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ARTICLE IN PRESSG ModelDDMA-87; No. of Pages 13

Additive Manufacturing xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

Additive Manufacturing

journa l homepage: www.e lsev ier .com/ locate /addma

ull length article

eniscus process optimization for smooth surface fabrication intereolithography

ayue Pan a,∗, Yong Chen b

Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor St., ERF 2039, Chicago, IL 60607, USAEpstein Department of Industrial and Systems Engineering, University of Southern California, 3715 McClintock Ave., GER 240, Los Angeles, CA 90089, USA

r t i c l e i n f o

rticle history:eceived 1 January 2016eceived in revised form 19 February 2016ccepted 8 May 2016vailable online xxx

eywords:tereolithographyurface finish

a b s t r a c t

In the layer-based additive manufacturing processes, it is well known that the surface finish is usuallypoor due to the stair-stepping effect. In our previous study, it was shown that the meniscus approachcan be used in the Stereolithography process to achieve much better surface finish. A related challengeis how to optimize parameters such that the formed meniscuses can lead to high surface finish andaccurate shape. In this paper, a systematic process planning method is presented for the Stereolithographyprocess that is integrated with the meniscus approach. Process planning, parameters characterizationand meniscus parameters optimization are presented with experimental verifications. To demonstratethe systematic process, surface profile simulation, meniscus database construction, meniscus forming

eniscusccuracyrocess optimization

parameter optimization, and shape accuracy prediction have been performed using a test case based onconvex surfaces. A comparison between the experimental results with and without the process planningmethod is presented, illustrating the effectiveness of the developed meniscus optimization method insimultaneously controlling the shape and surface finish of fabricated objects. Subsequently the simulationresults are also verified with experimental results.

© 2016 Elsevier B.V. All rights reserved.

. Introduction

Stereolithography (SL) Apparatus is the first commercializeddditive Manufacturing (AM) technology and also one of the mostommonly used AM technologies [1–3]. In SL process, liquid pho-osensitive resin is cured by light irradiation, usually through these of a UV laser beam or a Digital Light Projection (DLP) systemhat uses a Digital Micromirror Device (DMD). Controlled light irra-iation induces a curing reaction, forming a highly cross-linkedolymer. Compared to other polymer-based AM technologies suchs the extrusion or jetting based processes, the SL process canroduce parts with fine features, good accuracy, and using vari-us polymers [4–9]. SL also surpasses the processing speed andields of subtractive-type mask-based processing [10] as well asther micro-patterning processes [11]. Its potential applications inhree dimensional (3D) structure fabrications include prototyping,ooling and manufacturing, medical devices, artwork, and archi-

Please cite this article in press as: Y. Pan, Y. Chen, Meniscus process oAddit Manuf (2016), http://dx.doi.org/10.1016/j.addma.2016.05.004

ectures, to name a few. A schematic diagram of a Mask Imagerojection based Stereolithography (MIP-SL) system is shown inig. 1a.

∗ Corresponding author.E-mail address: yayuepan@uic.edu (Y. Pan).

ttp://dx.doi.org/10.1016/j.addma.2016.05.004214-8604/© 2016 Elsevier B.V. All rights reserved.

Similar to other additive manufacturing processes, the objectsfabricated by the SL process also have stair-stepping effect dueto the use of two-dimensional (2D) layers [11–13]. As shown inFig. 1b–d, a given 3D model is first sliced into a set of 2D layers. Bystacking the sliced 2D layers together, a physical object can be fab-ricated to approximate the original Computer-aided Design (CAD)model. Due to the stacking of 2D layers, the fabricated curved sur-faces especially the ones whose normals are close to the buildingdirection (Z axis) may have large approximation errors. This stair-stepping effect greatly limits the use of AM in the applications thatrequire smooth surfaces, for example, the optical lens, the rotatingshaft and socket in mechanism design, etc.

Approaches such as controlled curing depth [14,15] methodshave been developed for the fabrication of smooth down-facingsurfaces. As to the up-facing surfaces fabrication, a meniscusapproach for achieving improved surface finish in the SL processwas presented in our previous work [16]. The developed meniscusapproach for the Mask Image Projection based Stereolithography(MIP-SL) system is shown in Fig. 2. Supposing a CAD model that isto be built has a curved surface, as illustrated in Fig. 2a, the menis-

ptimization for smooth surface fabrication in Stereolithography,

cus approach could be used. The first step is to solidify liquid resinlayer by layer using the conventional Stereolithography process, asshown in Step-1 in Fig. 2b. The cured layers form intersecting planesin corners. Due to surface tension, meniscuses will be formed after

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2 Y. Pan, Y. Chen / Additive Manufacturing xxx (2016) xxx–xxx

Fig. 1. An illustration of a Stereolithography system and the stair-stepping effect.

h for i

rutptbascmcbsnbbc

mbc

Fig. 2. An illustration of the meniscus approac

aising the built layers out of the liquid resin by moving the platformp using the Z-stage with a proper velocity. The proper velocity set-ing for different resins has been identified by experiments in ourrevious work [16]. Accordingly, another mask image can be usedo solidify the resin in the meniscus areas as Step-2 in Fig. 2b. Inrief, using the meniscus approach, a set of 2D layers is solidifiednd the meniscuses are cured afterwards. By adjusting the slicingtrategy and when to raise the built layers out of liquid resin, wean change the height and width of corners, thus manipulating theeniscuses to better suit the given shape. To conclude, in SL pro-

ess with meniscus approach, the 3D design is still approximatedy stacking 2D slices as the conventional SL process. However, thetaircase effect problem caused by the 2D layer stacking mecha-ism is eliminated by coating the 2D layers and filling the cornersetween layers using meniscus principle. Hence, this approach cane used to fabricate smooth curved surfaces no matter convex oroncave.

