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Surface reconstruction for mid-slice generation on variable lamination manufacturing
Journal of Materials Processing Technology 130±131 (2002) 384±389
Surface reconstruction for mid-slice generation on variable lamination manufacturing S.H. Lee, D.G. Ahn, D.Y. Yang* Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Science Town, Daejon 305-701, South Korea
Abstract A new rapid prototyping process, variable lamination manufacturing (VLM), has been developed to reduce building time and to improve the surface ®nish of parts by using a 4-axis-controlled hotwire cutter and expandable polystyrene foam sheet as a laminating material of the part (VLM-S). The objective of this study is to reconstruct the surface of the original 3D computer-aided design (CAD) model in order to generate mid-slice data using the advancing front technique (AFT). The generation of 3D layers by a 4-axis-controlled hotwire cutter requires a completely different procedure to generate toolpath data unlike the conventional RP CAD systems. The cutting path data for VLM-S are created by VLM-Slicer, which is a special CAD/CAM software with automatic generation of 3D toolpath. For the conventional sheet type system like LOM, the STL ®le would be sliced into 2D data only. However, due to the use of thick layers and a sloping edge with the ®rst-order approximation between the top and bottom layers, VLM-Slicer requires surface reconstruction, mid-slice, and toolpath data generation as well as 2D slicing. Surface reconstruction demands that the connection between the two neighboring cross-sectional contours use the triangular facets. VLM-S employs thick layers with ®nite thickness, so that surface reconstruction is necessary to obtain a sloping angle of a side surface and the point data at a half of the sheet thickness. In the process of the toolpath data generation the surface reconstruction algorithm is expected to minimize the error between the ruled surface and the original parts. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Rapid prototyping; VLM-Slicer; Cutting path; Surface reconstruction; Advancing front technique
1. Introduction In 1987, stereolithography (SL), a rapid prototyping (RP) process that solidi®es layers of ultraviolet (UV) light-sensitive liquid polymer using light from a laser, was ®rst developed by 3D systems in the USA [1]. Since then, various RP processes have been developed and commercialized [2]. In all RP processes, a computer-aided design (CAD) solid model is sliced into thin layers of uniform, but not necessarily constant, thickness in the building direction. Each crosssectional layer is successively deposited and, at the same time, bonded onto the previous layer; the stacked layers form a physical part of the model. RP techniques have unique disadvantages caused by their working principles: stairstepped surface of parts due to layer-by-layer stacking, low build speed caused by line-by-line solidi®cation to ®nish each layer, and post-processing to improve surface ®nish, etc. A new rapid prototyping process, variable lamination manufacturing (VLM), which uses a 4-axis-controlled * Corresponding author. Tel.: 82-42-869-3214; fax: 82-42-869-3210. E-mail address: [email protected] (D.Y. Yang).
hotwire cutter and expandable polystyrene foam sheet as a laminating material of the part [3,4], has been developed to reduce building time and to decrease the staircase effect of parts with the thick layer and the ruled surfaces. In the VLMS process, a material in the form of strip or sheet is cut into a thick layer with the sloping surface (unit shape part or unit shape layer) by a 4-axis-controlled hotwire cutter according to the cutting path data generated from a CAD model. Then each layer is sequentially glued and built up. Finally, a 3D part is fabricated rapidly. The VLM-S process requires a procedure to generate toolpath data for the purpose of making 3D layers by a 4axis-controlled hotwire cutter. The cutting path data for VLM-S are created by VLM-Slicer [5], which is a special CAD/CAM software with automatic generation of 3D toolpath. The cutting path data include data such as positional coordinates (X, Y) of each layer and rotation angles of the 4-axis linear hotwire cutter (yx, yy) of each position for all the layers. The objective of this study is to reconstruct the surface from 2D slice data in order to generate mid-slice data using the advancing front technique (AFT). Surface reconstruction
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 7 3 0 - 6
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
