The purpose of the slicer is to subdivide the mesh(es) into a se-ries of distinct layers, and to compute the numeric control (NC)commands issued to the additive manufacturing machine in orderto produce the distinct toolpaths making up each layer. The buildlayout and slicer tools are often combined into a single commer-cial software product, that largely behaves as a “black box.” Oncethe toolpath has been produced and the proper G-code sequenceshave been generated, the motion control software and hardwaresystems present in the manufacturing environment are used todrive the additive manufacturing machine in order to produce theoutput object.

The various stages of the additive manufacturing digitalthread have been studied and developed for decades, and in manysenses have reached a high level of development sophistication[12, 13]. One important shortcoming of the current state of theart is the aforementioned loss of design information that occursat the interface between the design environment and the prepro-

FIGURE 2: Illustration of toolpath concepts. Note the perimetershells (black) that enclose a sparse infill pattern (red); this combi-nation of path types is commonly used in additive manufacturingapplications.

cessing environment. This effectively reduces the additive man-ufacturing process to a purely geometric exercise. While this isacceptable for established uses of additive manufacturing such asthe production of “look and feel” prototypes, it is of primary im-portance to ensure that the AM-produced parts can satisfy theirfunctional performance requirements in terms of exhibiting spe-cific functional properties. Examples of functional properties in-clude yield and ultimate strengths, elastic anisotropy constants,residual strains, and thermal or electrical conductivities. Initialefforts documenting such activities can be found in [14, 15].

The primary focus of this paper is the development of a newtype of slicing algorithm that enables control of the mesoscaletopology of the toolpaths in a manner that enables control ofthe functional performance of the final part. The motivation forthis development is to reduce the deficiency of design data inthe preprocessing domain, and facilitate ongoing efforts to de-velop functionally imbued additively manufactured objects. Inorder to do so, we adopt a fundamentally different approach tosolving the slicing problem. Unlike existing algorithms, that op-erate on the basis of explicit geometric transforms applied to theinput geometry, we employ a novel implicit method based onthe computation of level sets of field functions. We explore theuse of field functions defined upon these regions in order to re-introduce design intent into the preprocessing environment. Inparticular, we develop a methodology by which the results of aphysics-based perfromance specification that can be driven by aFinite Element Analyses (FEA) may be used to dictate the com-putation of toolpaths in order to improve functional performancefields of interest such as strain and stress distributions, generatedwithin additively manufactured components.

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BACKGROUNDSlicers are a class of algorithms in the domain of computationalgeometry that are used to convert input 3D geometry into a seriesof motion commands (a “toolpath”) for an additive manufactur-ing machine. The slicer is required to both process the input ge-ometry into a suitable toolpath for additive manufacturing, andexport this toolpath as a series of numeric control (NC) com-mands (e.g. G-code sequences) that are subsequently conveyedto the additive manufacturing hardware. As the second stage ofthis process depends heavily on the specific additive manufactur-ing process and device employed, we restrict our discussion tothat of toolpath generation. Figure 2 demonstrates a hypotheticaltoolpath generated by a slicing algorithm.

Slicers and Associated TerminologyBeyond the similarities to existing NC toolpath generation, thereare several unique aspects of additive manufacturing slicing thatrender it a distinct research field. A review of early work inthis vein may be found in [16–19]. Various improvements tothese early algorithms, generally termed “adaptive” slicing canbe found in [20–25]. Efforts to improve the results of slicing,for instance by eliminating voids in the infill portion of the tool-path, are explored by [26, 27]. Recent research into slicing, forinstance the work of [28, 29], has primarily been focused on in-creasing computational performance. Additionally, some workincluding [18, 29] has focused on “direct” slicers, that bypassthe model triangularization step and convert the input CAD ge-ometry directly into toolpaths. Most recently, image-space ap-proaches to slicing have emerged [30, 31], that operate on point-cloud or volumetric image data in a reverse-engineering context.One such slicer utilizes General-Purpose Graphics ProcessingUnit (GPGPU) computing [32]. Additionally, slicers intended toproduce objects with spatially-varying stiffness properties havealso been developed [33].

