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Heterogeneous object modeling based on multi-color distance field
Materials and Design 30 (2009) 939–946
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Materials and Design journal homepage: www.elsevier.com/locate/matdes
Heterogeneous object modeling based on multi-color distance field Zhou Hongmei *, Liu Zhigang, Lu Bingheng State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Shaanxi, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 11 May 2008 Accepted 1 July 2008 Available online 5 July 2008 Keywords: Heterogeneous object Multi-color distance field Layered manufacturing Boolean operations
a b s t r a c t With the development of layered manufacturing or rapid prototyping, modeling heterogeneous object with varying material distribution has become more and more important. Different from homogeneous object modeling, the heterogeneous object modeling contains not only geometry and topology information but also material information. In this paper, a new method of modeling heterogeneous object is presented. In this method, firstly, the geometric model of an object is represented by the signed distance field; next, each grid is assigned with material property and material distance field which has the same resolution with geometric distance field are calculated according to material feature; the geometric and material distance field are then unified, named as multi-color distance field. The complex heterogeneous object can also be obtained via the boolean operations of multi-color distance field. For a heterogeneous object represented by the distance field, the most significant advantage is that its geometry model and material information can easily be sliced into a series of 2D layers, so it is convenient for the heterogeneous object to be fabricated using layered manufacturing machine. Several examples show the validity of our modeling method. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Heterogeneous objects [1] are defined to be made up of several kinds of material compositions within an object. Generally, heterogeneous objects can be classified into three types: multi-material objects with clear material boundary [2], functionally graded materials objects [3,4] without clear material boundary and with varying continuously material composition, and composite materials which consist of one or more discontinuous phases distributed in one continuous phase. In order to systemically represent, design and fabricate the heterogeneous objects, a lot of former researchers have introduced many solutions [1–15]. Voxel based model [6] and volume mesh based model [7] describe heterogeneous material distribution through intensive space decompositions with a collection of voxels and sub-volumes/polyhedrons, respectively. Although it is suitable for representing highly heterogeneous objects whose material distributions are strongly irregular, the memory consumptions are a heavy burden [11]. Kumar and Dutta [1,2] employed the extended r-sets (including material composition rmsets) to model and represent heterogeneous objects and Shin et al.[5,8] presented a representation scheme for heterogeneous objects based on the constructive representation. Siu and Tan [9] and Kou and Tan [12] proposed a kind of control feature based model. They made use of the reference feature (like point, line and regular plane) to compute material composition, thus, the * Corresponding author. E-mail address: [email protected] (Z. Hongmei). 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.07.002
method is limited to FGM object with relatively simple geometry and composition gradients. Yang and Qian [10] presented the heterogeneous design and analysis method based on B-spline, however, it depends heavily on spatial parameterizations of control points and such parameterizations are a rather troublesome task for arbitrary 3D objects. Shapiro et al. [13,14] proposed the construction of field modeling using sampling distance field and interpolated physics field by way of extended R-functions, however, the limitations of R-functions results in the limitations in complexshaped heterogeneous object modeling. Apart from the above modeling schemes, some researchers also studied the fabrication of FDM objects, such as solid freeform fabrication (SFF) [3] or layered manufacturing (LM) [4]. LM is a process in which material is selectively deposited, layer by layer, to manufacture an object. When the selective material of each layer is varying material component, a heterogeneous object can be fabricated. Shin and Dutta [15] described a sequential procedure from material modeling using the extended r-sets [1,2] to processing that comprise discretization, building direction selection and adaptive slicing of heterogeneous objects. These modeling methods have mainly focused on simple topology heterogeneous objects with simple gradient material. It is difficult to use these methods to process arbitrary-shaped complex objects. Moreover, these methods have mainly been used to design and fabricate heterogeneous objects based on forward engineering. Therefore, it is necessary to develop a new representation and processing methods for modeling and fabricating 3D complex heterogeneous objects.
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So in this paper, a new method of complex heterogeneous object modeling to synthetically describe the geometry and material information is proposed. The main contributions of this paper are (1) A unified multi-color distance field between geometric model and material model is introduced. Not only the geometry model but also the material information can easily be sliced to fabricate for the object represented by the unified multi-color distance field. Moreover, the method is suitable not only for modeling known CAD model, but also the method may start from arbitrary sampled point cloud. (2) The important advantage of distance field is that arbitrary topology structure object can be modeled and represented. Furthermore, the boolean operations of distance field can generate complex heterogeneous objects through simple max/min operations on the distance field. 2. Heterogeneous objects modeling based on multi-color distance field Implicit function modeling method is widely used in arbitrary topology structure object, such as [16–19]. However, at present the implicit modeling is limited in modeling homogeneous object and few references are found in modeling heterogeneous object. In this paper the implicit modeling method of heterogeneous object is presented. The modeling process is depicted in the flowchart in Fig. 1. At the outset, a model is scanned or sampled to produce a 2D or 3D point cloud. The point cloud is then enclosed by a box which is uniformly subdivided to the prescribed resolution. The signed distance field of the model is then calculated by solving Eikonal equation [16] and iso-surface of the model is extracted by marching cube algorithm [18] for representing the model. At the same time, the material feature is collected by design intent or known material properties and then material distance field is calculated with the same resolution as the above signed distance field. The multi-color distance field is then formed by unifying the geometry field with material field. 2.1. Geometric distance field modeling 2.1.1. Computing unsigned distance field Compute the distance field within a bounding box represented as an axis-aligned uniform 3D grid. Let the size of each voxel in the
Fig. 1. Flowchart of heterogeneous object modeling.
