4 - surface roughness optimisation for selective laser melting...

Surface roughness optimisationfor selective laser melting (SLM):accommodating relevantand irrelevant surfaces

4

M. LearyRMIT University, Centre for Additive Manufacturing, Melbourne, VIC, Australia

4.1 Introduction

Additive manufacturing (AM) provides a significant opportunity for the manufactureof innovative products. In particular, AM allows the manufacture of geometries thatinclude complex features that are not possible with traditional manufacturing methods,and the opportunity to economically manufacture short production runs withoutsignificant cost penalty. In particular, AM is suited to the manufacture of:

• prototype components, for design validation before large-volume manufacture• low-volume production components, especially during the initial deployment of a specific

design, to allow rapid implementation of design changes, as required• high-volume customised components for which the geometry of each component is subject to

variation and is therefore not compatible with fixed-tooling methods• components that are not technically feasible with traditional methods, for example, complex

lattice structures and components with internal voids

For these scenarios, it is likely that surface roughness is a highly relevant designattribute. For AM, surface roughness is highly dependent on the specific attributesof the AM process used, the associated process parameters, and the orientation ofthe component within the available build volume.

To provide AM components that are acceptable for a specific purpose, a quantifiedsurface roughness is highly desirable; the available literature includes work that seeksto address this issue, including experimental investigation of surface roughness ofselective laser melting (SLM) [1e6], fused deposition modelling [7e10], and selectivelaser sintering [11]. The orientation of the component can be used to optimise theaverage surface roughness. Orientation is optimised by reference to either pre-selected [7,12] or continuously variable orientations [13e15]. This work contributesto the available literature by providing:

• New material roughness data: To provide useful design data, the surface roughness versusbuild inclination of SLM Ti64 is experimentally assessed and reported in detail, includinghigh-magnification microscopy and associated surface roughness statistics of the upperand lower surfaces for a comprehensive array of specimen inclinations.

Laser Additive Manufacturing. http://dx.doi.org/10.1016/B978-0-08-100433-3.00004-XCopyright © 2017 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/B978-0-08-100433-3.00004-X

• Visualisation of roughness objectives versus orientation: An infinite number of feasibleorientations of a component exist within the build volume. Visualisation of the roughnessobjectives within the assessed orientations aids identification of the optimal orientation(s)within the available optimisation time.

• Novel surface roughness objectives: The available literature on orientation optimisationconsiders only the minimisation of average surface roughness. This research proposes novelsurface roughness objectives, including:• peak roughness avoidance, which is appropriate when a specific roughness type violates

an inflexible design constraint.• weighted roughness, which is appropriate when a non-linear correlation exists between

specific roughness types and performance.• Filtering of irrelevant surfaces: It is common for a manufactured product to include surfaces

that are irrelevant in terms of associated surface roughness. For example, commercial prod-ucts consist of visible and hidden surfaces. Hidden surfaces are not relevant to the aestheticvalue associated with the product. Engineering components may have critical roughnesstolerances associated with key surfaces, but not all surfaces are functionally relevant. Themethod proposed in this research enables the discrimination of relevant and irrelevant sur-faces when assessing the associated surface roughness objective.

These contributions have been used to enable novel design outcomes for a numberof engineering applications of SLM technologies (Section 4.4):

• A surgical implant, whereby an orientation map of average roughness is presented forfeasible orientations to identify the optimal implant orientation for surgical objectives

• Low-volume prototype cover, whereby a custom roughness weighting is applied todiscourage facets with high roughness from occurring on visible surfaces

• Hydraulic valve assembly, whereby surfaces of relevance to functional objectives areassessed to enable robust identification of optimal build orientation

4.2 Additive manufacturing

AM is defined by the ASTM [16] as the ‘process of joining materials to make objectsfrom 3Dmodel data, usually layer upon layer, as opposed to subtractive manufacturingmethodologies, such as traditional machining’. AM begins with a computer-assisteddesign (CAD) representation of the intended component geometry. This data arethen converted to stereolithographic (STL) format to be compatible with AM processes(Fig. 4.1). The STL format represents the CAD data with a discrete number of trian-gular facets and their associated surface norms (Table 4.1). The STL representation isprocessed as required to provide specific tool path and processing instructions for thespecific AM method [17,18].

