A hierarchical representation for heterogeneous object modeling
Computer-Aided Design 37 (2005) 307–319 www.elsevier.com/locate/cad
A hierarchical representation for heterogeneous object modeling X.Y. Kou, S.T. Tan* Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Received 30 September 2003; received in revised form 15 March 2004; accepted 25 March 2004
Abstract A hierarchical representation for heterogeneous object modeling is presented in this paper. To model a heterogeneous object, Boundary representation is used for geometry representation, and a novel Heterogeneous Feature Tree (HFT) structure is proposed to represent the material distributions. HFT structure hierarchically organizes the material variation dependency relationships and is intuitive in modeling different types of material gradations. Based on the HFT structure, a recursive material evaluation algorithm is proposed to dynamically evaluate the material compositions at a specific location. Such a hierarchical representation guarantees complex material gradations and the user’s design intent can be intuitively represented. Example heterogeneous objects modeled with this scheme are provided and potential applications are discussed. q 2004 Elsevier Ltd. All rights reserved. Keywords: Hierarchical representation; Heterogeneous object modeling; Functionally graded material; Heterogeneous feature tree; Recursive material evaluation
1. Introduction Studies on heterogeneous object modeling (HOM) have been a hotspot in recent years [1–7]. In contrast to traditional solid modeling which assumes the material inside a solid is homogeneous, HOM allows material definition and variation inside the solids. It has been widely accepted that in some aspects heterogeneous components have key advantages over homogeneous objects: anisotropic properties can be obtained; different properties and advantages of various materials can be achieved; traditional limitations due to material incompatibility (stress concentration, non-uniform thermal expansion et al) can be solved by gradual material variations. With the latest development in Layered Manufacturing (LM) and modern CAD/CAE systems, analysis and fabrication of solids made of heterogeneous materials have been proved possible. As one of the most fundamental processes, HOM has been a major concern in recent years because such heterogeneous models are the prerequisites for * Corresponding author. Tel.: C852-2859-7909; fax: C852-2858-5415. E-mail addresses: [email protected] (X.Y. Kou), sttan@ hku.hk (S.T. Tan). 0010-4485//$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2004.03.006
downstream applications in both analysis and fabrications. For example, in finite element analysis (FEA), local material information must be derived from the heterogeneous models prior to further computation; in LM, the sliced representation (containing both geometry and material information) of the heterogeneous solid is also retrieved directly from the heterogeneous model. Recent studies show that an effective heterogeneous CAD modeling system should at least meet the following specifications: † Intuitive in representing both geometry, topology and material information simultaneously; † Capable of representing complex solids: the solids to be modeled may be complex in geometry as well as in material variations [5–7]; † Compact and exact: the representation should be compact, and both the geometry and material information can be retrieved accurately and efficiently [7–10]; This paper is motivated to provide an intuitive representation that can model complex material gradations in FGM solid modeling. The subsequent sections of this paper are organized as follows: Section 2 briefly reviews existing
308
X.Y. Kou, S.T. Tan / Computer-Aided Design 37 (2005) 307–319
HOM methods, and detail analysis on these methods are discussed; Section 3 presents a novel representation based on the Heterogeneous Feature Tree (HFT) structure. A recursive material evaluation algorithm is proposed to evaluate the material information at runtime; Section 4 applies this presented method in the development of a heterogeneous heat pump component, and detailed examples are illustrated. Finally, the paper is summarized in Section 5.
