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Compound scan mode developed from subarea and contour scan mode for selective laser sintering
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International Journal of Machine Tools & Manufacture 47 (2007) 873–883 www.elsevier.com/locate/ijmactool
Compound scan mode developed from subarea and contour scan mode for selective laser sintering Y. Shi, W. Zhang, Y. Cheng, S. Huang State Key Laboratory of Plastic Forming Simulation and Die & Mould Technology, School of Material Science and Engineering, Huazhong University of Science and Technology, Wuhan-Hubei 430074, PR China Received 25 May 2006; received in revised form 31 July 2006; accepted 14 August 2006 Available online 6 October 2006
Abstract Scan mode is an important parameter for selective laser sintering (SLS) processing. The improved mode will optimize scan path and improve the precision, strength and fabrication efficiency of a SLS part. A compound scan mode, which combines subarea scan mode and contour scan mode, is proposed. The principle of its hatch (path-planning) algorithm and implementation are presented. To testify the effectiveness of this compound mode compared to that of subarea scan mode, it has been utilized for researches at a SLS machine developed at Huangzhong University of Science and Technique (HUST). The results from the researches indicate that the degree of precision of a SLS part with the compound scan mode is higher than that with subarea one. There is little difference in the tensile strength, flexural strength, shock strength and fabrication efficiency of a SLS part under the compound scan pattern and the subarea scan mode. Therefore, implementation of the compound scan mode is of importance to improve the precision of a SLS part. r 2006 Elsevier Ltd. All rights reserved. Keywords: Selective laser sintering; Improved subarea scan mode; Compound scan mode; Fabrication precision
1. Introduction Scan mode is one of the important parameters that affect the precision, strength and fabrication efficiency of a selective laser sintering (SLS) part. Therefore, it is very important to optimize scan mode. Much study on scan mode has been done and many scan modes have been created, and they belong to parallel-line scan mode or broken-line scan mode. 1.1. Parallel-line scan mode Parallel-line scan mode [1] is also called as type Z scan mode whose scanning paths are parallel to x- and y-axis. Its principle is similar to that of filled regions in computer graphics. Though its algorithm can be simply and easily Corresponding authors. Tel.: +86 027 87557042; fax: +86 027 87548581. E-mail addresses: [email protected] (Y. Shi), [email protected] (W. Zhang).
0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.08.013
implemented, it has disadvantages. Firstly, laser has to be continually turned off and on, which shortens its life. Moreover, for a part with holes, cavities have to be spanned frequently and it causes scanners run emptily. Secondly, for the scan direction in layers is the same and there exists shrinkage stress in the same direction, which causes the warp and distortion of a SLS part. Finally, a SLS part with parallel scan mode has anisotropy strength. For subarea scan mode [2,3], scan lines are parallel lines. Every slice plane of three-dimensional (3D) CAD model is divided into smaller areas and then they are filled by parallel lines. Empty runs are less than those for parallelline scan mode. Its algorithm is simple, so it is widely used in the SLS process, and it has the following disadvantages: First, some micro-ladders appear on a SLS part with this mode, which causes the precision dependent of the spot size of laser beam. What is more, there exists the warp and distortion of a SLS part with this mode. Especially, in a part with a larger slice plane, there appears great shrinkage stresses and even cracks along the scanning direction because of longer scanning lines.
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For emanative starlike scan mode and angled emanative starlike scan mode [4], slice planes are divided into two areas from the center, which are filled with scan lines parallel to x or y axis or at an angle of 451 to the axis from the center to outside. Compared to that with the two scan modes above, the SLS part with these two modes causes less warp and distortion to some extent, but they have disadvantages such as that of parallel-line scan mode. 1.2. Fold-line scan mode For fractal scan mode [5] scan path has the features of local and whole comparability. When fractal dimensions are 2, the whole slice planes can be filled with fractal scan path. This scan mode overcomes the shortcomings of parallel-line scan mode and makes the temperature field more uniform, which reduces warp and distortion. But it has the following disadvantages: low scanning speed, laser scanners frequently accelerated and decelerated, a SLS part being getting fabricated with low precision, and frequent spanning of the cavities of the slice planes. Scan paths are spiral lines for spiral scan mode [6]. Though it overcomes the shortcomings of type z scan mode, and shrinkage stresses can be reduced, the cavities have to be spanned frequently. For contour equidistance path [7,8] no empty run appears which prolongs scanners’ life. The frequent changing scan direction and shorter scanning lines disperse the shrinkage stresses and reduce warp and distortion. Its algorithm is complex and it needs more CPU time. Besides, the algorithm is not reliable and some unfilled regions appear, which affects the strength of the SLS part. This algorithm of the scan path based on Voronoi map [9,10] is suitable for multi-connected domains in slice planes, and enhances the scan efficiency to some degree. But it has the same shortcomings as the contour equidistance path. As for scan mode using Pythagorean hodograph as filling lines [11], the intersecting parts of slices are separated into some subareas by the Delaunay triangle, and then they are filled with Hilbert curves [12]. However, it is suitable only for regular parts, except the fabrication of parts with complex contour. Besides, there are also other similar slice filling modes presented by Wasser et al. [13], Tiller and Hanson [14], Ganesan and Fadel [15], Pham [16] and Takashi [17]. To overcome the shortcomings of the scan modes above, a compound scan mode combining subarea scan mode with contour scan mode is presented in this paper, and it has not been reported in the literature.
we firstly discuss these two scan modes before the compound one. 2.1. Subarea scan mode A subarea scan mode is presented in Ref. [2] in order to improve quality of a SLS part. Strategies for it are as follows: The first step is that the slice planes are divided into subareas without holes shown in Fig. 1. The second one is that the slice planes are divided into subareas in terms of the number of intersecting points that the scan lines parallel to x or y-axis intersect with their contours. Fig. 1 is taken for an example. The scan lines within area 1 form two intersecting points with the contour. The number of intersecting points increases into four when they are within area 2. In the same way, the number in area 4 is six, the number in area 7 is eight, and the number in area 11 is six. So the whole slice area is subdivided into 16 subareas. The third one is that subareas are scanned in the fixed order from top to bottom, and from left to right. For example, the scan order in Fig. 1 is: area 1- area 2- area 3- area 4- area 5- area 6- area 7? ?. Finally, these subareas are filled with parallel lines, but there is an a angle between two neighboring scan layers in order to reduce shrinkage stress along the same direction. Generally a ¼ 90 . According to the above strategies, we can implement it by the following procedure: Initially, intersecting points where (at which) the scan lines are intersecting with the contours of a slice plane are obtained, and in the order they can be stored in pairs, such asðP0 ; P1 Þ,ðP2 ; P3 Þ, ðP4 ; P5 Þ,?,ðP2n1 ; P2n Þ. If these scan lines are in a connected domain, it can be filled with lines consisting of point pairs. And those lines are called the filling lines. Then the filling lines can be grouped. For the setfC i ; 1oioNg, the filling lines in the whole slice can be grouped into m groups from Eq. (1).
2. Subarea scan mode and contour scan mode The compound scan mode, based on the subarea scan mode and cotour scan mode, is presented. Therefore,
Fig. 1. Subarea scan mode.
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8 C 1 ¼ C 2 ¼ ¼ C k1 aC k1 þ1 ; > > > > > > < C k1 þ1 ¼ C k1 þ2 ¼ ¼ C k2 aC k2 þ1 ; > > > C km1 þ1 ¼ C km1 þ2 ¼ ¼ C km ; > > > : km ¼ N;
(1)
where Ci is the number of segments as filling lines, i.e., the number of pairs of intersecting point, and N is the number of grouped filling lines. So in Fig. 1 there are seven groups of filling lines. Though subareas are attained according to the number of crossing points scan lines intersect with the contours of a slice plane, its principle can be optimized. So the subarea scan mode can be developed into an improved subarea scan mode. The strategies of improved subarea scan mode are explained as follows: First of all, subarea zone is grouped as much as possible. Next the numbers of intersection points vary when two neighboring scanning lines intersect with the contours in a slice plane. According to the principle of the scan order, two filling lines are in a connected domain. According to the strategies of the improved subarea scan mode, we can implement it as follows: To start with, we could judge whether the neighboring scan lines are in a connected domain. Once they are projected into the scanning direction and there are some overlaps, they must be in a connected domain. Otherwise, they are not in it. According to the above methods, the algorithm of improved subarea scan mode needs frequent judgment whether the filling lines belong to a connected domain section. For this reason, much time will be taken by CPU to do this. To spare calculating time, complex polygons in a slice plane in Fig. 2a can be grouped according to the depth tree of the polygons’ contour cycles.