is expected to obtain the ®rst-order approximated surface data through the connection between the two neighboring cross-sectional contours using the triangular facets. VLM-S employs thick layers with ®nite thickness, so that surface reconstruction is necessary to obtain the point data and a sloping angle of a side surface at a half of the sheet thickness. In order to examine the applicability and the effectiveness of the algorithm in the procedure of generating the cutting path data, the surface reconstruction algorithm is applied to general 3D shapes and the error between the original CAD data (the STL ®le) and the surface reconstruction data is analyzed. 2. VLM-Slicer The generation of 3D layers by a 4-axis-controlled hotwire cutter requires a procedure to generate toolpath data that is completely different from that of conventional RP CAD systems. For the conventional sheet type system like LOM, the STL ®le would be sliced into 2D data only. However, owing to the use of thick layers and a sloping edge with the ®rst-order approximation between two layers, the cutting path data for VLM-S are created by VLM-Slicer [5], which is a special CAD/CAM software with automatic generation of the 3D toolpath. The procedure of generating the cutting path data for VLM-S consists of ®ve steps: STL ®le input, 2D slicing, surface reconstruction, mid-slice, and the cutting path data generation as shown in Fig. 1(a). Step 1: A 3D CAD model is converted into a triangular faceted file format, the STL file. The triangular facets of the STL file are inputted. Step 2: The facets of the STL file are sliced into layers with a constant thickness of more than 1 mm in the building direction in order to create 2D slice data. Step 3: The first-order approximated surface is reconstructed by connecting the two adjacent crosssectional contours using the triangular facets from 2D slice data. Step 4: Slicing the reconstructed surface at a half of the layer thickness creates the middle slice data so that the middle slice (mid-slice) exists on the same plane with the hotwire center of the cutter. The mid-slice data include the point data (X, Y) of each contour and the normal vector (Nx, Ny, Nz) of the triangular facet belonging to each point. Step 5: From the given mid-slice data, the rotation angle of the linear hotwire cutter is calculated knowing that a cross-product ~ T current , the unit tangential ~facet of the vector, of the unit normal vector N reconstructed facets and the unit direction vector ~cutting of the mid-slice edges in the hotwire U cutting direction is equal to the rotation transformations [R]y, and [R]x about the y-axis and the xaxis of the tangent vector ~ T initial at the initial
385
position of the cutter. Equations are given as Eqs. (1), (2a) and (2b), respectively ~facet U ~cutting ~ T initial N T current Rx Ry~ yy sin 1 Ny Uz Nz Uy Nx Uz Nz Ux yx sin 1 cos yy
(1) (2a) (2b)
~cutting Ux ; Uy ; ~facet Nx ; Ny ; Nz , and U where N Uz . As a result, the cutting path data are generated including positional coordinates (X, Y) of each layer and rotation angles of the 4-axis linear hotwire cutter (yx, yy) of each position for all the layers. The procedure of generating the cutting path data by VLM-Slicer is illustrated with the extruded cross shape in Fig. 1(b).
Fig. 1. The procedure of VLM-Slicer.
386
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
3. Surface reconstruction 3.1. Concept and characteristics Surface reconstruction [6,7] from a set of contours is very important in various ®elds such as biology, medical imaging and CAD. The surface reconstruction is de®ned as producing the ®rst-order approximated surface data through the connection between the two neighboring cross-sectional contours using the triangulated facets. This paper describes the surface reconstruction method to produce a triangulated facet between a pair of contours at adjacent levels using the AFT, which is one of the mesh generation methods for the ®nite element method. The surface reconstruction in the procedure of producing the cutting path data has the following characteristics: (i) The upper and lower contours at intervals of a layer thickness are linearly interpolated. (ii) The surface reconstruction of generating linear approximated surface data corresponds to the operating mechanism of the linear hotwire cutter to be able to produce the ruled surface as shown in Fig. 2. (iii) The rotation angle of the linear hotwire cutter can be calculated from the normal vector of a reconstructed triangular facet. (iv) The reconstructed surfaces form more regular surface patches than the surface facets of the STL file. (v) When the mid-slice is generated by slicing directly at a half of the layer thickness from a 3D CAD model without surface reconstruction, the boundary of fabricated part has a saw-toothed shape at the layer join. When the mid-slice is generated by slicing the reconstructed surface, the boundary of the fabricated part has continuity at the layer join.