AM processes can be characterised according to the method by which support isprovided, the resolution of the manufactured geometry to the intended geometry,and the relevant processing parameters of manufacture [18e20]. Of the AM processesof commercial interest, SLM is extremely important because it allows the manufactureof large-scale functional components with engineering-grade metals with highcomplexity and a relatively low unit cost [21]. This research focuses specifically onthe optimisation of SLM components with specific surface finish objectives; however,the associated methods are generally applicable to any AM processes.

100 Laser Additive Manufacturing

Non-visible“B-class” surface

Visible “A-class”surface

Error in representationof curvilinear geometry

Prismaticgeometry

representedwithout error

Figure 4.1 Computer-assisted design model (shaded) and associated stereolithographicrepresentation (wireframe facets). This model of a low-volume prototype cover indicatesvisible (A-class) and non-visible (B-class) surfaces, which inherently have different surfaceroughness objectives.

Table 4.1 Binary stereolithographic file standard

Element Content Size

Header Component information 80 bytes

n Number of facets 32-bit unsigned integer

Facet 1 Vertex 1 Cartesian coordinates 32-bit floating point

Vertex 2 Cartesian coordinates 32-bit floating point


Facet normal vector 32-bit floating point

Attribute byte count 2-byte unsigned integer

. . .

Facet n Vertex 1 Cartesian coordinates 32-bit floating point



Facet normal vector 32-bit floating point

Attribute byte count 2-byte unsigned integer

Surface roughness optimisation for selective laser melting (SLM) 101

4.2.1 Selective laser melting

SLM refers to the fusion of metal powder with a laser power source to incrementallymanufacture an intended product [22]. SLM is an important commercial AM processbecause it [21,23]:

• enables the manufacture of a range of fusible ferrous and non-ferrous metals• readily accommodates support structures to enable manufacture of acute overhang geometry• provides high structural integrity of manufactured components• has relatively low fixed and variable costs• has a reasonably short build time

Despite the particular advantages associated with SLM, the associated surfaceroughness may introduce technical challenges because of the layered nature of AMand the associated process parameters and powder morphology.

Control of surface roughness is of particular importance for commercial applica-tions because of the associated implications on product aesthetics, structural integrityand geometric tolerance. Surface roughness is fundamentally the result of some sourceof geometric error.

4.2.2 Sources of geometric error

Geometric error refers to a discrepancy between the CAD representation of theintended product and the geometry of the actual manufactured product. Geometricerror in AM can be attributed to either error in the STL representation of the CADgeometry or error in the AM manufacture of the STL representation (Fig. 4.2).

4.2.2.1 Error in STL representation of CAD data

Because of the discrete nature of the STL format, error may occur in the representationof the original CAD data. If the original CAD data are prismatic, the discrete facets ofthe STL represent the CAD data without error (Fig. 4.1). If the original CAD data are

Z

X

l

t

ϕacuteϕobtuse

Figure 4.2 Layered approximation of computer-assisted design (CAD) geometry, indicatinga loss of fidelity between the CAD and the manufactured geometry as a result of staircaseerror [24].


curvilinear, the STL representation of the original CAD data includes some approxi-mation error (Fig. 4.2); this error diminishes with an increasing number of facets[6]. It is desirable to minimise the number of facets to reduce computational demands;if possible, however, the number of facets should be sufficiently high that the associ-ated STL representation error is negligibly small.

4.2.2.2 Error in manufacture of the STL representation

AM involves the successive addition of material to manufacture an intended geometry.The AM process introduces manufacturing error in the representation of the STL dataas a result of:

1. machine precision and repeatability [25]2. distortion and part shrinkage [6]3. staircase errors [9,26], which introduce roughness caused by the discrete layers associated

with AM (Fig. 4.2)4. Roughness introduced by process parameters, the laser scan path [6], and the powder

morphology [4]

4.3 Roughness

Roughness refers to a geometric deviation from a reference datum. Numerous mea-sures of roughness have been proposed [27]. To be compatible with other contributionsin this field, the arithmetic average roughness, Ra, is used, where Ra measures thearithmetic average deviation of the measured profile from the centre line of themeasured profile (Eq. (4.1)):

Ra ¼ 1n

Xn

i¼1

��yj�� (4.1)

where yj is the vertical distance from the mean line to the jth data point and i is the facetof interest.

4.3.1 Experimental results

Specimens were fabricated to quantify the roughness of a commercial SLM methodversus inclination angle1 (Table 4.2, Fig. 4.3). Optical macroscopy was recorded(Fig. 4.5) and Ra recorded for the upper and lower surfaces (Fig. 4.6).

