computational quality measures for evaluation of

COMPUTATIONAL QUALITY MEASURES FOR EVALUATION OF PART ORIENTATION IN FREEFORM FABRICATION David C. Thompson Mechanical Engineering The University of Texas Austin, Texas 78712 USA (512) 471-7347 [email protected]

Richard H. Crawford Mechanical Engineering The University of Texas Austin, Texas 78712 USA (512) 471-3030 [email protected]

ABSTRACT Parts formed using freeform fabrication processes can vary significantly in quality depending on the orientation of the workpiece during manufacture. This paper presents four measures of quality for three performance measures in which this occurs, along with computational methods for computing their expected effects on faceted representations of a workpiece. The optimization of these quality measures is an important step in the automation of manufacturing straight from CAD models. The quality metrics developed quantify build time, material strength, and surface approximation error as functions of part orientation. A series of experiments performed on polycarbonate tensile specimens produced with selective laser sintering (SLS) show how significant orientation can be. Although SLS and selective area laser deposition (SALD) are used in the examples, the measures are applicable to other technologies as well. A preliminary investigation into the optimization of orientation with these measures is discussed, showing how properties of the measures developed can be used to generate trial orientations for the global optimization problem where several measures are combined.

KEYWORDS

solid freeform fabrication (SFF), process planning, workpiece orientation

INTRODUCTION A Solid Freeform Fabrication (SFF) process is any manufacturing technique that can be used to form a solid part of arbitrary geometry without part-specific tooling [1]. Ideally, processes should [1] • use design geometry directly, rather than relying on a faceted representation, and • run without human intervention. Currently, most SFF processes require the user to at least specify a set of operating parameters, and possibly change the designed geometry for manufacture. This requires significant expertise to produce parts of consistent quality. The goal of this research is to develop measures that quantify a process' performance for a given part so that prototyping and manufacturing process choices can be made by a designer who would otherwise not have the knowledge required. 1

DEVELOPING MEASURES Measures of quality are established for a process by examining the distinguishing characteristics of the process. When choosing a manufacturing technique, these characteristics may include the advantages or disadvantages compared to other techniques. When the measures are to be used in process optimization, the characteristics may simply be physical attributes of the parts produced. For SFF, these measures can be used to evaluate how significant orientation is, as well as to provide physically-based models for the objective functions used to optimize orientation. The following guidelines should be used when selecting a performance measure. 1. Determine the factors that differentiate the process from others (i.e., what distinguishing characteristics are associated with the process). 2. Find the set of operating parameters that affect the factors above. 3. Decide which factors are the most important to customers, and which of these are expected to vary significantly over the range of the parameters in step 2. 4. Choose a set of parameters with which to control the part quality. 5. Verify that varying the chosen parameters produces a significant change in the measures chosen in step 3. 6. Physically (as opposed to experimentally) model the measures in terms of the significant parameters. In the remainder of this section, we illustrate steps 1 through 4 for two SFF processes. Selective Laser Sintering As an example, consider selective laser sintering (SLS). SLS is a layered manufacturing technique that forms cross-sections of a part from powder. By melting only powder inside the cross-section with a laser, a layer is formed. The next layer of powder is spread on top of the existing layer by a roller, as shown in Figure 1. Since even the unsintered powder can support a load, no supports are necessary in SLS. The first step in determining performance measures is to list the differences between SLS and other manufacturing processes. A fishbone diagram of these differences is shown in Figure 2. For SLS, the fact that the part is built in layers is key because this leads to anisotropy. The operating parameters that have an effect on the performance measures of Figure 2 are shown in Figure 3. The performance measures of Figure 2 are broken into major areas where there is a difference from other manufacturing techniques; the density, thermal properties, and surface finish branches are present mainly due to the use of powder as a precursor material while the dimensional accuracy and mechanical properties can be attributed to the layered nature of SLS. Because of this, the parameters affecting SLS performance can be categorized as powder, orientation (or layer effects). The other branches in Figure 3 are due to the design of the machine and the physical principles used to carry out SLS. Of the effects in Figure 2, three will be mapped into objective functions: build time, surface finish (the “stair-step” effect), and material strength. These are chosen because of their importance to potential customers of SLS, such as service bureaus. Service bureaus are companies that purchase rapid prototyping machines and use them to manufacture parts for clients. Build time is vital to service bureaus because the more parts they can run in the same amount of time, the more profit can be made. Build time is also important to the customers of service bureaus who are charged for the amount of time it takes to manufacture their parts. Using the service bureau as a customer again, surface finish and part strength affect the price that parts made with SLS can yield. Surface finish is important to customers who intend to use the parts for sales, dimensional verification, or as patterns for molds. Good surface finish can also reduce the cost and time associated with post-processing of a model. Material strength is key for customers who are interested in functional parts. Other part properties, such as conduction coefficients or density may increase the value to the customer, but usually indirectly: for instance, density is related to part strength.

2

SIDE VIEW

Band Heaters Roller Feed Screw

Galvanometer mirrors

CO2 Laser

AAA

AAAA

Thermopile Laser Beam Powder Bed

Belt

Machine Frame

Motor Part Cylinder

Powder Cylinder TOP VIEW

Feed Rod

Scanned Part

Roller

Figure 1: Schematic of an SLS machine.

Dimensional Accuracy

Difference from Other Mfg. Techniques

In-plane accuracy Shrinkage Beam width Build direction accuracy Aliasing (layer thickness) Shrinkage

Thermal properties Density Local variation (Pwr. fluct.)

Heat capacity (density) Conductivity (density)

In-plane In-plane (porosity) Shear str. (interlayer bond) Build direction (aliasing) Local variation (SCSP, Vec. Length) Surface Finish Build direction Tensile str. (delamination) Local variation (Layer thickness, Pwr. fluct.) Fatigue str. (delamination)

Mechanical properties

Figure 2: Fishbone diagram of distinguishing physical effects, SLS

3

Setup time Build time

Time

Powder Particle size

Average Distribution Bed density Layer Thickness

Machine Preheat Temp. Force to roll powder(down) Layer thickness

Inaccuracy in piston step

Composition

Glass Transition Temp.