A review of the available literature reveals that existing ap-proaches may be broadly characterized as explicit transforma-tions applied to input geometry that is usually explicitly defined(although the work of [30,31] considers implicit input geometry).Figure 3 illustrates the variable definitions associated with slic-ing algorithms. We denote the input geometric domain Ω ∈ R3,and its boundary Γ. A series of slicing planes P are intersectedwith Ω in order to create planar subdomains ω = Ω∩P that arebounded by γ . The subdomains and boundaries associated withthe ith slicing plane Pi are thus denoted ωi and γi, respectively.The explicit slicers discussed earlier can be represented by theform

g =nl⊎

i=0

no⊎j=0

Fj (γi) (1)

FIGURE 3: Visualization of slicer variables.

where g represents the singleton set containing the output tool-path, and Fj are the various geometric transformations applied tothe boundary γi of each layer. The number of layers is nl and thenumber of transformation operations per layer is no. To provide aconvenient framework for elucidating the similar and dissimilaraspects of explicit and implicit slicers, we introduced the opera-tor to express a special combination of the union operator forbags (in the set-theoretic context) with an ordering optimizationfrom the time efficiency perspective. That is to say that or-ders the results of the Fj (γi) transformations so that g may beproduced by the additive manufacturing system in minimal time.This optimization is beyond the scope of the present work, butfurther information on the topic can be found in [34].

As seen in Eq. (1), The previously mentioned slicing algo-rithms typically operate on γ only, due to its direct availabilityby computing simple triangle-plane intersections. While this isa sensible choice when the purpose of slicing is viewed from astrictly geometric viewpoint, it is problematic from the stand-point of producing components that imbue a functional purpose.The functional responses of interest, for example mechanicalstress, are defined as fields over Ω and thus over ω . In orderto develop a slicing procedure that incorporates these fields, andthus allows the re-introduction of design intent into the digitalthread, we turn towards an implicit method detailed in the nextsection.

PROPOSED METHODOLOGYOne method by which relevant functional fields may be incorpo-rated into the slicing procedure is by the adoption of an implicitmethodology. We consider the formulation

g ≡nl⊎

i=0

no⊎j=0

nk⊎

k=0

Hj (x) = ck,x ∈ ωi (2)

where Hj are the field functions, x is a position vector of a pointin the volume Ω defined in R3, and ck is a discrete value of thefield functions defining the kth level set. Our implementationof this formulation is represented by the multi-step procedure

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FIGURE 4: Sequence of operations for the implicit slicer.

shown in Fig. 4. The steps outlined in this figure are detailed inthe sections below.

Model Parsing and PreparationIn the present work a standard STL file is used for input, dueto its widespread use in the additive manufacturing industry. Thealgorithms presented in the following sections are trivially adapt-able to other input types. The only precondition placed on theSTL input files is that it must describe a manifold or finite set ofmanifolds, without gaps or degenerate geometric features.

The input STL file explicitly defines Γ as a set of triangularfacets (F), each defined by three vertices in R3,

Γ =

n f⋃j=0

F j =

n f⋃j=0

v j1,v j2,v j3

(3)

where n f is the number of facets and v are the vectors describingthe vertex coordinates. Because of the limitations of the STLfile format, that does not contain any long range ordering of thefacets nor any inter-facet connectivity information, Γ is knownonly in a piecewise, discontinuous fashion.

Computing Layer BoundariesThe subdomain boundaries γi are computed directly from Γ, bysequential intersections with the slicing planes Pi,

γi = Γ∩Pi =

n f⋃j=1

Pi ∩F j (4)

The triangle-plane intersection algorithm of [35] is utilized tocompute these intersections. Upon the completion of these in-tersection tests, γi is a set of line segments without organization.This renders it difficult to directly reconstruct ωi from γi; with-out further organization γ contains no information regarding the

FIGURE 5: Illustration of the boundary γi computed from a sliceof model of a valve body. Note how the boundary subsets γi1, γi2,and γi3 define “outer” edges; while γi4 and γi5 define “holes.”

topology of ω . This difficulty is illustrated in Fig. 5, that showsthat some subsets of γi define outer boundaries of ωi, while othersdefine internal “holes.”