It is known that depth of the cycle, external and internal polygon, and direction of the polygon are defined as follows: Depth of the cycle refers to the number of cycles in the same contour enclosing it. The maximum depth in a contour is called the depth of a contour; the depth of a contour is also an indication of the planar complexity of the contour. As illustrated in Fig. 2b, the depth tree shows the relationship of the cycles to a contour. The depth of this contour is 3. For a given contour, we employ the depth tree to describe its complexity. Based on the definition of depth, we can also define the attribute of a cycle: if the depth of a cycle is even, it is an external cycle; otherwise, it is an internal cycle. In Fig. 2a, cycles A, D and E are external cycles, and the rest are internal cycles. Like most of definitions in the relevant literature, we define the direction as positive if the solid area is always to the left when walking along the boundary. Obviously, for external cycles, the positive direction is counterclockwise; for the internal cycles, the positive direction is clockwise. The Fig. 1 is taken as a case in point for the improved subarea scan mode. According to the above principles, Fig. 1 can be united into four subareas: a subarea I including area 1, area 2, area 4, area 7, area 11, area 14 and area 16; a subarea II consisting of area 3, area 6, area 9 and area 12; a subarea III including area 10, area 13 and area 15; a subarea IV consisting of area 5 and area 8. Therefore, the number of spanning cavity and subareas can be reduced, which is of benefit to SLS process. 2.2. Contour scan mode Contour scan mode belongs to fold-line scan mode that the two neighboring filling lines are not parallel. Filling lines with equidistance are parallel to the lines of a contour polygon. For the curves of a contour in a slice plane, they are approximated by line segments, to which the filling lines with equidistance are parallel. The principles of Contour
Depth 0
A C
E
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A
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G F B
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E
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F (a)
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(b) Fig. 2. (a) Contour polygons in a slice plane and (b) tree structure of polygons.
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contour scan mode are explained in the following Section 3.2. Because the solid area in a slice plane is filled with laser beam spot with constant size, filling lines are considered to be equidistant from lines consisting of a contour polygon.
Vi-1 V' i-1
R
y
Li
3. Compound scan mode L'i
Vi
Y
The compound scan mode is studied, which combines the algorithms of subarea scan mode and contour scan mode.
o
3.1. Strategies of compound scan mode Most slice planes of a 3D model consist of complex concave polygons, and most of them compose polygons. For the algorithm of contour scan mode, it happens that polygons may intersect each other and lines of concave polygons may intersect themselves. That is the main reason why the algorithm of a contour scan mode is complex. In slice planes the contour scan mode cannot be stopped or transformed into the improved subarea scan until filling polygons intersect each other or intersect themselves. These are the strategies of this mode.
x L i+ 1
V'i
R
Vi+ 1
L'i+1
V' i+1 O X
Fig. 3. Offset generations.
where 3.2. Implementation of compound scan mode According to the strategies, we can implement the algorithm by the following steps. The first step is that the contours in a slice plane are grouped into connected domains and then every one of connected domains is filled with lines with the compound scan mode once. The second one is that filling lines offset with equidistance are generated. The third one is that in scan space, the number of contour offset, is defined. The last one is that after the contours are filled with contour scanning mode, the unfilled parts are to be filled with the improved subarea scan mode. 3.2.1. Creation of contour scan path 3.2.1.1. Equidistance contour generation. When the solid part of a slice plane is filled with a series of equidistance beam paths, the offset directions to inner loops and external loops are opposite. According to the definition of the direction of a loop [18], the offsets of* inner and external loops are presented with the vector R shown in Fig. 3. Therefore, the endpoint coordinates of their lines can be calculated according to the following methods: As shown in Fig. 3, it is assumed that the origin of the temporary coordinate system xoy is the arbitrary intersecting point Vi .*Li and*Liþ1 are the vector formed by V i1 V i and V i V iþ1 . l i and l iþ1 are the unit vectors of Li and Liþ1 . They are given by 8* * * < l i ¼ ai i þb j ; i (2) * * :* l iþ1 ¼ aiþ1 i þbiþ1 j ;
xi xi1 ai ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðxi xi1 Þ2 þ ðyi yi1 Þ2 yi yi1 ffi, bi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xi1 Þ2 þ ðyi yi1 Þ2 xiþ1 xi aiþ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðxiþ1 xi Þ2 þ ðyiþ1 yi Þ2 yiþ1 yi ffi, biþ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxiþ1 xi Þ2 þ ðyiþ1 yi Þ2 V i1 ðxi1 ; yi1 Þ, V i ðxi ; yi Þ and V iþ1 ðxiþ1 ; yiþ1 Þ are, respectively, the points of V i1 , Vi and V iþ1 in the coordinate * system XOY. If lines Li and Liþ1 are offset with R, the following equation is derived: ( bi x þ ai y ¼ R; (3) biþ1 x þ aiþ1 y ¼ R: Because L0i is not parallel to L0iþ1 , ai biþ1 aiþ1 bi a0. After Eq. (3) is solved, we get the point V 0i by 8 ðaiþ1 ai ÞR > ; > x ¼ > > bi ai > > > > > aiþ1 biþ1 < ðb bi ÞR > > : y ¼ iþ1 > > > bi ai > > > > : aiþ1 biþ1
(4)
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The points in temporary coordinate system xoy can be transformed into the points in XOY with the transformation equation: ( X ¼ xi þ x; (5) Y ¼ yi þ y:
877
Vi
Thus we can get the coordinate values of all offset points. 3.2.1.2. Self-intersection and sharp angle. After the boundary cycle is offset inwardly, self-intersection may happen in Fig. 4. It causes the distortion of contours, so self-intersection should be eliminated [19]. To calculate the intersecting points, the method is to go around the offset polygon edge by edge, and check whether there are two no-adjacent edges intersecting each other, and store a list of intersection points along the path in the order, in which they appear in their parent cycle. Then check whether the vector direction of a newly offsetting line is the same as that of the offset line. Otherwise, it is considered that the newly offset lines are self-intersections, which cause polygon distortion; and laser beam cannot reach and therefore, they should be eliminated. For instance, in Fig. 4 the vector directions for contour lines 10 and 40 are the same as those of the original contour lines 1 and 4, respectively. Contour lines 20 and 30 , whose directions are opposite to the lines 2 and 3, should be removed. Then a newly intersecting point d00 between noadjacent contour lines 10 and line 40 is added. Selfintersections can be eliminated in this way. Furthermore, sharp angles must be processed before further offsetting. If the lines form a sharp angle (angle V i1 V i V iþ1 in Fig. 5), which means ai biþ1 aiþ1 bi 0, the values of x and y in Eq. (4) are so large that a newly offset contour will be distorted. The sharp angle can be processed as follows: the endpoint Vi+1 of the shorter edge of sharp angle V i1 V i V iþ1 is gone through by the line V iþ1 V 00i perpendicular to the line V i V 0 in Fig. 5. Substituting V 00i for V 0i may eliminate the distortion of the newly offset sharp angle.
c 2 b
c′ d′
b′
d
2′ d′′
4
1 1′ a
a′
4′ e′
Fig. 4. Self-intersections.
Vi′′
Vi+1′
Vi-1
Vi-1′ Vi′
Fig. 5. Sharp angle.
The rules of judging the sharp angle are as follows: Firstly, judge whether angle V i1 V i V iþ1 value is small enough to form a sharp angle. Secondly, area of DV i1 V i V iþ1 is also chosen as the criteria, for sometimes the sharp angle can be the feature of the contour geometry. Lastly, the points of a newly offset contour are stored in the order in which they appear in their parent cycles. 3.2.2. Transformation of scan modes The rules of stopping contour scanning are as follows: First, two contours intersect each other. Second, a contour self-intersects. Last, the direction of external contour is changed or the contour scan ends. If one of the first two rules is attained, stop contour scan mode and start improved subarea scan mode in slices. 3.2.2.1. Contour intersection. Contour intersection may be divided into the intersection among contours and selfintersection. Actually, these two kinds of intersections can be processed in the way as we judge whether lines are intersecting. 3.2.2.2. Intersection of lines. If two points of a line lie in two sides of the other line, the two lines intersect [20,21]. Lines CD and AB intersect, points C and D lie in the two areas of the line AB and vice versa. As for a line AB y ¼ f ðxÞ, if Dðx; yÞ ¼ f ðxÞ y, the following expressions hold: 8 > < Dðx; yÞ40 above a line; Dðx; yÞ ¼ 0 on a line; (6) > : Dðx; yÞo0 below a line:
3
3′
Vi+1
e
We can now derive the following relation, where C:x andC:y derive the coordination x and y of endpoint C of
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line CD respectively, and D:x and D:y stands for the coordination x and y of endpoint D of line CD, respectively: 8 > < DAB ðC:x; C:yÞ DAB ðD:x; D:yÞ40 DAB ðC:x; C:yÞ DAB ðD:x; D:yÞ ¼ 0 > : D ðC:x; C:yÞ D ðD:x; D:yÞo0 AB
AB
The line CD is in the same side of the line AB; The point C or D is on the line AB;
(7)
The line CD is in two side of the line AB:
For the same reason, for which point A and point B on two sides of the line CD, the following equation is valid. DCD ðA:x; A:yÞnDCD ðB:x; B:yÞo0.