Uneven spreading of powder

Part Quality Vector length Energy density

Orientation Laser power Uneven heating BS SCSP Beam width, ω

Laser

Figure 3: Fishbone diagram of parameters affecting quality, SLS. Now that the performance measures have been selected, a set of parameters must be chosen. In this paper, orientation is used because it is a parameter common to all manufacturing processes, and often requires expert determination. This example using SLS will be continued throughout the paper. However, we will also consider a measure for SFF processes that must deposit supports for parts that cannot be defined as a single-valued height field in the build direction. Examples include selective area laser deposition (SALD), shape deposition manufacturing (SDM), or other deposition techniques where volumetric supports are required. The quality measure developed for these processes is related to build time, since it is critical in both SALD and SDM. Selective Area Laser Deposition Selective Area Laser Deposition (SALD) is a process that uses a laser to pyrolitically or photolytically decompose a gas near a substrate surface. Pyrolitic reactions occur at the surface, wherever it exceeds a given temperature. Photolytic reactions occur in a region near the focal point of the beam, where the photon density is above a critical level. Both methods require the movable lens shown in Figure 4 to change the focal length of the beam since the platform is not moved vertically (as is done with stereolithography or SLS). As the beam moves across the surface, material is deposited in its path. Material can be built upwards by leaving the beam in one position and shortening the focal length of the laser at the rate of deposition. The rate of deposition is dependent on the conductivity and absorptivity of the substrate, product, and reactant, as well as the pressure and flow rate of gas in the chamber. Current deposition rates are approximately 1.4(8.24 × 10−5 ) 3 in.3 to 2.7(1.65 × 10−4 ) mm min. ( min. ) [2]. These deposition rates may seem small, but the materials available with this process include carbon, silicon carbide, silicon nitride, and titanium oxide. The ability to build fully dense parts of arbitrary shape out of these materials can easily justify the processing time. Fishbone diagrams similar to those presented for SLS lead to the selection of a single quality measure for SALD: build time. This is selected because of the ceramic material systems SALD uses. Traditional manufacturing techniques have turnaround times as high as a year for parts such as turbine blades. SLS and other rapid prototyping processes use polymeric binders that reduce the final density of the ceramic and require postprocessing that results in shrinkage. SALD is then an ideal process for parts that require low turnaround times and high density ceramics. To be competitive with other rapid prototyping technologies that produce ceramics by postprocessing, the SALD process must concentrate on making the build times for parts less than the total processing time for other methods and on achieving dimensional accuracy better than the other methods. Because SALD is not necessarily a layered manufacturing process, there is no time penalty for building in the

4

Mirror Concave lens CO2 or YAG Laser

Convex lens

Laser Beam

Workpiece X-Y Table Support Gas

Figure 4: Schematic of a SALD machine. direction normal to the substrate, b. This means the build time is independent of the “height” of the part. Another consequence, though, is that there is no material to support overhangs, such as those shown in Figure 5; some sacrificial material must be deposited to form supports for every face whose normal, n, satisfies b · n < 0. This is accomplished by using two different precursor gases: one to create the part and one to deposit supports. Building supports imposes two time penalties: 1. the amount of time required to deposit supports and 2. the amount of time required to flush the chamber of the part forming gas and insert the sacrificial forming gas. A breakdown of the SALD process time requirements is shown in Figure 6. Based on this, we can develop an expression for the build time of a part using the SALD process.

Support volumes

Faces needing support are marked with dashed lines.

Figure 5: Overhangs required for downward-pointing faces.

Purge chamber Fill chamber with new material Time

Scan first material Purge chamber Fill chamber with new material Scan second material Done?

No

Figure 6: Process time breakdown for SALD. Since the rate of deposition is independent of direction, the build time will have terms proportional to the volume of the part and the volume of the supports. All of the computations required to produce scan patterns must be performed beforehand so that the number of switches between material systems can be kept to a minimum. The only other terms

5

Downward-pointing faces Support Volume

Workpiece

Projected base

Figure 7: Each face is projected to the bottom bound of the part. will be the purging and filling times for switching materials. The total build time is then Tbuild

= Tscan,part + Tscan,support + Tpurge + Tfill = Rd,p Vp + Rd,s Vs + Np Tp + Nf Tf ,

(1)

3

where Rd,p and Rd,s are the volumetric deposition rates, in [ LT ], for the part and support, respectively; Vp and Vs are the total volumes of the part and support; Np and Nf are the number of purges and fills required; and Tp and Tf are the times required for a single purge and fill. Note that if there is an initial purge to remove oxygen from the chamber, Np = Nf = Nc , where Nc becomes the number of purge-fill cycles. The number of times that the chamber must be cycled depends on the number of times that the lowermost, unbuilt, downward-pointing face switches from one material to another. Now we have selected a performance measure for SALD (build time through the volume of supports required), and a parameter to control it (workpiece orientation). The next step is to verify that orientation produces a significant change in support volume. However, this is largely dependent on the geometry of the workpiece. A sphere, for example, exhibits the same support volume for any orientation. A cube, however, produces no support volume when a face is aligned with the substrate, or a support half its own volume when the cube is rotated 45 degrees. Assuming that most parts will produce a significant change in the required supports, we can now proceed to model the relationship between support volume and orientation in a manner that will allow minimization. VOLUMETRIC SUPPORTS In deposition processes such as SALD, where material is added to the part by causing a gas-phase reaction whose products solidify onto the part or substrate, supports must often be formed so that material above overhangs and cavities can be deposited. This section presents a method to generate candidate orientations that can be evaluated to determine the orientation that minimizes the support volume. Volumetric Support Generation Once a set of candidate directions has been determined, supports must be generated for each direction so that they may be compared. Conceptually, generating the support volumes involves sweeping each downward-pointing face on the part to the bottom bound of the part, and then subtracting the original part from the swept volumes, as shown in Figure 7. This is not the technique used, since it leads to round-off error and superfluous use of expensive boolean operations on the solid model. The actual implementation uses far fewer boolean operations, but is still expensive. Generating support volumes is an expensive process since it requires solid modeling operations. This makes numerical minimization routines inefficient since they typically require hundreds of evaluations to find a minimum. Also, support volume is not expected to be an easy criterion to minimize; because the supports are generated from 6