Please cite this article in press as: Y. Pan, Y. Chen, Meniscus process oAddit Manuf (2016), http://dx.doi.org/10.1016/j.addma.2016.05.004

Preliminary studies were presented in [16] and the use of theeniscus approach in fabricating smooth up-facing surfaces has

een verified. The primary challenge in planning meniscus pro-ess is also identified, that is, how to achieve the required accuracy

mproving surface finish in the MIP-SL process.

and build time when using the meniscus approach to improvesurface finish. Such manufacturing process optimization is a multi-objective problem with nonlinear functions. How to solve such anoptimization problem in the meniscus process planning is still anopen question. In this paper, we present a systematic process plan-ning method to optimize the meniscus forming parameters basedon a simulation of predicted meniscus shape.

The remainder of the paper is organized as follows. Section 2discusses the models of formed meniscus for various cases, andchallenges in planning meniscus forming process. Based on them,a meniscus process planning approach is presented in Section 3 toconsider the meniscus of multiple cured layers. Important processparameters related to forming the meniscus are also identified andcharacterized in the section. Section 4 presents the formulation ofthe optimization problem and a general process to solve the formu-lated optimization problem. The experimental setup for performingphysical experiments and test results are discussed in Section 5.

ptimization for smooth surface fabrication in Stereolithography,

Finally, conclusions with future work are drawn in Section 6.

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F

2

tdgb

�

wafrgtifll

acwfmcdca

Cb

y

Cca

y

y

Cca

y

y

Cbb

Fig. 4. Parts fabricated with conventional method and meniscus method: (a) Sideview of the region A1 of the part fabricated with layer-by-layer Stereolithographymethod (M1); (b) Side view of the region A1 of the part fabricated with meniscus

ig. 3. Meniscus wetting to intersecting plane surfaces and fluid interface profile.

. Challenges in meniscus process planning

As shown in Fig. 3, meniscus is the liquid residue that exists inhe corner of two intersecting planes. As presented in [16,17], theeveloped meniscus shape equation relates liquid interface withravitational influence and interfacial tension that is representedy contact angle � and capillary height hc , as shown in Eq. (1).

gy − 12

h2c × �gy

(1 − sin�)(1 + y2)2/3

= 0; (1)

here � is the density of the liquid, g is gravitational acceleration,nd y is the height of the meniscus above the horizontal plane sur-ace. The contact angle � in Eq. (1) is the angle at which the liquidesin interface meets the solidified resin surface. Specific to theiven liquid and solid system, the contact angle is determined byhe interactions between the liquid resin, solidified resin and airnterfaces. The capillary height hc is the maximum height that theuid can reach on an infinite vertical wall. hc is a characteristic

ength for the fluid subject to gravity and surface tension.The value of h and b of a corner determines the meniscus profile,

ccordingly, determining the geometrical accuracy of the fabri-ated curved surface. When and where to form and cure meniscusesill not only affect surface accuracy, but also influence other per-

ormance metrics such as build time. Therefore the prediction ofeniscus profiles and the optimization of process parameters are

ritical for the meniscus method. To model the meniscus shape,ifferent boundary conditions have been used in the following fiveases [16,17]. For completeness, the boundary conditions are listeds follows:

ase 1. both |h| and b can be considered infinite. The relatedoundary conditions are:

˙ (x = 0) = −tan−1 �; y (x = ∞) = 0 (2)

ase 2. |h| is smaller than hc , and b is bigger than b0. Hence theurvature of the meniscus is decided by h. The boundary conditionsre as follows.

˙ (x = 0) = −tan−1 �; y (x = 0) = −h;

˙ (y = 0) ≤ −tan−1(�

2− �

); (3)

ase 3. |h| is bigger than hc , and b is smaller than b0. Thus theurvature of the meniscus is decided by b. The boundary conditionsre as follows.

˙ (x = 0) = −tan−1 �; y (x = b) = 0

˙ (y = 0) ≤ −tan−1(� − �

); (4)

Please cite this article in press as: Y. Pan, Y. Chen, Meniscus process oAddit Manuf (2016), http://dx.doi.org/10.1016/j.addma.2016.05.004

2

ase 4. h is smaller than hc , and the length of the horizontal plane is smaller than b0. Hence the curvature of the meniscus is decidedy both b and h. The problem follows the boundary conditions:

method (M2), which has improved surface finish but large approximation error inArea A1; (c) Top view of the region A2 of the part fabricated with method M1; (d)top view of the region A2 of the part fabricated with method M2.

y (0) = −h, y (b) = 0; (5)

Case 5. |h| is smaller than hc , and the discontinuity is connectedwith a horizontal plane that the size is smaller than the liq-uid droplet size. The problem turns into a wetting puddle in thebottom-up projection system and the boundary conditions are:

y (x = 0) = 0, y (b) = 0; (6)

By using the governing equation Eq. (1) and boundary condi-tions Eqs. (2)–(6), a meniscus profile could be modelled by solvingthe mathematical problem. However, to build a part, the manufac-turing process could involve multiple meniscuses, with an infinitenumber of choices on forming meniscus segments to fabricate thefinal part. Hence, how to plan the meniscus-based SL process basedon surface shape prediction is quite challenging. Improper menis-cus forming process parameters will cause large approximationerrors or even surface defects of the fabricated parts. An exampleis shown in Fig. 4. The CAD model shown in Fig. 4 (Left) was fab-ricated with the conventional layer-by-layer SL method (M1) andthe meniscus-based SL method (M2). As shown in the microscopicimages of Area A1 and A2 in Fig. 4, the meniscus approach improvesthe high surface finish, but leads to a much larger approximationerror in the fabricated part in the area A1.