To enable the recovery of ωi, γi is partitioned into contiguousordered subsets denoted γi j as shown in Fig. 5. This is triviallyequatable to the problem of finding the connected components ofa graph. We parse the lines and edges of γi to an edge-list graphand apply the classical algorithm of [36], that is based on depth-first searching. While this algorithms operates in linear time onany graph, it is particularly fast in the present application becausethe graph defined by γ exclusively contains vertices of order twoonly. At the conclusion of this operation, the γi j are cohesivelyordered loops of line segments. From these loops, we proceed toreconstruct the corresponding domain.

Computing Layer Domains

Once γi has been properly organized, the reconstruction of thedomain ωi may be achieved using several well-known algo-rithms. In the present work, we utilize constrained Delaunaytriangulation for this purpose. This approach is given in detailby [37, 38], implemented in several high-quality freely availablesoftware libraries, [39,40], and also available in commonly-usedcomputer algebra tools such as Mathematica and Matlab. Theresult of computing the ωi corresponding to the γi j of Fig. 5 isshown in Fig. 6. The procedure outlined in this, and the previous,section is repeated for every slicing plane in order to compute allelements of ω . Once this process has been completed, these do-mains are defined as a set of planar facets F . These domains areused to bound the implicit field functions and compute the slicertoolpath, as described in the next section.

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FIGURE 6: The domain ωi corresponding to the γi j of Fig. 5calculated using constrained Delauny triangulation.

Implicit Field ComputationOnce the boundaries and domains for each layer have been de-fined, the additive manufacturing machine toolpath is determinedby the calculation of level sets of implicit functions upon thesedomains and boundaries. Equation (2) gives the most generalform of this procedure, in practice a decomposed form is used

g = gpr⊎

gin⊎

gother⊎

. . . (5)

where again indicates an operator representing the optimizedinterleaved sequencing of toolpath segments. gpr and gin are as-sociated with the perimeter and infill portions of the toolpath, asseen in Fig. 2. The gother term reflects other toolpath compo-nents, such as support material, that are commonly required butoften manufacturing technology specific. In the present work,we focus on the perimeter and infill toolpaths. For simplicity,we examine the case where the perimeters and infill are dictatedeach dictated by one field function and an associated set of levels

gpr ≡nl⊎

i=0

nk⊎

k=0

Hpr (x) = ck,x ∈ ωi (6)

gin ≡nl⊎

i=0

nk⊎

k=0

Hin (x) = ck,x ∈ ωi (7)

In general, several different fields could be weighted andcombined to weigh different design functional objectives in or-der to produce either gpr and gin. In the following sections, wediscuss the formulation of these functions, and the associated se-lection of values for ck in greater detail.

Computing Perimeter Shells The most fundamentalrequirement of the slicer is that the output toolpath must repli-cate the geometry of the input model. Naively interpreted, thissuggests that

Hpr (x) = c0,x ∈ γi (8)

FIGURE 7: Results of the application of the distance transformto a portion of the domain seen in Fig. 6.

Since the shape of γ must be strongly encoded in Hpr, the signeddistance transform is the most logical selection for this function.With this choice,

Hpr (x) =

min‖x−xγ‖,xγ ∈ γi x ∈ ωi

−min‖x−xγ‖,xγ ∈ γi otherwise(9)

Figure 7 illustrates the perimeters generated by this approachwhen applied to a portion of the domain seen in Fig. 6. Fourperimeters are computed, with ck = 0.0,0.5,1.0,1.5 mm. It isseen that the implicit formulation can handle cases of vanishingand subdividing inner perimeters without any difficulty. In prac-tice, it may not be desirable to run the outermost perimeter atHpr (x) = 0.0. For instance, in FDM additive manufacturing, thepolymer extrusion has a width ε . In this case, a sensible schemeis to choose ck = ε 1/2,3/2, . . ..