a piecewise faceted (STL) representation of the geometry, the volume generated will not necessarily be continuous as a function of build direction. Where faces switch from needing support in one orientation to needing none in another, the volume is discontinuous. This creates many local minima which cannot be detected by techniques that rely on approximating the curvature of the objective function. To avoid the problems of a numerical optimization routine, some analytical technique is needed to provide a set of candidate directions. This set may be determined with visibility maps [3, 4, 5, 6]. Visibility Maps The visibility map is a tool developed by Gan, Woo, et al. [5] to orient parts for NC machining. They consider processes that machine a point at a time (such as NC machining, where the mill contacts the part at only one point), a line at a time (as in wire EDM operations), or an entire part surface at a time (such as casting). SALD and SLS both create a plane at a time. Woo [6] points out that processes such as stereolithography (SLA) and SLS do not need orientation for visibility because they only require one setup to machine an entire part. SALD, however, can require multiple supports to be formed, each of which requires a setup (i.e., a purge and fill cycle, and formation of a new support). Most manufacturing processes cannot construct a surface of any arbitrary geometry without repositioning the workpiece or forming supports of some kind. It is useful to think of the machine in terms of the set of surface normals that it can produce without repositioning or supports. For SALD, or any process with plane visibility, this set is a hemisphere whose pole is the unit normal of the substrate, b [6]. This means that any face whose normal, when projected onto b, is codirectional, does not need support. Any other downward-pointing faces will require solid supports to be built beneath them before they may be constructed. The only exception to this is any section of the part' s base that is in contact with the substrate. Unlike the interactions between the part and the support, there are not interactions between faces within the same part; as long as the part is formed from the outside towards the center, there is no need to consider self-occlusion. Our target, then, is to find a hemisphere that contains all of the normals of the part, save these sections on the base. If all the part normals can be contained in a hemisphere, then the part can be manufactured using the SALD process without any supports (see Figure 8). Rather than perform a search for one hemisphere that will meet the criterion, we can transform the collection of normals into a visibility map, which contains the solution to the problem. The visibility map is derived from the Gaussian map, which is simply the collection of all normals from the part on a unit sphere [4]. To transform the Gaussian map into the visibility map, consider a set of hemispheres, one for each point in the Gaussian map and each with its pole centered at that point, as shown for the simple case of two faces in Figure 9. By intersecting each of these hemispheres, we obtain the visibility map. Each hemisphere represents the possible set of directions from which the manufacturing process can form the face being considered. By intersecting two of these, we are left with the possible directions from which both faces can be formed simultaneously. Thus, any point in the visibility map indicates an orientation for the part where the manufacturing process can “see” every face. This means that if b is contained in the region on the far right of Figure 9, both faces would be visible. When attempting to orient a part for SALD so that no supports are required, the base faces should be excluded from the visibility map, or else no orientation will leave any points in the visibility map for any closed surface. However, if the orientation is treated as an unknown, then there is no way to determine which faces form the base of the part (i.e, which faces are in contact with the substrate). By treating any face on the convex hull of the part as a base face, we are guaranteed to find all the base faces. Discarding every face on the convex hull may eliminate some faces that point downwards (toward the substrate), but will leave a candidate list of directions which may be evaluated quickly. Figure 8 shows a simple example of the process of generating a visibility map. The direction marked “A” in the figure and its negative are the only possible build directions for the part that do not require support volumes. The convex hull is shown in dashed lines when it is not coincident with the part. For this case, because there is no curvature on the faces that form the base along direction “A,” the removal of all of the faces on the convex hull has no effect on the outcome. Because this method does not guarantee there will be a direction where no supports are required, it would help if there was a way to generate candidate directions that can have supports generated and their volumes

7

A Two regions not on convex hull

A Gaussian Map

Visibility Map

Figure 8: Obtaining the visibility map of a part. Note that the Gaussian map includes the boundaries of the hemisphere, so that two antipodal hemispheres intersect along a great circle. 1

2

2

1

Figure 9: The construction of a visibility map. compared. This can be accomplished by constructing a visibility map for each set of connected faces once those on the convex hull have been discarded. The visibility map represents the directions from which that group of faces can be made without supports. These directions can serve as trial directions even if the visibility map for the entire part is empty. EXPERIMENTS This section describes a set of experiments that were designed to test the significance of orientation compared to two other factors. As part of the development of the performance measures, several example parts will be used. The simplest of these is a tensile specimen, as shown in Figure 10. Two more complex examples are shown in Figure 11. The first is an electrical connector housing, and the second is a turbine blade core. The turbine blade core is especially pertinent since the optimum orientation is not obvious; the compound curvature of the surfaces does not allow the directions for minimum build time or alias volume to be determined visually. Factorial experiments were used because they minimize the number of tests, use all the data collected to form a model, and reveal interactions between factors [7]. The two outputs for this set of experiments were surface roughness and tensile strength. Polycarbonate powder was used to form the tensile specimens in Figure 10. A contact profilometer was used to obtain surface profiles before each specimen was fracture tested on an Instron machine. The parameters varied in the experiments were laser power, P , layer thickness, lt , and build angle, θb . Laser power was chosen because it allows easy control over the Andrew number, AN . The Andrew number is the amount of incident radiation per unit surface area, and has been correlated to the penetration depth of melting in SLS [8]: AN =

P , BS · SCSP

(2)

where P is laser power, BS is beam speed, and SCSP is scan spacing. Layer thickness was taken as a second parameter because it is independent of AN . Orientation is quantified in terms of the build angle. The ranges for each parameter are shown in Table 1.

8

)

70

.8 17 )

12.7 (0.50)

00

.2

76

(0.

(3.

0 7.0

)

28

(0.

5.1

0(

0.2

0)

12

.7

Dimensions are in mm (inches ). (0.

50

)

Figure 10: Tensile specimen geometry.

(a) A DB25 connector housing.

(b) A turbine blade core.

Figure 11: More example parts.