3D Cartesian grid be Nx Ny Nz. The optimal grid size is determined by a specified reconstruction tolerance. For a given point cloud S 2 R3, an unsigned distance function is defined as the function that calculate the distance from point p (grid point) to the closest point x in S (see Fig. 2). The signed distance function is the distance to the boundary oS of the model, and the sign is used to denote inside or outside the model.
dS ðpÞ ¼ signðpÞinf x2oS kx pk;
ð1Þ
where
signðpÞ ¼
1
if p inside the model;
1; otherwise:
2.1.2. Estimating the signs of distance field The calculation of unsigned distance is easy to implement but the sign assignment is very difficult for unorganized point cloud. However, the sign assignment is important for extracting iso-surface of model exactly. There are two commonly used methods to estimate the signs of distance field, that is, oriented tangent plane method [19] and solving Eikonal equation method [16]. Although the former is the most direct method to determine the signs of distance field, the computing process will be problematic and errorprone especially for regions with high curvature. In this paper, the latter is used to estimate the signs. The bounding cube of point cloud S is regularly subdivided into the desired accuracy sub-cubes and the distance function d(x) to S satisfies the Eikonal equation with boundary condition.
jrdðxÞj ¼ 1;
ð2Þ
where, d(x 2 S) = 0. The Eikonal equation is solved using upwind schemes and Gauss-Seidel iterations with various orderings of the sweepings. Details of solving these PDEs can be found in [17]. 2.2. Material distance field modeling and fabricating How to arrange the material distribution is determined by design intent or object performance. There are different material features for different objects. Similar to [13], in terms of the quantity of material feature, heterogeneous objects can be classified into two types: single-material feature and multiple-material feature objects. The single-material feature as the reference datum for defining material distributions throughout the solid is the simplest approach in capturing the designers’ intents. However, a more typical situation with heterogeneous objects involves several material features. Details of describing multiple-material feature object will be discussed in Section 2.2.2. In terms of the relationship between material feature and model surface the heterogeneous objects can
Fig. 2. 2D model (30 40 Cartesian grid).
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also be classified into two types: consistent-feature (in Section 2.2.1) and inconsistent-feature (in Section 2.2.2) objects. To represent the heterogeneous objects, the traditional Euclidean space R3 is extended to a tensor space: T = R3 Mn, where Mn denotes n-dim material space. The vector of material composition: M = [m1,m2 ,. . . , mn], where, mi denotes primary material and P satisfies ni¼1 mi ¼ 1, that is, the total volume fraction of all material composition is summed to 1. In this paper, the material distributions are defined in terms of the distances from the grid points to the material features. The total distance values of each grid point make up of material distance field. There are three major steps involved in calculating material distance field. In the first step, the suitable primary materials and material features are selected for the model to improve its performance. The second step involves calculating distance values from each grid point to material features. In the third step, for the multiple-material object, the several distance fields are blended. In other words, the distance field of material feature is selected as material distribution function, which is added to the above signed distance field of geometry model, then the multi-color distance field is formed. It is described as
(
R3 ! dðSÞ; M n ! FðrÞ;
ðNx Ny NzÞ;
ð3Þ
where, d(S) is the signed distance to model surface and F(r) is the distance to material feature in grid size Nx Ny Nz. 2.2.1. Consistent-feature objects The material feature of consistent-feature object coincides with object surface, that is, the material composition is linear or nonlinear gradients along the object thickness direction. The material distribution is the function of distance field for heterogeneous object that is composed of two materials.