According to the characteristics of these two scan modes, the scan speed and laser energy should not be the same. In this paper, the scan speed and laser power for the
(8)
To increase the efficiency of the algorithm, a rectangle window is constituted by the points in Eq. (9). If a line spans the window or its endpoints are in it, we can further judge whether they are intersecting lines according to Eq. (7): 8 ymax ¼ maxðyA ; yB Þ; > > > > < y ¼ minðy ; y Þ; min A B (9) x ¼ maxðx > max A ; xB Þ; > > > : xmin ¼ minðxA ; xB Þ: The mean time complexity of the algorithm is Oðn log nÞ. For the object consisting of convex polygon, when the external loop offsets inward, it does not happen that lines intersect each other. But the direction of the external loop might change into the counterclockwise when the external loop offsets to a certain. If it continues to be offset, it can be amplified in the opposite direction. To avoid the situation, whether the direction of a loop is in the opposite direction should be judged. If the direction of a loop is changed into the opposite direction, contour scan must be stopped. 3.2.3. Filling after contour loop scan After a solid area in a slice plane is filled with contour scan path, the remainder will be filled with the improved subarea scan mode presented in Section 2.1. 3.2.4. Laser parameters and scan order for this mode 3.2.4.1. Scan speed and laser power. When complex contour curves are scanned, laser scan speed for contour scan mode is required to be less than that for improved scanning mode in order to increase the scan precision. Besides, because the scan time interval between two neighboring contour cycles is longer, there much heat loss occured. For these reasons laser power for contour scanning mode is required to be high to improve the sintering density. Because the center part of a slice plane is scanned with the improved subarea scan mode, the scan paths are short, which shortens the scan time interval between two neighboring scanning paths and reduces heat loss. Therefore, the laser power for improved subarea scan mode may be lower to avoid the center parts to be overly sintered [22].
two scan modes are summarized from the experiments. If reference scan speed and laser power is set as laser speed and laser power, the scan speed and laser power for the improved subarea scan mode are laser speed and 0.85 laser-power, respectively, while for the contour scan mode they are laser-speed and 1.2 laser-power, respectively. 3.2.4.2. Scan order. Outside powder material is firstly sintered when laser scans powder inward, which causes inner heat stress. So the scan order inward causes warp and distortion, and even crack in the SLS parts [23]. Therefore, in order to release the inner heat stress, compound scan order is outward. 3.3. The results of simulation The results of simulating the process of compound scan mode are shown in Fig. 6. Fig. 6a is the 3D CAD standard test piece. The result of simulating filling the slice plane at the height of 7 mm with contour scan mode is in Fig. 6b. Fig. 6c is the result of simulating filling the remainder part of the slice with improved scan mode. While the Figs. 7a–c are the amplified filling result of parts I, II, and III in Fig. 6c. In the process of SLS polystyrene (PS) powder is sintered outward with the compound scan mode. 4. Experiment and results We have tested the effect of subarea scan mode and compound scan mode on fabrication efficiency or scan efficiency, strength and precision of SLS parts. 4.1. Scan efficiency 4.1.1. Experimental equipment and conditions Experimental equipment was a HRPS-IIIA type SLS machine built at Huazhong University of Science and Technology (HUST), Wuhan, PR China. A CO2 laser source with a wavelength of 10.6 mm and maximum output power of 50 W is used. Other conditions, such as the laser power of 14 W, scan space of 0.1 mm, layer thickness of 0.2 mm and the scanning speed of 2000 mm/s, are carefully set. The powder used in this study was PS powder with grain size below 74 mm. It was sintered at 95–110 1C.
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Fig. 6. Simulating the process of a slice with compound scan mode: (a) 3D CAD mode of standard test piece with the size of 200 200 20 mm3; (b) filling the slice plane at the height of 7 mm with contour scan mode, and the dashed presenting the tracks of beam spanning cavities; (c) filling the remainder cycles 1–6 of the slice with improved scan mode.
4.1.2. Experimental methods General standard test parts in Fig. 8 were sintered. Two scan modes are used to fabricate the test parts with the height of 6 mm. They have the area of valid fabrication layers of 9080 mm2. The scan efficiency Z can be defined as Z ¼ ðvalid sintering areas S=laser scan areaÞ. The number of scanning location point is defined as the number of endpoints of filling lines, and the area of laser empty run is equal to: mean run length the number of empty run scan spacing.
strengths of simple shock test piece, simple flexural test piece and simple tensile test piece have little difference. So the complex structure test pieces with scan modes have large difference in their characteristics. For this reason, to simulate the strength of complex SLS parts, standard test pieces were made as in Figs. 9 and 10, in the light of the experimental standards of the strength of plastic parts. In Fig. 10, the SLS standard test pieces for flexural strength have the height of 10 mm, while they have the height of 5 mm for shock strength.
4.1.3. Experimental results Table 1 shows the results of the scan efficiency of the same test part with the compound scan modes and the subarea scan mode. In Table 1 for the same test part the scan efficiency of the compound scan mode is less than that of the subarea scan mode. The fabrication time taken for the standard test part in Fig. 8 with the compound scan modes and the subarea scan mode are shown in Table 2. For the same test part fabricated with two scan modes, the fabrication time with the compound scan mode is longer than that with the subarea scan mode.
4.2.2. Results of strength tests Test results for tensile strength, flexural strength and shock strength of SLS PS testing parts are shown in Tables 3–5.
4.2. Strength tests 4.2.1. Experimental methods and test instruments Experimental equipment and conditions are the same as those of testing scan efficiency in Section 4.1.1. The test instruments include the MDM-20 electron pulling force machine and the XJJ-5 freely supported beam hammer strength machine with a resolution of 0.005J. They are used for testing tensile strength, flexural strength and shock strength under 20 1C air temperature with 60% humidity. SLS test part with simple structure cannot reflect the effect of scan modes on the strength of SLS part. For example, effects of different scan modes, such as the compound scan mode and the subarea scan mode, on the
4.2.3. Analysis of test results The strength of a SLS part is related to the energy absorbed by powder [24]. Within some range, the more energy powder absorbs, the more strength of a SLS part increases. The energy absorbed by powder mainly depends on the energy directly supplied and the dissipated energy. The supplied energy depends on laser energy and preheating temperature, while the dissipated energy depends on the sintering time in a layer. It can be seen from Tables 3–5 that the compound scan mode and the subarea scan mode have little difference on the absorbed energy. 4.3. Dimensional accuracy 4.3.1. Experimental methods Experimental equipment and conditions are the same as those of testing scan efficiency in Section 4.1.1. The standard test pieces in Fig. 8 are sintered with the compound scan mode and the subarea scan mode. 4.3.2. Test results Test results of the dimensional accuracy of the SLS standard test piece in Fig. 8 with the subarea scan mode and the compound scan mode are listed in Tables 6 and 7.
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-97
-92
-87
-82
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-72 -72100
-7795
-8290
-8785
-9280
-9775
-10270 (b) 70
(a)
75
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100
70
65
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55
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40
35
30 (c) 30
35
40
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50
55
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Fig. 7. Amplification of Fig. 6c: (a), (b), and (c) are the amplification of part I, II, III, respectively.
4.3.3. Analysis of experimental results 4.3.3.1. Dimensional accuracy. Tables 6 and 7 show that the dimensional accuracy of the SLS part with the compound scan mode is higher than that of the SLS part with the subarea scan mode. Therefore, scan modes have a great effect on the dimensional accuracy. According to the principle of the working laser scan system, the laser beam scanning on the edge of a SLS part will cause great error between its practical position and the theoretical position due to the inertia force of the system. Areas with the subarea scan mode are filled with parallel scan lines, so the laser beam will produce a position error to some degree. However, areas with the compound scan mode may have an error only on the turning point of the borderline of the SLS part. Because the whole contours are filled with line
segments with the compound scan mode, there exists no inertia force. Therefore, its dimensional accuracy is higher. 4.3.3.2. Shape accuracy. Shape error is mainly caused by the shape change or warp and distortion of a SLS part. Tables 6 and 7 show that the diameter of circular hole of the standard test piece fabricated by the subarea scan mode has great variable errors at different positions, but a good circular degree can be guaranteed for the parts with the compound scan mode. It is obvious that the compound scan mode is of benefit to reducing the shape error. Warp and distortion of the SLS part are dependent on the temperature. Scan modes directly affect the temperature field, so the warp and distortion of the SLS part vary with different scan modes. From the results the warp and
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Fig. 8. Standard test pieces.
Table 1 Scan efficiency with the compound scan mode and the subarea scan mode
Subarea scan Compound scan
Laser locating points
Number of laser on–off
Area of scan empty run (mm2)
Area of practical scan (mm2)
Laser scan efficiency
8820 11872
16 87
640 3480
9720 12560
0.93 0.72
Table 2 The fabrication time with the compound scan mode and the subarea scan mode Scan mode
Fabricated height (mm)
Layered number
Laser power (W)
Scan spacing (mm)
Fabrication time (h:m:s)
Subarea Scan Compound scan
15.03 15.00
75 75
14 14
0.10 0.10
02:18:22 03:06:56
Fig. 9. SLS test pieces for tensile strength.