generation procedure using the AFT is simply a repetition of the creation of a facet on the current mesh front using available entities and the advancement of the generation front from the creation of facets. The differences between speci®c algorithms arise in detail schemes such as the criteria in determining the advancing generation front, choosing the facet among available candidates. The AFT used in this work is limited to the contour triangulation for contours of the same number in each cross-section. As shown in Fig. 3, the procedure of the AFT in this study is as follows: (i) Select a starting point P(Xp, Yp, Zp) on the ith layer, and search the closest point Q(Xq, Yq, Zq) on the i 1th layer from this starting point. Then choose the line segment connecting two points as the base front. (ii) Choose the next points of the base front between the upper and lower contours as each check point C(Xc, Yc, Zc)I and C Xc ; Yc ; Zc i1 . Compute the distance from the middle point M(Xm, Ym, Zm) of the base front. At this time, the distance between the check point C(Xc, Yc, Zc) and the middle point M(Xm, Ym, Zm) is given in Eq. (3): q d Xc Xm 2 Yc Ym 2 Zc Zm 2 (3) (iii) Select the shorter check point of the two distances from the middle point of the base front to each check point as the third vertex of a triangular facet. (iv) Advance the front after the generation of facet. (v) Repeat the creation of facets until the complete surface is reconstructed. The application of the surface reconstruction algorithm for the various 3D CAD models is illustrated in Fig. 4.
3.2. Advancing front technique
3.3. Error analysis
VLM-Slicer makes use of the AFT [8,9] that is one of the automatic mesh generation methods for the ®nite element method with a view to reconstructing the triangulated surface from a series of contours. In general, the mesh
The error analysis model [10] is shown in Fig. 5(a). In this model, the dashed line represents the surface boundary of the STL ®le, while the solid line represents the boundary of the reconstructed surface. The cusp height error l between
Fig. 2. Surface reconstruction and linear hotwire cutting.
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
Fig. 3. Advancing front technique.
Fig. 4. Applications of surface reconstruction.
387
388
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
Fig. 5. Error analysis.
the STL ®le and the surface reconstruction in the middle of the layer is computed in Eq. (4): q l R R 12 t R 12 t (4) Fig. 5(b) illustrates the relationship between the cusp height error l and the radius of curvature r according to various layer thickness such as 2, 3.9, and 10 mm. In Fig. 5(b), as the radius of curvature r increases the error rapidly decreases. In case of a layer thickness of 3.9 mm, the maximum value of cusp height error is 0.52 mm when the radius of the surface curvature is the same as the layer thickness, and the error is rapidly decreased when the radius of the surface curvature is increased. This error l 0:52 mm is within the range of the admissible error considering that the error in the commercialized RP process is between 0.1 and 0.5 mm [11]. In case of a layer thickness of 2 mm, the maximum value of cusp height error is 0.27 mm when the
radius of the surface curvature is the same as the layer thickness. This error l 0:27 mm is also within the range of the admissible error considering that the error in the commercialized RP process is between 0.1 and 0.5 mm. Because one of the main characteristics in the VLM-S process is to reduce building time with thick layers, the layer thickness is selected as 3.9 mm. In case the value of r is relatively large, the reconstructed surface closely approximates to the STL ®le in spite of the layer thickness of 3.9 mm. 4. Conclusions The procedure of generating the cutting path data of the linear hotwire cutter by VLM-Slicer requires that the linear interpolated surface be reconstructed by forming a connection between the two adjacent cross-sectional contours using the triangular facets from 2D slice data. The surface
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
reconstruction not only corresponds to the operating mechanism of the linear hotwire cutter to produce the ruled surface, but it also conserves the continuity at the layer join. The purposes for which this study has been carried out are as follows: (i) The surface reconstruction algorithm has been developed to produce a triangulated facet between a pair of contours at adjacent levels from 2D slice data using the AFT that is one of the mesh generation methods for the finite element method. (ii) Through the application of several 3D CAD models, the validity of the proposed surface reconstruction algorithm is demonstrated. (iii) Through an error analysis between the STL file and the surface reconstruction it has been shown that the reconstructed surface closely approximates to the STL file in spite of the layer thickness of 3.9 mm.