9

The ranges for layer thickness and laser power are set by practical considerations. To decrease layer thickness below 127µm (.005in.) is physically impractical since powder particles are approximately 50 − 70µm (.002 − .003in.). The roller is unable to spread single-particle layers reliably. Increasing layer thickness much more than 178µm (.007in.) makes for extremely rough parts with poor interlayer bonding because laser energy absorption is mostly at the surface, and the transient heat input will not melt powder through the entire layer. Increasing laser power degrades (burns) the powder at the top of the layer [9]. Decreasing power below 10W results in fragile parts that are difficult to remove from the bed; the parts crumble because powder was not melted enough to bond sufficiently. Parameter Power, P = X1 Layer Thickness, lt = X2 Build Angle, θb = X3

10 127 0

Low [W ] [µm] [deg]

15 178 90

High [W ] [µm] [deg]

Table 1: Parameters and their ranges. Test 1 2 3 4 5 6 7 8 9 10 11 12

Run A1, A2

B1, B2

C1, C2

D1, D2

X1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1

X2 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1

X3 -1 0 1 -1 0 1 -1 0 1 -1 0 1

Strength 2.2 2.12 0.9 0.825 0.625 0.575 1.7 1.56 0.32 0.18 0.2 0.12 3.3 3.3 2.35 2.5 1.85 1.85 2.6 2.6 0.925 0.8 0.775 0.675

RQ 34.4 34.8 31.4 36.1 23.3 27.1 35.0 35.0 47.2 46.6 30.4 31.7 41.2 43.7 32.9 32.5 25.0 22.0 34.9 43.9 43.6 39.7 30.9 29.0

Table 2: Experiment conditions and results. A three-level factor was used for build angle to capture the “stair-step” effect that occurs between 0◦ and 90◦ . Linear approximations were used for the other parameters, since we only wish to show the significance of the other parameters. Midpoint tests will determine whether or not there is curvature in these variables. Each test has two outputs – maximum tensile force, S, in [kN ] and roughness, RQ, in [µm]. Because the amount of time required to perform the experiments would become prohibitive otherwise, each run includes a tensile specimen at every build angle in the test plus two in-betweens, which are produced for verifying the surface finish model. The resulting trials are shown in Table 2. Two data points were taken at each trial level. Wolf and Tauber' s [10] rule of thumb indicates this should be adequate for showing an effect is significant to within a standard deviation. Tensile Strength An analysis of variance (ANOVA) was performed on the strength data with the results shown in Table 3. For a 99% level of confidence, an F -test comparing the contribution of an effect to that of the pure error must be above F.01,1,12 = 9.33 [7]. All of the main and interaction effects except for two are significant. The interaction terms X1 X3 (power and build angle) and X2 X3 (layer thickness and build angle) are insignificant. Based on this analysis, the reduced model is: y

= 1.1 + 0.545X1 − 0.545X2 − 0.794X3 − 0.24X1 X2 +0.529X32 − 0.054X1 X32 + 0.196X2X32 −0.064X1 X2 X3 + 0.133X1 X2 X32

(3)

To verify the model, a lack of fit test was performed [11]. The sum of squares of the residual was less than the sum of the squares of the pure error. Thus, there is no lack of fit since the terms contribute less to the model than the pure error. Another test to verify the model is the residual analysis. The model shows strength increasing as power increases, layer thickness decreases, and build angle decreases. Plots of the strength response surface are presented in Figure 12. The physical significance of these trends can be explained by the fracture surfaces. 10

Source Main Effect X1 Main Effect X2 Main Effect X3 Interaction Effect X1 X2 Interaction Effect X1 X3 Interaction Effect X2 X3 Interaction Effect X32 Interaction Effect X1 X32 Interaction Effect X2 X32 Interaction Effect X1 X2 X3 Interaction Effect X1 X2 X32 Within Error

SS 4.75 4.75 10.1 0.92 0.02 0.03 4.47 0.046 0.61 0.065 0.286 0.054

DOF 1 1 1 1 1 1 1 1 1 1 1 12

MS 4.75 4.75 10.1 0.92 0.02 0.03 4.47 0.046 0.61 0.065 0.286

F 1056 1056 2240 205 4.67 6.05 994 10.27 136.9 14.45 63.60

Table 3: Tensile load ANOVA results. During testing, several interesting phenomena were observed. In general, fracture occurred along the direction of the layers. Because of this, the part strength decreases with increasing build angle since the layer area decreases with build angle. This explains why a quadratic model fits the data well; the area of fracture varies with the cosine of the 2 build angle, whose first two terms in a Taylor series expansion are 1 − x2 . However, some of the specimens built at high power have tensile strengths independent of build angle. This is caused by good bonding between layers – the fractures do not occur along a layer boundary in runs such as D2 (see Table 2). This alone does not explain the trends in tensile strength because several parts that did not fracture between build planes did show a decrease in strength with increased build angle. Badrinaryan and Barlow [12] and Vail [9] note that high surface temperatures induced by excessive laser power and/or short scan vectors can cause polymer degradation. Because the cross-sectional area of the tensile specimen decreases with build angle, the vectors used to scan the area will be shorter and may cause enough of a temperature increase to burn the polymer, thus lowering strength. Surface Roughness In addition to material strength, the surface roughness of the specimens was studied. The roughness measure RQ , chosen because it is a reliable measure of amplitude variations [13], is defined as: v u n u1 X RQ = t y2 , (4) n i=1 i where yi is the distance of the ith measurement from the average linear trend of the entire profile, and n is the total number of samples. Profile measurements were made along the long axis of the tensile specimen, on the face pointing “upwards,” over a length of 40mm. Roughness measures on parallel sets of faces were tested and found to be similar. As with strength, nonlinear trends are expected in the variation of surface roughness with build angle. At 0◦ and ◦ 90 , there will be no “stair-step” effects on any of the measured surfaces. At 45◦ , however, the effect should be prominent. This mode of surface roughness should also depend on the layer thickness. These trends are shown to be significant in Table 4 and appear in the reduced model, y

= 38.72 + 5.544X2 − 5.222X3 − 6.097X32 +2.741X1 X32 − 4.347X2 X32 .

(5)

As with the tensile strength model, there was no lack of fit. Figure 13 shows the response surfaces. Note that in Figure 13(b) the two surfaces are almost coincident; this indicates that power is nearly insignificant in determining surface roughness. As anticipated, build angle shows high curvature with a peak near X3 = 0, and an increasing curvature as layer thickness increases due to more pronounced aliasing. Summary of Experiments The significance of orientation is clearly demonstrated with these experiments. In both the strength and surface roughness models, build angle is significant and the main effect for build angle is the 11

Layer Thickness 127 µm 178 µm Load [kN] 4 3 2 1 0 -1

1 -0.5

0.5 0

Power

0 -0.5 Build Angle

0.5

(a) Strength vs. Power and Build Angle

Power 10 W 15 W Load [kN] 4 3 2 1 0 -1

1 -0.5

0.5

0 Thickness

0 -0.5 Build Angle

0.5

(b) Strength vs. Layer Thickness and Build Angle

Figure 12: Tensile strength response surfaces.