Inspection of the sample macroscopy (Fig. 4.5) and measured surface roughness(Fig. 4.6) identifies distinct topology regions associated with the inclination angle F:

• For inclinations approaching horizontal (f/ 0 degree), staircase effects dominate, andthe observed roughness asymptotes towards peak values. For inclinations below the

1 Note, the inclination angle, f, used in this work seems to be the most common nomenclature reported inthe literature; however, q ¼ p/2 � f, for example, is also used [8].


Table 4.2 Process parameters associated with the samplesshown in Fig. 4.4

Parameter Value

Machine Specification SLM Solutions 250HL

Layer thickness 30 mm

Ambient temperature 90�C

Process parameters As in Ref. [28]

Powder Material Ti64

Size distribution See Fig. 4.4

3

3

40

15

α

β

B

φ

Figure 4.3 Standard specimens to assess surface roughness versus inclination angle, f. a and brefer to the perpendicular direction of roughness measurement.

25

20

15

10

5

00 10 20 30 40 50 60

Size (μm)70 80 90 100 110

% V

olum

e de

nsity

Dv10 Dv50 Dv90

DV10 = 32.7 μmDV50 = 49.5 μmDV90 = 71.9 μmD(3,2) = 42.2 μmD(4,3) = 50.6 μm

Figure 4.4 Ti64 powder distribution.


self-supporting inclination angle, supporting structures may be desirable to reduce errorduring manufacturing.

• As the surface tends towards a vertical orientation, f/ 90 degrees, the staircase effects coa-lesce, and observed roughness seems to be predominantly the result of the presence ofadhered, partially melted particles.

• Of the curve fit options considered (linear, exponential, power, and logarithmic), thecorrelation of the logarithmic curve provides the best match for both the upper and lowersurfaces (Table 4.3); although the variability associated with the logarithmic curve fitshown in Fig. 4.6 is relatively high, it does provide a useful insight into observed rough-ness trends.

• The observed roughness in the lower surface is always larger than that of the upper surface,and often significantly larger. The rationale for this observation is that heat transfer throughthe inclined specimen causes elevated temperatures at downward-facing (lower) surfaces,resulting in greater thermal distortion and particle adhesion. Surface damage todownward-facing surfaces can also occur where supporting structures are physicallyremoved from the surface, although no supporting structures were used to generate thecoupons used in this research.

Deg Upper Lower

5°

10°

20°

30°

40°

AdheredparticlesLayer effect

Deg Upper Lower

50°

60°

70°

80°

90°

(a) (b)

β

α

β

α

Figure 4.5 Macroscopy of SLM samples at inclination angles f of 0e40 degrees (a) and50e90 degrees (b). Orientation a and b are as defined in Fig. 4.4.


By accommodating these observations of surface roughness versus surface inclina-tion and by rotating the intended component according to the spatial rotations shown inFig. 4.7, the associated surface finish can be optimised for the specific designobjectives.

4.3.2 Roughness objectives

To achieve robust surface roughness outcomes in AM, it is critically important to applya surface roughness objective that is appropriate for the intended functional andaesthetic objectives. In addition to the area-weighted average roughness, Ra_ave, thatis reported in the literature, this section presents a series of novel surface roughnessobjectives that are of specific relevance to AM.

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Angle, φ (degrees)

Ave

rage

roug

hnes

s, R

a (m

icro

ns)

Typical non self-supportingregion, ie, 25° ≤

Upper surface

Lower surface

y = –8.767In(x) + 52.109R2 = 0.4249

y = –4.675In(x) + 32.947R2 = 0.3183

φ

Figure 4.6 Surface roughness Ra versus the specimen inclination angle f for the upper andlower surfaces with directions a and b (Fig. 4.4).

Table 4.3 Linear, exponential, power, and logarithmic curve fitoptions and associated correlation coefficients (R2) for upper andlower surfaces shown in Fig. 4.6

Linear Exponential Power Logarithmica

Upper surface 0.167 0.117 0.234 0.318

Lower surface 0.318 0.350 0.372 0.425

aLogarithmic provides the highest correlation coefficient for both the upper and lower scenarios.


4.3.2.1 Average roughness

Average roughness, Ra_ave, refers to the area-weighted average roughness of each facetwithin the component geometry (Eq. (4.2)). The area, A, associated with each facetmay be assessed by the Cartesian coordinates of the associated vertices (Eq. (4.3))or by Heron’s rule with reference to the facet side lengths (a, b, and c) (Eq. (4.4)).