Computing Volumetric Infill While the form of Hpris necessarily dictated by γ , the infill function Hin is not simi-larly constrained. This is the principal advantage of the implicitmethodology, as the infill may be dictated by an arbitrary fieldof functional or design interest. Several examples are given inFig. 8. Parts (a) and (b) of Fig. 8 show simple closed-form func-tions that replicate infill strategies commonly employed by exist-ing commercial slicers. Parts (c) and (d) show alternate closed-form selections, that demonstrate the flexibility of the implicitapproach. Part (e) shows an infill pattern based on a stochas-tic pseudo-random number generator, and part (f) shows an infillpattern based on the solution to Poisson’s equation with a Dirich-let boundary condition on γ .

The infill patterns of Fig. 8 are representative of one layerωi of the entire toolpath. It is typical to modulate the infill pat-tern between layers, for instance to rotate the linear infill by 90

between layers. In Fig. 8 it is assumed that the ωi are parallelto the xy-plane. In this frame of reference, the modulation wouldtake the form of a functional dependence of Hin on z.

The only restriction on Hin is that it must be a function that

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(a) Concentric infillHin (x) = Hpr (x)

(b) Linear infillHin (x) = axx +bxy

(c) Wave infillHin (x) = asin(bxx)+xy

(d) Eggcrate infillHin (x) = sin(axx)

2 + sin(bxy)2

(e) Randomized infillHin (x) = min‖x−xrand‖

(f) Poisson infill∇2Hin = a,Hin (x) = 0∀x ∈ γ

FIGURE 8: Various selections for the infill function. xx and xydenote the Cartesian components of x, while a and b denote freeparameters that may be changed to vary the infill pattern (e.g.scaling, density, or rotation).

returns a real-valued result. This opens the possibility of usingthe infill to couple the performance properties of the additivelymanufactured component to some field of engineering or designinterest. For example, the stress or strain fields computed by FEAcould be used as the basis for the infill geometry. This is illus-trated in greater detail in the ‘Experimental Validation’ section.However, there are several other details of the toolpath genera-tion that must be discussed first.

FIGURE 9: Illustration of the contour computing process. Thedomain wi transformed into R3 is show at top, intersections withthe slicing planes defined by ck are highlighted. These contours,projected back into R2, are shown in the lower portion of thefigure.

Computing Contours of Hin Due to the widespread useof contour plots in a great variety of computer tools , as well asthe usage of level sets in simulation methods such as [41], al-gorithms for computing level sets and contours of functions/dataare very well developed. Various fast algorithms for the calcu-lation of contours have been developed [42–44], and there havebeen some recent improvements in such algorithms [45]. Thesealgorithms are embodied in virtually all modern computer alge-bra tools, as well as in libraries for most computer programminglanguages.

Any of the existing algorithms for computing contour plotsmay be employed to calculate the contours of Hin in order toproduce the corresponding toolpath components. In the presentwork, the following method is employed. Recalling the defini-tion of ω, the layer domains are each encoded as a set planarfacets F in R2. We take Hin as a transform

T : R2→ R3,T (x) = (xx,xy,Hin (xx,xy)) ,x ∈ ωi (10)

This transform is applied to the vertex coordinates of the facetsin ωi, effectively carrying the transformation of F to F . Theprocedure for intersecting slicing planes with F is employed tocompute the contours in R3 in precisely the same fashion as theoriginal computation of γi. The slicing planes are positioned atthe values of ck. The resultant series of line segments are thenprojected back into the slicing plane via

T : R3→ R2,T (x) = (xx,xy) ,x ∈ ωi (11)

This process is illustrated in Fig. 9.