Source Main Effect X1 Main Effect X2 Main Effect X3 Interaction Effect X1 X2 Interaction Effect X1 X3 Interaction Effect X2 X3 Interaction Effect X32 Interaction Effect X1 X32 Interaction Effect X2 X32 Interaction Effect X1 X2 X3 Interaction Effect X1 X2 X32 Within Error

SS 59.06 737.6 436.3 26.15 56.81 55.69 594.8 120.2 302.3 4.050 7.631 77.13

DOF 1 1 1 1 1 1 1 1 1 1 1 12

MS 236.3 2590 1745 104.6 227.3 222.8 2379 480.7 1209 16.20 30.53

Table 4: Roughness ANOVA results.

12

F 9 115 68 4.07 8.84 8.66 92.5 18.7 47 0.63 1.19

Layer Thickness 127 µm 178 µm RQ [µm] 40 30 20 -1

1 -0.5

0.5 0

Power

0 -0.5 Build Angle

0.5

(a) RQ vs. Power and Build Angle

Power 10 W 15 W RQ [µm] 40 30 20 -1

1 -0.5

0.5

0 Thickness

0 -0.5 Build Angle

0.5

(b) RQ vs. Layer Thickness and Build Angle

Figure 13: Roughness response surfaces.

13

same order of magnitude as the other main effects. Orientation is clearly significant for build time as well, although there is no way to make accurate measurements because manufacturing parts at various build angles at the same time makes it impossible to separate the time spent on each. However, the magnitude of the change in build time is easily modeled for SLS. Using only the time spent on moving the roller, the horizontal tensile specimen composed of 71 slices of lt = 0.178mm(.007in.) takes approximately 90 fewer minutes of machine time than the vertical specimen composed of 428 slices. The horizontal part alone would only require 17 minutes of machine time, so this represents a factor of 5 increase in build time. Now that the significance of orientation has been established for SLS, an attempt to model these effects in a way amenable to optimization can be undertaken. MINIMUM HEIGHT On some occasions, the amount of time required to manufacture a part will be significant enough that it must be considered as a motive for changing orientation. For instance, if a large number of the same part is being made, the amount of time the process takes may demand optimization.

Lower part piston

Raise powder piston

Time Move roller across bed Compute face/build-plane intersections Scan cross-section Done?

Figure 14: Process time breakdown for SLS. Figure 14 shows how the time taken to build a single layer is used for SLS. Assume that the computation time to process each slice is fixed and small. In practice, this is a good assumption for all but the most complex parts since the control program is optimized to process only faces in the current layer. For SLS, the amount of time required to build a part is related to the volume of the part, and the total number of layers required to build the part. Of these two contributions, the volume is independent of orientation. Because the amount of time for a roller sweep is fixed, the total time used for roller motion is Troller = Tl Nl , where Tl is the amount of time required to spread one layer of powder and Nl is the total number of layers in the part. To model the time taken in scanning each cross-section, we need the average cross-sectional area of each slice: V A¯s = , Nl lt where A¯s is the average area, V is the volume of the part, and lt is layer thickness. Using this average area, we can add a second factor to the build time: Tbuild

0 = Tpistons + Troller + T% compute +Tscan = (Tp + Tl + A¯s Ta )Nl

= (Tp + Tl )Nl +

V Ta , lt

(6)

where Ta is the time required to scan a unit area, and Tp is the time required to move the pistons by one layer increment. The computation time is neglected because it is small enough that it cannot be measured without modifying the program and is undetectable in all but the largest files. Note that because the cross-section area is inversely related to the number of layers, the time required to scan the part is independent of orientation; only layer thickness affects it.

14

The key is then to find the orientation with the least number of layers, which will be the orientation with the minimum height in the build direction. The minimum height can be defined as the shortest distance between two parallel planes, P1 and P2. These planes are defined with a root point, Pir , and a normal, n ˆ, so that the inequalities (P − P1 r ) · n ˆ

≥ 0

(P − P2r ) · n ˆ

≤ 0

define a volume that contains every point in the workpiece. Note that since the planes are parallel, the two planes have equal but opposite normals. In the equations above, n ˆ denotes the normal to P1. Clearly, the two planes should each contain at least one point from the workpiece, or a new plane can be defined that makes the distance between the two shorter without violating our constraints. Also, note that only geometry on the convex hull of the workpiece may be coincident with either of the bounding planes. The next question is: how many points should be contained in each of the two planes so that the distance between them is minimized? The theorem below answers this question. Theorem 1 Consider two parallel planes, P1 and P2 , and a closed, convex planar-faceted surface with linear edges, S. If P1 and P2 are constrained to touch the surface S so that the entire surface is contained between the planes, then, for any surface S, either P2 n P1

S

1. one of the planes (P1 or P2) must be coincident with a face of S (as opposed to a vertex or edge) when the distance between the planes is at a minimum, or, 2. each of the planes P1 and P2 must contain exactly one edge of S when the distance between the planes is at a minimum, and the edges in P1 and P2 may not be codirectional.

Proof Assume there is an orientation such that the distance between P1 and P2 is at a minimum and both planes are in contact with S at only a vertex or edge, but not two edges. Rotate S about the edge (or vertex in the case of two vertices contacting the bounding planes) in contact with the bounding plane, say P1. If the vertex (or codirectional edge) on P2 is directly above the edge or vertex on P1 , then any rotation about the edge or vertex on P1 will move the geometry coincident on P2 closer to P1. If the vertex or edge on P2 is not directly above the coincident geometry on P1 , there must still be at least one direction in which the part may be rotated about the edge or vertex on P1 to bring the geometry on P2 closer to P1 . For some angle of rotation, no other geometry on S will interfere with P1 or the newly defined P2 . This is because S is convex, so there may only be one connected group of points in contact with a plane, and it must be convex. So, any points that contact the bounding planes must be connected to the vertex or edge previously on these planes. If there were an angle where such interference did exist, then more than a single vertex would be in contact with P2 initially, or more than an edge (or vertex) would be in contact with P1 initially. Both of these cases violate the hypothesis. 2

15

(a) DB25 connector.

(b) Turbine blade core.

