- Email: [email protected]
Medial surface generation using chordal axis transformation in shell structures
Computers and Structures 84 (2006) 1673–1683 www.elsevier.com/locate/compstruc
Medial surface generation using chordal axis transformation in shell structures K.Y. Kwon
a,*
, B.C. Lee a, S.W. Chae
b
a
b
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusong-Dong, Yusong-Gu, Daejeon 305-701, Republic of Korea Department of Mechanical Engineering, Korea University, Anam-Dong, Seongbuk-Gu, Seoul 136-713, Republic of Korea Received 4 May 2005; accepted 12 April 2006 Available online 5 July 2006
Abstract This paper describes the generation of chordal surfaces for various shell structures, such as automobile bodies, plastic injection mold components, and sheet metal parts. After a single-layered tetrahedral mesh is generated by the advancing front method, the chordal surface is generated by cutting the tetrahedral mesh. One or more shell elements are generated at each tetrahedral element, and the chordal surface is constructed with triangular or quadrilateral shell elements. This algorithm has been tested on several models with rib structures. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: CAT (chordal axis transformation); MAT (medial axis transformation); Medial surface generation; Tetrahedral meshing; Shell structure; Surface classification
1. Introduction Because solid models are easier to handle than surface models when using CAD systems, they are generally utilized for modeling thin objects such as automobile bodies, plastic injection mold components, and sheet metal parts. Shell finite elements are preferred to 3D solid finite elements for computational efficiency and accuracy in the analysis of thin structures. Given tetrahedral and hexahedral elements, hexahedral elements are generally preferred in the finite element analysis. However, hexahedral mesh generation is usually more difficult than a tetrahedral mesh generation [1,2]. In order to model with plate or shell finite elements for thin objects, it is necessary to determine the medial surfaces from thin solid objects. Hexahedral elements can be created by offsetting shell elements [3] or by sweeping the divided
*
Corresponding author. Tel.: +82 42 869 3071; fax: +82 42 869 3210. E-mail address: [email protected] (K.Y. Kwon).
0045-7949/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.04.003
simple areas [4–7], if necessary. The medial axis transform (MAT), as proposed by Blum, is used to create a medial surface from a solid object [8]. By using MAT, the locus of the center of a circle (2D) or sphere (3D) is obtained as it rolls inside a solid object. MAT can be applied to the following: pattern recognition [9,10], motion planning [11], dimensional reduction [12,13], and mesh generation [3–7,14–16]. Similar to the medial surface generation by MAT, the approximation of a medial surface, called a chordal axis transform (CAT), has been proposed by Prasad [17] for the representation the skeleton of 3D objects. The approximated medial surface is constructed by cutting the midplanes of the tetrahedral elements. The generation of a medial surface by CAT is computationally less expensive than that by MAT. Quadros [3] proposed a hexahedral mesh generation method for thin objects by using CAT, in which the chordal surfaces are obtained by cutting the mid-plane of each tetrahedral element constructed for a thin shell structure. With respect to the generation chordal surfaces, previous works [3] presented results of ambiguous conditions for the cutting of mid-planes. In addition to these results,
1674
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
Fig. 1. Tri and quad elements of a chordal surface.
chordal axis
split1 split2
tri element
(a)
(b)
Fig. 2. 2D example of an incorrect chordal surface generation. (a) Type with two cutting planes; (b) impossible type with dangling edge.
walled solids have small wall thicknesses and relatively large surface areas, as illustrated in Fig. 3. The surfaces of these solids are classified into the top, bottom and lateral surfaces. Lateral surfaces have a relatively smaller area than the other surfaces. If a shape such as a sphere does not have lateral surfaces, the proposed algorithm cannot be applied. In addition, triangular elements are used as the initial input data, and interior nodes are not allowed to be located on the lateral surfaces, as illustrated in Fig. 4. The procedures utilized to create a medial surface by CAT are illustrated in the following paragraphs. Fig. 5(a) shows the initial triangular elements. Step 1. Sort the top, bottom and lateral surfaces (Fig. 5(b)). A medial surface is composed of the cutting planes of tetrahedral elements. The criterion to determine the cutting direction is based on the information of the top, bottom, and lateral surfaces. In this paper, the surfaces are classified into the top, bottom, and lateral surfaces by using the dihedral angle of triangular elements as well as the criterion of the non-existence of an interior node on a lateral surface. Step 2. Generate tetrahedral elements from the triangulated surface meshes (Fig. 5(c)). In order for the cutting planes of the tetrahedral elements to become the medial surfaces of a given 3D object,
they failed to cut the mid-planes of tetrahedral elements. Additionally, no more than one triangular or quadrilateral element is generally constructed for a tetrahedral element, as illustrated in Fig. 1; however, there are certain cases when chordal surfaces cannot be obtained by cutting the midplanes of tetrahedral elements especially at the corner of an object. Fig. 2(a) and (b) shows that improbable chordal edges can be generated at the corners of two-dimensional regions; this result implies that similar three-dimensional examples can be obtained. As shown in Fig. 2(a), two cutting edges are required, and the nodes must be relocated to correctly create a chordal axis. If triangular elements are constructed by using a conventional method at a corner as shown in Fig. 2(b), a dangling chordal edge is generated. In this paper, the cutting direction of tetrahedral elements for the generation of a medial surface is clearly classified. New types that can be applied to the case of various thin objects with a rib are also introduced. In addition, a tetrahedral meshing method is employed, which creates as few interior nodes as possible during the mesh generation. Finally, the concept of the top or bottom surface data in tetrahedral meshing is introduced and applied to prevent the occurrence of an abnormal case such as the one shown in Fig. 2(b).
Top surface
Hole
Bottom surface
Lateral surface Fig. 3. Thin sectioned object.
(a)
2. Overview The approach for the creation of shell elements on a thin section solid is based on the method of Quadros [3]. Thin
Mid surface
(b)
Fig. 4. Improbable second case applied with the proposed algorithm (an internal node does not exist on the lateral surface). (a) Appropriate mesh of triangular elements on a lateral surface; (b) inappropriate mesh of triangular elements on a lateral surface.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
1675
Fig. 5. Overview of medial surface generation. (a) Initial surface triangular elements; (b) surface classification; (c) tetra mesh generation; (d) medial surface generation (tri/quad mesh).
minimum number of new nodes should be created inside the 3D object. In addition, four nodes of a tetrahedral element should not belong to the same surface to avoid the abnormal case, as shown in Fig. 2(b). To generate tetrahedral elements, the advancing front method that uses trimming, wedging, splitting, rearranging and finishing operators is utilized [20,21]. Digging operators are not utilized so that interior nodes are not created in 3D objects. Step 3. Construct medial surfaces using the cutting planes of tetrahedral elements (chordal surface) (Fig. 5(d)). The cutting direction of a tetrahedron determines the shape of the sliced section, which is either triangular or quadrilateral. In order to determine the cutting directions, tetrahedral elements are classified into the basic, modified and inner nodes types discussed in Section 5. The numbers of faces and external edges of a tetrahedral element belonging to the top, bottom or lateral surfaces are used as the criteria for classification.