material 1 : f1 ðdÞ ¼ ðadÞk ; material 2 : f2 ðdÞ ¼ 1 ðadÞk ;
) ð4Þ
f2 ðdÞ ¼ 1 di;j;k = maxðdÞ;
FðxÞ ¼
n X N X j¼1
ð5Þ
where di,j,k denotes the distance value from a query point (i, j, k) to outer surface, max(d) is maximum distance value of distance field. The material composition of arbitrary point inside of model can be computed through interpolating the adjacent grid points. The points on each iso-contour extracted from the signed distance field own the same material composition in Fig. 3. 2.2.2. Inconsistent-feature objects The material distance field is easy to be computed for the above consistent-feature objects. However, sometimes the material feature does not coincide with the object surface. In this case, the material distance field from points inside the object to material feature has to be computed additionally. Commonly used material feature is composed of face feature, curve feature, line feature and point feature. Single-material feature is easy to implement like Fig. 4a and b. Multiple-material features based model need blend several material distance fields. The steps are as follows:
xij f ðrij Þ;
ð6Þ
i¼1
where n X N X j¼1
n X N X j¼1
xij ¼ 1;
ð7Þ
f ðrij Þ ¼ 1;
ð8Þ
i¼1
i¼1
where xij denotes the material blending weight or the contribution of Si, N is the total number of material feature of this heterogeneous object, n corresponds to the number of predefined primary materials. The feature weight of the material blending function can be determined in different ways. Designers can directly specify appropriate feature weight based on the design intents and their experiences. Arbitrary prescribed modeling can also be gained through a graphical user interface to input different weight and feature. Fig. 4 is an example for single-feature and multiple-feature. 2.2.3. Fabrication by LM machine Laser-based LM is based on a layer by layer cladding approach and can deposit different materials by changing the powder delivery into the cladding zone as discussed in [5]. Fig. 5a shows the sketch of laser-based LM process. For a 3D heterogeneous object described by multi-color distance field in Eq. (3), the distance field is sliced by the slice planes XY to obtain 2D layers Li(z)
(
where d is normalized distance from a query point to model surface (thickness of the heterogeneous object); a is the scaling factor; k is the distribution exponent to describe gradient property. If k is 1, material property is linear distribution. Usually a and k are determined by design intent or performance test. In Fig. 2, for each grid point (i, j, k) inside of S, the material composition is
f1 ðdÞ ¼ di;j;k = maxðdÞ;
(1) The material feature is defined as Si(i = 1 ,. . . , N). (2) The material field of the jth primary material is calculated from point x inside the object to each material feature, defined as rij. That is, rij = dist(x,Si). Material composition function is the mapping function: f(rij) = mj. (3) The material composition at any point x is
Li ðzÞ ¼
R2 ! di ðSÞ; M n ! F i ðrÞ;
ðNx NyÞ:
ð9Þ
Thus, each layer is defined by a layer thickness and a set of contours each of which refers to its own material composition. For a slice (see Fig. 5b) of a 3D model, the fabrication process using LM machine is as follows: (1) Assume the diameter of the extrusion nozzle is a; (2) Symmetrical contours (Fig. 5b) can be gained through extracting iso-contour of distance field. The space of adjacent contours is a. The tool paths for material deposition will move along these iso-contours; (3) The volume ratios of primary materials are calculated using Eq. (5) and the primary materials are mixed by powder mixer according to the volume ratios.
3. Boolean operations of distance field In current constructive solid geometry (CSG) modeling systems, regularized boolean operations are used to create complex objects from simple shapes (cube, sphere and cylinder etc.). Similarly, boolean operations of distance field need to be defined in order to create and manipulate complex heterogeneous solid models. The boolean operations on object models depend on the operations that are defined for manipulating the geometry models as well as the material models.
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Fig. 3. Consistent-feature objects: (a) Sphere of point feature, (b) Cylinder of line feature, (c) Box of axis feature and (d) Surface of arbitrary curve feature.
Fig. 4. (a–b) Single-feature, (c) Multiple-feature (the weight of point feature is 0.3 and that of line feature is 0.7) and (d) Multiple-feature (the weight of point feature is 0.7 and that of line feature is 0.3).
Fig. 5. (a) Laser-based LM process and (b) Iso-contours of multi-color distance field.
For the purpose of convenient observation, the boolean operations are described in 2D example. Boolean operations of distance field are performed for two simple models: T1 (A, m1) (Fig. 6 a), T2 (B, m2) (Fig. 6b). According to the position between A and B, it can be divided into two cases (as discussed in Sections 3.1 and 3.2, respectively). In the two cases, firstly, a bounding box C which encloses the two models (A and B) is obtained. The bounding box C is then uniformly subdivided into sub-squares and the distance fields (A and B) are calculated, respectively. Note that the distance fields of A and B must be calculated at the same resolution, or the boolean operations between A and B will be a troublesome work. Thus, in order to satisfy the precision require-
ment, the bounding box C has to be subdivided to the prescribed size. 3.1. Case I In Fig. 6c, A \ B = B, the signed distance fields of A and B are calculated, denoted as dA and dB, respectively. The geometry model can be gained through the following boolean operations. Geometric intersection operation:
dA\B ¼ dA \ dB ¼ dB : Geometric union operation:
Fig. 6. (a) Model A, (b) Model B, (c) A \ B = B and (d) A \ B = D.