Figure 15: Convex hulls of the example parts. Rank 1 2 3 4 5 6 7 8 9 10

Distance 29.01 29.07 29.60 29.79 29.95 30.00 30.04 30.09 30.65 30.81

x 0.6668 -0.6756 0.6963 0.6581 -0.6956 -0.7124 -0.7133 -0.7144 -0.6497 -0.6480

y 0.7440 -0.7355 0.7142 0.7521 -0.7175 -0.6963 -0.6972 -0.6978 -0.7597 -0.7608

z -0.0414 0.0499 -0.0703 -0.0332 0.0341 0.0868 0.0704 0.0506 0.0253 0.0330

Table 5: The ten smallest build heights and build directions for the turbine example. Now it is easy to develop an algorithm that will find the minimum height of the object. Since we know that at least one plane must contain either a face or an edge from the convex hull of the workpiece, we can simply sift through all of the faces and all of the edges on the convex hull of the workpiece. For the tensile specimen and DB25 connector, the minimum build heights can be determined visually. For the turbine blade core, however, this measure is very useful. Figure 15 shows the convex hulls of the example parts, and Table 5 shows the values returned from a minimization of build height. MATERIAL STRENGTH According to the maximum stress theory, a layered part will fail when any one of the maximum stresses (shear or normal) is exceeded [14]. Thus, a part can fail from the normal strength of a layer being exceeded, from shear between layers, and from transverse normal failure. However, this model does not allow for interactions between stresses and is difficult to code. A more accurate model is the Tsai-Wu interactive tensor polynomial model [15], Fi σi + Fij σi σj ≤ 1

(7)

where σ = [σ1 σ2 σ3 τ23 τ13 τ12 ] is the stress tensor, and Fi and Fij are coefficients for the fit. The Tsai-Wu model is a quadratic model of the failure surface. Although a cubic model might be used so that interaction terms between all three principal stresses could be present, a cubic model could possibly have open failure surfaces. Through a series of simplifications, the strength of a laminated object can be represented in its local coordinate system as [16]:

16



Fij

     =    

1 Xt Xc

− 12

q

1 Xt Xc Yt Yc 1 Yt Yc

− 12

q

− 12

1

qZt Zc Xt Xc 1 Yt Yc Zc Zc 1 Zt Zc

0

0

0

0

0

0

0

0 0

0 0

1 R2

0

1 Q2

          

(8)

1 S2

where X2 , Y2 , and Z2 are tensile (2 = t) and compressive (2 = c) strengths in the three axes of the local coordinate system; and Q, R, and S are the shear strengths in the local coordinate system. Particularly, Q is shear strength in the 2-3 plane, R in the 1-3 plane, and S in the 1-2 plane. If tensile and compressive strengths are assumed equivalent, then Fi = 0. Otherwise,   1 1 Xt − Xc  1 − 1   Yt Yc    1 − 1   (9) Fi =  Zt Zc    0     0 0 The last three entries must be zero if the material is to be insensitive to the sign of shear [16]. Next, for a given orientation, a change of basis must be applied to the stress tensor to transform it into the coordinate system of the material. The test for failure is then a simple inequality: Fi Tik σk + Fij Tik σk Tjl σl ≤ 1,

(10)

where T is the transformation between local and global coordinates. The transformation matrix, T, is [17]   [A11 ] [A12 ] T= [A21 ] [A22 ]

(11)



A11

A12

A21

A22

 l12 l22 l32 =  m21 m22 m23  n21 n22 n23   l3 l2 l1 l3 l1 l2 =  m1 m2 m3 m2 m1 m3  n1 n2 n3 n2 n1 n3   2l1 m1 2l2 m2 2l3 m3 =  2n1 m1 2n2 m2 2n3 m3  2l1 n1 2l2 n2 2l3 n3  l1 m2 + l2 m1 l2 m3 + l3 m2 =  n1 m2 + n2 m1 n2 m3 + n3 m2 l1 n2 + l2 n1 l2 n3 + l3 n2

 l3 m1 + l1 m3 n3 m1 + n1 m3  l3 n1 + l1 n3

where l, m, and n are direction cosines for the new coordinate system. Specifically, when given two angles, φ and θ, as shown in Figure 16, that define the rotation to move into the new coordinate system, l1 m1 n1

= = =

cos θ cos φ sin θ cos φ sin φ

l2 m2 n2

= = =

− sin θ cos φ cos θ cos φ sin φ

l3 m3 n3

= = =

cos θ sin φ sin θ sin φ cos φ

(12)

The failure surface described by the Tsai-Wu polynomial is an ellipsoid. For the case where tensile and compressive strengths are equal and opposite in sign, all the entries in Fi will be zero, and Equation 7 represents an ellipsoid centered at the origin. 17

Material Build 3

3 φ

2

2

1 1

θ

Figure 16: Transformation to a material' s local coordinate system. This model is incorporated into the optimization strategy by checking candidate points for failure as described below. Stresses must be known to the designer from some model analysis, such as a finite element simulation. By including maximum stress points from several loading conditions, a designer can determine whether the part will survive all of the intended loads. SURFACE APPROXIMATION Another criterion for optimization is the difference in volume between the intended part and the layered approximation. This can be approximated on a face by face basis for a given orientation, and then summed to give a total “alias” volume, Va,t . There are two cases to consider: when the height, h, of a face projected onto the build direction, b, is less than a layer thickness, lt , and when it is larger. When the projected height is larger than a layer, the alias volume can be written as )   ( 2  lt L cos θ As Va = tan θ (13) L 2 lt As lt sin θ , 4 where As is the surface area of the face, L is the length of the line formed by projecting the face onto a plane normal to the face and the build direction, and θ is the angle between the face and the build direction. All of these dimensions are shown in Figure 17. The first term in Equation 13 is the effective width of the face. The second is the area of two triangular alias volumes enclosed inside one layer thickness. Finally, the third term is the number of triangular alias volumes on the surface, divided by two to account for the second term. The equation above may be taken as a minimum volume since the conditions at the boundaries of the face may add volume. Clearly, the larger a face is, the more accurate the equation is. =

lt L

q

θ

Figure 17: Alias volume when the face is larger than lt When a face is smaller than a layer thickness for a given build direction, the alias volume is minimally    As (L cos θ)(L sin θ) Va = L 2 As L sin 2θ , = 2 18

(14)

with the dimensions shown in Figure 18.