Ra_ave ¼Pn

i¼1 Ra;iðØÞAPni¼1 A

(4.2)

A ¼ 12jx1y2 � x1y3 þ x2y3 � x2y1 þ x3y1 � x3y2j (4.3)

A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=16½ðaþ bþ cÞð� aþ bþ cÞða� bþ cÞðaþ b� cÞ�

p(4.4)

The literature associated with the surface roughness of AM specimens universallyreports Ra_ave as the objective function (for example, Refs. [9,10,14,15,29]). Althoughthe average roughness is useful in determining the quality of a manufactured compo-nent in terms of roughness, it is not the only parameter of relevance. This work extendsthe available literature associated with surface roughness by accommodating addi-tional roughness objectives, including peak roughness avoidance, the roughness ofrelevant surfaces only and weighted average roughness.

4.3.2.2 Peak roughness avoidance

Peak roughness avoidance, Ra_pa, returnsNwhen the orientation results in any facetroughness value that exceeds the reference limit, Ra_limit, and thereby identifies

Y1, Y2

Y0

Z0

Z1 Z2

X0, X1X2

α1

α1

α2

α2

Figure 4.7 Standard spatial rotations, a1 and a2, used to define the rotation of a component’sgeometry in space [24].


orientations (if they exist) that avoid the inclusion of excessive roughness (Eq. (4.5)).Ra_pa is useful to avoid roughness types that violate some fundamental designconstraint. For example, Ra_pa may be used to identify orientations that do not requiresupporting structures by setting Ra_limit to the minimum self-supporting build angle(Fig. 4.5).

Ra_pa¼ N; for max

�Ra;i

�ni¼1 > Ra_limit

¼ Ra-ave; otherwise(4.5)

4.3.2.3 Roughness of relevant surfaces

The roughness of relevant surfaces, Ra_r, is dependent only on facets that are relevant(ri ¼ 1) to the functionality of the intended component. Facets that are functionally oraesthetically irrelevant (ri ¼ 0) do not contribute to Ra_r (Eq. (4.6)). The roughness ofrelevant surfaces provides a useful surface roughness objective for scenarios that havegeometric tolerances assigned to mating surfaces (Section 4.4.1) or aesthetic restrictionsof surface roughness on visible surfaces only. The Ra_r approach is applied in case study2 to identify orientations that meet the functional requirement for selected surfaceswithin a prototype cover (Section 4.4.2) and a hydraulic housing (Section 4.4.3).

Ra r ¼Pn

i¼1 riRa;iðØÞAiPni¼1 Airi

(4.6)

4.3.2.4 Weighted roughness

Weighted roughness, Ra_w, applies a roughness weighting l(f) to the measured surfaceroughness of each facet to optimise the desired surface roughness distribution in thefinished component (Eq. (4.7)).

Ra_w ¼Pn

i¼1 lðØÞRa;iðØÞAiPni¼1 Ai

(4.7)

Weighted roughness is a useful roughness objective because it allows the presenceof high roughness facets (unlike for Ra_pa), on the condition that their associatedsurface area is correspondingly low. For example, in response to the roughness profileobserved in Fig. 4.6, the hypothetical roughness weighting shown in Fig. 4.8 is appliedin Section 4.4.3 to discourage facets that have high roughness or are not robustlyself-supporting.

4.3.2.5 Weighted roughness of relevant surfaces

Weighted roughness of relevant surfaces, Ra_w_r, applies a roughness weighting l(f) tothe measured surface roughness of each facet to promote the desired surface roughnessdistribution in the finished component (Eq. (4.8)). As for Ra_r, facets are defined aseither relevant (ri ¼ 1) or irrelevant (ri ¼ 0) to the associated surface roughness


objective. The Ra_w_r approach is applied in case study 2 to identify orientations thatavoid high roughness values for relevant surfaces (Section 4.4.2).

Ra_w_r ¼Pn

i¼1 ril Øð ÞRa Øð ÞAiPni¼1 Airi

(4.8)

4.4 Case studies

Case studies are presented for a surgical implant, a low-volume prototype cover, and afunctional hydraulic housing. These case studies demonstrate the enhanced design out-comes that are achievable by the material data and surface roughness objectives devel-oped in this research.