Fig. 10. SLS test pieces with H ¼ 10 mm for flexural strength, SLS test pieces with H ¼ 5 mm for shock strength.
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distortion with the compound scan mode are less than with other scan modes.
Table 3 The results of tensile tests Scan mode
Subarea scan Compound scan
Tensile strengths (MPa)
Mean tensile strengths (MPa)
Test 1
Test 2
Test 3
3.057 1.823
2.862 1.450
2.667 1.928
2.862 1.734
5. Conclusions
Table 4 The results of bending tests Scan mode
Subarea scan Compound scan
Flexural strengths (MPa)
Mean flexural strengths (MPa)
Test 1
Test 2
Test 3
0.453963 0.316846
0.454774 0.396634
0.492338 0.442507
0.467025 0.385329
Table 5 The results of shock tests Scan mode
Subarea scan Compound scan
4.3.3.3. Surface quality. The sintered seam occurs only in the place where two scans meet. For the subarea scan mode, there are subareas where the melted seams occurs, and for the compound scan mode there are less subareas, and they are in the center of slice planes. Therefore, the surface finish of the SLS part with compound scan mode is higher.
Shock strengths (J/m2)
Mean shock strengths (J/m2)
Test 1
Test 2
Test 3
933.33 866.67
1000.00 933.33 1333.33 933.33
Test 4 1000.00 966.67 933.33 1016.67
The thesis presents the principle of a compound scan mode and its implementation. The compound scan path generation combines the algorithms of contour path generation and subarea path generation. The results of this compound scan mode for a standard test piece are given. It proves that the compound scan mode is available. Some experiment on the effectiveness of the compound scan mode and subarea scan mode on fabrication efficiency, strength and precision were made. The results restate that (1) For the same part the former (compound scan mode) has less fabrication efficiency than the latter (subarea scan mode) does; i.e., it needs more time to be fabricated with the former than the latter. (2) The part with the former has slightly less tensile strength, flexural strength, and shock strength, than
Table 6 Test results of the dimensional accuracy of the SLS standard test piece with the subarea scan mode Items
Theoretical values (mm)
Measured values (mm)
Length of part Width of part Inner diameter of central hole External diameter of central hole Inner diameter of external circle Inner length of square hole External length of square hole Width of external rib Width of inner rib
200 200 20 30 20 20 30 5 5
199.80 199.20 19.50 29.82 19.60 20.00 30.00 5.30 5.00
199.40 199.40 19.48 29.72 19.62 19.70 30.00 5.00 5.30
199.90 199.70 19.62 29.80 19.80 19.78 30.40 5.00 5.94
199.80 199.50 19.54 29.82 19.90 19.92 30.20 5.50 5.30
Average values (mm)
Dimensional errors (mm)
199.72 199.45 19.53 29.79 19.73 19.85 30.15 5.20 5.38
0.28 0.55 0.47 0.21 0.27 0.15 0.15 0.20 0.38
Average values (mm)
Dimensional errors (mm)
200.12 199.91 19.71 30.02 19.79 19.72 29.82 5.07 4.90
0.12 0.09 0.29 0.02 0.21 0.28 0.18 0.07 0.10
Table 7 Test results of the dimensional accuracy of the SLS standard test piece with the compound scan mode Items
Theoretical values (mm)
Measured values (mm)
Length of part Width of part Inner diameter of central hole External diameter of central hole Inner diameter of external circle Inner length of square hole External length of square hole Width of external rib Width of inner rib
200 200 20 30 20 20 30 5 5
200.00 200.20 19.60 30.00 19.80 19.60 29.82 5.00 4.80
200.10 200.00 19.80 30.00 19.80 19.80 29.68 5.00 4.98
200.20 199.80 19.60 30.00 19.80 19.60 29.80 5.10 4.82
200.20 199.64 19.84 30.10 19.76 19.88 30.00 5.20 5.00
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that of the latter; i.e., the former has no superiority on the strength of a SLS part. (3) The part with the former has more precise on the dimensional accuracy, shape accuracy and surface quality, than the latter. (4) The SLS part with high fabrication precision should be scanned with the compound scan mode, and the SLS part with high fabrication efficiency and lower fabrication precision should be scanned with the subarea scan mode. The presented compound scan mode has been utilized on SLS system built at HUST. It can also be applied to other RP systems. Though the algorithm in this paper can improve the quality of SLS parts to some extent, there are algorithms to be developed for SLS processing.
Acknowledgments This work was supported by the key project under the 863 program of China (2002AA6Z3083), the key R&D program of Hubei Province of China (2001A107B02), and the key technological project under the tenth five-year plan of Guangdong Province of China (A1040201). References [1] J. Wang, Z. Li, B. Lu, The study of scan path on rapid prototyping, Mechanical Science and Technology 16 (2) (1997) 303–305. [2] Y. Shi, Q. Zhong, X. Chen, Research and implementation of a new kind of scan mode for selective laser sintering, Chinese Journal of Mechanical Engineer 38 (2) (2002) 35–39. [3] D. Cai, Y. Shi, G. Chen, S. Huang, The study of the optimization of SLS rapid prototyping scan path, Metal Forming Machinery (2002) 18–20. [4] S.O. Onuh, K.K.B. Hon, Application of the taguchi method and new hatch styles for quality improvement in stereolithography. In: Proceedings of the Institution of Mechanical Engineers, vol. 212, 1998, pp. 461–471. [5] Z. Liu, H. Bin, X. Zhang, H. Gao, The influence of the fractal scan path on the temperature field of the layer in material increase manufacturing, Journal of Huazhong University of Science and Technology 26 (8) (1998) 32–34. [6] R. Zhang, Z. Shan, G. Su, Y. Yang, Study on laser scan process of selective laser sintering for powder material, Applied Laser 19 (5) (1999) 299–302.
883
[7] H. Chen, Z. Zhang, J. Cheng, Laser scan path stratagems on SLS rapid prototyping craft, Journal of Basic Science and Engineering 9 (2–3) (2001) 202–207. [8] Y. Yang, H.T. Loh, F.Y.H. Fuh, Y.G. Wang, Equidistant path generation for improving scan efficiency in layered manufacturing, Rapid Prototyping Journal 8 (1) (2002) 30–37. [9] D. Kim, Polygon offsetting using a Voronoi diagram and two stacks, Computer Aided Design 30 (14) (1998) 1069–1075. [10] J. Chen, P. Ma, J. Tian, Z. Liu, Scan path algorithm for rapid prototyping (RP) based on Voronoi diagrams, Mechanical Science and Technology 22 (5) (2003) 728–731. [11] R.T. Farouki, T. Sakkalis, Pythagorean hodographs, IBM Journal of Research and Development 34 (5) (1990) 736–752. [12] M. Bertoldi, C.M. Yardimci, Domain decomposition and surface filling curves in toolpath planning and generation. In: Solid Freeform Fabrication Symposium Proceedings, 1998. pp. 267–274. [13] T. Wasser, A. Dhair, Jayal, C. Pistor, Implementation and evaluation of novel buildstyles in fused deposition modeling (FDM). In: Solid Freeform Fabrication Symposium Proceedings, Austin, 1999, pp. 95–102. [14] W. Tiller, E.G. Hanson, Offsets of two-dimensional profiles, IEEE Computer Graphics and Application (1984) 36–46. [15] M. Ganesan, G. Fadel, Hollowing rapid prototyping parts using offsetting techniques, In: Proceedings of the Fifth International Conference on Rapid Prototyping, University of Dayton, 1994, pp. 241–251. [16] B. Pham, Offset curves and surfaces: a brief survey, Computer Aided Design 24 (4) (1992) 223–229. [17] M. Takashi, An overview of offset curves and surfaces, Computer Aided Design 31 (4) (1999) 165–173. [18] Y. Cheng. The Study and Development of Scan Path for Selective Laser Sintering [M], Huangzhong University of Science and Technology, Wuhan, April 2004. [19] L. Zhang, M. Han, S. Huang, The study of radius compensation algorithm on radius of laser spot in selective laser sintering, Journal of Huazhong University of Science and Technology 30 (6) (1998) 16–18. [20] D. XIA, X. Xiong, New algorithm and principle for intersection of line segments, Journal of Cad and Computer Graphics 8 (2) (1996) 95–100. [21] C. Cheng, T. Yan, Optimize algorithm of intersecting n lines, Engineering of Surveying and Mapping 10 (3) (2001) 29–31. [22] X. Li, M. Han, Y. Shi, S. Huang, Modal of shrinking and curl distortion for SLS prototypes, Chinese Journal of Mechanical Engineer 12 (8) (2001) 887–889. [23] J. Zhang, The Study of Material STP1 and Process for SLS Prototypes [M], North China University of Science and Technology, Taiyuan, 2002. [24] J. Bai, J. Zhao, Y. Tang, C. Yu, Study on the strength of SLS rapid prototyping test-pieces, Aviation Precision Manufacturing Technology 33 (5) (1997) 29–31.