2. A brief review on heterogeneous object modeling schemes Methods for HOM have been extensively studied in recent years [1–18]. Dutta and Kumar [12] introduced the the rm-set and rm-object model for the representation of heterogeneous solids. An rm-set is defined as a set of points, whose material distribution can be represented by the same material mapping function F, such that MZF(P) holds true for every point within the set. A heterogeneous object is defined as a collection of rm-sets, namely the rm-object. Jackson et al. [13,14] divided complex solids with tetrahedron decomposition methods and applied control points (for geometry) and control compositions (for material) in HOM. In each decomposed sub-region, the material composition is obtained ‘in terms of a set of control points and control compositions blended with barycentric Bernstein polynomials’. Siu and Tan proposed a source based method for functionally graded material (FGM) modeling [3], and extended this idea in modeling heterogeneous structures in composite laminates [4]. In their method, the geometry features from which the material composition varies are saved in a list structure and by evaluating the distances from the source features, the material composition at a specific location is evaluated at runtime. Rvachev et al. [15] employed the theory of R-functions to construct smooth approximations to distance functions for closed semi-analytic features. The geometry G is described as a set of points constrained by an implicit R-function. This R-function is non-negative on or inside the boundary of the solid and negative outside the solid. Using this R-function based approach, the inverse distance weighting interpolations can be generated and applied in smooth material gradation. Pasko and Adzhiev et al. [16] model the heterogeneous objects as ‘multidimensional point sets with multiple attributes’ based on a function representation (FRep). FRep is used as the basic model for point set geometry, and the attributes (material compositions in the context of FGM object modeling) are represented independently using real-valued scalar functions. Qian [1,17] presented a diffusion based B-spline method to specify material compositions within the feature volumes. Boolean operators are also utilized to build complex heterogeneous objects. Siu and Tan [3] proposed new
operators such as heterogeneous insertion, merge and immersion in FGM modeling. Sun and Hu [2] proposed a complex-union operator in modeling multi-material objects. Shin [6] applied a constructive representation for HOM, in which heterogeneous primitives are first constructed and then Boolean operators are applied. Most of the above discussed heterogeneous objects modeling methods involve a two-step approach, or a sequential approach termed by Liu and Maekawa [18]: first on 3D geometric modeling, followed by addition of material properties to 3D models (material modeling) [2–5,7,12–15]. In the first step, no material information is defined and the solid model is homogeneous; this is typically a homogeneous solid modeling process. In the second step, the material information is incorporated and mapped onto the modeled 3D geometry. With this type of representation, the material information can be integrated into the model only after the 3D geometry is fully defined. Although the material distribution can be modeled at 3D level, however the material is undefined or assumed homogeneous at the lower 1D/2D level. With this assumption, the reference entities are generally homogeneous in feature based schemes, and only 1D dependent material gradation can be modeled. Take the source based method [3] as an example. Let M(P) be point Ps material definition, where P is an arbitrary point inside the object geometry, Mi be the ith source entity’s material composition(usually the source entities can be identified through interactive input), and Wi be the weight of material gradation from source entity i, then M(P) can be formulated as MðPÞ Z
nK1 X
Wi Mi
(1)
iZ0
where n is the total numbers of the source entities utilized in modeling the material gradation. Note that in Eq. (1), Mi is constant (homogeneous), and it remains the same no matter where P is located; only Wi changes with P’s location, as described in Eq. (2) Wi Z F ðiÞ ðPÞ;
Mi hC ðiÞ ;
nK1 X
Wi Z 1
(2)
iZ0
where F(i) is the weighting function related to point P’s location and the ith source feature, C(i) is the ith source entity’s constant material composition. In the context of this paper, we denote the material distribution modeled with the two-step approaches as simple material distribution and we term the material distribution formulated by Eq. (3) as compound or complex material distributions, in which the reference source entities’ material can be also heterogeneous Wi Z F ðiÞ ðPÞ;
Mi Z FðiÞ ðPÞ;
nK1 X iZ0
Wi Z 1
(3)
X.Y. Kou, S.T. Tan / Computer-Aided Design 37 (2005) 307–319
where F(i)(P) is a function to determine the ith source entity’s material composition, which is variant with input point P’s location. In the past, most FGM object modeling schemes focused on modeling simple material variations. Although objects with simple material gradation are straight forward in applications, however, their functional performances are often limited. Generally, the objects’ functions are multifolded in physical properties, for example, in fracture mechanics, objects are often subject to ‘mixed-mode’ loading (tensile and shear) [19], and 1D dependent material gradation may not be sufficient to get the optimal results in both aspects. In some applications, multiple material interfaces coexist and smooth material variations are expected on all of the material interfaces and throughout the object geometries. In this case, 1D dependent material gradation cannot guarantee such smooth material distributions in all directions. In aerospace applications, modern aerospace shuttles and craft are subjected to super high temperatures that have variations in two or three directions [20]. Nemat-Alla [20] points out that ‘conventional FGM may not be so effective in such design problems since all outer surface of the body will have the same composition distribution. Since, temperature distribution in such advanced machine element changes in two or three directions. Therefore, if the FGM has two-dimensional dependent material properties, more effective high-temperature resistant material can be obtained’. In nature, objects with compound material distributions are also commonplace. Examples of this type of objects are animal tissues (e.g. human bone and cartilage), plant structures (e.g. wood.) and geological materials (e.g. rocks, soil) [21,22]. The multiple function requirements for the heterogeneous objects call for a systematic approach in modeling complex material variations, however this problem has not been so thoroughly addressed in the past. To facilitate the modeling of compound material variations, we regard that the material property should be regarded as an integrated rather than an attached part of the object model. The material information should be accessible at all the 1D, 2D and 3D level. This paper is directed at providing a hierarchical representation that integrates material information into 1D and 2D features right at the outset. A HFT structure is proposed to model the material variation dependency relationships. One-dimensional heterogeneous features are first defined; 2D and 3D complex features are subsequently built hierarchically from the lower level 1D/2D heterogeneous features.