To address the challenges in predicting and optimizing menis-cus surfaces, a systematic process planning and modeling methodis presented in Section 3. First, a pipeline of the meniscus-basedprocess is reviewed in Section 3.1. Two important properties formodeling the meniscus surface profile and establishing parametercharacterization are presented in Sections 3.2 and 3.3. A databaseis built to store the shape information of all meniscus segments,and a discrete point-based representation method is developed torepresent 3D meniscus shapes. Based on the models presented inSection 3, meniscus optimization method is presented in Section 4in order to achieve desired building accuracy. Moreover, an exam-ple of the process optimization based on a convex lens is given inSection 4.

3. Meniscus process planning in the SL processes

3.1. Pipeline of the meniscus-based Stereolithography process

ptimization for smooth surface fabrication in Stereolithography,

Fig. 5 shows the pipeline of the meniscus-based Stereolithog-raphy process. An input CAD model is first divided into a set ofcurved segments using a given meniscus thickness D. Notice thatthe used meniscus thickness is determined by the capillary height

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s-bas

itl22s

sfiaeuowcascaAfibpor

3

mtlaTaclIs

immwhbt

Fig. 5. Pipeline of the meniscu

n the meniscus models and is usually much larger than the layerhickness as used in SLA, which is related to the curing depth of theiquid resin. Hence, a meniscus segment can be sliced into multipleD curing layers using the curing layer thickness. After curing all theD layers of each meniscus segment in a layer-by-layer fashion, thetacked layers will be raised out of liquid resin to form meniscuses.

For each divided meniscus segment, we could extract curvedurface information from the CAD model and save the surface pro-le as a set of points P(x,y,z,n), where x, y and z are coordinatesnd n is the norm vector. The process of point-based surface profilextraction is discussed in Section 4.2. The normal information n issed to determine the surface type. The meniscus approach willnly be applied to the up-facing surfaces whose normal is positiveith the Z axis [16]. If a meniscus segment does not contain any

urved surfaces, it will be built using the layer-by-layer fabricationpproach without forming meniscuses. Otherwise, if a meniscusegment has curved up-facing surfaces, the meniscus forming pro-ess will be planned and optimized. The meniscus process planningpproaches and related algorithms are presented in Section 4.3.fter the optimal meniscus process parameters have been identi-ed for one meniscus segment, the next meniscus segment woulde processed in the same way until the last one. Based on thelanned 2D layers and meniscus forming segments, a physicalbject can be fabricated using the meniscus-based Stereolithog-aphy process.

.2. Modeling of surface profile of a meniscus segment

As discussed in Section 2, two important parameters in deter-ining the shape of the formed meniscus profile are h and b. When

he meniscus is formed in the corner defined by two neighbouringayers, as shown in Fig. 3, the meniscus model is straightforwardnd the mathematic model in Section 2 can be directly applied.his method has been presented in [18], in which the meniscusesre formed and cured in each layer. However, this approach willause extremely slow build speed, that is, the build time will be ateast doubled due to the additional meniscus curing for each layer.n addition, such an approach was found to only work for up-facingurfaces that are 40◦ or less from the horizontal surface [18,19].

Different from all others’ work, in this research, the meniscuss not formed for each layer. Instead, meniscuses are formed after

ultiple layers have been cured. By forming meniscuses amongultiple 2D layers, we are able to improve the building speed as

Please cite this article in press as: Y. Pan, Y. Chen, Meniscus process oAddit Manuf (2016), http://dx.doi.org/10.1016/j.addma.2016.05.004

ell as to achieve better geometric accuracy. The challenge is thenow to model the meniscus when the corner structure is not formedy just two intersecting planes, but a set of small stairs. An illustra-ive example is shown in Fig. 6b and c, where the meniscuses are

ed Stereolithography process.

formed after two 2D layers are cured (1st–2nd). To use the equa-tions developed in Section 2 to model the meniscus, the primaryquestion that needs to be answered is:

What is height h and width b in the equations to describethe meniscuses formed by multiple cured layers?

To simplify the discussion of the problem, we will present ourwork for a 2D case. The work could be extended to 3D cases asshown in Section 4. Suppose that a meniscus segment starting fromz0 to zn(zn − z0 = �), is further sliced into multiple 2D layers andthe ith layer is at the level z = zi. In order to model the meniscus,we have the following observation with two properties to addressthe above question.

Observation. If a layer is inside the meniscus, the layer has smallervalue of z than the meniscus M(x,z), that is, p(x′) = zi < M(x′) where p is aboundary point of the layer. If a layer is outside the meniscus, the layerhas bigger value of z than the meniscus M(x,z), that is, p(x′) > M(x′)where p is a boundary point of the layer.