Acknowledgements The authors would like to acknowledge the ®nancial assistance of the Ministry of Science and Technology.
389
References [1] P.F. Jacobs, Stereolithography and Other RP&M Technologies from Rapid Prototyping to Rapid Tooling, ASME Press, 1996. [2] T.T. Wohlers, Wohlers Report 2001, Rapid Prototyping and Tooling State of the Industry, Wohlers Associates Inc., 2001. [3] Variable Lamination Manufacturing (VLM) Process and Apparatus, US Patent 09/804,175 (2001). [4] D.G. Ahn, S.H. Lee, D.Y. Yang, B.S. Shin, S.K. Park, Y.I. Lee, Development of variable lamination manufacturing (VLM) process and apparatus by using expandable polystyrene foam, KSPE 18 (8) (2001) 54±63. [5] S.H. Lee, T.H. Kim, D.G. Ahn, D.Y. Yang, H.C. Chae, Software development for automatic generation of unit shape part for variable lamination manufacturing process, KSPE 18 (8) (2001) 64±70. [6] E. Keppel, Approximating complex surfaces by triangulation of contour lines, IBM J. Res. Dev. 19 (1) (1975) 2±11. [7] D. Meyers, S. Skinner, K. Sloan, Surfaces from contours, ACM Trans. Graphics 11 (3) (1992) 228±258. [8] C.K. Lee, R.E. Hobbs, Automatic adaptive finite element mesh generation over arbitrary two dimensional domain using advancing front technique, Comput. Struct. 71 (1999) 9±34. [9] P.L. George, Automatic Mesh Generation, Wiley, New York, 1991, pp. 137±158. [10] R.L. Hope, P.A. Jacobs, R.N. Roth, Rapid prototyping with sloping surfaces, Rapid Prototyping J. 3 (1) (1997) 12±19. [11] T. Muller, Fundamentals of rapid prototyping, RP&M Conference Tutorial Notes SME (Society of Manufacturing Engineers), Chicago, USA, 2000.
Surface reconstruction for mid-slice generation on variable lamination manufacturing S.H. Lee, D.G. Ahn, D.Y. Yang* Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Science Town, Daejon 305-701, South Korea
Abstract A new rapid prototyping process, variable lamination manufacturing (VLM), has been developed to reduce building time and to improve the surface ®nish of parts by using a 4-axis-controlled hotwire cutter and expandable polystyrene foam sheet as a laminating material of the part (VLM-S). The objective of this study is to reconstruct the surface of the original 3D computer-aided design (CAD) model in order to generate mid-slice data using the advancing front technique (AFT). The generation of 3D layers by a 4-axis-controlled hotwire cutter requires a completely different procedure to generate toolpath data unlike the conventional RP CAD systems. The cutting path data for VLM-S are created by VLM-Slicer, which is a special CAD/CAM software with automatic generation of 3D toolpath. For the conventional sheet type system like LOM, the STL ®le would be sliced into 2D data only. However, due to the use of thick layers and a sloping edge with the ®rst-order approximation between the top and bottom layers, VLM-Slicer requires surface reconstruction, mid-slice, and toolpath data generation as well as 2D slicing. Surface reconstruction demands that the connection between the two neighboring cross-sectional contours use the triangular facets. VLM-S employs thick layers with ®nite thickness, so that surface reconstruction is necessary to obtain a sloping angle of a side surface and the point data at a half of the sheet thickness. In the process of the toolpath data generation the surface reconstruction algorithm is expected to minimize the error between the ruled surface and the original parts. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Rapid prototyping; VLM-Slicer; Cutting path; Surface reconstruction; Advancing front technique
1. Introduction In 1987, stereolithography (SL), a rapid prototyping (RP) process that solidi®es layers of ultraviolet (UV) light-sensitive liquid polymer using light from a laser, was ®rst developed by 3D systems in the USA [1]. Since then, various RP processes have been developed and commercialized [2]. In all RP processes, a computer-aided design (CAD) solid model is sliced into thin layers of uniform, but not necessarily constant, thickness in the building direction. Each crosssectional layer is successively deposited and, at the same time, bonded onto the previous layer; the stacked layers form a physical part of the model. RP techniques have unique disadvantages caused by their working principles: stairstepped surface of parts due to layer-by-layer stacking, low build speed caused by line-by-line solidi®cation to ®nish each layer, and post-processing to improve surface ®nish, etc. A new rapid prototyping process, variable lamination manufacturing (VLM), which uses a 4-axis-controlled * Corresponding author. Tel.: 82-42-869-3214; fax: 82-42-869-3210. E-mail address: [email protected] (D.Y. Yang).