309
composition M is represented by an n-dimensional vector whose elements represent the volume fractions of a predefined primary material [3], where n is total number of primary materials. 3.1. Heterogeneous features In this paper, we define the heterogeneous features as entities with heterogeneous material distributions A heterogeneous feature is represented by its geometries (rigorously this includes the geometrical and topological information, for simplicity reasons, we use geometry in this paper) and material distributions. Boundary representation (B-Rep) is used to describe the geometry information, and the material information is represented by a HFT structure. 3.2. One-dimensional heterogeneous features One-dimensional heterogeneous features are extensions of the traditional 1D curve features, with the material composition of every point on these features explicitly defined. Fig. 1(a) illustrates a heterogeneous line with a linear material gradation from one endpoint to the other, which is formulated as P Z ð1 K tÞPs C tPe ;
tZ
jPPs j ; jPs Pe j
0% t% 1
where P is an arbitrary point on the line, Ps(Xs, Ys, Zs, Ms) and Pe(Xe, Ye, Ze, Me) are the starting and ending point. Fig. 1(b) illustrates another heterogeneous B-spline curve which is defined as PðuÞ Z
n X
Ni;p ðuÞPi ;
iZ0
Ni;0 ðuÞ Z Ni;p ðuÞ Z
8
A hierarchical representation for heterogeneous object modeling X.Y. Kou, S.T. Tan* Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Received 30 September 2003; received in revised form 15 March 2004; accepted 25 March 2004
Abstract A hierarchical representation for heterogeneous object modeling is presented in this paper. To model a heterogeneous object, Boundary representation is used for geometry representation, and a novel Heterogeneous Feature Tree (HFT) structure is proposed to represent the material distributions. HFT structure hierarchically organizes the material variation dependency relationships and is intuitive in modeling different types of material gradations. Based on the HFT structure, a recursive material evaluation algorithm is proposed to dynamically evaluate the material compositions at a specific location. Such a hierarchical representation guarantees complex material gradations and the user’s design intent can be intuitively represented. Example heterogeneous objects modeled with this scheme are provided and potential applications are discussed. q 2004 Elsevier Ltd. All rights reserved. Keywords: Hierarchical representation; Heterogeneous object modeling; Functionally graded material; Heterogeneous feature tree; Recursive material evaluation
1. Introduction Studies on heterogeneous object modeling (HOM) have been a hotspot in recent years [1–7]. In contrast to traditional solid modeling which assumes the material inside a solid is homogeneous, HOM allows material definition and variation inside the solids. It has been widely accepted that in some aspects heterogeneous components have key advantages over homogeneous objects: anisotropic properties can be obtained; different properties and advantages of various materials can be achieved; traditional limitations due to material incompatibility (stress concentration, non-uniform thermal expansion et al) can be solved by gradual material variations. With the latest development in Layered Manufacturing (LM) and modern CAD/CAE systems, analysis and fabrication of solids made of heterogeneous materials have been proved possible. As one of the most fundamental processes, HOM has been a major concern in recent years because such heterogeneous models are the prerequisites for * Corresponding author. Tel.: C852-2859-7909; fax: C852-2858-5415. E-mail addresses: [email protected] (X.Y. Kou), sttan@ hku.hk (S.T. Tan). 0010-4485//$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2004.03.006
downstream applications in both analysis and fabrications. For example, in finite element analysis (FEA), local material information must be derived from the heterogeneous models prior to further computation; in LM, the sliced representation (containing both geometry and material information) of the heterogeneous solid is also retrieved directly from the heterogeneous model. Recent studies show that an effective heterogeneous CAD modeling system should at least meet the following specifications: † Intuitive in representing both geometry, topology and material information simultaneously; † Capable of representing complex solids: the solids to be modeled may be complex in geometry as well as in material variations [5–7]; † Compact and exact: the representation should be compact, and both the geometry and material information can be retrieved accurately and efficiently [7–10]; This paper is motivated to provide an intuitive representation that can model complex material gradations in FGM solid modeling. The subsequent sections of this paper are organized as follows: Section 2 briefly reviews existing
308
X.Y. Kou, S.T. Tan / Computer-Aided Design 37 (2005) 307–319
HOM methods, and detail analysis on these methods are discussed; Section 3 presents a novel representation based on the Heterogeneous Feature Tree (HFT) structure. A recursive material evaluation algorithm is proposed to evaluate the material information at runtime; Section 4 applies this presented method in the development of a heterogeneous heat pump component, and detailed examples are illustrated. Finally, the paper is summarized in Section 5.