4.4.1 Case study 1: surgical implant design

AM provides significant application in the manufacture of high-value, patient-specificsurgical implants [28,30]. Surgical implants have both surgical and biological require-ments for specific surface roughness targets. For example, the proposed surgicalimplant is to be fastened with a surgical-grade screw of predetermined geometry ata vector defined a priori by the surgical team. Based on these inputs, correspondingscrew clearance envelopes are specified geometrically (Fig. 4.9). It is highly desirablethat the implant surface roughness be determined versus orientation within the buildvolume such that orientations that avoid high roughness values are avoided, andthat screw clearance envelopes be self-supporting such that no post-processing

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Inclination angle (φ)

Roughness weighting designed todiscourage facets with inclinationangle requiring support structures,

ie, ≤ 25°φ

Rou

ghne

ss w

eigh

ting

λ(φ)

Figure 4.8 Roughness weighting l(f) to discourage facets with and inclination angle requiringsupport structures (ie, f � 25 degrees) (Fig. 4.6).


operation is required to remove internal support material. These orientation maps(Fig. 4.10) allow the identification of optimal implant orientation to meet surgical re-quirements; however, it is apparent that the identification of global optima requiresanalysis with a relatively small orientation angle increment, Da.

4.4.2 Case study 2: low-volume prototype cover

A low-volume prototype cover is to be manufactured for an automotive conceptvehicle (Fig. 4.2). Minimisation of average surface roughness, Ra_ave, is a usefulroughness objective; however, it does not fully represent the requirements of this appli-cation. In particular, it is necessary to accommodate aesthetic and post-processingroughness requirements:

• The prototype housing is an interior product with visible ‘A-class’ surfaces that are critical tocustomer perception. Surfaces that are not visible to the customer are not relevant in terms of

Proximal femur(patient CT scan)

Patient-specificTi64 AM implant

Screw clearanceenvelopes (×4)

Figure 4.9 Rear, side and isometric views of the proximal femur, including the zone identifiedfor patient-specific Ti64 additive manufacturing (AM) implant and associated screw clearanceenvelopes (�4). CT, computed tomography.


surface roughness and are defined as ‘B-class’ surfaces. To meet this roughness requirement,B-class surfaces are not included in the surface roughness analysis (Fig. 4.2). The associatedroughness of relevant surfaces, Ra_r, of the prototype component is presented in Fig. 4.11 (left).

• It is necessary to avoid high roughness values that occur as a result of low inclinationangles and geometry that is not self-supporting. To meet this roughness objective, a customroughness weighting was developed (Fig. 4.8). Weighted roughness of relevant surfaces,Ra_w_r, is presented for the prototype component in Fig. 4.11 (right).

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Figure 4.10 Orientation maps of weighted roughness, Ra_w, of selective laser melting titaniumimplant versus orientation for Da ¼ 90, 60, 30, 15, and 5 degrees. Note that local optima arenot identified until the number of orientations (and associated computational time) is increased.


The orientation map of Ra_r and Ra_w_r provides design guidance on the effect ofcomponent orientation on the surface roughness objective. The orientation map clearlyidentifies global minima and maxima and enables an informed trade-off when compro-mising between competing objectives such as build time and support materialconsumption. It is apparent that the minima for both Ra_r (a1 ¼ 170 degrees,a2 ¼ 0 degree) and Ra_w_r (a1 ¼ 160 degrees, a2 ¼ 0 degree) occur at slightlydifferent locations, indicating that the Ra_w_r successfully identifies a more favourableorientation for the design objectives.

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α α

α α

α α

Figure 4.11 Orientation map of weighted average roughness, Ra_w (left), and weightedroughness of relevant surfaces, Ra_w_r (right), for all facets of the prototype housing (Section4.4.2) for Da ¼ 30,15, and 5 degrees.


4.4.3 Case study 3: functional hydraulic housingfor an engineering application

The hydraulic housing shown in Fig. 4.13 is to be manufactured by SLM for low- tomedium-volume production. For this functional engineering application, there are noaesthetic requirements for surface finish. However, it is technically necessary that thesurface finish of the mating features and mounting holes be minimised to ensure robusttechnical function.

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Figure 4.12 Cover component orientation for minimum weighted roughness, Ra_w (left).For comparison, the orientation for the maximum Ra_w_r is included (right). Note the differencein scale.