270o
3. Classification of the top, bottom, and lateral surfaces The surfaces are classified into top, bottom and lateral by using the dihedral angles of the initial triangular elements. Quadros [3] classified relatively narrow surfaces as lateral surfaces and classified other surfaces as either top or bottom surfaces after classifying the surfaces with the use of dihedral angles. In Fig. 6(a), A1, A2, and A3 are actually lateral surfaces; however, A4 was incorrectly classified as a lateral surface according to the criterion developed by Quadros. In this paper, the normal directions of the triangular elements are used for calculating dihedral angles between 0° and 360°. Because of this, the normal directions of the triangular elements should be directed outward. Surface A4 is at a 90° angle with the adjacent surfaces, as shown in Fig. 6(b). And this angle is less than the predefined threshold value, which in the present case is set as 225°. After the surfaces are classified in this way, the
180 o Same surface
270o
A2
90o Same surface
A4
270 o
A3 A1
Different surface
270
270o Different surface (a)
(b) Fig. 6. Surface classification. (a) By area; (b) by angle.
o
1676
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
Fig. 7. Example of a surface classification. (a) Tri elements; (b) top/bottom/lateral surfaces; (c) top/bottom surfaces.
lateral surface is considered one that does not have interior nodes on it. Fig. 7(b) shows an example of a surface classification by the proposed method. There are four top or
bottom surfaces and four lateral surfaces. It is not initially clear which surfaces are top or bottom surfaces. Eventually, top and bottom surfaces are defined by using the con-
Fig. 8. 3D operators (IE: current edge, LF: left face, RF: right face). (a) Trimming operator; (b) wedging operator; (c) rearranging operator; (d) splitting operator.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683 N4
1677
N4
N3
N3
Type 1
N1
N1
N2
N2
(a)
Type 2
Top/Bottom Surface
Edge on top/bottom surface (b)
Fig. 9. Cutting plane of a tetrahedral element. (a) Tri element; (b) quad element.
nectivity of adjacent surfaces, which is not considered for lateral surfaces. Fig. 7(c) shows one top surface and one bottom surface. 4. Tetrahedral meshing Tetrahedral meshing is by far the most common form of unstructured mesh generation, generally involving one of
Face on top/bottom surface Fig. 10. Basic types of cutting.
three main approaches: (1) advancing front method [18– 21], (2) Delaunay method [22,23] or (3) Octree based method [24]. In this paper, an advancing front method is utilized. In this method, tetrahedral elements are created from the outside surfaces toward the inside by using 3D operators cutting the sharp corner edges of a 3D triangulated object. Fig. 8 shows the 3D operators that were utilized for the tetrahedral meshing. Trimming and wedging
Fig. 11. Application of basic types. (a) Tetrahedron elements; (b) tri and quad elements with basic types; (c) uncut tetrahedrons.
1678
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
operators are used for generating one tetrahedral element. For enhancing the quality of the mesh, a rearranging operator is introduced in this work. This operator enables the operation of the trimming operators by rearranging the order of the node numberings in the elements, as illustrated in Fig. 8(c). The splitting operator is identical to the wedging operator, but the 3D object is divided into two objects [20,21], as shown in Fig. 8(d). Trimming, wedging and splitting operators are utilized for the tetrahedral mesh generation, and a rearranging operator is introduced to improve the mesh quality. In the beginning, nodes are checked to decide which method
(a)
(b)
(c)
(d)
Fig. 12. Modified types. (a) Modified Type 1; (b) modified Type 2; (c) modified Type 3; (d) modified Type 4.
Table 1 Criteria to classify the modified types NtopEdgea
NmidNodeb
NmidEdgec
NcutFaced
Applied types
1 1 1
4 5 5
2 4 6
2 3 4
TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE
1 1 1 2 2 2 2 2
5 4 4 4 4 4 3 3 a
5 3 2 4 2 3 0 2
3 2 2 3 2 2 0 2
1 1, 1, 2, 1, 1, 1, 3 3 3 3 3
to generate the tetrahedral elements. In order to avoid the abnormal case shown in Fig. 2(b), four nodes of a tetrahedral element cannot be located on the same surface. If further tetrahedral elements are not generated because of the nodes checks, then a mesh generation without a node check is performed. 5. Medial surface generation Medial surfaces are constructed by cutting the tetrahedral elements of a 3D object at their mid-planes. The surfaces are composed of triangular or quadrilateral elements (Fig. 9). When a medial triangular element is to be generated, a tetrahedral element can be divided into two along four different directions. Correspondingly, when a medial quadrilateral element is to be generated, a tetrahedral element can be divided into two along six different directions. Therefore, specific criteria are needed to determine the cutting directions for the top, bottom, and lateral surfaces. In previous work [3], the numbers of exterior faces and edges of a tetrahedral element were used as the criterion to classify tetrahedral elements. A tetrahedral element has four faces and six edges. For all incidences of a face, there is the set (0, 1, 2, 3 and 4); thus, there are five possible instances in the case of a face. For all instances of an edge, there is the set (0, 1, 2, 3, 4, 5 and 6); thus, there are seven possible instances in the case of an edge. Multiplying seven and five gives 35 cases in total, which were extremely complicated. In addition, if only one triangular or quadrilateral element is to be generated in a tetrahedral element, the medial surface occasionally cannot be precisely expressed. Fig. 2 shows a 2D example in which two cutting planes are generated. When two cutting planes are used, nodes need to be relocated to create a correct medial surface. We clarify the cutting direction by using the numbers of faces and edges of tetrahedral elements that belong to the top, bottom or lateral surfaces.
edge
TYPE 2 TYPE TYPE TYPE TYPE
mid-node
4 2 2 2
NtopEdge: Number of element edges belonging to the top/bottom surface. b NnodEdge: Number of nodes generated at the middle of an element edge. c NmidEdge: Number of element edges generated on an element face. d NsplFace: Number of element faces with a generated element edge.
tetra mid-edge face
Fig. 13. Data structure for modified types.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
1679
used to generate only a single triangular or quadrilateral element, while the modified types are used to generate two or three elements. Additionally, if some nodes of a tetrahedral element are located in the interior of the object, the medial surface can be generated using the inner-node types. The modified and inner-node types were proposed for the generation of medial surfaces for the various shapes presented in this paper. 5.1. Basic types
Fig. 14. Medial surface with modified types. (a) Before node movement; (b) after node movement.
current edge
NM
NN
N R : Reference node N N : Mid−node on edge
V NR
f1
N M : Modified node with vector V
adjacent tetra V
external edge
Fig. 15. Calculation of new node position.