lt

lt

Va > predicted

Va = predicted

Figure 18: Alias volume when the face is smaller than lt The surface area, As , is easily calculated for a triangular facet. The “height” of the surface is one of the facet edges projected onto the build direction, for a triangular facet. The distance we must compare to the layer thickness, lt , is then L cos θ = max {Vi · b − Vj · b, ∀i, j ∈ [0, 1, 2] }, where Vi is the ith vertex of the surface, and b is the build direction. To ensure that the alias volume modeled is a significant phenomenon on the surface of SLS parts, a series of tensile specimens were made at varying angles to the build direction and the resulting surfaces measured with a profilometer. The power spectra of the profiles were then compared to those of a surface produced with the aliasing model. Two such spectra are shown in Figure 19. The frequencies of the peaks match to within 0.10 cycles , the resolution of the mm data. OPTIMIZING MEASURES Now that the physical models of the performance measures are complete, the question of how such models can be used should be addressed. To automate the manufacturing process, some sort of optimization should be used to find a suitable orientation. Since there are multiple objective functions to be optimized, this is not trivial. If a weighted sum of objective functions is used, then the designer must set the weights based on the intended function of the part. An alternative to this approach is the use of a Pareto optimum [18], which only tests the curvature of a candidate point to ensure that every objective function is increasing in the neighborhood of the candidate. A third approach would be to optimize a single performance measure and use others as constraints. Of the three measures above, only the Pareto minimum does not require a numerical optimization. Simply applying numerical optimization without any consideration of the data used to describe surfaces during manufacturing (triangular faceted models) can lead to problems. As an example, consider the model of a coffee mug shown in Figure 20(a). To reduce the computation time of the performance measures, the number of vertices in the faceted model was lowered, as shown in Figure 20(b). Once this was done, a brute force search of alias volume was performed. The Model Results

Experimental Results

0.007

0.005 30 45 60

0.006

30 45 60

0.0045

Magnitude squared [mm^2]

Magnitude squared [mm^2]

0.004 0.005

0.004 0.003

0.002

0.0035 0.003 0.0025 0.002 0.0015 0.001

0.001 0.0005 0

0 0

1

2

3 4 Frequency [cycles/mm]

5

6

7

0

1

2

3 4 Frequency [cycles/mm]

Figure 19: Theoretical and measured power spectra. 19

5

6

7

resulting alias volume as a function of orientation (given by two angles) is shown in Figure 21. The surface in this figure is marked by a large number of local minima and maxima, which makes numerical optimization intractable. In some cases, instead of relying on numerical optimization, it is possible to use a property of the quality measure to confine the search to a finite set of candidates that can be evaluated to determine the optimum. Note that the Pareto minimum requires a set of candidate points, so either this approach or a stochastic search must be used for the Pareto minimum. An example of this technique is the minimum build time measure. By showing that the minimum could only occur on planes defined by a finite set of geometric combinations, the search for a minimum build time is reduced to a search through the edges and faces of the convex hull of the workpiece.

φ

θ

(a) Initial model, 1213 vertices.

(b) Reduced model, 180 vertices.

3

Vol [mm ]

Figure 20: A coffee mug.

25 20 15 10 5 0

4 3 2

0

1 0.5

0 1

1.5

φ [r ad

]

2

2.5

3

d]

-1

Degeneracies

-2 -3

θ

[ra

-4

Figure 21: Alias volume as a function of orientation for the mug.

APPLICATION TO OTHER PROCESSES Although only SLS and SALD are used as examples, other SFF techniques can make direct use of several of these criteria. Table 6 shows some of the possibilities, categorized by the technology used to form parts [19, 20]: • Sheet. These processes stack sheets of material on top of one another, cutting and bonding them selectively to form cross-sections of the model. • Deposition. These processes deposit material to form the model, either through a moving nozzle or by inducing a gas phase reaction at a surface. 20

• Photoreactive. These processes cure liquid photor eactive polymer resins selectively to form cross-sections of the model. • Powder. These processes form cross-sections of a computer model by bonding powder particles together in a layer. Then, layers of powder are distributed to form successive cross-sections. SFF Proc.

Examples

LOM

Sheet

SALD, FDM, SDM

Deposition

3D Systems, Cubital,Formigraphics

Photoreactive

SLS, 3DP, SALDVI

Powder

Description

Possible Application

Laminated Object Manufacturing uses sheets of paper, plastic, or ceramic to form cross-sections, each of which is cut out by a laser [21].

Build time should be dependent on sheet feed and lamination time, as SLS is related to powder spreading. Aliasing is also applicable, but may not be significant due to lower layer thickness.

Fused Deposition Modeling pushes a semi-fluid thermoplastic through a nozzle which scans crosssections of a part to form the model [22].Shape deposition manufacturing (SDM) uses “weld-based” metal deposition to form layers followed by NC machining to finish the out surfaces. Supports are cast out of brass [23].

FDM does not require volumetric supports, nor does it form material more slowly in one direction than another. So, our build time criteria are not applicable. Alias volume will be insignificant for the thin layers used. Material strength is anisotropic, though, if the newly formed material does not join well with previous layers.The SDM process requires volumetric supports and has a time penalty involved in changing from support generation to part deposition. Minimizing the volume of supports should be applicable.

3D Systems(3DS) and Cubital selectively cure a photoreactive polymer resin, 3DS with a laser and Cubital with a mask and UV lamp. 3DS lowers the platform further into a vat of resin after each layer is cured so the next layer can be processed. Cubital flushes the chamber after each layer and adds wax to fill in gaps in the cross-section. The wax is milled level and then resin is introduced into the chamber for the next layer [1]. Formigraphics uses two intersecting lasers to cure resin at a point [1].

Since Cubital's process requires several postprocessing steps per layer, minimizing the number of layers is extremely important. Neither 3DS or Formigraphics use processes that are amenable to the quality measures developed in this research. Allen [24] and Wozny [25] have addressed this area.

3-D Printing involves spreading a layer of powder and then selectively applying binder material with an ink-jet print head to glue the powder together into a cross-section [26]. SALDVI is a layered technique like SLS, except that the powder is not melted; rather, the decomposition of the gas in the powder and at the surface fills the voids between particles.

Since this process requires powder to be spread, the same measure of build time as SLS may be used. Also, since layer thicknesses are approximately 0.175–0.380mm [27, 28], aliasing will be as noticeable as with SLS. If binder penetration is not consistent, material anisotropy will make strength a concern.

Table 6: Possible applications of this work to other SFF processes.