ð10Þ
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dA[B ¼ dA [ dB ¼ dA :
ð11Þ
(1) If p is inside of D (dA(i,j,k) > 0 and dB(i,j,k) > 0)
Geometric difference operation:
dA\Bði;j;kÞ ¼ minðdAði;j;kÞ ; dBði;j;kÞ Þ;
dA=B ¼ dA =dB :
dA=Bði;j;kÞ ¼ dBði;j;kÞ ;
ð12Þ
Obviously the above boolean operations (intersection and union) need not compute a new distance field. In order to obtain the difference result (Eq. (12)), grid point p (i, j, k) inside of C is divided into three types.
dA[Bði;j;kÞ ¼ minðdðp; L3Þ; dðp; L4ÞÞ;
dB=Aði;j;kÞ ¼ dAði;j;kÞ , where, dðp; L3Þ and dðp; L4Þ denote the distance value from p to L3 and from p to L4, respectively. (2) If p is inside of A and outside of B (dA(i,j,k) > 0 and dB(i,j,k) < 0)
dA\Bði;j;kÞ ¼ dBði;j;kÞ ;
dA[Bði;j;kÞ ¼ minðdðp; L3Þ; dðp; L4ÞÞ;
dA=Bði;j;kÞ ¼ minðdAði;j;kÞ ; jdBði;j;kÞ jÞ;
(1) If p is inside of B (dB(i,j,k) > 0)
dB=Aði;j;kÞ ¼ dAði;j;kÞ :
(3) If p is inside of B and outside of A (dA(i,j,k) < 0 and dB(i,j,k) > 0)
dA=Bði;j;kÞ ¼ dBði;j;kÞ : (2) If p is outside of B and inside of A (dA(i,j,k) > 0 and dB(i,j,k) < 0)
dA=Bði;j;kÞ ¼ minðdAði;j;kÞ ; jdBði;j;kÞ jÞ; where min(dA(i,j,k),jdB(i,j,k)j) denotes the smaller value between dA(i,j,k) and jdB(i,j,k)j. (3) If p is outside of A (dA(i,j,k) < 0)
dA=Bði;j;kÞ ¼ dAði;j;kÞ : All above distance values of grid points are combined together and a new distance field of difference operation is formed in the Fig. 7a. The material boolean operator () is consistently applicable to modeling operations such as material intersection, union and differm1 in Fig. 7b, ence. The material difference operation of A and B is dA=B where m1denotes its material composition. The material intersecm1 m2 in Fig. 7c, where m1 m2 denotes tion operation of A and B is dA\B the grading varying material component. The material union operm1 m2 m1 m2 m1 m1 m2 in Fig. 7d, where dA[B ¼ dA=B þ dA\B . Of ation of A and B is dA[B course, the new geometry model may redistribute new material feature to model new material composition according to user’s requirement.
dA\Bði;j;kÞ ¼ dAði;j;kÞ ;
dA[Bði;j;kÞ ¼ minðdðp; L3Þ; dðp; L4ÞÞ;
dA=Bði;j;kÞ ¼ minðdðp; L1Þ;
ðp; L3ÞÞ; dB=Aði;j;kÞ ¼ minðdBði;j;kÞ ; jdAði;j;kÞ jÞ:
(4) If p is outside of A and B (dA(i,j,k) < 0 and dB(i,j,k) < 0)
dA\Bði;j;kÞ ¼ minðdðp;L1Þ;dðp;L3ÞÞ;dA[Bði;j;kÞ ¼ minðdðp;L3Þ;dðp;L4ÞÞ; dA=Bði;j;kÞ ¼ minðdðp;L1Þ;dðp;L3ÞÞ;dB=Aði;j;kÞ ¼ minðdðp;L2Þ;dðp;L4ÞÞ: All above distance values of grid points are combined together and the results are shown in Fig. 8. Then the material information is incorporated into the above m1 and geometry model. The material difference operations are dA=B m2 dB=A between A and B in Fig. 9a-b, where m1 and m2denote their material compositions. The material intersection operation of A m1 m2 in Fig. 9c, where, m1 m2 denotes the grading varyand B is dA\B ing material component and the boundary L1 and L2 are material m1 m2 in Fig. features. The material union operation of A and B is dA[B m1 m2 m1 m2 9d, which comprise three parts: dA=B , dB=A and dA\B , where, the material feature of intersection section is boundary L2. 4. Engineering applications
3.2. Case II 4.1. Example 1 In Fig. 6d, A \ B = D, the two models are subdivided into L1, L2, L3 and L4. The signed distance fields of A and B are calculated, defined as dA and dB, respectively. The geometry model can be gained through the following boolean operations. Geometric intersection operation:
dA\B ¼ dA \ dB ¼ dD :
For a homogeneous gear there is always a possibility that the heat from the outer surface transmits to the shaft causing adverse effects on its function and efficiency as discussed in [20]. To prevent heat from being transferred to the shaft and improve the
ð13Þ
Geometric union operation:
dA[B ¼ dA [ dB :
ð14Þ
Geometric difference operation:
dA=B ¼ dA =dB ;
ð15Þ
dB=A ¼ dB =dA :
ð16Þ
In order to obtain the above operation results (Eqs. (13)–(16)), grid point p (i,j,k) inside of C is divided into four types.
Fig. 8. Geometric boolean operations: (a)dA[B, (b) dA\B, (c) dA/B and (d) dB/A.
m
m m2
1 1 Fig. 7. Material boolean operations: (a) dA/B, (b) dA=B , (c) dA\B
m m2
1 and (d) dA[B
.