As shown in Fig. 6b and c, the dotted orange curve denotes themeniscus which is formed among two layers (1st and 2nd layers). InFig. 6b, the boundary point p of the 1st layer is inside the meniscus,while in Fig. 6c, p is outside the original meniscus curve.

Property 1. If the 2D layer (z = zi) is inside the meniscus curve({p(x′) < M(x′)|for all p(x′) = zi}), its effect on the formed meniscusshape is negligible.

As shown in Fig. 6b, the 1st layer is inside the meniscus curve.The meniscus M could be modelled by choosing h = L1 + L2 = D, andb = t1 + t2 = t. For some resins and some geometry where the 2D layercontour inside of the meniscus is very close to the meniscus profile,the effect of the 2D layer on the meniscus curve may not be neg-ligible. An undesired bulge may occur due to the surface tensionand affect the accuracy. To keep Property 1 true in such cases, wewill manipulate the mask image for that layer curing, so that theboundary of the problem layer will align with the boundary of thetop layer of the meniscus and have no effect on the meniscus curve.For example, supposing the p(x) in Fig. 6b will affect the meniscuscurve greatly, the geometry of this layer, L1, will be modified byediting its mask image so that the fabricated layer L1 will alignwith the top layer L2, and the resulted geometry will be the sameas Fig. 6a.

ptimization for smooth surface fabrication in Stereolithography,

Property 2. If the cured layer (z = zi) is outside of the meniscus curve({p(x′) > M(x′)|for all p(x′) = zi}), the formed meniscus profile M issplit by this layer. Let p(x′, zi) be a boundary point on this layer, and(x0, z0), (xn, zn) are the boundary points of the bottom and top 2D

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Meniscus

t

(a) Meniscus M: f(L1, t)= f(D, t)

D L1

Curing layerthickness

(b) Meniscus M: f (L1+L2

L2

L1

t1

t2

Meniscus Mx p(x’)

1st

2ndM

M(x’)Curing layer

thickness

L2

L1

t1

t2

f2

f1

Meniscus segment

x

p’(x’)

1st

2nd

M(x’)

Fig. 6. Two properties for modeling meniscus t

F2

lb

olt

3

mfotlrbmm

1

2

3

ig. 7. Variables in modeling of meniscus segments that are formed among multipleD layers.

ayers in the meniscus segment. Then the meniscus could be modelledy: M′ = f(h1, b1) + f(h2, b2) = f(zi − z0, x′ − x0) + f(zn− zi, xn, − x′).

For example, in Fig. 6c, the 1st layer boundary p’ extends outf the meniscus profile. So the meniscus formed between the 0th

ayer and 2nd layer is constructed by two curves f1(L1, t1) and f2(L2,2), instead of the curve f (h = D, b = t) in Fig. 6b.

.3. Meniscus process parameters characterization

With the two aforementioned properties and the mathematicalodels in Section 2, we can model the meniscus profiles that are

ormed among multiple 2D slicing layers. Fig. 7 is a 2D illustrationf a meniscus formed over multiple layers. In this example, afterhe 1st meniscus profile is formed and cured for D1, assume n 2Dayers (i.e. 1st–nth) are built using the conventional Stereolithog-aphy process for 2nd meniscus segment D2. After the nth layer isuilt, all the layers will be raised out of the liquid resin to form aeniscus profile to be cured. The process parameters that affect theeniscus shape in such a multiple layers case include:

. Starting and ending positions: p0, pn. They are called the menis-cus points, where the meniscus forming process is performed.That is, the object is fabricated layer by layer until the nth layerhas been built and the fabricated object is then raised out of liq-uid resin to form meniscus. All the layers from the 1st layer tothe nth layer are called a meniscus segment, with a thickness ofD. It is the thickness that is used to divide curved surfaces intosegments, as described in Section 3.1.

. Boundaries of each cured layer in the meniscus segment: {pi|i =1, 2, . . ., n}. Fig. 7 is an illustration of a 2D case. In this 2D exam-ple, p1, . . ., pn are the boundaries of the 1st, . . ., nth layers,repectively. Based on the two aforementioned properties, theboundary positions of the sliced 2D layers will determine themeniscus profiles.

. Stair dimensions: ti, Li (i = 1,. . .n). They can be derived from the

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boundary positions of each 2D layer. Based on the two proper-ties, if all the layers (i = 1, 2, . . ., n − 1) sit inside of the meniscusprofile f

(h = D, b =

∑n1ti

), the meniscus is this simple curve

f(h = D, b =

∑n1ti

). However, if the ith layer breaks the menis-

, t1+t2) = f(D, t) (c) Meniscus M: f1 (L1, t1)+f2( L2, t2)

hat is formed among multiple 2D layers.

cus curve f , the meniscus will be calculated separately on the two

sides of the ith layer, leading to f = f 1(h =

∑i1Li, b =

∑i1ti

)∪

f 2(h =

∑ni Li, b =

∑ni ti

). If there are other layers that break the

meniscus curves, the same rule will be applied to split the menis-cus curves into smaller segments. The meniscus shape is definedas a series of those connected meniscus segments.