hotwire cutter and expandable polystyrene foam sheet as a laminating material of the part [3,4], has been developed to reduce building time and to decrease the staircase effect of parts with the thick layer and the ruled surfaces. In the VLMS process, a material in the form of strip or sheet is cut into a thick layer with the sloping surface (unit shape part or unit shape layer) by a 4-axis-controlled hotwire cutter according to the cutting path data generated from a CAD model. Then each layer is sequentially glued and built up. Finally, a 3D part is fabricated rapidly. The VLM-S process requires a procedure to generate toolpath data for the purpose of making 3D layers by a 4axis-controlled hotwire cutter. The cutting path data for VLM-S are created by VLM-Slicer [5], which is a special CAD/CAM software with automatic generation of 3D toolpath. The cutting path data include data such as positional coordinates (X, Y) of each layer and rotation angles of the 4-axis linear hotwire cutter (yx, yy) of each position for all the layers. The objective of this study is to reconstruct the surface from 2D slice data in order to generate mid-slice data using the advancing front technique (AFT). Surface reconstruction
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 7 3 0 - 6
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
is expected to obtain the ®rst-order approximated surface data through the connection between the two neighboring cross-sectional contours using the triangular facets. VLM-S employs thick layers with ®nite thickness, so that surface reconstruction is necessary to obtain the point data and a sloping angle of a side surface at a half of the sheet thickness. In order to examine the applicability and the effectiveness of the algorithm in the procedure of generating the cutting path data, the surface reconstruction algorithm is applied to general 3D shapes and the error between the original CAD data (the STL ®le) and the surface reconstruction data is analyzed. 2. VLM-Slicer The generation of 3D layers by a 4-axis-controlled hotwire cutter requires a procedure to generate toolpath data that is completely different from that of conventional RP CAD systems. For the conventional sheet type system like LOM, the STL ®le would be sliced into 2D data only. However, owing to the use of thick layers and a sloping edge with the ®rst-order approximation between two layers, the cutting path data for VLM-S are created by VLM-Slicer [5], which is a special CAD/CAM software with automatic generation of the 3D toolpath. The procedure of generating the cutting path data for VLM-S consists of ®ve steps: STL ®le input, 2D slicing, surface reconstruction, mid-slice, and the cutting path data generation as shown in Fig. 1(a). Step 1: A 3D CAD model is converted into a triangular faceted file format, the STL file. The triangular facets of the STL file are inputted. Step 2: The facets of the STL file are sliced into layers with a constant thickness of more than 1 mm in the building direction in order to create 2D slice data. Step 3: The first-order approximated surface is reconstructed by connecting the two adjacent crosssectional contours using the triangular facets from 2D slice data. Step 4: Slicing the reconstructed surface at a half of the layer thickness creates the middle slice data so that the middle slice (mid-slice) exists on the same plane with the hotwire center of the cutter. The mid-slice data include the point data (X, Y) of each contour and the normal vector (Nx, Ny, Nz) of the triangular facet belonging to each point. Step 5: From the given mid-slice data, the rotation angle of the linear hotwire cutter is calculated knowing that a cross-product ~ T current , the unit tangential ~facet of the vector, of the unit normal vector N reconstructed facets and the unit direction vector ~cutting of the mid-slice edges in the hotwire U cutting direction is equal to the rotation transformations [R]y, and [R]x about the y-axis and the xaxis of the tangent vector ~ T initial at the initial
385
position of the cutter. Equations are given as Eqs. (1), (2a) and (2b), respectively ~facet U ~cutting ~ T initial N T current Rx Ry~ yy sin 1 Ny Uz Nz Uy Nx Uz Nz Ux yx sin 1 cos yy
(1) (2a) (2b)
~cutting Ux ; Uy ; ~facet Nx ; Ny ; Nz , and U where N Uz . As a result, the cutting path data are generated including positional coordinates (X, Y) of each layer and rotation angles of the 4-axis linear hotwire cutter (yx, yy) of each position for all the layers. The procedure of generating the cutting path data by VLM-Slicer is illustrated with the extruded cross shape in Fig. 1(b).