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m
m
m m2
1 2 1 Fig. 9. Material boolean operations: (a) dA=B , (b) dB=A , (c) dA\B
m m2
1 and (d) dA[B
.
performance of the gear, the gear can be made of FGM with ceramic at the outer surface for its high hardness and fine wearing resistance properties, metal at the inner surface for its fine mechanical properties. The point cloud (Fig. 10b) can be gained through scanning the surface of spur gear (Fig. 10a) using coordinate measuring machine Leitz PMM866. The signed distance field (Fig. 10c) can be obtained by the above geometry modeling method (in Section 2.1). The material feature is the gear axis and the material ratio is shown in Fig. 11a. Thus, the inconsistent-feature object needs to compute the material field (Fig. 11b) from a query point inside the object to gear axis. Then, the layers (Fig. 11c) can be gained by slicing the material field model. The gap between two adjacent slices defines the layer thickness.
shank interface [21], and which generates stress concentration near such interface. Furthermore, this stress concentration could lead to the undesired functional loss of tool. Thus, the FGM tool is designed to improve the stress concentration. Fig. 12 shows a heterogeneous boring cutter tool. The FGM graded region is inserted between shank and the tip in order to enforce the material composition continuity. The object is partitioned into a cemented carbide cutter and a shank that is manufactured from FGM of steel and cemented carbide. Assume that the material composition of the cutter tool is varied with explicit function along the x-direction. The modeling of inconsistent-feature material field is gained through computing the distance from the query point to the left face feature. The two kinds of distribution are as follows: (1) Linear material distribution (Fig. 13a)
8 0 6 x 6 40; > < 1; Steel : 1 ðx 40Þ=24; 40 < x < 64; > : 0; 64 6 x 6 76: 8 0 6 x 6 40; > < 0; Cemented carbide : ðx 40Þ=24; 40 < x < 64; > : 1; 64 6 x 6 76: (2) Parabolic material distribution (Fig. 13b)
4.2. Example 2 As is widely known, conventional tools used in metal or glass cutting machineries are manufactured by welding together tip and shank of different materials. As a result, there is the sharp discontinuity in the material composition distribution across the tip-
8 0 6 x 6 40; > < 1; Steel : 1 ððx 40Þ=24Þ2 ; 40 < x < 64; > : 0; 64 6 x 6 76:
Fig. 10. (a) Model of a spur gear, (b) Point cloud, (c) Geometric distance field and (d) Slices of (c).
Fig. 11. (a) Material distribution function, (b) Multi-color distance field model and (c) Slices of (b).
Fig. 12. Heterogeneous boring cutter tool.
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Fig. 13. (a) Material field modeling of linear material distribution and (b) Material field modeling of non-linear material distribution.
Fig. 14. (a) Material distribution of turbine blade, (b) Ceramic part, (c) FGM part (d1) (h1 + h2)-contour (d2–d4) inner surface (d5) Metal part (difference operations: d1/d2/d3/ d4) and (e) The union model of (b), (c) and (d5).
8 0 6 x 6 40; > < 0; 2 Cemented carbide : ððx 40Þ=24Þ ; 40 < x < 64; > : 1; 64 6 x 6 76: Because the material composition varies along the x-direction, the object is sliced with planes at different x-heights (planes perpendicular to the x direction and parallel to YZ plane) to obtain the layers. The composition ratios of each layer are gained through the above material distribution and the tool paths for material deposition are determined by the slicing geometric models. 4.3. Example 3 In aerospace applications, in order to reduce the mechanical stress and thermal stress at the same time, metals on the low temperature side and ceramic on the high temperature side are used. Moreover, in order to reduce the high stress concentration at the interfaces of metal and ceramics, a mixture of metal and ceramic with varying graded material are designed. The material composition of turbine blade is varying along the thickness direction of model surface. The material composition ratio and stress property are shown in Fig. 14a. The thickness of cera-
mic part is h1, then, the model of ceramic part (Fig. 14b) is gained through extracting the zero-contour and h1-contour from the signed distance field of the outer surface. For the FDM part (Fig. 14c, its thickness is h2), the geometric model is gained through extracting the h1-contour and (h1 + h2)-contour from the signed distance field of the outer surface. However the material composition can be directly obtained in Eq. (4) because of the consistentfeature material. Finally metal part (Fig. 14d5) is constructed via difference operations (in Section 3.1) of several distance fields: (h1 + h2)-contour (Fig. 14d1) and inner surface (Fig. 14d2–d4). The whole turbine blade (Fig. 14e) is obtained by the union operation of three models (Fig. 14b, c, and d5). 5. Conclusion A novel approach of modeling heterogeneous objects based on multi-color distance field has been developed in this study. Most material modeling applications may be reformulated in a terms of distance fields, which serve as convenient and intuitive parameters for specifying variation in material properties. In this method, the geometry modeling and material modeling are unified with multi-color distance field. The unified distance field can be also
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used to model much more complex heterogeneous object through the extended boolean operations of distance field. Finally, we list some issues that were briefly discussed or not discussed in this paper but need to be addressed in details forming the basis of out ongoing work. (1) Boolean operations for those models with different resolution will be a challenge work in the future research. (2) The precision of geometry modeling must be improved further on the basis of same resolution. (3) The optimization and simulation object’s performances by selecting material parameter is a valuable issue.