2. A brief review on heterogeneous object modeling schemes Methods for HOM have been extensively studied in recent years [1–18]. Dutta and Kumar [12] introduced the the rm-set and rm-object model for the representation of heterogeneous solids. An rm-set is defined as a set of points, whose material distribution can be represented by the same material mapping function F, such that MZF(P) holds true for every point within the set. A heterogeneous object is defined as a collection of rm-sets, namely the rm-object. Jackson et al. [13,14] divided complex solids with tetrahedron decomposition methods and applied control points (for geometry) and control compositions (for material) in HOM. In each decomposed sub-region, the material composition is obtained ‘in terms of a set of control points and control compositions blended with barycentric Bernstein polynomials’. Siu and Tan proposed a source based method for functionally graded material (FGM) modeling [3], and extended this idea in modeling heterogeneous structures in composite laminates [4]. In their method, the geometry features from which the material composition varies are saved in a list structure and by evaluating the distances from the source features, the material composition at a specific location is evaluated at runtime. Rvachev et al. [15] employed the theory of R-functions to construct smooth approximations to distance functions for closed semi-analytic features. The geometry G is described as a set of points constrained by an implicit R-function. This R-function is non-negative on or inside the boundary of the solid and negative outside the solid. Using this R-function based approach, the inverse distance weighting interpolations can be generated and applied in smooth material gradation. Pasko and Adzhiev et al. [16] model the heterogeneous objects as ‘multidimensional point sets with multiple attributes’ based on a function representation (FRep). FRep is used as the basic model for point set geometry, and the attributes (material compositions in the context of FGM object modeling) are represented independently using real-valued scalar functions. Qian [1,17] presented a diffusion based B-spline method to specify material compositions within the feature volumes. Boolean operators are also utilized to build complex heterogeneous objects. Siu and Tan [3] proposed new
operators such as heterogeneous insertion, merge and immersion in FGM modeling. Sun and Hu [2] proposed a complex-union operator in modeling multi-material objects. Shin [6] applied a constructive representation for HOM, in which heterogeneous primitives are first constructed and then Boolean operators are applied. Most of the above discussed heterogeneous objects modeling methods involve a two-step approach, or a sequential approach termed by Liu and Maekawa [18]: first on 3D geometric modeling, followed by addition of material properties to 3D models (material modeling) [2–5,7,12–15]. In the first step, no material information is defined and the solid model is homogeneous; this is typically a homogeneous solid modeling process. In the second step, the material information is incorporated and mapped onto the modeled 3D geometry. With this type of representation, the material information can be integrated into the model only after the 3D geometry is fully defined. Although the material distribution can be modeled at 3D level, however the material is undefined or assumed homogeneous at the lower 1D/2D level. With this assumption, the reference entities are generally homogeneous in feature based schemes, and only 1D dependent material gradation can be modeled. Take the source based method [3] as an example. Let M(P) be point Ps material definition, where P is an arbitrary point inside the object geometry, Mi be the ith source entity’s material composition(usually the source entities can be identified through interactive input), and Wi be the weight of material gradation from source entity i, then M(P) can be formulated as MðPÞ Z
nK1 X
Wi Mi
(1)
iZ0
where n is the total numbers of the source entities utilized in modeling the material gradation. Note that in Eq. (1), Mi is constant (homogeneous), and it remains the same no matter where P is located; only Wi changes with P’s location, as described in Eq. (2) Wi Z F ðiÞ ðPÞ;
Mi hC ðiÞ ;
nK1 X
Wi Z 1
(2)
iZ0