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Collecting Layers & OutputOnce the perimeter and infill toolpath components have beencomputed individually for each layer, they must be assembledinto a single cohesive NC program. The first stage of this processis to ensure that the infill and perimeter toolpaths for each layerare compatible and do not overlap. This is achieved by pruningthe infill contours and discarding any portions of the infill thatfall outside of the contracted domain

ω′ = x|Hpr (x)> max(ck)+ ε (12)

where ε is a small constant that introduces a minor overlap orclearance between infill and perimeters. This variable accomo-dates the physics of some additive manufacturing processes; forinstance in FDM it is typical to overlap the infill and perimetersslightly in order to realize improved structural performance. Thesequencing of the contours within each layer is a general combi-natorial optimization problem that is constrained by the physicsof the additive manufacturing process that is being targeted bythe slicer. Large difference in machine movement speed, as wellas different times associated with stopping, repositioning, andrestarting the deposition of material/energy, dictate that optimaltoolpaths for different additive manufacturing technologies willbe structured in different fashions. For the purposes of demon-strating the implicit approach in the present work, we use a sim-ple scheme for ordering the toolpath components, as opposed tosolving the full optimization problem. For every layer:

1. Begin with the outermost perimeter, traverse it in a clock-wise fashion, starting/ending at a randomly selected point.

2. Select the nearest point on the next perimeter, traverse inclockwise fashion. Repeat for all perimeters.

3. Select the endpoint of an infill curve that is nearest to thestart/end point of the final perimeter. Traverse it to the otherendpoint.

4. Select the endpoint of an untraversed infill curve that is clos-est to the current endpoint. Traverse this new infill curve.Repeat until all infill curves are traversed.

Once these steps have been completed for every layer, the finaloutput toolpath is simply the sequence of these layers from firstto last (typically from lowest to highest in the z-direction). Aprocess-specific, possibly machine-specific parser must be usedto translate the purely numerical toolpath into a format under-stood by the additive manufacturing machine, but such develop-ments are beyond the scope of the present work.

Support StructureMany components that are additively manufactured require so-called “support material.” This support material is deposited bythe additive manufacturing process outside of the component do-main, in order to prevent the sagging or curling of the component

FIGURE 10: Input geometry for the first test problem.

itself under gravitational or other forces during the manufactur-ing process. The proposed implicit methodology may also beused to compute the toolpaths associated with the support mate-rial, in a very simple fashion. The support material domain ωsi

for any layer is defined as the difference between a layer’s re-gion ωi and the union of all layer regions above ωi. The distancetransform may be used to compute perimeters for this supportregion, and any function may be used to compute the supportregion infill, in a fashion identical to that described previously.

DEMONSTRATION RESULTSA prototype of the implicit slicer described in the previous sec-tion was implemented in the Mathematica symbolic mathemat-ics system [46]. A laptop computer with a dual-core proces-sor and 4 GB of memory was used to execute this program andconduct the slicing procedure. Two specific test geometrieswere sliced using this implementation and hardware in order todemonstrate the capabilities of the implicit slicing methodology.

Test Case 1: Engine Cylinder HeadThe goal of the first test problem is to apply the implicit slicerto a highly complex input geometry, and to demonstrate that byusing a linear function for Hin toolpaths equivalent to those pro-duced by an explicit slicer are produced. The input geometryselected is a sufficiently complex model of of an automobile en-gine cylinder head [47]. The geometry of this model is shown inFig. 10. The implicit slicer was applied with a layer thicknessl = 0.2 mm, and Hin (x)= xx+xy with xx and xy corresponding tothe Cartesian coordinates of the slicing planes. The values of ckfor the perimeters were 0,0.3 and the corresponding values forthe infill were −50,−49, ...,50. In order to validate the resultsproduced by the implicit slicer, an existing explicit slicer was ap-plied to the same geometry with equivalent settings. In this casethe open-source “Slic3r” [48] software package was used. It wasselected due to its widespread use in AM by FDM. A compari-

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FIGURE 11: Comparison of the results of the implicit (top) andexplicit (bottom) slicer results on the cylinder head test geometry

FIGURE 12: Magnified sections of Fig. 11 with comparison ofthe results of the implicit (left) and explicit (right) slicers.

son of the results produced by the implicit and explicit slicers isshown in Fig. 11. The particular layer seen in this comparisonwas chosen as it exhibits the highest topological complexity ofthe slices. A magnified portion of this comparison is shown inFig. 12. A comparison of Figs. 11 and 12 reveals that the im-plicit methodology produces an output that is equivalent to thatproduced by the explicit slicer. However, some minor differencesare noted. First, the explicit slicer resorts to concentric shells tofill narrow gaps. Second, it can be seen that the explicit slicer isjoining the infill segments in an attempt to optimize the toolpath.Examination of the implicit output, clearly indicates that similarprovisions could be made.