Hence, the meniscus formed among multiple cured layers {pi|i =1, 2, ..., n} could be written as:

M (x) = f (pi|i = 0, 1, .., n} =∑m=M

m=1f(∑i=km+1

i=kmLi,

∑i=km+1

i=kmti

);

(7)

k1 = 1; kM+1 = n; M < n; (8)

ti = x (pi−1) − x (pi) ; Li = y (pi) − y (pi−1) ; i ∈{

1, 2, .., n}

(9)

As denoted in Eq. (7), the meniscus formed among layers {pi|i =1, 2, ..n} is a combination of M segments. The mth segment covers(km+1 − km) cured layers, i.e., from the pkm layer to the pkm+1

layer.

3.4. Meniscus database construction

In a SLA system, the achievable resolution is limited by the res-olution of mask image and material deposition. Ideally, it is desiredto achieve infinitely small resolution for material deposition usingthe meniscus approach. However, regardless of whether the inputenergy is based on mask image projection or laser beam scanning,the light spot size is limited. Therefore, there is no need to cal-culate the continuous mathematic model of the formed meniscus.Instead, the performance will not be affected if the meniscus profileis sampled to a set of points with a controlled sampling density. Bysampling the continuous meniscus curve into discrete points, theNP-hard optimization problem that is to be presented in the fol-lowing sections can be solved approximately. Also the computationtime can be greatly reduced based on sampling data.

A meniscus database is constructed to store the sampled pointsof functions f (h, b). As shown in Eq. (7), any meniscus formedamong multiple cured layers is a combination of segments whichare represented by function (h, b) with different values of h and b.With the constructed database of the meniscus sampling points,the process planning will not need to repeatedly solve and samplea meniscus segment. Instead, it can just search the correspondingpoint set in the database with a given setting (h, b). Fig. 8 shows themeniscus database structure used in our study. The material prop-erties and model parameters of the two materials that are used inthe experimental tests (E-shell and SI 500 from EnvisionTEC Inc.)are characterized in Table 1.

ptimization for smooth surface fabrication in Stereolithography,

In our tests, the two materials have residue modeling param-eters “Maximum wetting b0” and “Capillary height” as calibratedin Table 1. Also the Stereolithography system used in our testshas 5 �m XY plane resolution and 10 �m minimum curing layer

Please cite this article in press as: Y. Pan, Y. Chen, Meniscus process optimization for smooth surface fabrication in Stereolithography,Addit Manuf (2016), http://dx.doi.org/10.1016/j.addma.2016.05.004

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Fig. 8. Database of the meniscus profiles M(x) with different h and b.

Fig. 9. An illustration of the curved surface information extraction process.

Fig. 10. Approximation error check process in optimized meniscus approach.

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Y. Pan, Y. Chen / Additive Manufacturing xxx (2016) xxx–xxx 7

Input: Meniscus segment modelinformation C(x,y), h=D, b=t,

Approximation errorrequirements: {-e_left, e_right}

Exceeds the left boundry?

Simulate theMeniscus profile

M(x,y)

Generate the error profileE(x,y)

Exceeds the rightboundry?

N

Y

Y

Insert new sub-slice,update {pi}, h,b

Insert a new meniscus point to split thecurrent meniscus curve, update {p0, pn}, h,b

OptimizedMeniscus

fabrication planN

Fig. 11. The framework of the optimization process of meniscus segment fabrication plan.

(A)

Pn (bn, y0)

P0 (b0, 0)x

x

(a)

M0

M1

(b)

M0

M1

M0-1*

(c)

M0

M1

M0-1* S0-1*

Fig. 12. An illustration of a meniscus planning proce

Table 1Residue modeling parameters.

Materials Viscosity Density Contact angle Capillaryheight

Maximumwetting b0

ttfmTi

tndai

3

ncs

E-shell 339.8cP 180cP 21◦ 1.72 mm 2.25 mmSI 500 1.19 g/cm3 1.10 g/cm3 25◦ 1.88 mm 1.567 mm

hickness in the Z axis. So in the database, the meniscus segmenthickness is considered to be 1500 �m of b and 1700 �m of h. It isurther sampled into a set of segments M (x) = f (h, b) by an incre-

ent of 5 �m in b direction and 10 �m in the building direction h.hen each segment is sampled into a list of points (x, M (x)) by an

ncrement of 5 �m in the X direction.Fig. 8 shows the table of all meniscus segments f (h, b). Each

able element contains a list of sampling points. Therefore, theonlinear equation solving problem is substituted with a simpleatabase searching problem. Due to the limited system resolutionnd the material properties, the meniscus sampling point databases relatively small, and the data searching time can be rather fast.

.5. Sampling CAD model into point-based meniscus segments

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Similar to representing formed meniscus profiles, there is noeed to use a continuous representation model for storing theurved surface information. A discrete point-based model repre-entation can be used to meet the accuracy requirement of the

(d)

ss by inserting meniscus points and sub-slices.

meniscus-based SLA processes. Moreover, a point-based represen-tation could be processed faster and more easily.

Since both boundary and normal information are required forplanning the meniscus forming process, the Layered Depth-normalImages (LDNI) representation [20,21] is used to slice the originalCAD input and generate the point set of a segment of curved sur-faces with normal information. LDNI is an implicit representationof solid models that sparsely encodes the shape boundary in threeorthogonal directions [20,21].