Basic, modified and inner-node types are considered for the generation of the medial surface. The basic types are
If three edges and one face of a tetrahedral element are on a top or bottom surface, basic Type 1 is applied, and a triangular element is generated on the medial surface. If there are two edges on the top or bottom surface and these are not connected, basic Type 2 is applied, and one quadrilateral element is generated. Fig. 10 shows both types. BASIC TYPE 3: top/bottom edges + 1 external face ! tri element; BASIC TYPE 2: 2 top/bottom edges without a shared node ! quad element. If, however, the aforementioned two basic types are considered, then some tetrahedral elements will remain undefined, and the medial surface will have empty regions, as shown in Fig. 11(b). The tetrahedral elements in Fig. 11(c) are undefined in terms of basic types. One of these elements has one exterior edge and lacks an exterior face, which was not considered in the previous research [3] and cannot be categorized as either basic Type 1 or basic
Fig. 16. Thin model with rib. (a) Tetrahedron elements; (b) generation of a medial surface; (c) node relocation; (d) local smoothing.
1680
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
edge’’ and the ‘‘mid-node’’ are updated on the element face. If the basic types are not applicable, modified types are applied using the information of the element faces, and this information is also updated. When several types can be applied simultaneously, a suitable type is selected according to the structure in which that the ‘‘mid-edge’’ is linked. Fig. 14(a) shows the modified types applied to the medial surface generation for an object of Fig. 11. However, some nodes are not located exactly in the middle of the shape. Therefore, the nodes created by the modified types as in Fig. 14(b) must be relocated. To move the nodes into the middle of the shape, the node (NR) (created by the basic types), as well as the adjacent elements, is used. Fig. 15 shows how a node is moved into a new position, using the steps outlined below.
Type 2. To generate a medial surface for such a shape and for more complex thin shapes, four modified types are adopted in the present work. These are shown in Fig. 12. 5.2. Modified types Modified Types 1 and 3 create two triangular elements. Modified Type 2 creates one triangular element and one quadrilateral element. Modified Type 4, which uses the shape of a rib, creates three triangular elements. The information obtained from the basic types is used to establish the classification criteria of the modified types, and is given in Table 1. Fig. 13 shows the data structure that is employed to gather and update data in an efficient manner. As the cutting plane of a tetrahedral element is generated, the ‘‘mid-
Face 2
Face 1
E2 E1 Intersected line (a)
(b) Fig. 19. A channel model. (a) Initial surface triangular elements (2081 tetrahedral elements, no inside node); (b) a medial surface (Basic Type 1: 1298, Type 2 : 609, Modified Type 1: 72, Type 2: 72, Type 3: 64).
Fig. 17. Calculation of new node position (Rib model). (a) Find intersected line of two faces; (b) node relocation.
Top/Bottom Surface Edge on top/bottom surface
Inner node
(a)
(b)
(c)
Fig. 18. Inner-node types: some nodes of a tetrahedral element are located in the model. (a) Inner-node Type 1: one node located in the model; (b) innernode Type 2: two nodes located in the model; (c) Inner-node Type 3: three nodes located in the model.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
1. Find an adjacent element to the element applied with the modified type, and find the exterior edges of this adjacent element. Among these edges, the edge located at the corner is utilized. The dihedral angle of this edge should be nearly 90° or 270° if it is a corner edge. The deviation angle of this case is set to 30°.
1681
2. Calculate the direction vector V using two nodes of the edge identified in Step 1. 3. Find a tetrahedral element that includes the current edge. Face f1 is one of these tetrahedral element’s faces, and this face also includes the current edge. The new node is now located at the position (NM) that
Fig. 20. Thin object with a rib (Basic Type 1: 393, Basic Type 2: 631, Modified Type 1: 21, Modified Type 2: 64, Modified Type 3: 12). (a) Initial surface triangular elements (1546 triangular elements; 2036 tetrahedral elements are generated without generating an interior node); (b) medial surface (451 triangular and 1164 quadrilateral elements); (c) medial surface (after improving mesh quality: triangular elements).
Fig. 21. Panel model. (a) Initial surface triangular elements (13,820 triangular elements; 19,751 tetrahedral elements are generated without generating the interior node). (b) medial surface (13,166 triangular and 6580 quadrilateral elements); (c) rendered view of a medial surface.
1682
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
intersects the face f1 with the vector V originating at NR. The shape with a rib, such as the one shown in Fig. 16(a), requires a different method to move a node into the middle surface. The nodes of three elements connected on the edge should be moved in this case. After determining which element to move, Face 1 is constructed by using element E1, which belongs on the base part, as shown in Fig. 17. Additionally, Face 2 is constructed by using element E2, which belongs on the rib part. After calculating the intersecting line of these two faces, the new positions for these nodes are calculated by projecting them onto this line. Fig. 16(c) shows the medial surface after moving the nodes, and Fig. 16(d) shows the final medial surface after local smoothing, which was performed to improve the shapes of the elements around the rib. 5.3. Inner node types Several nodes can sometimes be created inside the domain in the tetrahedral mesh generation. When this case occurs, a conventional method cannot be applied for the medial surface generation. A new method to generate a medial surface is proposed in this case, and this is shown in Fig. 18. Fig. 18(a) shows that one node of a tetrahedral element is located inside the domain. If one edge lies on the top or bottom surface, one triangular element is generated; otherwise, the tetrahedral element is not cut. Fig. 18(b) shows that two nodes of a tetrahedral element are located inside the domain. If no edge lies on the top or bottom surface, one triangular element is generated; otherwise the tetrahedral element is not cut. Fig. 18(c) shows that three nodes of a tetrahedral element are located inside the domain. One triangular element is generated by using these three nodes. 6. Example problems Four example problems demonstrate the various features of the proposed algorithm. Figs. 19–21 show the medial surface generation process using the input surface triangular meshes, which include the tetrahedral mesh generation. Fig. 22 shows the medial surface generation process using the input tetrahedral meshes. Fig. 19 shows the medial surface generation process for a channel section model. Using the input triangular mesh, tetrahedral elements are generated as shown in Fig. 19(a). During the medial surface generation process, modified Type 1 is applied at 72 tetrahedral elements, modified Type 2 at 72 tetrahedral elements and modified Type 3 at 64 tetrahedral elements (as shown in Fig. 19(b)). Fig. 20 shows the medial surface of a thin-walled shape with a rib, in which 1546 triangular elements are utilized for the tetrahedral meshing and medial surface generation, resulting in 451 triangular and 1164 quadrilateral elements. Fig. 20(c) shows the creation of triangular elements after improving the element
Fig. 22. Phone case model (Inner Type 1: 1187, Inner Type 2: 45, Inner Type 3: 4, Basic Type 1: 1790, Basic Type 2: 4057, Modified Type 1: 1, Modified Type 3: 16). (a) Initial tetrahedral mesh (15,806 tetrahedral elements, 493 inside nodes); (b) shell elements (3153 triangular, 7942 quadrilateral elements).