Matingfeatures ( 4)

Mountingholes ( 4)×

×

Figure 4.13 Functional housing for an engineering application, indicating mounting holes (�4)and features (�4) that contribute to the surface roughness objective.


For this scenario, the average surface roughness, Ra_ave, is not an appropriatesurface roughness objective because the surface roughness of features other than themating features is inconsequential to component function. For example, minimisingaverage surface roughness of the entire component results in a minima of Ra_ave ata1 ¼ a2 ¼ 0 degrees (Fig. 4.14). This minima repeats with a period of 180 degreesand results in low Ra facets on the large planar surfaces and mounting holes (howeverwith large Ra facets on cylindrical mounting holes). However, when only relevant sur-faces are considered, the global minima for Ra_r occurs at a1 ¼ 90 degrees anda2 ¼ 0 degree (ie, the minima orientation for Ra_ave does not exist), and low Ra facetsoccur on the cylindrical mating features (Fig. 4.12, lower). Note that for the orientationthat is optimal for minimising average surface roughness over the entire component(a1 ¼ a2 ¼ 0 degree; Fig. 4.14, top), the Ra_ave of relevant facets only is not optimal

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including only relevent surfaces

Ra for housing facets at min (Rave)

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Ra_ave minima

Ra_r non-optimalfor α1 = 0°, α2 = 0°

Ra_r minimaα1 = 90°, α2 = 0°

α1 = 0°, α2 = 0°

000000000000000000000

)) 0000000000000000

000000000000

Figure 4.14 Average surface roughness, Ra_ave, and facet Ra at min (Ra_ave) for functionalhousing: all surfaces (upper), relevant surfaces only (lower). Da ¼ 2 degree.


(Fig. 4.11, bottom). This case study demonstrates that the use of Ra_ave as the surfaceroughness objective can result in erroneous design guidance when the componentconsists of surfaces that are irrelevant to component function.

4.5 Conclusions

To provide greater insight into the optimal component orientation for AM, this workcontributes various outcomes to the available literature:

• To ensure robust design data, the surface roughness versus build inclination of SLM-manufactured Ti64 was assessed experimentally and reported in detail, including measuredroughness data and high-resolution macroscopy for upper and lower surfaces. Based on thesedata, correlations between the orientation and the observed surface roughness are presented,including the dominance of staircase effects and adhered particles on observed roughnesswhen inclination tends towards horizontal (f/ 0 degree) and vertical (f/ 90 degrees),respectively.

• Manufactured products typically include surfaces that are not relevant to customer satisfac-tion or function. For example, consumer products consist of visible and hidden surfaces, andengineered components have critical roughness tolerances associated with key surfaces. Themethods proposed in this research enable the discrimination of relevant and irrelevant sur-faces when assessing the associated surface roughness objective. It was shown that thismethod can avoid erroneous design guidance when the component consists of surfacesthat are irrelevant to component function.

• The available literature considers only the minimisation of average surface roughness. Thiswork proposes novel surface roughness objectives, including penalty avoidance and penaltyweighting, to avoid specific roughness types. When applied to optimise the orientation of alow-volume prototype cover, the proposed penalty-weighting method identified a morefavourable orientation than optimisation methods based on average surface roughness only.

• An infinite number of component orientations exist. The orientation maps presented in thiswork clearly identify global minima and associated trends, enabling an informed trade-offwhen compromising between competing objectives such as build time and support materialconsumption.

Nomenclature

a1 (degrees) Rotation angle about the X0 axis between {X0, Y0, Z0} and{X1, Y1, Z1}

a2 (degrees) Rotation angle about the Y1 axis between {X1, Y1, Z1} and{X2, Y2, Z2}

Da (degrees) Orientation angle increment

f (degrees) Filament inclination angle

i e Facet index

Continued


n e Number of facets within a component

A (m2) Facet area

Ra,i (mm) Average roughness of facet i

Ra_ave (mm) Area-average roughness

Ra_pa (mm) Peak avoidance surface roughness

Ra_limit (mm) Maximum allowable Ra value for peak avoidance

Ra_r (mm) Surface roughness accommodating relevant surfaces

Ra_w (mm) Weighted surface roughness

Ra_w_r (mm) Weighted surface roughness accommodating relevant surfaces

t (m) Layer thickness

l (m) Overhang distance

yj (m) Vertical distance from the mean line to the data point

Norient e Number of assessed orientations

p e Unit vector oriented vertically downwards to the platen

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