International Journal of Machine Tools & Manufacture 47 (2007) 873–883 www.elsevier.com/locate/ijmactool
Compound scan mode developed from subarea and contour scan mode for selective laser sintering Y. Shi, W. Zhang, Y. Cheng, S. Huang State Key Laboratory of Plastic Forming Simulation and Die & Mould Technology, School of Material Science and Engineering, Huazhong University of Science and Technology, Wuhan-Hubei 430074, PR China Received 25 May 2006; received in revised form 31 July 2006; accepted 14 August 2006 Available online 6 October 2006
Abstract Scan mode is an important parameter for selective laser sintering (SLS) processing. The improved mode will optimize scan path and improve the precision, strength and fabrication efficiency of a SLS part. A compound scan mode, which combines subarea scan mode and contour scan mode, is proposed. The principle of its hatch (path-planning) algorithm and implementation are presented. To testify the effectiveness of this compound mode compared to that of subarea scan mode, it has been utilized for researches at a SLS machine developed at Huangzhong University of Science and Technique (HUST). The results from the researches indicate that the degree of precision of a SLS part with the compound scan mode is higher than that with subarea one. There is little difference in the tensile strength, flexural strength, shock strength and fabrication efficiency of a SLS part under the compound scan pattern and the subarea scan mode. Therefore, implementation of the compound scan mode is of importance to improve the precision of a SLS part. r 2006 Elsevier Ltd. All rights reserved. Keywords: Selective laser sintering; Improved subarea scan mode; Compound scan mode; Fabrication precision
1. Introduction Scan mode is one of the important parameters that affect the precision, strength and fabrication efficiency of a selective laser sintering (SLS) part. Therefore, it is very important to optimize scan mode. Much study on scan mode has been done and many scan modes have been created, and they belong to parallel-line scan mode or broken-line scan mode. 1.1. Parallel-line scan mode Parallel-line scan mode [1] is also called as type Z scan mode whose scanning paths are parallel to x- and y-axis. Its principle is similar to that of filled regions in computer graphics. Though its algorithm can be simply and easily Corresponding authors. Tel.: +86 027 87557042; fax: +86 027 87548581. E-mail addresses: [email protected] (Y. Shi), [email protected] (W. Zhang).
0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.08.013
implemented, it has disadvantages. Firstly, laser has to be continually turned off and on, which shortens its life. Moreover, for a part with holes, cavities have to be spanned frequently and it causes scanners run emptily. Secondly, for the scan direction in layers is the same and there exists shrinkage stress in the same direction, which causes the warp and distortion of a SLS part. Finally, a SLS part with parallel scan mode has anisotropy strength. For subarea scan mode [2,3], scan lines are parallel lines. Every slice plane of three-dimensional (3D) CAD model is divided into smaller areas and then they are filled by parallel lines. Empty runs are less than those for parallelline scan mode. Its algorithm is simple, so it is widely used in the SLS process, and it has the following disadvantages: First, some micro-ladders appear on a SLS part with this mode, which causes the precision dependent of the spot size of laser beam. What is more, there exists the warp and distortion of a SLS part with this mode. Especially, in a part with a larger slice plane, there appears great shrinkage stresses and even cracks along the scanning direction because of longer scanning lines.
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For emanative starlike scan mode and angled emanative starlike scan mode [4], slice planes are divided into two areas from the center, which are filled with scan lines parallel to x or y axis or at an angle of 451 to the axis from the center to outside. Compared to that with the two scan modes above, the SLS part with these two modes causes less warp and distortion to some extent, but they have disadvantages such as that of parallel-line scan mode. 1.2. Fold-line scan mode For fractal scan mode [5] scan path has the features of local and whole comparability. When fractal dimensions are 2, the whole slice planes can be filled with fractal scan path. This scan mode overcomes the shortcomings of parallel-line scan mode and makes the temperature field more uniform, which reduces warp and distortion. But it has the following disadvantages: low scanning speed, laser scanners frequently accelerated and decelerated, a SLS part being getting fabricated with low precision, and frequent spanning of the cavities of the slice planes. Scan paths are spiral lines for spiral scan mode [6]. Though it overcomes the shortcomings of type z scan mode, and shrinkage stresses can be reduced, the cavities have to be spanned frequently. For contour equidistance path [7,8] no empty run appears which prolongs scanners’ life. The frequent changing scan direction and shorter scanning lines disperse the shrinkage stresses and reduce warp and distortion. Its algorithm is complex and it needs more CPU time. Besides, the algorithm is not reliable and some unfilled regions appear, which affects the strength of the SLS part. This algorithm of the scan path based on Voronoi map [9,10] is suitable for multi-connected domains in slice planes, and enhances the scan efficiency to some degree. But it has the same shortcomings as the contour equidistance path. As for scan mode using Pythagorean hodograph as filling lines [11], the intersecting parts of slices are separated into some subareas by the Delaunay triangle, and then they are filled with Hilbert curves [12]. However, it is suitable only for regular parts, except the fabrication of parts with complex contour. Besides, there are also other similar slice filling modes presented by Wasser et al. [13], Tiller and Hanson [14], Ganesan and Fadel [15], Pham [16] and Takashi [17]. To overcome the shortcomings of the scan modes above, a compound scan mode combining subarea scan mode with contour scan mode is presented in this paper, and it has not been reported in the literature.
we firstly discuss these two scan modes before the compound one. 2.1. Subarea scan mode A subarea scan mode is presented in Ref. [2] in order to improve quality of a SLS part. Strategies for it are as follows: The first step is that the slice planes are divided into subareas without holes shown in Fig. 1. The second one is that the slice planes are divided into subareas in terms of the number of intersecting points that the scan lines parallel to x or y-axis intersect with their contours. Fig. 1 is taken for an example. The scan lines within area 1 form two intersecting points with the contour. The number of intersecting points increases into four when they are within area 2. In the same way, the number in area 4 is six, the number in area 7 is eight, and the number in area 11 is six. So the whole slice area is subdivided into 16 subareas. The third one is that subareas are scanned in the fixed order from top to bottom, and from left to right. For example, the scan order in Fig. 1 is: area 1- area 2- area 3- area 4- area 5- area 6- area 7? ?. Finally, these subareas are filled with parallel lines, but there is an a angle between two neighboring scan layers in order to reduce shrinkage stress along the same direction. Generally a ¼ 90 . According to the above strategies, we can implement it by the following procedure: Initially, intersecting points where (at which) the scan lines are intersecting with the contours of a slice plane are obtained, and in the order they can be stored in pairs, such asðP0 ; P1 Þ,ðP2 ; P3 Þ, ðP4 ; P5 Þ,?,ðP2n1 ; P2n Þ. If these scan lines are in a connected domain, it can be filled with lines consisting of point pairs. And those lines are called the filling lines. Then the filling lines can be grouped. For the setfC i ; 1oioNg, the filling lines in the whole slice can be grouped into m groups from Eq. (1).
2. Subarea scan mode and contour scan mode The compound scan mode, based on the subarea scan mode and cotour scan mode, is presented. Therefore,
Fig. 1. Subarea scan mode.
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8 C 1 ¼ C 2 ¼ ¼ C k1 aC k1 þ1 ; > > > > > > < C k1 þ1 ¼ C k1 þ2 ¼ ¼ C k2 aC k2 þ1 ; > > > C km1 þ1 ¼ C km1 þ2 ¼ ¼ C km ; > > > : km ¼ N;
(1)
where Ci is the number of segments as filling lines, i.e., the number of pairs of intersecting point, and N is the number of grouped filling lines. So in Fig. 1 there are seven groups of filling lines. Though subareas are attained according to the number of crossing points scan lines intersect with the contours of a slice plane, its principle can be optimized. So the subarea scan mode can be developed into an improved subarea scan mode. The strategies of improved subarea scan mode are explained as follows: First of all, subarea zone is grouped as much as possible. Next the numbers of intersection points vary when two neighboring scanning lines intersect with the contours in a slice plane. According to the principle of the scan order, two filling lines are in a connected domain. According to the strategies of the improved subarea scan mode, we can implement it as follows: To start with, we could judge whether the neighboring scan lines are in a connected domain. Once they are projected into the scanning direction and there are some overlaps, they must be in a connected domain. Otherwise, they are not in it. According to the above methods, the algorithm of improved subarea scan mode needs frequent judgment whether the filling lines belong to a connected domain section. For this reason, much time will be taken by CPU to do this. To spare calculating time, complex polygons in a slice plane in Fig. 2a can be grouped according to the depth tree of the polygons’ contour cycles.