According to the definition of LDNI, an orthogonal projection isused in extracting LDNI points of a certain Z segment from inputpolygonal models. That is, the coordinate of a query point used inthe LDNI projection procedure is calculated as following:

x′ = minExt (0) + Ix × �x (10)

y′ = minExt (1) + Iy × �y (11)

z′ = minExt (2) + i × �z (12)

�z = ceil (maxExt (2) − minExt (2))/ (s − 1) (13)

where �x, �y denote the resolutions in the X and Y directions,

ptimization for smooth surface fabrication in Stereolithography,

respectively, and {minExt (k) , k = 1, 2, 3} and {maxExt (k) , k =1, 2, 3} are vectors that represent the minimum and maximum val-ues of the X, Y, and Z coordinates of the calculated boundary box. Ixand Iy are the index of the X and Y axis, respectively. s is the number

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Fig. 13. An illustration of the CAD model of a divided meniscus segment and its original meniscus model.

icated

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Fig. 14. Approximation errors of the fabr

f points per ray, and i is an integer between 0 and S. Because onlyhe current layer is of interest, z follows the boundary condition:

inExt (2) = n × D; maxExt (2) = (n + 1) × D (14)

here n is the current segment number, and D is the meniscusegment thickness. Notice that D does not have to be the normallicing thickness in AM processes. As mentioned in Section 4.1, thehickness of a meniscus segment is defined by the capillary heightf the liquid, which is usually larger than the curing thickness in

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he SLA process.By adjusting the values of �x, �y and �z, the accuracy of the

xtracted LDNIs sampling points can be controlled accordingly. Its reported that even for very complex geometries and high accu-

part by using the meniscus in Fig. 13c–d.

racy requirements, the construction process is fast with the aid ofgraphics hardware [22]. Fig. 9 is an illustration of the point-basedmeniscus segment of a given CAD model using the LDNI algorithms.Similar research in efficient slicing using LDNI has been reportedbefore [23,24].

Positive normal denotes that the meniscus segment containsup-facing surfaces (N·Z > 0). Let ε be the threshold that indicates thecurves that need the meniscus method. If (N·Z > ε), meniscus will beformed in order to better approximate the up-facing surfaces. The

ptimization for smooth surface fabrication in Stereolithography,

meniscus planning and process optimization will be presented inthe following section.

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Fig. 15. Randomly modified menisc

. Meniscus planning and process optimization

.1. Optimization objectives

As discussed in Section 3, the optimization problem is a multi-bjective problem:

1) Smoothness. This is achieved by the choice of the curve indi-cator ε that was discussed in Section 3.5. The smaller ε is, thebigger portion of the up-facing surfaces will be taken care of byusing the meniscus approach.

2) Approximation errors: (−emin, emax). The dimensional accuracyis not guaranteed, and in some cases, may be worse if theformed meniscus is not planned well. Therefore, the dimen-sional accuracy is an important objective when we plan themeniscus-based SLA process. The required accuracy should bemet together with the smoothness by applying the meniscusapproach. In this research, −emin and emax is used to denote thebounds of the allowed approximation error.

3) Minimum build time. After the planned 2D layers are cured, theplatform is raised up above the liquid surface to form menis-cuses. After a certain waiting time, light will be projected to

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cure the formed meniscuses. This additional meniscus formingand curing process will elongate the build time. Hence using aminimum number of meniscus segments is desired for reducingthe required build time.

dels and the resulting error models.

4.2. Optimization strategies

Fig. 10 shows the meniscus process planning framework. Basedon the ith meniscus segment information that is extracted fromCAD model as discussed in Section 3, the Z height map of curvedsurfaces is generated as C(x,y). Accordingly, the Z height map of theplanned meniscus is shown as M(x,y). By deducting the C(x,y) fromM(x,y), an error Z height map E(x,y) could be obtained.

The initial meniscus plan is a simple meniscus curve with h = Dand b = t, as the example shown in Fig. 6a. If the error Z height mapis bounded between {−emin, emax}, it will be saved; otherwise, themeniscus process parameters as discussed in Section 3.3 will beadjusted. The following two strategies are used in modifying themeniscus shape:

(1) If the left bound −emin is exceeded, new cured layer slices willbe inserted to adjust the process parameters {pi|i = 1, 2, ..., n}to push the meniscus closer to the CAD profile.

(2) If the right bound emax is exceeded, new meniscus starting pointand ending point will be inserted to adjust the process param-eters p0, pn to pull the meniscus back to the CAD profile.

Fig. 11 shows the meniscus segment curing plan optimizationalgorithm.

Fig. 12 is a graphic illustration based on a simple 2D example.The orange curve represents formed meniscus profiles; the dot-

ptimization for smooth surface fabrication in Stereolithography,

ted orange curves represent sliced layers; and the black curve isthe given surface profile defined in the CAD model. The meniscusshape in the area of y ∈ (0, y0) is first estimated and then com-pared with the input geometry. If the error map E(x) is within the

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cceptable range, points p0 (b0, 0) , pn (bn, y0) are selected as theeniscus points M0 and M1. If the meniscus curve is outside of the

AD profile and the error is represented by a positive �a, a new pointill be selected as an additional meniscus point, as M0–1 in Fig. 12c.n the contrary, if the meniscus curve is inside the CAD profile,

or example, the orange curve between M0–1 and M0 in Fig. 12c,ew slices would be added to push the meniscus curve outwardo better approximate the CAD profile. Curves S0–1 is an example.bviously, by selecting different positions for the inserted meniscusoint M0–1, the modified meniscus would have different accuracy.ith the same meniscus curing times and smoothness, it is desired

o have the minimum approximation error.Therefore, when the current meniscus plan cannot meet the

ccuracy requirements, how to identify the optimal position tonsert new slices or new meniscus points, is the key to the meniscusrocess planning and optimization problem. Methods like brutalearching, trial and error, or guessing are not efficient and mayot work for more complicated geometries. Consequently an opti-ization algorithm is required for the meniscus process planning

roblem.