Test Case 2: Stanford DragonTo demonstrate the capability of the implicit slicer to generatesupport structure a second test case is considered, utilizing thegeometry of the well-known Stanford Dragon which is shown inFig. 13. In this case, it is clear that the generation of support ma-terial is required for satisfactory production using FDM. Using

FIGURE 13: Input geometry for the second test problem.

the same frame of reference as the previous test problem, theimplicit slicer is applied with a layer thickness of l = 0.5 mm.The values of ck for the perimeters were 0,0.2,0.4 mm andthe corresponding values for the infill were −20,−19, ...,20mm. The “wave” infill of Fig. 8(c) was arbitrarily selected forthe main body of the component.

The support material for this model was generated with asingle perimeter at ck = 0.2. This creates a small gap between thesupport material and the main body of the component, and alsoprevents the generation of support material for shallow-angleoverhangs. The infill of the support material was generated us-ing the “eggcrate” infill of Fig. 8(d), at levels of −0.5,0.0,0.5mm. The output produced by the implicit slicer using these set-tings is shown in Fig. 14.

The results of shown in Fig. 14 indicate that the implicitslicer may be used to produce objects with infill defined by anyfield function of interest. In the next section we employ this abil-ity in order to incorporate functional fields of design interest intothe additive manufacturing process.

FUNCTIONAL TAILORING AND EXPERIMENTALVALIDATIONThe examples given in the previous section demonstrate that theimplicit slicing methodology is capable of producing highly non-uniform toolpaths. The purpose of this section is twofold; firstto demonstrate that the implicit methodology may be used toproduce components with significantly improved physical prop-erties and performance for the purpose of functional tailoring;and second to validate this capability by comparing its perfor-mance response for various ways of applying the slicer to afffectthe mesoscale, via experimental data produced from physical ex-perimentation.

A “dogbone” test specimen intended to be subjected to uni-axial tension loads is used for this purpose. The input geometry

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FIGURE 14: Results produced by the implicit slicer for the sec-ond test problem (top). Several layers are shown individually(bottom).

of the test specimen is shown in Fig. 15, along with the associ-ated γ and ω . The test specimens are 100 mm tall, 30 mm wide,and 5 mm thick. The gage area used for strain measurement is10 mm wide and 20 mm tall, centered on the test specimen. The“COMSOL Multiphysics” FEA software tool [49] was used tosolve the structural mechanics PDE governing the physics of thesystem under the influence of a 1 kN tension load. In additiona clamping load of 1 kN (total) was applied to each end of thespecimens, in an area coinciding with the application of tensileforces, as shown in Fig. 15. The regions of force applicationare 12 mm tall and cover the full width of the specimen. Theclamping load magnitude was estimated according to our priorexperience with the system employed for physical testing. Fig.16 shows the resulting von-Mises stress distribution (σvm).

We first compute a purely rectilinear grid infill, as used inmany existing slicers, as

Hlin (x,y,z;θ) = xsin(θ)+ ycos(θ)(−1(z/lt )%2

), (13)

where x and y are the Cartesian coordinates within each slice,z is the Cartesian coordinate normal to the slices, lt is the layer

(a) (b)

FIGURE 15: (a) Input dogbone geometry in STL format. Notethe triangular facets that constitute Γ. The red-shaded areas indi-cate the grip area at which the boundary loads are applied. Thegage area used for strain measurements is shaded green. (b) Thetriangulated domain ω1 and associated γi computed for the firstlayer. Because the dogbone shape is uniform through the thick-ness, all other layers have similar ω and γ with small differencescaused by the faceted nature of the input model.

thickness, and % indicates the arithmetic modulo operator. Thisfunction creates a grid of parallel diagonal lines, rotated by θfrom the x-axis for each slice’s infill, which flip about the y axisevery other layer. The linear infill is subsequently modulatedaccording to the value of σvm in order to produce infill that isdenser in high-stress regions. As is common in existing slicers,a value of θ = π/4 is used. The infill function is composed fromσvm and Hlin, taking the form

Hin (x,y,z) = σvm (x,y,z)Hlin (x,y,z;π/4)k. (14)

where k is a scalar parameter that controls the relative infill den-sity. A visualization of the field functions σvm, Hlin, and Hin isgiven in Fig. 16.