CONCLUSIONS AND FUTURE WORK Automation can be used to reduce the amount of expert knowledge required to set operating parameters for solid freeform manufacturing. This is done through the development of measures of part quality that can be computed for parts of arbitrary geometry to help a designer make an objective decision about either which process to use, or how to best use a process to produce a given part. The quality measures are developed, using the example of SLS, by identifying distinguishing characteristics of the manufacturing process. The parameters affecting these measures are considered, and then several characteristics are chosen as important: as examples, build time, surface finish, and part strength are considered. Next the parameters used to exert control over quality measures are selected, with orientation as the example. To verify the significance of the parameters testing is performed. In this paper, the test is a design of experiments analyzed with ANOVA. Finally physical models of the performance measures are produced. A preliminary study of numerical optimization indicates that, due to excessive numbers of local minima, some other means of finding an optimum is required. Using properties of the performance measures to find candidate directions is presented as an alternative to numerical optimization. A future improvement would be to add the ability to select weights for sets of faces during the computation of some performance measures such as alias volume, rather than uniform weighting for each face. This would allow a

21

user to identify key surface features and tailor the optimization to ensure those surfaces are formed as precisely as possible. ACKNOWLEDGMENTS The support for this work comes from ONR grant N00014-92-J-1514 and Texas Advanced Technology Program grant CSME-ATPD-157, which is gratefully acknowledged. REFERENCES [1] D. L. Bourell, J. J. Beaman, H. L. Marcus, and J. W. Barlow, “Solid freeform fabrication: An advanced manufacturing approach,” in SFF Symposium Proceedings, pp. 1–7, The University of Texas at Austin, 1990. [2] J. V. Tompkins, R. Laabi, B. R. Birmingham, and H. L. Marcus, “Advances in selective area deposition of silicon carbide,” in SFF Symposium Proceedings, (Austin), pp. 412–421, The University of Texas at Austin, August 1994. [3] L.-l. Chen and T. C. Woo, “Computational geometry on the sphere with application to automated machining,” Journal of Mechanical Design, vol. 114, pp. 288–295, June 1992. [4] J. G. Gan, T. C. Woo, and K. Tang, “Spherical maps: Their construction, properties, and approximation,” Journal of Mechanical Design, vol. 116, pp. 357–363, June 1994. [5] K. Tang, T. Woo, and J. Gan, “Maximum intersection of spherical polygons and workpiece orientation for 4- and 5-axis machining,” Journal of Mechanical Design, vol. 114, pp. 477–485, September 1992. [6] T. C. Woo, “Visibility maps and spherical algorithms,” Computer-Aided Design, vol. 26, pp. 6–16, January 1994. [7] D. C. Montgomery, Design and Analysis of Experiments. New York: John Wiley & Sons, 3rd ed., 1991. [8] J. C. Nelson, Selective Laser Sintering: A Definition of the Process and an Empirical Sintering Model. PhD thesis, The University of Texas at Austin, May 1993. [9] N. K. Vail and J. W. Barlow, “Modeling of polymer degradation in SLS,” in SFF Symposium Proceedings, pp. 387–395, The University of Texas at Austin, 1994. [10] S. Wolf and R. N. Tauber, Process Technology, vol. 1 of Silicon Processing for the VLSI Era. Sunset Beach, California: Lattice Press, 1986. [11] R. E. DeVor, T. how Chang, and J. W. Sutherland, Statistical Quality Design and Control. New York: Macmillan Co., 1992. [12] B. Badrinarayan and J. W. Barlow, “Effect of processing parameters in SLS of metal-polymer powders,” in SFF Symposium Proceedings, (Austin), The University of Texas at Austin, August 1995. [13] D. Whitehouse, Handbook of Surface Metrology. Bristol, UK: Institute of Physics Publishing, 1994. [14] S. W. Tsai, “Three-dimensional effective moduli of orthotropic and symmetric laminates,” Journal of Applied Mechanics, vol. 59, p. 39, March 1992. [15] I. M. Daniel and O. Ishai, Engineering Mechanics of Composite Materials. New York: Oxford University Press, 1994. [16] S. W. Tsai and E. M. Wu, “A general theory of strength for anisotropic materials,” Journal of Composite Materials, vol. 5, pp. 58–80, 1971.

22

[17] R. D. Cook, D. S. Malkus, and M. E. Plesha, Concepts and Applications of Finite Element Analysis. New York: John Wiley & Sons, 3rd ed., 1989. [18] A. Osyczka, Multicriterion Optimization in Engineering. Chichester: Ellis Horwood Limited, 1984. [19] J. J. Beaman, J. W. Barlow, D. L. Bourell, R. H. Crawford, and K. McAlea, Solid Freeform Fabrication: A New Direction in Manufacturing. Kluwer, not yet published. [20] R. I. Campbell and P. M. Dickens, “Rapid prototyping: A global view,” in SFF Symposium Proceedings, pp. 110– 117, The University of Texas at Austin, 1994. [21] C. Griffin, J. Daufenbach, and S. McMillen, “Solid freeform fabrication of functional ceramic components using a laminated object manufacturing technique,” in SFF Symposium Proceedings, pp. 17–24, The University of Texas at Austin, 1994. [22] J. W. Comb, W. R. Priedeman, and P. W. Turley, “FDM technology process improvements,” in SFF Symposium Proceedings, pp. 42–49, The University of Texas at Austin, 1994. [23] R. Merz, F. B. Prinz, K. Ramaswami, M. Terk, and L. E. Weiss, “Shape Deposition Manufacturing,” in SFF Symposium Proceedings, pp. 1–8, The University of Texas at Austin, 1994. [24] S. Allen and D. Dutta, “On the computation of part orientation and support structures for layered manufacturing,” in SFF Symposium Proceedings, August 1994. [25] Y. S. Suh and M. Wozny, “Integration of a solid freeform fabrication process into a feature-based CAD system environment,” in SFF Symposium Proceedings, pp. 334–341, The University of Texas at Austin, 1995. [26] R. F. Aubin, “A world wide assessment of rapid prototyping technologies,” in SFF Symposium Proceedings, pp. 118–145, The University of Texas at Austin, 1994. [27] M. J. Cima, A. Lauder, S. Khanuja, and E. M. Sachs, “Microstructural elements of components derived from 3–D printing,” in SFF Symposium Proceedings, pp. 220–227, The University of Texas at Austin, 1992. [28] S. Michaels, E. M. Sachs, and M. J. Cima, “Metal parts generation by three dimensional printing,” in SFF Symposium Proceedings, pp. 244–250, The University of Texas at Austin, 1992.

23