quality. Fig. 21 shows the medial surface of a panel and Fig. 21(c) shows a rendered view of its medial surface. Fig. 22 shows a FE mesh model of a phone case with 493 nodes inside the domain. The proposed inner-node types are applied to create the medial surface. Fig. 22(b) shows the results of the quadrilateral elements after improving the element quality. 7. Conclusion A modified method was proposed for the generation of a medial surface for various shell structures. After generating the tetrahedral elements with the initial triangular elements, medial surfaces were generated by using the mid-cutting planes of tetrahedral elements. To prevent the creation of interior nodes, the advancing front method was used for tetrahedral meshing. This method also used the top and bottom surface data to suppress abnormal
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
cases. In this paper, the cutting directions of tetrahedral elements were classified clearly and applied for the stable creation of the medial surfaces. New types to generate triangular or quadrilateral elements on the medial surface were also introduced so that this method can be applied to various shapes of 3D objects. Acknowledgement The support of the ITEP (Korea Institute of Industrial Technology Evaluation and Planning) is gratefully acknowledged. References [1] Cifuentes AO, Kalbag A. A performance study of tetrahedral and hexahedral elements in 3-D Finite Element Structural Analysis. Finite Elem Anal Des 1992;12:313–8. [2] Benzley SE, Perry E, Karl M, Brett C. A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis. In: Proceedings of the fourth international meshing roundtable. Albuquerque, NM; October, 1995. p. 179–91. [3] Quadros WR, Shimada K. Hex-layer: layered all-hex mesh generation on thin section solids via chordal surface transformation. In: Proceedings of the 11th international meshing roundtable. Ithaca, NY; September, 2001. p. 169–82. [4] Price MA, Armstrong CG. Hexahedral mesh generation by medial surface subdivision: part II solids with flat and concave edges. Int J Numer Methods Eng 1997;40:111–36. [5] Sheffer A, Michel B. Hexahedral meshing of non-linear volumes using Voronoi faces and edges. Int J Numer Methods Eng 2000;49:329–51. [6] Sampl P. Semi-structured mesh generation based on medial axis. In: Proceedings of the ninth international meshing roundtable. New Orleans, CA; October, 2000. p. 21–32. [7] Li TS, McKeag RM, Armstring CG. Hexahedral meshing using midpoint subdivision and integer programming. Comput Methods Appl Mech Eng 1995;124:171–93. [8] Blum H. A transformation for extracting new descriptors of shape. In: Wathen-Dunn Weinant, editor. Models for the perception of speech and visual form. Cambridge, MA: MIT Press; 1967.
1683
[9] Ma CM, Wan SY. A medial-surface oriented 3-d two-subfield thinning algorithm. Pattern Recogn Lett 2001;22:1439–46. [10] Palagyi K. A 3D 6-subiteration thinning algorithm for extracting medial surfaces. Pattern Recogn Lett 2002;23:663–75. [11] Held M, Lukacs G, Andor L. Pocket machining based on contourparallel tool path generated by means of proximity maps. Comput Aided Des 1994;26(3):189–203. [12] Donaghy RJ, McCune W, Robinson DJ, McCkeag RM. Dimensional reduction of analysis models. In: Proceedings of the fifth international meshing roundtable. Pittsburgh, PA; October, 1996. p. 307–20. [13] Suresh K. Generalization of the Kantorovich method of dimensional reduction. In: Proceedings of the 12th international meshing roundtable. Santa Fe, NM; September, 2002. p. 261–76. [14] Ang PY, Armstrong CG. Adaptive curvature-sensitive meshing of the medial axis. In: Proceedings of the 10th international meshing roundtable. Newport Beach, CA; October, 2001. p. 155–65. [15] Quadros WR, Ramaswami K, Prinz FB, Gurumoorthy B. LayTracks: a new approach to automated quadrilateral mesh generation using MAT. In: Proceedings of the ninth international meshing roundtable. New Orleans, CA; October, 2000. p. 239–50. [16] Yamakawa S, Shimada K. Quad-layer: layered quadrilateral meshing of narrow two-dimensional domain by bubble packing and chordal axis transformation. J Mech Des 2002;124(3):564–73. [17] Prasad L. Morphological analysis of shapes. In http://cnls.lanl.gov/ Highlights/1997-07/, Los Alamos National Laboratory; 1997. [18] Moller P. On advancing front mesh generation in three dimensions. Int J Numer Methods Eng 1995;38:3551–69. [19] Chan CT, Anastasiou K. An automatic tetrahedral mesh generation scheme by advancing front method. Commun Numer Meth Eng 1997;13:33–46. [20] Chae SW, Bathe KJ. On automatic mesh construction and mesh refinement in finite element analysis. Comput Struct 1989;32(3):911– 36. [21] Kwon KY, Chae SW, Lee BC. Automatic tetrahedral mesh generation using 3-D operators. Soc CAD/CAM Eng 2004;9(1):27–34. [22] Borouchaki H, George PL, Lo SH. Optimal Delaunay point insertion. Int J Numer Methods Eng 1996;39:3407–37. [23] Weatherill NP, Hassan O. Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary conditions. Int J Numer Methods Eng 1994;37:2005–39. [24] Shephard MS, George MK. Automatic three-dimensional mesh generation by the finite Octree technique. Int J Numer Methods Eng 1991;32:709–49.