Acknowledgements The work described in this paper was supported by a Grant from the National Natural Science Foundation of China (Projects No. 50575177 and 50305027). The correlative members of the projects are hereby acknowledged.
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[5] Shin KH. Representation and process planning for layered manufacturing of heterogeneous object. PhD thesis. University of Michigan, Michigan, USA; 2002. [6] Chen M, Tucker JV. Constructive volume geometry. Comput Graph Forum 2000;19:281–93. [7] Jackson TR. Analysis of functionally graded material object representation methods. PhD thesis, Clemson Univerisity, South Carolina, USA; 2000. [8] Shin KH, Natu H, Dutta D, Mazumder J. A method for the design and fabrication of heterogeneous objects. Mater Des 2003;24:339–53. [9] Siu YK, Tan ST. Source-based heterogeneous solid modeling. Comput Aided Design 2002;34:41–55. [10] Yang Pinghai, Qian Xiaoping. A B-spline-based approach to heterogeneous objects design and analysis. Comput Aided Design 2007;39:95–111. [11] Kou XY, Tan ST. Heterogeneous object modeling: a review. Comput Aided Design 2007;39:284–301. [12] Kou XY, Tan ST. A hierarchical representation for heterogeneous object modeling. Comput Aided Design 2005;37:307–19. [13] Biswas A, Shapiro V, Tsukanov I. Heterogeneous material modeling with distance fields. Comput Aided Design 2004;21:215–42. [14] Tsukanov I, Shapiro V. Meshfree modeling and analysis of physical fields in heterogeneous media. Adv Comput Math 2005;23:95–124. [15] Shin Ki-Hoon, Dutta Debasish. Process-planning for layered manufacturing of heterogeneous objects using direct metal deposition. Trans ASME 2002;2: 330–44. [16] Zhao Hong-Kai, Osher Stanley, Merriman Barry, Kang Myungjoo. Implicit and non-parametric shape reconstruction from unorganized data using a variational level set method. Comput Vis Image Understand 2001;80:295–314. [17] Zeng HF, Liu ZG, Lin ZH. PDE-Driven implicit reconstruction of 3D object. In: Proceedings of the computer graphics, imaging and vision: New Trends (CGIV’05); 2005. p. 251–6. [18] Theisel H. Exact isosurfaces for marching cubes. Comput Graph forum 2002;21:19–31. [19] Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W. Surface reconstruction from unorganized points. In: SIGGRAPH’92 proceedings, vol. 26; 1992. p. 71–8. [20] Bayat Mehdi, Saleem M, Sahari BB, Hamouda AMS, Mahdi E. Analysis of functionally graded rotating disks with variable thickness. Mech Res Commun 2008. doi:10.1016/ j. mechrescom.2008.02.007. [21] Nomura T, Moriguchi H, Tsuda K, Isobe K, Ikegaya A, Moriyama K. Material design method for the functionally graded cemented carbide tool. Int J Refract Met Hard Mater 1999;17:397–404.
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Materials and Design journal homepage: www.elsevier.com/locate/matdes
Heterogeneous object modeling based on multi-color distance field Zhou Hongmei *, Liu Zhigang, Lu Bingheng State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Shaanxi, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 11 May 2008 Accepted 1 July 2008 Available online 5 July 2008 Keywords: Heterogeneous object Multi-color distance field Layered manufacturing Boolean operations
a b s t r a c t With the development of layered manufacturing or rapid prototyping, modeling heterogeneous object with varying material distribution has become more and more important. Different from homogeneous object modeling, the heterogeneous object modeling contains not only geometry and topology information but also material information. In this paper, a new method of modeling heterogeneous object is presented. In this method, firstly, the geometric model of an object is represented by the signed distance field; next, each grid is assigned with material property and material distance field which has the same resolution with geometric distance field are calculated according to material feature; the geometric and material distance field are then unified, named as multi-color distance field. The complex heterogeneous object can also be obtained via the boolean operations of multi-color distance field. For a heterogeneous object represented by the distance field, the most significant advantage is that its geometry model and material information can easily be sliced into a series of 2D layers, so it is convenient for the heterogeneous object to be fabricated using layered manufacturing machine. Several examples show the validity of our modeling method. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Heterogeneous objects [1] are defined to be made up of several kinds of material compositions within an object. Generally, heterogeneous objects can be classified into three types: multi-material objects with clear material boundary [2], functionally graded materials objects [3,4] without clear material boundary and with varying continuously material composition, and composite materials which consist of one or more discontinuous phases distributed in one continuous phase. In order to systemically represent, design and fabricate the heterogeneous objects, a lot of former researchers have introduced many solutions [1–15]. Voxel based model [6] and volume mesh based model [7] describe heterogeneous material distribution through intensive space decompositions with a collection of voxels and sub-volumes/polyhedrons, respectively. Although it is suitable for representing highly heterogeneous objects whose material distributions are strongly irregular, the memory consumptions are a heavy burden [11]. Kumar and Dutta [1,2] employed the extended r-sets (including material composition rmsets) to model and represent heterogeneous objects and Shin et al.[5,8] presented a representation scheme for heterogeneous objects based on the constructive representation. Siu and Tan [9] and Kou and Tan [12] proposed a kind of control feature based model. They made use of the reference feature (like point, line and regular plane) to compute material composition, thus, the * Corresponding author. E-mail address: [email protected] (Z. Hongmei). 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.07.002