It is known that depth of the cycle, external and internal polygon, and direction of the polygon are defined as follows: Depth of the cycle refers to the number of cycles in the same contour enclosing it. The maximum depth in a contour is called the depth of a contour; the depth of a contour is also an indication of the planar complexity of the contour. As illustrated in Fig. 2b, the depth tree shows the relationship of the cycles to a contour. The depth of this contour is 3. For a given contour, we employ the depth tree to describe its complexity. Based on the definition of depth, we can also define the attribute of a cycle: if the depth of a cycle is even, it is an external cycle; otherwise, it is an internal cycle. In Fig. 2a, cycles A, D and E are external cycles, and the rest are internal cycles. Like most of definitions in the relevant literature, we define the direction as positive if the solid area is always to the left when walking along the boundary. Obviously, for external cycles, the positive direction is counterclockwise; for the internal cycles, the positive direction is clockwise. The Fig. 1 is taken as a case in point for the improved subarea scan mode. According to the above principles, Fig. 1 can be united into four subareas: a subarea I including area 1, area 2, area 4, area 7, area 11, area 14 and area 16; a subarea II consisting of area 3, area 6, area 9 and area 12; a subarea III including area 10, area 13 and area 15; a subarea IV consisting of area 5 and area 8. Therefore, the number of spanning cavity and subareas can be reduced, which is of benefit to SLS process. 2.2. Contour scan mode Contour scan mode belongs to fold-line scan mode that the two neighboring filling lines are not parallel. Filling lines with equidistance are parallel to the lines of a contour polygon. For the curves of a contour in a slice plane, they are approximated by line segments, to which the filling lines with equidistance are parallel. The principles of Contour
Depth 0
A C
E
D
B
A
1
C
G F B
2
E
D
F (a)
875
G
(b) Fig. 2. (a) Contour polygons in a slice plane and (b) tree structure of polygons.
3
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contour scan mode are explained in the following Section 3.2. Because the solid area in a slice plane is filled with laser beam spot with constant size, filling lines are considered to be equidistant from lines consisting of a contour polygon.
Vi-1 V' i-1
R
y
Li
3. Compound scan mode L'i
Vi
Y
The compound scan mode is studied, which combines the algorithms of subarea scan mode and contour scan mode.
o
3.1. Strategies of compound scan mode Most slice planes of a 3D model consist of complex concave polygons, and most of them compose polygons. For the algorithm of contour scan mode, it happens that polygons may intersect each other and lines of concave polygons may intersect themselves. That is the main reason why the algorithm of a contour scan mode is complex. In slice planes the contour scan mode cannot be stopped or transformed into the improved subarea scan until filling polygons intersect each other or intersect themselves. These are the strategies of this mode.
x L i+ 1
V'i
R
Vi+ 1
L'i+1
V' i+1 O X
Fig. 3. Offset generations.
where 3.2. Implementation of compound scan mode According to the strategies, we can implement the algorithm by the following steps. The first step is that the contours in a slice plane are grouped into connected domains and then every one of connected domains is filled with lines with the compound scan mode once. The second one is that filling lines offset with equidistance are generated. The third one is that in scan space, the number of contour offset, is defined. The last one is that after the contours are filled with contour scanning mode, the unfilled parts are to be filled with the improved subarea scan mode. 3.2.1. Creation of contour scan path 3.2.1.1. Equidistance contour generation. When the solid part of a slice plane is filled with a series of equidistance beam paths, the offset directions to inner loops and external loops are opposite. According to the definition of the direction of a loop [18], the offsets of* inner and external loops are presented with the vector R shown in Fig. 3. Therefore, the endpoint coordinates of their lines can be calculated according to the following methods: As shown in Fig. 3, it is assumed that the origin of the temporary coordinate system xoy is the arbitrary intersecting point Vi .*Li and*Liþ1 are the vector formed by V i1 V i and V i V iþ1 . l i and l iþ1 are the unit vectors of Li and Liþ1 . They are given by 8* * * < l i ¼ ai i þb j ; i (2) * * :* l iþ1 ¼ aiþ1 i þbiþ1 j ;
xi xi1 ai ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðxi xi1 Þ2 þ ðyi yi1 Þ2 yi yi1 ffi, bi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xi1 Þ2 þ ðyi yi1 Þ2 xiþ1 xi aiþ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðxiþ1 xi Þ2 þ ðyiþ1 yi Þ2 yiþ1 yi ffi, biþ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxiþ1 xi Þ2 þ ðyiþ1 yi Þ2 V i1 ðxi1 ; yi1 Þ, V i ðxi ; yi Þ and V iþ1 ðxiþ1 ; yiþ1 Þ are, respectively, the points of V i1 , Vi and V iþ1 in the coordinate * system XOY. If lines Li and Liþ1 are offset with R, the following equation is derived: ( bi x þ ai y ¼ R; (3) biþ1 x þ aiþ1 y ¼ R: Because L0i is not parallel to L0iþ1 , ai biþ1 aiþ1 bi a0. After Eq. (3) is solved, we get the point V 0i by 8 ðaiþ1 ai ÞR > ; > x ¼ > > bi ai > > > > > aiþ1 biþ1 < ðb bi ÞR > > : y ¼ iþ1 > > > bi ai > > > > : aiþ1 biþ1
(4)
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The points in temporary coordinate system xoy can be transformed into the points in XOY with the transformation equation: ( X ¼ xi þ x; (5) Y ¼ yi þ y:
877
Vi
Thus we can get the coordinate values of all offset points. 3.2.1.2. Self-intersection and sharp angle. After the boundary cycle is offset inwardly, self-intersection may happen in Fig. 4. It causes the distortion of contours, so self-intersection should be eliminated [19]. To calculate the intersecting points, the method is to go around the offset polygon edge by edge, and check whether there are two no-adjacent edges intersecting each other, and store a list of intersection points along the path in the order, in which they appear in their parent cycle. Then check whether the vector direction of a newly offsetting line is the same as that of the offset line. Otherwise, it is considered that the newly offset lines are self-intersections, which cause polygon distortion; and laser beam cannot reach and therefore, they should be eliminated. For instance, in Fig. 4 the vector directions for contour lines 10 and 40 are the same as those of the original contour lines 1 and 4, respectively. Contour lines 20 and 30 , whose directions are opposite to the lines 2 and 3, should be removed. Then a newly intersecting point d00 between noadjacent contour lines 10 and line 40 is added. Selfintersections can be eliminated in this way. Furthermore, sharp angles must be processed before further offsetting. If the lines form a sharp angle (angle V i1 V i V iþ1 in Fig. 5), which means ai biþ1 aiþ1 bi 0, the values of x and y in Eq. (4) are so large that a newly offset contour will be distorted. The sharp angle can be processed as follows: the endpoint Vi+1 of the shorter edge of sharp angle V i1 V i V iþ1 is gone through by the line V iþ1 V 00i perpendicular to the line V i V 0 in Fig. 5. Substituting V 00i for V 0i may eliminate the distortion of the newly offset sharp angle.
c 2 b
c′ d′
b′
d
2′ d′′
4
1 1′ a
a′
4′ e′
Fig. 4. Self-intersections.
Vi′′
Vi+1′
Vi-1
Vi-1′ Vi′
Fig. 5. Sharp angle.
The rules of judging the sharp angle are as follows: Firstly, judge whether angle V i1 V i V iþ1 value is small enough to form a sharp angle. Secondly, area of DV i1 V i V iþ1 is also chosen as the criteria, for sometimes the sharp angle can be the feature of the contour geometry. Lastly, the points of a newly offset contour are stored in the order in which they appear in their parent cycles. 3.2.2. Transformation of scan modes The rules of stopping contour scanning are as follows: First, two contours intersect each other. Second, a contour self-intersects. Last, the direction of external contour is changed or the contour scan ends. If one of the first two rules is attained, stop contour scan mode and start improved subarea scan mode in slices. 3.2.2.1. Contour intersection. Contour intersection may be divided into the intersection among contours and selfintersection. Actually, these two kinds of intersections can be processed in the way as we judge whether lines are intersecting. 3.2.2.2. Intersection of lines. If two points of a line lie in two sides of the other line, the two lines intersect [20,21]. Lines CD and AB intersect, points C and D lie in the two areas of the line AB and vice versa. As for a line AB y ¼ f ðxÞ, if Dðx; yÞ ¼ f ðxÞ y, the following expressions hold: 8 > < Dðx; yÞ40 above a line; Dðx; yÞ ¼ 0 on a line; (6) > : Dðx; yÞo0 below a line:
3
3′
Vi+1
e
We can now derive the following relation, where C:x andC:y derive the coordination x and y of endpoint C of
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line CD respectively, and D:x and D:y stands for the coordination x and y of endpoint D of line CD, respectively: 8 > < DAB ðC:x; C:yÞ DAB ðD:x; D:yÞ40 DAB ðC:x; C:yÞ DAB ðD:x; D:yÞ ¼ 0 > : D ðC:x; C:yÞ D ðD:x; D:yÞo0 AB
AB
The line CD is in the same side of the line AB; The point C or D is on the line AB;
(7)
The line CD is in two side of the line AB:
For the same reason, for which point A and point B on two sides of the line CD, the following equation is valid. DCD ðA:x; A:yÞnDCD ðB:x; B:yÞo0.