Medial surface generation using chordal axis transformation in shell structures K.Y. Kwon
a,*
, B.C. Lee a, S.W. Chae
b
a
b
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusong-Dong, Yusong-Gu, Daejeon 305-701, Republic of Korea Department of Mechanical Engineering, Korea University, Anam-Dong, Seongbuk-Gu, Seoul 136-713, Republic of Korea Received 4 May 2005; accepted 12 April 2006 Available online 5 July 2006
Abstract This paper describes the generation of chordal surfaces for various shell structures, such as automobile bodies, plastic injection mold components, and sheet metal parts. After a single-layered tetrahedral mesh is generated by the advancing front method, the chordal surface is generated by cutting the tetrahedral mesh. One or more shell elements are generated at each tetrahedral element, and the chordal surface is constructed with triangular or quadrilateral shell elements. This algorithm has been tested on several models with rib structures. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: CAT (chordal axis transformation); MAT (medial axis transformation); Medial surface generation; Tetrahedral meshing; Shell structure; Surface classification
1. Introduction Because solid models are easier to handle than surface models when using CAD systems, they are generally utilized for modeling thin objects such as automobile bodies, plastic injection mold components, and sheet metal parts. Shell finite elements are preferred to 3D solid finite elements for computational efficiency and accuracy in the analysis of thin structures. Given tetrahedral and hexahedral elements, hexahedral elements are generally preferred in the finite element analysis. However, hexahedral mesh generation is usually more difficult than a tetrahedral mesh generation [1,2]. In order to model with plate or shell finite elements for thin objects, it is necessary to determine the medial surfaces from thin solid objects. Hexahedral elements can be created by offsetting shell elements [3] or by sweeping the divided
*
Corresponding author. Tel.: +82 42 869 3071; fax: +82 42 869 3210. E-mail address: [email protected] (K.Y. Kwon).
0045-7949/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.04.003
simple areas [4–7], if necessary. The medial axis transform (MAT), as proposed by Blum, is used to create a medial surface from a solid object [8]. By using MAT, the locus of the center of a circle (2D) or sphere (3D) is obtained as it rolls inside a solid object. MAT can be applied to the following: pattern recognition [9,10], motion planning [11], dimensional reduction [12,13], and mesh generation [3–7,14–16]. Similar to the medial surface generation by MAT, the approximation of a medial surface, called a chordal axis transform (CAT), has been proposed by Prasad [17] for the representation the skeleton of 3D objects. The approximated medial surface is constructed by cutting the midplanes of the tetrahedral elements. The generation of a medial surface by CAT is computationally less expensive than that by MAT. Quadros [3] proposed a hexahedral mesh generation method for thin objects by using CAT, in which the chordal surfaces are obtained by cutting the mid-plane of each tetrahedral element constructed for a thin shell structure. With respect to the generation chordal surfaces, previous works [3] presented results of ambiguous conditions for the cutting of mid-planes. In addition to these results,
1674
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
Fig. 1. Tri and quad elements of a chordal surface.
chordal axis
split1 split2
tri element
(a)
(b)
Fig. 2. 2D example of an incorrect chordal surface generation. (a) Type with two cutting planes; (b) impossible type with dangling edge.
walled solids have small wall thicknesses and relatively large surface areas, as illustrated in Fig. 3. The surfaces of these solids are classified into the top, bottom and lateral surfaces. Lateral surfaces have a relatively smaller area than the other surfaces. If a shape such as a sphere does not have lateral surfaces, the proposed algorithm cannot be applied. In addition, triangular elements are used as the initial input data, and interior nodes are not allowed to be located on the lateral surfaces, as illustrated in Fig. 4. The procedures utilized to create a medial surface by CAT are illustrated in the following paragraphs. Fig. 5(a) shows the initial triangular elements. Step 1. Sort the top, bottom and lateral surfaces (Fig. 5(b)). A medial surface is composed of the cutting planes of tetrahedral elements. The criterion to determine the cutting direction is based on the information of the top, bottom, and lateral surfaces. In this paper, the surfaces are classified into the top, bottom, and lateral surfaces by using the dihedral angle of triangular elements as well as the criterion of the non-existence of an interior node on a lateral surface. Step 2. Generate tetrahedral elements from the triangulated surface meshes (Fig. 5(c)). In order for the cutting planes of the tetrahedral elements to become the medial surfaces of a given 3D object,
they failed to cut the mid-planes of tetrahedral elements. Additionally, no more than one triangular or quadrilateral element is generally constructed for a tetrahedral element, as illustrated in Fig. 1; however, there are certain cases when chordal surfaces cannot be obtained by cutting the midplanes of tetrahedral elements especially at the corner of an object. Fig. 2(a) and (b) shows that improbable chordal edges can be generated at the corners of two-dimensional regions; this result implies that similar three-dimensional examples can be obtained. As shown in Fig. 2(a), two cutting edges are required, and the nodes must be relocated to correctly create a chordal axis. If triangular elements are constructed by using a conventional method at a corner as shown in Fig. 2(b), a dangling chordal edge is generated. In this paper, the cutting direction of tetrahedral elements for the generation of a medial surface is clearly classified. New types that can be applied to the case of various thin objects with a rib are also introduced. In addition, a tetrahedral meshing method is employed, which creates as few interior nodes as possible during the mesh generation. Finally, the concept of the top or bottom surface data in tetrahedral meshing is introduced and applied to prevent the occurrence of an abnormal case such as the one shown in Fig. 2(b).
Top surface
Hole
Bottom surface
Lateral surface Fig. 3. Thin sectioned object.
(a)
2. Overview The approach for the creation of shell elements on a thin section solid is based on the method of Quadros [3]. Thin
Mid surface
(b)
Fig. 4. Improbable second case applied with the proposed algorithm (an internal node does not exist on the lateral surface). (a) Appropriate mesh of triangular elements on a lateral surface; (b) inappropriate mesh of triangular elements on a lateral surface.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
1675
Fig. 5. Overview of medial surface generation. (a) Initial surface triangular elements; (b) surface classification; (c) tetra mesh generation; (d) medial surface generation (tri/quad mesh).
minimum number of new nodes should be created inside the 3D object. In addition, four nodes of a tetrahedral element should not belong to the same surface to avoid the abnormal case, as shown in Fig. 2(b). To generate tetrahedral elements, the advancing front method that uses trimming, wedging, splitting, rearranging and finishing operators is utilized [20,21]. Digging operators are not utilized so that interior nodes are not created in 3D objects. Step 3. Construct medial surfaces using the cutting planes of tetrahedral elements (chordal surface) (Fig. 5(d)). The cutting direction of a tetrahedron determines the shape of the sliced section, which is either triangular or quadrilateral. In order to determine the cutting directions, tetrahedral elements are classified into the basic, modified and inner nodes types discussed in Section 5. The numbers of faces and external edges of a tetrahedral element belonging to the top, bottom or lateral surfaces are used as the criteria for classification.