method is limited to FGM object with relatively simple geometry and composition gradients. Yang and Qian [10] presented the heterogeneous design and analysis method based on B-spline, however, it depends heavily on spatial parameterizations of control points and such parameterizations are a rather troublesome task for arbitrary 3D objects. Shapiro et al. [13,14] proposed the construction of field modeling using sampling distance field and interpolated physics field by way of extended R-functions, however, the limitations of R-functions results in the limitations in complexshaped heterogeneous object modeling. Apart from the above modeling schemes, some researchers also studied the fabrication of FDM objects, such as solid freeform fabrication (SFF) [3] or layered manufacturing (LM) [4]. LM is a process in which material is selectively deposited, layer by layer, to manufacture an object. When the selective material of each layer is varying material component, a heterogeneous object can be fabricated. Shin and Dutta [15] described a sequential procedure from material modeling using the extended r-sets [1,2] to processing that comprise discretization, building direction selection and adaptive slicing of heterogeneous objects. These modeling methods have mainly focused on simple topology heterogeneous objects with simple gradient material. It is difficult to use these methods to process arbitrary-shaped complex objects. Moreover, these methods have mainly been used to design and fabricate heterogeneous objects based on forward engineering. Therefore, it is necessary to develop a new representation and processing methods for modeling and fabricating 3D complex heterogeneous objects.
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So in this paper, a new method of complex heterogeneous object modeling to synthetically describe the geometry and material information is proposed. The main contributions of this paper are (1) A unified multi-color distance field between geometric model and material model is introduced. Not only the geometry model but also the material information can easily be sliced to fabricate for the object represented by the unified multi-color distance field. Moreover, the method is suitable not only for modeling known CAD model, but also the method may start from arbitrary sampled point cloud. (2) The important advantage of distance field is that arbitrary topology structure object can be modeled and represented. Furthermore, the boolean operations of distance field can generate complex heterogeneous objects through simple max/min operations on the distance field. 2. Heterogeneous objects modeling based on multi-color distance field Implicit function modeling method is widely used in arbitrary topology structure object, such as [16–19]. However, at present the implicit modeling is limited in modeling homogeneous object and few references are found in modeling heterogeneous object. In this paper the implicit modeling method of heterogeneous object is presented. The modeling process is depicted in the flowchart in Fig. 1. At the outset, a model is scanned or sampled to produce a 2D or 3D point cloud. The point cloud is then enclosed by a box which is uniformly subdivided to the prescribed resolution. The signed distance field of the model is then calculated by solving Eikonal equation [16] and iso-surface of the model is extracted by marching cube algorithm [18] for representing the model. At the same time, the material feature is collected by design intent or known material properties and then material distance field is calculated with the same resolution as the above signed distance field. The multi-color distance field is then formed by unifying the geometry field with material field. 2.1. Geometric distance field modeling 2.1.1. Computing unsigned distance field Compute the distance field within a bounding box represented as an axis-aligned uniform 3D grid. Let the size of each voxel in the
Fig. 1. Flowchart of heterogeneous object modeling.
3D Cartesian grid be Nx Ny Nz. The optimal grid size is determined by a specified reconstruction tolerance. For a given point cloud S 2 R3, an unsigned distance function is defined as the function that calculate the distance from point p (grid point) to the closest point x in S (see Fig. 2). The signed distance function is the distance to the boundary oS of the model, and the sign is used to denote inside or outside the model.
dS ðpÞ ¼ signðpÞinf x2oS kx pk;
ð1Þ
where
signðpÞ ¼
1
if p inside the model;
1; otherwise:
2.1.2. Estimating the signs of distance field The calculation of unsigned distance is easy to implement but the sign assignment is very difficult for unorganized point cloud. However, the sign assignment is important for extracting iso-surface of model exactly. There are two commonly used methods to estimate the signs of distance field, that is, oriented tangent plane method [19] and solving Eikonal equation method [16]. Although the former is the most direct method to determine the signs of distance field, the computing process will be problematic and errorprone especially for regions with high curvature. In this paper, the latter is used to estimate the signs. The bounding cube of point cloud S is regularly subdivided into the desired accuracy sub-cubes and the distance function d(x) to S satisfies the Eikonal equation with boundary condition.
jrdðxÞj ¼ 1;
ð2Þ
where, d(x 2 S) = 0. The Eikonal equation is solved using upwind schemes and Gauss-Seidel iterations with various orderings of the sweepings. Details of solving these PDEs can be found in [17]. 2.2. Material distance field modeling and fabricating How to arrange the material distribution is determined by design intent or object performance. There are different material features for different objects. Similar to [13], in terms of the quantity of material feature, heterogeneous objects can be classified into two types: single-material feature and multiple-material feature objects. The single-material feature as the reference datum for defining material distributions throughout the solid is the simplest approach in capturing the designers’ intents. However, a more typical situation with heterogeneous objects involves several material features. Details of describing multiple-material feature object will be discussed in Section 2.2.2. In terms of the relationship between material feature and model surface the heterogeneous objects can
Fig. 2. 2D model (30 40 Cartesian grid).