According to the characteristics of these two scan modes, the scan speed and laser energy should not be the same. In this paper, the scan speed and laser power for the
(8)
To increase the efficiency of the algorithm, a rectangle window is constituted by the points in Eq. (9). If a line spans the window or its endpoints are in it, we can further judge whether they are intersecting lines according to Eq. (7): 8 ymax ¼ maxðyA ; yB Þ; > > > > < y ¼ minðy ; y Þ; min A B (9) x ¼ maxðx > max A ; xB Þ; > > > : xmin ¼ minðxA ; xB Þ: The mean time complexity of the algorithm is Oðn log nÞ. For the object consisting of convex polygon, when the external loop offsets inward, it does not happen that lines intersect each other. But the direction of the external loop might change into the counterclockwise when the external loop offsets to a certain. If it continues to be offset, it can be amplified in the opposite direction. To avoid the situation, whether the direction of a loop is in the opposite direction should be judged. If the direction of a loop is changed into the opposite direction, contour scan must be stopped. 3.2.3. Filling after contour loop scan After a solid area in a slice plane is filled with contour scan path, the remainder will be filled with the improved subarea scan mode presented in Section 2.1. 3.2.4. Laser parameters and scan order for this mode 3.2.4.1. Scan speed and laser power. When complex contour curves are scanned, laser scan speed for contour scan mode is required to be less than that for improved scanning mode in order to increase the scan precision. Besides, because the scan time interval between two neighboring contour cycles is longer, there much heat loss occured. For these reasons laser power for contour scanning mode is required to be high to improve the sintering density. Because the center part of a slice plane is scanned with the improved subarea scan mode, the scan paths are short, which shortens the scan time interval between two neighboring scanning paths and reduces heat loss. Therefore, the laser power for improved subarea scan mode may be lower to avoid the center parts to be overly sintered [22].
two scan modes are summarized from the experiments. If reference scan speed and laser power is set as laser speed and laser power, the scan speed and laser power for the improved subarea scan mode are laser speed and 0.85 laser-power, respectively, while for the contour scan mode they are laser-speed and 1.2 laser-power, respectively. 3.2.4.2. Scan order. Outside powder material is firstly sintered when laser scans powder inward, which causes inner heat stress. So the scan order inward causes warp and distortion, and even crack in the SLS parts [23]. Therefore, in order to release the inner heat stress, compound scan order is outward. 3.3. The results of simulation The results of simulating the process of compound scan mode are shown in Fig. 6. Fig. 6a is the 3D CAD standard test piece. The result of simulating filling the slice plane at the height of 7 mm with contour scan mode is in Fig. 6b. Fig. 6c is the result of simulating filling the remainder part of the slice with improved scan mode. While the Figs. 7a–c are the amplified filling result of parts I, II, and III in Fig. 6c. In the process of SLS polystyrene (PS) powder is sintered outward with the compound scan mode. 4. Experiment and results We have tested the effect of subarea scan mode and compound scan mode on fabrication efficiency or scan efficiency, strength and precision of SLS parts. 4.1. Scan efficiency 4.1.1. Experimental equipment and conditions Experimental equipment was a HRPS-IIIA type SLS machine built at Huazhong University of Science and Technology (HUST), Wuhan, PR China. A CO2 laser source with a wavelength of 10.6 mm and maximum output power of 50 W is used. Other conditions, such as the laser power of 14 W, scan space of 0.1 mm, layer thickness of 0.2 mm and the scanning speed of 2000 mm/s, are carefully set. The powder used in this study was PS powder with grain size below 74 mm. It was sintered at 95–110 1C.
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Fig. 6. Simulating the process of a slice with compound scan mode: (a) 3D CAD mode of standard test piece with the size of 200 200 20 mm3; (b) filling the slice plane at the height of 7 mm with contour scan mode, and the dashed presenting the tracks of beam spanning cavities; (c) filling the remainder cycles 1–6 of the slice with improved scan mode.
4.1.2. Experimental methods General standard test parts in Fig. 8 were sintered. Two scan modes are used to fabricate the test parts with the height of 6 mm. They have the area of valid fabrication layers of 9080 mm2. The scan efficiency Z can be defined as Z ¼ ðvalid sintering areas S=laser scan areaÞ. The number of scanning location point is defined as the number of endpoints of filling lines, and the area of laser empty run is equal to: mean run length the number of empty run scan spacing.
strengths of simple shock test piece, simple flexural test piece and simple tensile test piece have little difference. So the complex structure test pieces with scan modes have large difference in their characteristics. For this reason, to simulate the strength of complex SLS parts, standard test pieces were made as in Figs. 9 and 10, in the light of the experimental standards of the strength of plastic parts. In Fig. 10, the SLS standard test pieces for flexural strength have the height of 10 mm, while they have the height of 5 mm for shock strength.
4.1.3. Experimental results Table 1 shows the results of the scan efficiency of the same test part with the compound scan modes and the subarea scan mode. In Table 1 for the same test part the scan efficiency of the compound scan mode is less than that of the subarea scan mode. The fabrication time taken for the standard test part in Fig. 8 with the compound scan modes and the subarea scan mode are shown in Table 2. For the same test part fabricated with two scan modes, the fabrication time with the compound scan mode is longer than that with the subarea scan mode.
4.2.2. Results of strength tests Test results for tensile strength, flexural strength and shock strength of SLS PS testing parts are shown in Tables 3–5.
4.2. Strength tests 4.2.1. Experimental methods and test instruments Experimental equipment and conditions are the same as those of testing scan efficiency in Section 4.1.1. The test instruments include the MDM-20 electron pulling force machine and the XJJ-5 freely supported beam hammer strength machine with a resolution of 0.005J. They are used for testing tensile strength, flexural strength and shock strength under 20 1C air temperature with 60% humidity. SLS test part with simple structure cannot reflect the effect of scan modes on the strength of SLS part. For example, effects of different scan modes, such as the compound scan mode and the subarea scan mode, on the
4.2.3. Analysis of test results The strength of a SLS part is related to the energy absorbed by powder [24]. Within some range, the more energy powder absorbs, the more strength of a SLS part increases. The energy absorbed by powder mainly depends on the energy directly supplied and the dissipated energy. The supplied energy depends on laser energy and preheating temperature, while the dissipated energy depends on the sintering time in a layer. It can be seen from Tables 3–5 that the compound scan mode and the subarea scan mode have little difference on the absorbed energy. 4.3. Dimensional accuracy 4.3.1. Experimental methods Experimental equipment and conditions are the same as those of testing scan efficiency in Section 4.1.1. The standard test pieces in Fig. 8 are sintered with the compound scan mode and the subarea scan mode. 4.3.2. Test results Test results of the dimensional accuracy of the SLS standard test piece in Fig. 8 with the subarea scan mode and the compound scan mode are listed in Tables 6 and 7.
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880 -102
-97
-92
-87
-82
-77
-72 -72100
-7795
-8290
-8785
-9280
-9775
-10270 (b) 70
(a)
75
80
85
90
95
100
70
65
60
55
50
45
40
35
30 (c) 30
35
40
45
50
55
60
65
70
Fig. 7. Amplification of Fig. 6c: (a), (b), and (c) are the amplification of part I, II, III, respectively.
4.3.3. Analysis of experimental results 4.3.3.1. Dimensional accuracy. Tables 6 and 7 show that the dimensional accuracy of the SLS part with the compound scan mode is higher than that of the SLS part with the subarea scan mode. Therefore, scan modes have a great effect on the dimensional accuracy. According to the principle of the working laser scan system, the laser beam scanning on the edge of a SLS part will cause great error between its practical position and the theoretical position due to the inertia force of the system. Areas with the subarea scan mode are filled with parallel scan lines, so the laser beam will produce a position error to some degree. However, areas with the compound scan mode may have an error only on the turning point of the borderline of the SLS part. Because the whole contours are filled with line
segments with the compound scan mode, there exists no inertia force. Therefore, its dimensional accuracy is higher. 4.3.3.2. Shape accuracy. Shape error is mainly caused by the shape change or warp and distortion of a SLS part. Tables 6 and 7 show that the diameter of circular hole of the standard test piece fabricated by the subarea scan mode has great variable errors at different positions, but a good circular degree can be guaranteed for the parts with the compound scan mode. It is obvious that the compound scan mode is of benefit to reducing the shape error. Warp and distortion of the SLS part are dependent on the temperature. Scan modes directly affect the temperature field, so the warp and distortion of the SLS part vary with different scan modes. From the results the warp and
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Fig. 8. Standard test pieces.
Table 1 Scan efficiency with the compound scan mode and the subarea scan mode
Subarea scan Compound scan
Laser locating points
Number of laser on–off
Area of scan empty run (mm2)
Area of practical scan (mm2)
Laser scan efficiency
8820 11872
16 87
640 3480
9720 12560
0.93 0.72
Table 2 The fabrication time with the compound scan mode and the subarea scan mode Scan mode
Fabricated height (mm)
Layered number
Laser power (W)
Scan spacing (mm)
Fabrication time (h:m:s)
Subarea Scan Compound scan
15.03 15.00
75 75
14 14
0.10 0.10
02:18:22 03:06:56
Fig. 9. SLS test pieces for tensile strength.