270o
3. Classification of the top, bottom, and lateral surfaces The surfaces are classified into top, bottom and lateral by using the dihedral angles of the initial triangular elements. Quadros [3] classified relatively narrow surfaces as lateral surfaces and classified other surfaces as either top or bottom surfaces after classifying the surfaces with the use of dihedral angles. In Fig. 6(a), A1, A2, and A3 are actually lateral surfaces; however, A4 was incorrectly classified as a lateral surface according to the criterion developed by Quadros. In this paper, the normal directions of the triangular elements are used for calculating dihedral angles between 0° and 360°. Because of this, the normal directions of the triangular elements should be directed outward. Surface A4 is at a 90° angle with the adjacent surfaces, as shown in Fig. 6(b). And this angle is less than the predefined threshold value, which in the present case is set as 225°. After the surfaces are classified in this way, the
180 o Same surface
270o
A2
90o Same surface
A4
270 o
A3 A1
Different surface
270
270o Different surface (a)
(b) Fig. 6. Surface classification. (a) By area; (b) by angle.
o
1676
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
Fig. 7. Example of a surface classification. (a) Tri elements; (b) top/bottom/lateral surfaces; (c) top/bottom surfaces.
lateral surface is considered one that does not have interior nodes on it. Fig. 7(b) shows an example of a surface classification by the proposed method. There are four top or
bottom surfaces and four lateral surfaces. It is not initially clear which surfaces are top or bottom surfaces. Eventually, top and bottom surfaces are defined by using the con-
Fig. 8. 3D operators (IE: current edge, LF: left face, RF: right face). (a) Trimming operator; (b) wedging operator; (c) rearranging operator; (d) splitting operator.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683 N4
1677
N4
N3
N3
Type 1
N1
N1
N2
N2
(a)
Type 2
Top/Bottom Surface
Edge on top/bottom surface (b)
Fig. 9. Cutting plane of a tetrahedral element. (a) Tri element; (b) quad element.
nectivity of adjacent surfaces, which is not considered for lateral surfaces. Fig. 7(c) shows one top surface and one bottom surface. 4. Tetrahedral meshing Tetrahedral meshing is by far the most common form of unstructured mesh generation, generally involving one of
Face on top/bottom surface Fig. 10. Basic types of cutting.
three main approaches: (1) advancing front method [18– 21], (2) Delaunay method [22,23] or (3) Octree based method [24]. In this paper, an advancing front method is utilized. In this method, tetrahedral elements are created from the outside surfaces toward the inside by using 3D operators cutting the sharp corner edges of a 3D triangulated object. Fig. 8 shows the 3D operators that were utilized for the tetrahedral meshing. Trimming and wedging
Fig. 11. Application of basic types. (a) Tetrahedron elements; (b) tri and quad elements with basic types; (c) uncut tetrahedrons.
1678
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
operators are used for generating one tetrahedral element. For enhancing the quality of the mesh, a rearranging operator is introduced in this work. This operator enables the operation of the trimming operators by rearranging the order of the node numberings in the elements, as illustrated in Fig. 8(c). The splitting operator is identical to the wedging operator, but the 3D object is divided into two objects [20,21], as shown in Fig. 8(d). Trimming, wedging and splitting operators are utilized for the tetrahedral mesh generation, and a rearranging operator is introduced to improve the mesh quality. In the beginning, nodes are checked to decide which method
(a)
(b)
(c)
(d)
Fig. 12. Modified types. (a) Modified Type 1; (b) modified Type 2; (c) modified Type 3; (d) modified Type 4.
Table 1 Criteria to classify the modified types NtopEdgea
NmidNodeb
NmidEdgec
NcutFaced
Applied types
1 1 1
4 5 5
2 4 6
2 3 4
TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE
1 1 1 2 2 2 2 2
5 4 4 4 4 4 3 3 a
5 3 2 4 2 3 0 2
3 2 2 3 2 2 0 2
1 1, 1, 2, 1, 1, 1, 3 3 3 3 3
to generate the tetrahedral elements. In order to avoid the abnormal case shown in Fig. 2(b), four nodes of a tetrahedral element cannot be located on the same surface. If further tetrahedral elements are not generated because of the nodes checks, then a mesh generation without a node check is performed. 5. Medial surface generation Medial surfaces are constructed by cutting the tetrahedral elements of a 3D object at their mid-planes. The surfaces are composed of triangular or quadrilateral elements (Fig. 9). When a medial triangular element is to be generated, a tetrahedral element can be divided into two along four different directions. Correspondingly, when a medial quadrilateral element is to be generated, a tetrahedral element can be divided into two along six different directions. Therefore, specific criteria are needed to determine the cutting directions for the top, bottom, and lateral surfaces. In previous work [3], the numbers of exterior faces and edges of a tetrahedral element were used as the criterion to classify tetrahedral elements. A tetrahedral element has four faces and six edges. For all incidences of a face, there is the set (0, 1, 2, 3 and 4); thus, there are five possible instances in the case of a face. For all instances of an edge, there is the set (0, 1, 2, 3, 4, 5 and 6); thus, there are seven possible instances in the case of an edge. Multiplying seven and five gives 35 cases in total, which were extremely complicated. In addition, if only one triangular or quadrilateral element is to be generated in a tetrahedral element, the medial surface occasionally cannot be precisely expressed. Fig. 2 shows a 2D example in which two cutting planes are generated. When two cutting planes are used, nodes need to be relocated to create a correct medial surface. We clarify the cutting direction by using the numbers of faces and edges of tetrahedral elements that belong to the top, bottom or lateral surfaces.
edge
TYPE 2 TYPE TYPE TYPE TYPE
mid-node
4 2 2 2
NtopEdge: Number of element edges belonging to the top/bottom surface. b NnodEdge: Number of nodes generated at the middle of an element edge. c NmidEdge: Number of element edges generated on an element face. d NsplFace: Number of element faces with a generated element edge.
tetra mid-edge face
Fig. 13. Data structure for modified types.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
1679
used to generate only a single triangular or quadrilateral element, while the modified types are used to generate two or three elements. Additionally, if some nodes of a tetrahedral element are located in the interior of the object, the medial surface can be generated using the inner-node types. The modified and inner-node types were proposed for the generation of medial surfaces for the various shapes presented in this paper. 5.1. Basic types
Fig. 14. Medial surface with modified types. (a) Before node movement; (b) after node movement.
current edge
NM
NN
N R : Reference node N N : Mid−node on edge
V NR
f1
N M : Modified node with vector V
adjacent tetra V
external edge
Fig. 15. Calculation of new node position.