.3. Meniscus process optimization algorithms

To find the optimal position for inserting a new slice, the mini-ization problem can be formulated as follows.

Policy for determining sub-meniscus point position:

roblem 1.1. Input : C (x, y) ,{P0, P1, . . ., Pi, . . ., Pn

}

in{(C (x, y) − M′ (x, y)

)|y ∈ (y0, yn)}

′ (x, y) =∑n

1ki ∗ (f (Pb|b = 0, 1, . . .i) + f (Pt |t = i, i + 1, . . .., n));

n

1ki = 1, 0 ≤ ki ≤ 1, ki : integer

here M′ (x, y) is the modified meniscus among the 0th layer P0 andhe nth layer Pn as shown in Fig. 7, so the portion of the meniscusegment where y ∈ (y0, yn) is considered here. ki is the indicatorhat whether the point Pi on the ith layer is selected as an inserted

eniscus point M0–1. If ki = 1, Pi is selected as a meniscus point. Lind ti are the height and width of the step created by the ith layernd the (i–1)th layer, as shown in Fig. 7.

Similarly, the position of the inserted additional sub-slice coulde optimized by solving the following minimization problem:

Policy for determining sub-slice position:

roblem 1.2. Input : C (x, y) , l,{P0 (x0, y0) , P1 (x1, y1)

}

in{(C (x, y) − M

′′(x, y)

)|y ∈ (y0, y1)}

′′(x, y) =

∑u

1si ∗ (f (i ∗ l, |xi − x0|) + f ((u − i) ∗ l, |x1 − xi|)) ;

u

1si = 1, 0 ≤ si ≤ 1, si : integer

here M′′(x,y) is the modified meniscus among the 0th layer and 1stayer after having inserted a new slice, as the orange dotted curvehown in Fig. 12d. There are u candidate sub slices generated byurther slicing the 1st layer (from P0 to P1) with a layer thickness l.

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f si equals to 1, ith sub-slice is selected to be inserted between layer0 and layer P1, at y = yi = y0 + i ∗ l. With the above two policies,he optimized meniscus segment fabrication plan could be gen-rated for one meniscus segment and then the next one can be

PRESScturing xxx (2016) xxx–xxx

repeated until the whole CAD model is processed. The algorithmsare described as following.

Algorithm 1.

Algorithm 1.1. Insert meniscus point.

Algorithm 1.2. Insert new curing layer.

4.4. Meniscus process optimization example

To better demonstrate the presented optimization algorithms,an example of the meniscus process planning and optimizing pro-cess is shown in Figs. 13–16. Suppose the maximum allowedapproximation error is (−30 �m, 30 �m). Fig. 13a is the extractedCAD model of a sliced meniscus segment with D = 250 �m andFig. 13b is its Z height map of the CAD model. We first set the cur-ing layer thickness L the same as D, which is the case as illustratedin Fig. 6a, with D = L1 = 250 �m and b = 90 �m. The meniscus curveinformation f(250,90) relates to the element (i = 24, j = 17) can befound in the table shown in Fig. 8. The meniscus model and therelated meniscus height map are generated by using the samplingpoints in the corresponding list, as shown in Fig. 13c and d.

By deducting Fig. 13b from d, the error model and the relatederror height map can be generated as shown in Fig. 14a and b. It isfound that the approximation error is in the range of (−56 �m, 0)

ptimization for smooth surface fabrication in Stereolithography,

after applying the planned meniscus. To demonstrate the benefitof the optimization algorithms, the meniscus process parameterswere first modified randomly 20 times to find the meniscus modelthat gives best result. The optimal meniscus model generated by the

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Y. Pan, Y. Chen / Additive Manufacturing xxx (2016) xxx–xxx 11

Fig. 16. Optimal meniscus models and the resulting error models.

Table 2Accuracy of the built geometries with different building strategies.

Measurements Accuracy (�m) Roughness (�m)

Maximum offset Average offset Ra Rq Rz

Fig. 4a—M1: Part fabricated by conventional SL method 12 4 6.7 8.8 16Fig. 18—M2-1: Part fabricated by previous meniscus-based SL method 375 258 NA NA NAFig. 18—M2-2: Part fabricated by optimized meniscus-based SL method 6 2.3 0.7 0.9 1.7

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Fig. 19b—Lens A: Part fabricated by conventional SL method 45Fig. 19d—Lens B: Part fabricated by optimized meniscus−based SL method To

rutal search and its relevant error models are shown in Fig. 15a–d.t is shown that only a slight improvement was made with a randomearch with a pool size of 20. The approximation error is in the range−41 �m, 26 �m) with the modified meniscus.

Instead of the brutal search, the optimization algorithm asescribed in Section 4.3 was used to find the optimal meniscusrocess parameters. The original meniscus result gives a perfectrror upper bound, but the lower bound of the approximation errorxceeds the given error bound, which leads to the second “if” condi-ion in “Algorithm 1” being executed. Hence, Algorithm 1.2 is usedo insert a new sliced 2D layer to modify the meniscus. The optimal

eniscus process plan was identified after iterating once. The opti-ized meniscus models and its resulting error models are shown

n Fig. 16. It is shown that the result is significantly improved byeducing the approximation error to (−20 �m, 0 �m), which has

et the given accuracy requirement. Compared to the brutal search

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ptimization, it returns much better results with less computationime.