Figure 17 shows the final toolpath design produced by theimplicit slicer. Also shown is a physical realization of this tool-path, produced on a consumer FDM 3D-printer. In order to pro-vide a comparison, the implicit slicer was also used to producedogbone test specimens using a different modulation pattern,with Hin (x,y,z) = σvm (x,0,0)Hlin (x,y,z;π/4)k. Finally, refer-ence specimens were produced using the unmodulated, uniforminfill function Hin = Hlin. The density parameter k was tuned sothat all three types specimens were of identical mass.

Three replicates of each specimen photographed in Fig. 17were additively manufactured from ABS polymer using a desk-top FDM 3D printer. Each was printed sequentially, using an

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(a) (b) (c) (d) (e)

FIGURE 16: (a) COMSOL-derived stress field σvm. (b) Linearinfill field Hlin for the first (and all odd-numbered) layers. (c)Linear infill field Hlin for the second (and all even-numbered)layers. (d) Stress-modulated infill field Hin for the first (and allodd-numbered) layers. (e) Stress-modulated infill field Hin forthe second (and all even-numbered) layers.

(a) (b) (c) (d) (e) (f)

FIGURE 17: (a) Toolpath and (b) corresponding physical spec-imen with Hin (x,y,z) = σvm (x,y,z)Hlin (x,y,z;π/4)k, termed“Stress Infill A.” (c) Toolpath and (d) corresponding physi-cal specimen with Hin (x,y,z) = σvm (x,0,0)Hlin (x,y,z;π/4)k,termed “Stress Infill B.” (e) Toolpath and (f) corresponding phys-ical specimen with Hin (x,y,z) = Hlin (x,y,z;π/4)k, termed “Lin-ear Infill.”

identical set of instructions sent to the additive manufacturingmachine so as to minimize differences between replicates. Thespecimens were gripped and loaded in tension until failure usinga small uniaxial test frame. Full-field techniques [50, 51] wereused to measure the specimen strains during testing. The experi-mental apparatus and setup are shown in Fig. 18.

The results of the testing are shown in Fig. 19. It is seenthat the properties of the specimens featuring stress-modulatedinfill are markedly different than those of the specimens withconventional linear infill and the same mass. The ultimate failurestress is increased by 45%, and the apparent elastic modulus is

FIGURE 18: Photograph of the uniaxial test frame used for test-ing the additively manufactured specimens.

TABLE 1: Results from testing of additively manufactured spec-imens. All figures represent the average results of three replicateexperiments. Parenthetical values indicate relative performanceversus the Linear infill case.

Infill Type UltimateStress (MPa)

Apparent ElasticModulus (GPa)

Linear 13.7 0.79

Stress A 19.4 1.24(+ 41.6%) (+ 57.0%)

Stress B 19.8 1.71(+ 44.5%) (+ 116%)

increased by more than 100%. These data are given in detail inTable 1. These results clearly demonstrate that the implicit slicermay be used to alter the properties of an additively manufacturedobject in order to produce enhanced performance with respectto its intended function. It should also be noted that the stress-modulated specimens failed outside the gauge area, typically ator near the grips. This indicates that the ultimate strength ofthe specimens may be under-represented by the data of Fig. 19.This may indicate that additional performance increases may berealized by augmenting the formulation of Eq. (14) with an addi-tional component to compensate for such failure modes. Finally,the selection of θ = π/4 is effectively arbitrary; in the future itwill be beneficial to examine in more detail the effects of thisparameter as well as potential infill densification in the regionswithin the grips.

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FIGURE 19: Results from the testing of the additively manufac-tured specimens.