where F(i) is the weighting function related to point P’s location and the ith source feature, C(i) is the ith source entity’s constant material composition. In the context of this paper, we denote the material distribution modeled with the two-step approaches as simple material distribution and we term the material distribution formulated by Eq. (3) as compound or complex material distributions, in which the reference source entities’ material can be also heterogeneous Wi Z F ðiÞ ðPÞ;
Mi Z FðiÞ ðPÞ;
nK1 X iZ0
Wi Z 1
(3)
X.Y. Kou, S.T. Tan / Computer-Aided Design 37 (2005) 307–319
where F(i)(P) is a function to determine the ith source entity’s material composition, which is variant with input point P’s location. In the past, most FGM object modeling schemes focused on modeling simple material variations. Although objects with simple material gradation are straight forward in applications, however, their functional performances are often limited. Generally, the objects’ functions are multifolded in physical properties, for example, in fracture mechanics, objects are often subject to ‘mixed-mode’ loading (tensile and shear) [19], and 1D dependent material gradation may not be sufficient to get the optimal results in both aspects. In some applications, multiple material interfaces coexist and smooth material variations are expected on all of the material interfaces and throughout the object geometries. In this case, 1D dependent material gradation cannot guarantee such smooth material distributions in all directions. In aerospace applications, modern aerospace shuttles and craft are subjected to super high temperatures that have variations in two or three directions [20]. Nemat-Alla [20] points out that ‘conventional FGM may not be so effective in such design problems since all outer surface of the body will have the same composition distribution. Since, temperature distribution in such advanced machine element changes in two or three directions. Therefore, if the FGM has two-dimensional dependent material properties, more effective high-temperature resistant material can be obtained’. In nature, objects with compound material distributions are also commonplace. Examples of this type of objects are animal tissues (e.g. human bone and cartilage), plant structures (e.g. wood.) and geological materials (e.g. rocks, soil) [21,22]. The multiple function requirements for the heterogeneous objects call for a systematic approach in modeling complex material variations, however this problem has not been so thoroughly addressed in the past. To facilitate the modeling of compound material variations, we regard that the material property should be regarded as an integrated rather than an attached part of the object model. The material information should be accessible at all the 1D, 2D and 3D level. This paper is directed at providing a hierarchical representation that integrates material information into 1D and 2D features right at the outset. A HFT structure is proposed to model the material variation dependency relationships. One-dimensional heterogeneous features are first defined; 2D and 3D complex features are subsequently built hierarchically from the lower level 1D/2D heterogeneous features.
309
composition M is represented by an n-dimensional vector whose elements represent the volume fractions of a predefined primary material [3], where n is total number of primary materials. 3.1. Heterogeneous features In this paper, we define the heterogeneous features as entities with heterogeneous material distributions A heterogeneous feature is represented by its geometries (rigorously this includes the geometrical and topological information, for simplicity reasons, we use geometry in this paper) and material distributions. Boundary representation (B-Rep) is used to describe the geometry information, and the material information is represented by a HFT structure. 3.2. One-dimensional heterogeneous features One-dimensional heterogeneous features are extensions of the traditional 1D curve features, with the material composition of every point on these features explicitly defined. Fig. 1(a) illustrates a heterogeneous line with a linear material gradation from one endpoint to the other, which is formulated as P Z ð1 K tÞPs C tPe ;
tZ
jPPs j ; jPs Pe j
0% t% 1
where P is an arbitrary point on the line, Ps(Xs, Ys, Zs, Ms) and Pe(Xe, Ye, Ze, Me) are the starting and ending point. Fig. 1(b) illustrates another heterogeneous B-spline curve which is defined as PðuÞ Z
n X
Ni;p ðuÞPi ;
iZ0
Ni;0 ðuÞ Z Ni;p ðuÞ Z
8