Fig. 1. The procedure of VLM-Slicer.
386
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
3. Surface reconstruction 3.1. Concept and characteristics Surface reconstruction [6,7] from a set of contours is very important in various ®elds such as biology, medical imaging and CAD. The surface reconstruction is de®ned as producing the ®rst-order approximated surface data through the connection between the two neighboring cross-sectional contours using the triangulated facets. This paper describes the surface reconstruction method to produce a triangulated facet between a pair of contours at adjacent levels using the AFT, which is one of the mesh generation methods for the ®nite element method. The surface reconstruction in the procedure of producing the cutting path data has the following characteristics: (i) The upper and lower contours at intervals of a layer thickness are linearly interpolated. (ii) The surface reconstruction of generating linear approximated surface data corresponds to the operating mechanism of the linear hotwire cutter to be able to produce the ruled surface as shown in Fig. 2. (iii) The rotation angle of the linear hotwire cutter can be calculated from the normal vector of a reconstructed triangular facet. (iv) The reconstructed surfaces form more regular surface patches than the surface facets of the STL file. (v) When the mid-slice is generated by slicing directly at a half of the layer thickness from a 3D CAD model without surface reconstruction, the boundary of fabricated part has a saw-toothed shape at the layer join. When the mid-slice is generated by slicing the reconstructed surface, the boundary of the fabricated part has continuity at the layer join.
generation procedure using the AFT is simply a repetition of the creation of a facet on the current mesh front using available entities and the advancement of the generation front from the creation of facets. The differences between speci®c algorithms arise in detail schemes such as the criteria in determining the advancing generation front, choosing the facet among available candidates. The AFT used in this work is limited to the contour triangulation for contours of the same number in each cross-section. As shown in Fig. 3, the procedure of the AFT in this study is as follows: (i) Select a starting point P(Xp, Yp, Zp) on the ith layer, and search the closest point Q(Xq, Yq, Zq) on the i 1th layer from this starting point. Then choose the line segment connecting two points as the base front. (ii) Choose the next points of the base front between the upper and lower contours as each check point C(Xc, Yc, Zc)I and C Xc ; Yc ; Zc i1 . Compute the distance from the middle point M(Xm, Ym, Zm) of the base front. At this time, the distance between the check point C(Xc, Yc, Zc) and the middle point M(Xm, Ym, Zm) is given in Eq. (3): q d Xc Xm 2 Yc Ym 2 Zc Zm 2 (3) (iii) Select the shorter check point of the two distances from the middle point of the base front to each check point as the third vertex of a triangular facet. (iv) Advance the front after the generation of facet. (v) Repeat the creation of facets until the complete surface is reconstructed. The application of the surface reconstruction algorithm for the various 3D CAD models is illustrated in Fig. 4.
3.2. Advancing front technique
3.3. Error analysis
VLM-Slicer makes use of the AFT [8,9] that is one of the automatic mesh generation methods for the ®nite element method with a view to reconstructing the triangulated surface from a series of contours. In general, the mesh
The error analysis model [10] is shown in Fig. 5(a). In this model, the dashed line represents the surface boundary of the STL ®le, while the solid line represents the boundary of the reconstructed surface. The cusp height error l between
Fig. 2. Surface reconstruction and linear hotwire cutting.
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
Fig. 3. Advancing front technique.
Fig. 4. Applications of surface reconstruction.