CONCLUSIONSWe have developed and demonstrated a new approach for tool-path generation and slicing for AM. A review of existing slic-ing methodologies has shown that prior approaches may be cat-egorized as explicit slicers, as they operate by means of explicittransformations applied upon the input geometry. In contrast tothis, the present paper introduces an implicit approach in orderto address the loss of non-geometric information in the digitalthread, and to better convey the intent of designers throughoutthe additive manufacturing toolchain. The introduced methodol-ogy allows for a straightforward implementation that avoids thecomplexity associated with traditional polygon offsetting. It alsoallows great flexibility in the specification of the additive man-ufacturing toolpath, and for the generation of toolpaths basedon physical fields of interest for the purpose of controlling themesoscale structure in manner that enables functional tailoringof the to be manufactured parts.

The results demonstrated in this work show that the implicitslicer offers several important advantages. The first test case, onthe engine head geometry, demonstrates that the methodologycan be applied to generalized, and even very complex compo-nents, such as those that are increasingly produced using additivemanufacturing systems in practice. The second test case, on theStanford Dragon, demonstrates the ability of the implicit slicer togenerate support structure. The third and last test problem, withdogbone specimen, demonstrates that the infill toolpath may begenerated based on the solution to a partial differential equa-tion capturing the physics relevant to the original design intentsuch as the stress behavior of the system under tension. The re-sults of an FEA simulation, that captures the explicitly specifiedintended use of the dogbone specimen, are successfully used tocompute the infill toolpath geometry. Quantitative assesment

of the implicit slicer via actual mechanical testing of differentvariations of the mesoscale structure shows that significantly im-proved mechanical properties and component performance maybe achieved by use of this methodology to incorporate the resultsof an external structural mechanics simulation.

In the test cases presented, the toolpaths produced by the im-plicit slicer do not represent optimal structures. Rather, the pur-pose of these exercises is to demonstrate the great flexibility of-fered by the implicit methodology, and the possibility of includ-ing relevant physics or multi-physics results into the productionof additively manufactured components. This in turn opens thedoor to the production of components with performance proper-ties that are tailored towards the intended use of the component.

It must also be noted that the present work is exploratory innature, and that there is a great deal of future work that is re-quired to bring the implicit methodology to the state of maturityexhibited by its explicit counterparts. Our future plans include

1. Development of strategies in order to generate productiontime-optimized toolpaths.

2. Development of a methodology for producing functionallyoptimized components. This may require the developmentof constitutive models of additively manufactured objectswhich reflect the highly nonuniform and hierarchical natureof these processes.

3. Expansion of the implicit methodology to include featuressuch as bridge and gap fill routines, thin section detection,and other features present in current explicit slicers. The de-velopment of routines for producing “support” or scaffold-ing material is particularly important.

4. Exploration of the best methods for automatically determin-ing the contour intervals. For example, it may be necessaryto limit the gradient of the infill function in order to avoidover-dense contours, such as those seen in the second testproblem. Conversely, it may be desirable to increase thedensity of the contours in a localized sense in order to sat-isfy additional performance constraints.

5. Developing a methodology for either discrete or gradualtranslations between different fields for different portions ofthe toolpath. For example, to ensure that the outer surface ofthe component is solid when a sparse infill pattern is used.

6. The development of process and machine-specific inter-preters to translate the output toolpath into NC codes for avariety of additive manufacturing hardware.

The present work illustrates many of the advantages ofadopting an implicit slicing methodology. Augmented by thefuture work outlined above, this methodology may significantlyimprove the state of the additive manufacturing digital thread.By re-introducing design intent into the toolpath generation pro-cess, it is likely that the implicit methodology will allow the ad-ditive manufacture of complex components that are better suitedto their intended purpose. Additionally, it may be possible to

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achieve important improvements in production time, raw mate-rial, and energy consumption that will in turn reduce the cost ofadditively manufactured structures and components.

ACKNOWLEDGMENTThe authors acknowledge support for this work by the Office ofNaval Research through the Naval Research Laboratory’s corefunding, as well as the National Research Council’s ResearchAssociateship Program. This work was also supported in partby the National Institute of Standards and Technology throughInteragency Agreement NIST IAA-2018-0011-0.

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