Basic, modified and inner-node types are considered for the generation of the medial surface. The basic types are
If three edges and one face of a tetrahedral element are on a top or bottom surface, basic Type 1 is applied, and a triangular element is generated on the medial surface. If there are two edges on the top or bottom surface and these are not connected, basic Type 2 is applied, and one quadrilateral element is generated. Fig. 10 shows both types. BASIC TYPE 3: top/bottom edges + 1 external face ! tri element; BASIC TYPE 2: 2 top/bottom edges without a shared node ! quad element. If, however, the aforementioned two basic types are considered, then some tetrahedral elements will remain undefined, and the medial surface will have empty regions, as shown in Fig. 11(b). The tetrahedral elements in Fig. 11(c) are undefined in terms of basic types. One of these elements has one exterior edge and lacks an exterior face, which was not considered in the previous research [3] and cannot be categorized as either basic Type 1 or basic
Fig. 16. Thin model with rib. (a) Tetrahedron elements; (b) generation of a medial surface; (c) node relocation; (d) local smoothing.
1680
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
edge’’ and the ‘‘mid-node’’ are updated on the element face. If the basic types are not applicable, modified types are applied using the information of the element faces, and this information is also updated. When several types can be applied simultaneously, a suitable type is selected according to the structure in which that the ‘‘mid-edge’’ is linked. Fig. 14(a) shows the modified types applied to the medial surface generation for an object of Fig. 11. However, some nodes are not located exactly in the middle of the shape. Therefore, the nodes created by the modified types as in Fig. 14(b) must be relocated. To move the nodes into the middle of the shape, the node (NR) (created by the basic types), as well as the adjacent elements, is used. Fig. 15 shows how a node is moved into a new position, using the steps outlined below.
Type 2. To generate a medial surface for such a shape and for more complex thin shapes, four modified types are adopted in the present work. These are shown in Fig. 12. 5.2. Modified types Modified Types 1 and 3 create two triangular elements. Modified Type 2 creates one triangular element and one quadrilateral element. Modified Type 4, which uses the shape of a rib, creates three triangular elements. The information obtained from the basic types is used to establish the classification criteria of the modified types, and is given in Table 1. Fig. 13 shows the data structure that is employed to gather and update data in an efficient manner. As the cutting plane of a tetrahedral element is generated, the ‘‘mid-
Face 2
Face 1
E2 E1 Intersected line (a)
(b) Fig. 19. A channel model. (a) Initial surface triangular elements (2081 tetrahedral elements, no inside node); (b) a medial surface (Basic Type 1: 1298, Type 2 : 609, Modified Type 1: 72, Type 2: 72, Type 3: 64).
Fig. 17. Calculation of new node position (Rib model). (a) Find intersected line of two faces; (b) node relocation.
Top/Bottom Surface Edge on top/bottom surface
Inner node
(a)
(b)
(c)
Fig. 18. Inner-node types: some nodes of a tetrahedral element are located in the model. (a) Inner-node Type 1: one node located in the model; (b) innernode Type 2: two nodes located in the model; (c) Inner-node Type 3: three nodes located in the model.
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
1. Find an adjacent element to the element applied with the modified type, and find the exterior edges of this adjacent element. Among these edges, the edge located at the corner is utilized. The dihedral angle of this edge should be nearly 90° or 270° if it is a corner edge. The deviation angle of this case is set to 30°.
1681
2. Calculate the direction vector V using two nodes of the edge identified in Step 1. 3. Find a tetrahedral element that includes the current edge. Face f1 is one of these tetrahedral element’s faces, and this face also includes the current edge. The new node is now located at the position (NM) that
Fig. 20. Thin object with a rib (Basic Type 1: 393, Basic Type 2: 631, Modified Type 1: 21, Modified Type 2: 64, Modified Type 3: 12). (a) Initial surface triangular elements (1546 triangular elements; 2036 tetrahedral elements are generated without generating an interior node); (b) medial surface (451 triangular and 1164 quadrilateral elements); (c) medial surface (after improving mesh quality: triangular elements).
Fig. 21. Panel model. (a) Initial surface triangular elements (13,820 triangular elements; 19,751 tetrahedral elements are generated without generating the interior node). (b) medial surface (13,166 triangular and 6580 quadrilateral elements); (c) rendered view of a medial surface.
1682
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
intersects the face f1 with the vector V originating at NR. The shape with a rib, such as the one shown in Fig. 16(a), requires a different method to move a node into the middle surface. The nodes of three elements connected on the edge should be moved in this case. After determining which element to move, Face 1 is constructed by using element E1, which belongs on the base part, as shown in Fig. 17. Additionally, Face 2 is constructed by using element E2, which belongs on the rib part. After calculating the intersecting line of these two faces, the new positions for these nodes are calculated by projecting them onto this line. Fig. 16(c) shows the medial surface after moving the nodes, and Fig. 16(d) shows the final medial surface after local smoothing, which was performed to improve the shapes of the elements around the rib. 5.3. Inner node types Several nodes can sometimes be created inside the domain in the tetrahedral mesh generation. When this case occurs, a conventional method cannot be applied for the medial surface generation. A new method to generate a medial surface is proposed in this case, and this is shown in Fig. 18. Fig. 18(a) shows that one node of a tetrahedral element is located inside the domain. If one edge lies on the top or bottom surface, one triangular element is generated; otherwise, the tetrahedral element is not cut. Fig. 18(b) shows that two nodes of a tetrahedral element are located inside the domain. If no edge lies on the top or bottom surface, one triangular element is generated; otherwise the tetrahedral element is not cut. Fig. 18(c) shows that three nodes of a tetrahedral element are located inside the domain. One triangular element is generated by using these three nodes. 6. Example problems Four example problems demonstrate the various features of the proposed algorithm. Figs. 19–21 show the medial surface generation process using the input surface triangular meshes, which include the tetrahedral mesh generation. Fig. 22 shows the medial surface generation process using the input tetrahedral meshes. Fig. 19 shows the medial surface generation process for a channel section model. Using the input triangular mesh, tetrahedral elements are generated as shown in Fig. 19(a). During the medial surface generation process, modified Type 1 is applied at 72 tetrahedral elements, modified Type 2 at 72 tetrahedral elements and modified Type 3 at 64 tetrahedral elements (as shown in Fig. 19(b)). Fig. 20 shows the medial surface of a thin-walled shape with a rib, in which 1546 triangular elements are utilized for the tetrahedral meshing and medial surface generation, resulting in 451 triangular and 1164 quadrilateral elements. Fig. 20(c) shows the creation of triangular elements after improving the element
Fig. 22. Phone case model (Inner Type 1: 1187, Inner Type 2: 45, Inner Type 3: 4, Basic Type 1: 1790, Basic Type 2: 4057, Modified Type 1: 1, Modified Type 3: 16). (a) Initial tetrahedral mesh (15,806 tetrahedral elements, 493 inside nodes); (b) shell elements (3153 triangular, 7942 quadrilateral elements).