Fig. 10. SLS test pieces with H ¼ 10 mm for flexural strength, SLS test pieces with H ¼ 5 mm for shock strength.
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distortion with the compound scan mode are less than with other scan modes.
Table 3 The results of tensile tests Scan mode
Subarea scan Compound scan
Tensile strengths (MPa)
Mean tensile strengths (MPa)
Test 1
Test 2
Test 3
3.057 1.823
2.862 1.450
2.667 1.928
2.862 1.734
5. Conclusions
Table 4 The results of bending tests Scan mode
Subarea scan Compound scan
Flexural strengths (MPa)
Mean flexural strengths (MPa)
Test 1
Test 2
Test 3
0.453963 0.316846
0.454774 0.396634
0.492338 0.442507
0.467025 0.385329
Table 5 The results of shock tests Scan mode
Subarea scan Compound scan
4.3.3.3. Surface quality. The sintered seam occurs only in the place where two scans meet. For the subarea scan mode, there are subareas where the melted seams occurs, and for the compound scan mode there are less subareas, and they are in the center of slice planes. Therefore, the surface finish of the SLS part with compound scan mode is higher.
Shock strengths (J/m2)
Mean shock strengths (J/m2)
Test 1
Test 2
Test 3
933.33 866.67
1000.00 933.33 1333.33 933.33
Test 4 1000.00 966.67 933.33 1016.67
The thesis presents the principle of a compound scan mode and its implementation. The compound scan path generation combines the algorithms of contour path generation and subarea path generation. The results of this compound scan mode for a standard test piece are given. It proves that the compound scan mode is available. Some experiment on the effectiveness of the compound scan mode and subarea scan mode on fabrication efficiency, strength and precision were made. The results restate that (1) For the same part the former (compound scan mode) has less fabrication efficiency than the latter (subarea scan mode) does; i.e., it needs more time to be fabricated with the former than the latter. (2) The part with the former has slightly less tensile strength, flexural strength, and shock strength, than
Table 6 Test results of the dimensional accuracy of the SLS standard test piece with the subarea scan mode Items
Theoretical values (mm)
Measured values (mm)
Length of part Width of part Inner diameter of central hole External diameter of central hole Inner diameter of external circle Inner length of square hole External length of square hole Width of external rib Width of inner rib
200 200 20 30 20 20 30 5 5
199.80 199.20 19.50 29.82 19.60 20.00 30.00 5.30 5.00
199.40 199.40 19.48 29.72 19.62 19.70 30.00 5.00 5.30
199.90 199.70 19.62 29.80 19.80 19.78 30.40 5.00 5.94
199.80 199.50 19.54 29.82 19.90 19.92 30.20 5.50 5.30
Average values (mm)
Dimensional errors (mm)
199.72 199.45 19.53 29.79 19.73 19.85 30.15 5.20 5.38
0.28 0.55 0.47 0.21 0.27 0.15 0.15 0.20 0.38
Average values (mm)
Dimensional errors (mm)
200.12 199.91 19.71 30.02 19.79 19.72 29.82 5.07 4.90
0.12 0.09 0.29 0.02 0.21 0.28 0.18 0.07 0.10
Table 7 Test results of the dimensional accuracy of the SLS standard test piece with the compound scan mode Items
Theoretical values (mm)
Measured values (mm)
Length of part Width of part Inner diameter of central hole External diameter of central hole Inner diameter of external circle Inner length of square hole External length of square hole Width of external rib Width of inner rib
200 200 20 30 20 20 30 5 5
200.00 200.20 19.60 30.00 19.80 19.60 29.82 5.00 4.80
200.10 200.00 19.80 30.00 19.80 19.80 29.68 5.00 4.98
200.20 199.80 19.60 30.00 19.80 19.60 29.80 5.10 4.82
200.20 199.64 19.84 30.10 19.76 19.88 30.00 5.20 5.00
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that of the latter; i.e., the former has no superiority on the strength of a SLS part. (3) The part with the former has more precise on the dimensional accuracy, shape accuracy and surface quality, than the latter. (4) The SLS part with high fabrication precision should be scanned with the compound scan mode, and the SLS part with high fabrication efficiency and lower fabrication precision should be scanned with the subarea scan mode. The presented compound scan mode has been utilized on SLS system built at HUST. It can also be applied to other RP systems. Though the algorithm in this paper can improve the quality of SLS parts to some extent, there are algorithms to be developed for SLS processing.
Acknowledgments This work was supported by the key project under the 863 program of China (2002AA6Z3083), the key R&D program of Hubei Province of China (2001A107B02), and the key technological project under the tenth five-year plan of Guangdong Province of China (A1040201). References [1] J. Wang, Z. Li, B. Lu, The study of scan path on rapid prototyping, Mechanical Science and Technology 16 (2) (1997) 303–305. [2] Y. Shi, Q. Zhong, X. Chen, Research and implementation of a new kind of scan mode for selective laser sintering, Chinese Journal of Mechanical Engineer 38 (2) (2002) 35–39. [3] D. Cai, Y. Shi, G. Chen, S. Huang, The study of the optimization of SLS rapid prototyping scan path, Metal Forming Machinery (2002) 18–20. [4] S.O. Onuh, K.K.B. Hon, Application of the taguchi method and new hatch styles for quality improvement in stereolithography. In: Proceedings of the Institution of Mechanical Engineers, vol. 212, 1998, pp. 461–471. [5] Z. Liu, H. Bin, X. Zhang, H. Gao, The influence of the fractal scan path on the temperature field of the layer in material increase manufacturing, Journal of Huazhong University of Science and Technology 26 (8) (1998) 32–34. [6] R. Zhang, Z. Shan, G. Su, Y. Yang, Study on laser scan process of selective laser sintering for powder material, Applied Laser 19 (5) (1999) 299–302.
883
[7] H. Chen, Z. Zhang, J. Cheng, Laser scan path stratagems on SLS rapid prototyping craft, Journal of Basic Science and Engineering 9 (2–3) (2001) 202–207. [8] Y. Yang, H.T. Loh, F.Y.H. Fuh, Y.G. Wang, Equidistant path generation for improving scan efficiency in layered manufacturing, Rapid Prototyping Journal 8 (1) (2002) 30–37. [9] D. Kim, Polygon offsetting using a Voronoi diagram and two stacks, Computer Aided Design 30 (14) (1998) 1069–1075. [10] J. Chen, P. Ma, J. Tian, Z. Liu, Scan path algorithm for rapid prototyping (RP) based on Voronoi diagrams, Mechanical Science and Technology 22 (5) (2003) 728–731. [11] R.T. Farouki, T. Sakkalis, Pythagorean hodographs, IBM Journal of Research and Development 34 (5) (1990) 736–752. [12] M. Bertoldi, C.M. Yardimci, Domain decomposition and surface filling curves in toolpath planning and generation. In: Solid Freeform Fabrication Symposium Proceedings, 1998. pp. 267–274. [13] T. Wasser, A. Dhair, Jayal, C. Pistor, Implementation and evaluation of novel buildstyles in fused deposition modeling (FDM). In: Solid Freeform Fabrication Symposium Proceedings, Austin, 1999, pp. 95–102. [14] W. Tiller, E.G. Hanson, Offsets of two-dimensional profiles, IEEE Computer Graphics and Application (1984) 36–46. [15] M. Ganesan, G. Fadel, Hollowing rapid prototyping parts using offsetting techniques, In: Proceedings of the Fifth International Conference on Rapid Prototyping, University of Dayton, 1994, pp. 241–251. [16] B. Pham, Offset curves and surfaces: a brief survey, Computer Aided Design 24 (4) (1992) 223–229. [17] M. Takashi, An overview of offset curves and surfaces, Computer Aided Design 31 (4) (1999) 165–173. [18] Y. Cheng. The Study and Development of Scan Path for Selective Laser Sintering [M], Huangzhong University of Science and Technology, Wuhan, April 2004. [19] L. Zhang, M. Han, S. Huang, The study of radius compensation algorithm on radius of laser spot in selective laser sintering, Journal of Huazhong University of Science and Technology 30 (6) (1998) 16–18. [20] D. XIA, X. Xiong, New algorithm and principle for intersection of line segments, Journal of Cad and Computer Graphics 8 (2) (1996) 95–100. [21] C. Cheng, T. Yan, Optimize algorithm of intersecting n lines, Engineering of Surveying and Mapping 10 (3) (2001) 29–31. [22] X. Li, M. Han, Y. Shi, S. Huang, Modal of shrinking and curl distortion for SLS prototypes, Chinese Journal of Mechanical Engineer 12 (8) (2001) 887–889. [23] J. Zhang, The Study of Material STP1 and Process for SLS Prototypes [M], North China University of Science and Technology, Taiyuan, 2002. [24] J. Bai, J. Zhao, Y. Tang, C. Yu, Study on the strength of SLS rapid prototyping test-pieces, Aviation Precision Manufacturing Technology 33 (5) (1997) 29–31.