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S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
Fig. 5. Error analysis.
the STL ®le and the surface reconstruction in the middle of the layer is computed in Eq. (4): q l R R 12 t R 12 t (4) Fig. 5(b) illustrates the relationship between the cusp height error l and the radius of curvature r according to various layer thickness such as 2, 3.9, and 10 mm. In Fig. 5(b), as the radius of curvature r increases the error rapidly decreases. In case of a layer thickness of 3.9 mm, the maximum value of cusp height error is 0.52 mm when the radius of the surface curvature is the same as the layer thickness, and the error is rapidly decreased when the radius of the surface curvature is increased. This error l 0:52 mm is within the range of the admissible error considering that the error in the commercialized RP process is between 0.1 and 0.5 mm [11]. In case of a layer thickness of 2 mm, the maximum value of cusp height error is 0.27 mm when the
radius of the surface curvature is the same as the layer thickness. This error l 0:27 mm is also within the range of the admissible error considering that the error in the commercialized RP process is between 0.1 and 0.5 mm. Because one of the main characteristics in the VLM-S process is to reduce building time with thick layers, the layer thickness is selected as 3.9 mm. In case the value of r is relatively large, the reconstructed surface closely approximates to the STL ®le in spite of the layer thickness of 3.9 mm. 4. Conclusions The procedure of generating the cutting path data of the linear hotwire cutter by VLM-Slicer requires that the linear interpolated surface be reconstructed by forming a connection between the two adjacent cross-sectional contours using the triangular facets from 2D slice data. The surface
S.H. Lee et al. / Journal of Materials Processing Technology 130±131 (2002) 384±389
reconstruction not only corresponds to the operating mechanism of the linear hotwire cutter to produce the ruled surface, but it also conserves the continuity at the layer join. The purposes for which this study has been carried out are as follows: (i) The surface reconstruction algorithm has been developed to produce a triangulated facet between a pair of contours at adjacent levels from 2D slice data using the AFT that is one of the mesh generation methods for the finite element method. (ii) Through the application of several 3D CAD models, the validity of the proposed surface reconstruction algorithm is demonstrated. (iii) Through an error analysis between the STL file and the surface reconstruction it has been shown that the reconstructed surface closely approximates to the STL file in spite of the layer thickness of 3.9 mm.
Acknowledgements The authors would like to acknowledge the ®nancial assistance of the Ministry of Science and Technology.
389
References [1] P.F. Jacobs, Stereolithography and Other RP&M Technologies from Rapid Prototyping to Rapid Tooling, ASME Press, 1996. [2] T.T. Wohlers, Wohlers Report 2001, Rapid Prototyping and Tooling State of the Industry, Wohlers Associates Inc., 2001. [3] Variable Lamination Manufacturing (VLM) Process and Apparatus, US Patent 09/804,175 (2001). [4] D.G. Ahn, S.H. Lee, D.Y. Yang, B.S. Shin, S.K. Park, Y.I. Lee, Development of variable lamination manufacturing (VLM) process and apparatus by using expandable polystyrene foam, KSPE 18 (8) (2001) 54±63. [5] S.H. Lee, T.H. Kim, D.G. Ahn, D.Y. Yang, H.C. Chae, Software development for automatic generation of unit shape part for variable lamination manufacturing process, KSPE 18 (8) (2001) 64±70. [6] E. Keppel, Approximating complex surfaces by triangulation of contour lines, IBM J. Res. Dev. 19 (1) (1975) 2±11. [7] D. Meyers, S. Skinner, K. Sloan, Surfaces from contours, ACM Trans. Graphics 11 (3) (1992) 228±258. [8] C.K. Lee, R.E. Hobbs, Automatic adaptive finite element mesh generation over arbitrary two dimensional domain using advancing front technique, Comput. Struct. 71 (1999) 9±34. [9] P.L. George, Automatic Mesh Generation, Wiley, New York, 1991, pp. 137±158. [10] R.L. Hope, P.A. Jacobs, R.N. Roth, Rapid prototyping with sloping surfaces, Rapid Prototyping J. 3 (1) (1997) 12±19. [11] T. Muller, Fundamentals of rapid prototyping, RP&M Conference Tutorial Notes SME (Society of Manufacturing Engineers), Chicago, USA, 2000.