quality. Fig. 21 shows the medial surface of a panel and Fig. 21(c) shows a rendered view of its medial surface. Fig. 22 shows a FE mesh model of a phone case with 493 nodes inside the domain. The proposed inner-node types are applied to create the medial surface. Fig. 22(b) shows the results of the quadrilateral elements after improving the element quality. 7. Conclusion A modified method was proposed for the generation of a medial surface for various shell structures. After generating the tetrahedral elements with the initial triangular elements, medial surfaces were generated by using the mid-cutting planes of tetrahedral elements. To prevent the creation of interior nodes, the advancing front method was used for tetrahedral meshing. This method also used the top and bottom surface data to suppress abnormal
K.Y. Kwon et al. / Computers and Structures 84 (2006) 1673–1683
cases. In this paper, the cutting directions of tetrahedral elements were classified clearly and applied for the stable creation of the medial surfaces. New types to generate triangular or quadrilateral elements on the medial surface were also introduced so that this method can be applied to various shapes of 3D objects. Acknowledgement The support of the ITEP (Korea Institute of Industrial Technology Evaluation and Planning) is gratefully acknowledged. References [1] Cifuentes AO, Kalbag A. A performance study of tetrahedral and hexahedral elements in 3-D Finite Element Structural Analysis. Finite Elem Anal Des 1992;12:313–8. [2] Benzley SE, Perry E, Karl M, Brett C. A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis. In: Proceedings of the fourth international meshing roundtable. Albuquerque, NM; October, 1995. p. 179–91. [3] Quadros WR, Shimada K. Hex-layer: layered all-hex mesh generation on thin section solids via chordal surface transformation. In: Proceedings of the 11th international meshing roundtable. Ithaca, NY; September, 2001. p. 169–82. [4] Price MA, Armstrong CG. Hexahedral mesh generation by medial surface subdivision: part II solids with flat and concave edges. Int J Numer Methods Eng 1997;40:111–36. [5] Sheffer A, Michel B. Hexahedral meshing of non-linear volumes using Voronoi faces and edges. Int J Numer Methods Eng 2000;49:329–51. [6] Sampl P. Semi-structured mesh generation based on medial axis. In: Proceedings of the ninth international meshing roundtable. New Orleans, CA; October, 2000. p. 21–32. [7] Li TS, McKeag RM, Armstring CG. Hexahedral meshing using midpoint subdivision and integer programming. Comput Methods Appl Mech Eng 1995;124:171–93. [8] Blum H. A transformation for extracting new descriptors of shape. In: Wathen-Dunn Weinant, editor. Models for the perception of speech and visual form. Cambridge, MA: MIT Press; 1967.
1683
[9] Ma CM, Wan SY. A medial-surface oriented 3-d two-subfield thinning algorithm. Pattern Recogn Lett 2001;22:1439–46. [10] Palagyi K. A 3D 6-subiteration thinning algorithm for extracting medial surfaces. Pattern Recogn Lett 2002;23:663–75. [11] Held M, Lukacs G, Andor L. Pocket machining based on contourparallel tool path generated by means of proximity maps. Comput Aided Des 1994;26(3):189–203. [12] Donaghy RJ, McCune W, Robinson DJ, McCkeag RM. Dimensional reduction of analysis models. In: Proceedings of the fifth international meshing roundtable. Pittsburgh, PA; October, 1996. p. 307–20. [13] Suresh K. Generalization of the Kantorovich method of dimensional reduction. In: Proceedings of the 12th international meshing roundtable. Santa Fe, NM; September, 2002. p. 261–76. [14] Ang PY, Armstrong CG. Adaptive curvature-sensitive meshing of the medial axis. In: Proceedings of the 10th international meshing roundtable. Newport Beach, CA; October, 2001. p. 155–65. [15] Quadros WR, Ramaswami K, Prinz FB, Gurumoorthy B. LayTracks: a new approach to automated quadrilateral mesh generation using MAT. In: Proceedings of the ninth international meshing roundtable. New Orleans, CA; October, 2000. p. 239–50. [16] Yamakawa S, Shimada K. Quad-layer: layered quadrilateral meshing of narrow two-dimensional domain by bubble packing and chordal axis transformation. J Mech Des 2002;124(3):564–73. [17] Prasad L. Morphological analysis of shapes. In http://cnls.lanl.gov/ Highlights/1997-07/, Los Alamos National Laboratory; 1997. [18] Moller P. On advancing front mesh generation in three dimensions. Int J Numer Methods Eng 1995;38:3551–69. [19] Chan CT, Anastasiou K. An automatic tetrahedral mesh generation scheme by advancing front method. Commun Numer Meth Eng 1997;13:33–46. [20] Chae SW, Bathe KJ. On automatic mesh construction and mesh refinement in finite element analysis. Comput Struct 1989;32(3):911– 36. [21] Kwon KY, Chae SW, Lee BC. Automatic tetrahedral mesh generation using 3-D operators. Soc CAD/CAM Eng 2004;9(1):27–34. [22] Borouchaki H, George PL, Lo SH. Optimal Delaunay point insertion. Int J Numer Methods Eng 1996;39:3407–37. [23] Weatherill NP, Hassan O. Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary conditions. Int J Numer Methods Eng 1994;37:2005–39. [24] Shephard MS, George MK. Automatic three-dimensional mesh generation by the finite Octree technique. Int J Numer Methods Eng 1991;32:709–49.