32 15 13 29ll for the measurement Too small for the measurement 0.7 1.1 2.2

5. Experimental verification of the optimized meniscusmethod

A Mask Image Projection based Stereolithography (MIP-SL) sys-tem has been built to verify the meniscus-based SLA approachpresented in the paper. A photo of the prototype system is shown inFig. 17. In addition, a Matlab program has been built to simulate themeniscus profiles for different variables. Accordingly, a meniscusdatabase can be pre-computed and stored.

The model shown in Fig. 4 was fabricated with an optimizedmeniscus process method. In the optimized meniscus process, toreduce the positive approximation error, a new meniscus point wasadded into the previous meniscus point set {Mn}, which was usedfor fabrication of Fig. 4b. The new meniscus point pulls the meniscusprofile back to get closer to the CAD model profile. A graphical andquantitative comparison of the fabrication results using conven-

ptimization for smooth surface fabrication in Stereolithography,

tional method M1, previous meniscus method M2-1, and the newlyoptimized meniscus method M2-2, are shown in Fig. 18. It is shownthat the part fabricated with optimized meniscus method M2-2 is

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F

nbmirpt

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Fig. 18. Geometry profile of the models fabricated with the CAD model given in

Fp

ig. 17. The developed MIP-SL testbed for verifying optimized meniscus method.

ot only smoother than part fabricated with conventional method,ut also closely approximates the given CAD model. The approxi-ation error and roughness measurements of the parts are listed

n Table 2. Because of the large approximation error, the part fab-icated by previous meniscus method M2-1 is considered a failedart and its surface finish has not been measured quantitatively,hus the corresponding cell in Table 2 is denoted as NA.

In addition to the concave test case, a convex case was alsoested to verify the optimization algorithm. Fig. 19 shows the fab-

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icated results of a micro-lens with a convex surface. The lens sizes 2.54 × 2.54 × 0.5 m. Fig. 19a shows the CAD model and Fig. 19fhows a piece of paper with blue and red lines for testing. A layerhickness of 100 �m was used to fabricate the part. Fig. 19b and c are

ig. 19. A test case of a micro-lens using E-shell: (a) CAD model (c–d) microscopic imagart fabricated using the optimized meniscus method, (g) surface measurements of the tw

Fig. 4: using conventional layer-by-layer method M1, previous meniscus methodM2-1, and the newly optimized meniscus method M2-2.

the microscopic images of the lens that is fabricated by the conven-tional layer-based MIP-SL process (lens A), and Fig. 19g shows theoptical performance of the fabricated micro-lens. The paper imageis distorted by the micro-lens due to its rough surface. In compar-ison, Fig. 19d and e are the microscopic images of the lens that isfabricated by the optimized meniscus method (lens B). The opti-cal performance of lens B on the same piece of paper is shown inFig. 19h, which demonstrates a significant improvement over thatof lens A. Profiles of the two surfaces as showed in Fig. 19c ande were sampled. Quantitative measurements of the accuracy andsurface finish were performed.

Table 2 shows statistics of the approximation errors and surfacefinish of the fabricated parts using conventional method, previousmeniscus method, and the new meniscus method that is opti-mized. The experimental results agreed with simulation results,and verified the effectiveness of the optimized meniscus methodin improving the surface finish of curved surfaces and meetingaccuracy requirements.

ptimization for smooth surface fabrication in Stereolithography,

6. Conclusion

A meniscus-based Stereolithography process and the relatedprocess planning and optimization methods have been developed

e of the part fabricated using conventional method (e–f) microscopic image of theo parts.

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or the fabrication of smooth curved surfaces. Theoretical models,rocess parameters characterization and optimization algorithmsave been presented in the paper.

An example based on a convex lens has been presented tollustrate the meniscus optimization process. Based on the givenAD model, the meniscus models in a database, and the computedpproximation error models, optimized meniscus process parame-ers have been identified using the developed algorithms. The resultas been compared with the optimization results generated byandom search. The example shows the improved efficiency andffectiveness of the optimization algorithm.

In addition, both hardware and software systems have beeneveloped for physical experimental verifications. Test exam-les on up-facing surfaces have been performed. The comparisonetween experimental results with and without the process plan-ing method illustrated the effectiveness of the developed method

n simultaneously controlling the accuracy and surface finish ofhe fabricated objects. It also showed that the optimized menis-us approach can reduce the build time by minimizing the numberf meniscus segments.

Although the presented micro-lens test case does not have aomplex shape, the developed frame could be applied to meniscusrocess planning of any arbitrary complicated up-facing surfaces.otential challenges include computation cost and meniscus struc-ure construction. Moreover, by combining this approach withpproaches for fabricating smooth down-facing surfaces, such asontrolled curing depth approach, complicated geometries withurved surfaces could be fabricated with high smoothness by usingtereolithography equipment. Therefore, future work will include:1) to improve meniscus database structure, (2) to improve predic-ion accuracy and test more complicated free-form surfaces, and (3)o integrate meniscus approaches for fabrication of both up-facingnd down-facing curved surfaces.

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[

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