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also be classified into two types: consistent-feature (in Section 2.2.1) and inconsistent-feature (in Section 2.2.2) objects. To represent the heterogeneous objects, the traditional Euclidean space R3 is extended to a tensor space: T = R3 Mn, where Mn denotes n-dim material space. The vector of material composition: M = [m1,m2 ,. . . , mn], where, mi denotes primary material and P satisfies ni¼1 mi ¼ 1, that is, the total volume fraction of all material composition is summed to 1. In this paper, the material distributions are defined in terms of the distances from the grid points to the material features. The total distance values of each grid point make up of material distance field. There are three major steps involved in calculating material distance field. In the first step, the suitable primary materials and material features are selected for the model to improve its performance. The second step involves calculating distance values from each grid point to material features. In the third step, for the multiple-material object, the several distance fields are blended. In other words, the distance field of material feature is selected as material distribution function, which is added to the above signed distance field of geometry model, then the multi-color distance field is formed. It is described as
(
R3 ! dðSÞ; M n ! FðrÞ;
ðNx Ny NzÞ;
ð3Þ
where, d(S) is the signed distance to model surface and F(r) is the distance to material feature in grid size Nx Ny Nz. 2.2.1. Consistent-feature objects The material feature of consistent-feature object coincides with object surface, that is, the material composition is linear or nonlinear gradients along the object thickness direction. The material distribution is the function of distance field for heterogeneous object that is composed of two materials.
material 1 : f1 ðdÞ ¼ ðadÞk ; material 2 : f2 ðdÞ ¼ 1 ðadÞk ;
) ð4Þ
f2 ðdÞ ¼ 1 di;j;k = maxðdÞ;
FðxÞ ¼
n X N X j¼1
ð5Þ
where di,j,k denotes the distance value from a query point (i, j, k) to outer surface, max(d) is maximum distance value of distance field. The material composition of arbitrary point inside of model can be computed through interpolating the adjacent grid points. The points on each iso-contour extracted from the signed distance field own the same material composition in Fig. 3. 2.2.2. Inconsistent-feature objects The material distance field is easy to be computed for the above consistent-feature objects. However, sometimes the material feature does not coincide with the object surface. In this case, the material distance field from points inside the object to material feature has to be computed additionally. Commonly used material feature is composed of face feature, curve feature, line feature and point feature. Single-material feature is easy to implement like Fig. 4a and b. Multiple-material features based model need blend several material distance fields. The steps are as follows:
xij f ðrij Þ;
ð6Þ
i¼1
where n X N X j¼1
n X N X j¼1
xij ¼ 1;
ð7Þ
f ðrij Þ ¼ 1;
ð8Þ
i¼1
i¼1
where xij denotes the material blending weight or the contribution of Si, N is the total number of material feature of this heterogeneous object, n corresponds to the number of predefined primary materials. The feature weight of the material blending function can be determined in different ways. Designers can directly specify appropriate feature weight based on the design intents and their experiences. Arbitrary prescribed modeling can also be gained through a graphical user interface to input different weight and feature. Fig. 4 is an example for single-feature and multiple-feature. 2.2.3. Fabrication by LM machine Laser-based LM is based on a layer by layer cladding approach and can deposit different materials by changing the powder delivery into the cladding zone as discussed in [5]. Fig. 5a shows the sketch of laser-based LM process. For a 3D heterogeneous object described by multi-color distance field in Eq. (3), the distance field is sliced by the slice planes XY to obtain 2D layers Li(z)
(
where d is normalized distance from a query point to model surface (thickness of the heterogeneous object); a is the scaling factor; k is the distribution exponent to describe gradient property. If k is 1, material property is linear distribution. Usually a and k are determined by design intent or performance test. In Fig. 2, for each grid point (i, j, k) inside of S, the material composition is
f1 ðdÞ ¼ di;j;k = maxðdÞ;
(1) The material feature is defined as Si(i = 1 ,. . . , N). (2) The material field of the jth primary material is calculated from point x inside the object to each material feature, defined as rij. That is, rij = dist(x,Si). Material composition function is the mapping function: f(rij) = mj. (3) The material composition at any point x is
Li ðzÞ ¼
R2 ! di ðSÞ; M n ! F i ðrÞ;
ðNx NyÞ:
ð9Þ
Thus, each layer is defined by a layer thickness and a set of contours each of which refers to its own material composition. For a slice (see Fig. 5b) of a 3D model, the fabrication process using LM machine is as follows: (1) Assume the diameter of the extrusion nozzle is a; (2) Symmetrical contours (Fig. 5b) can be gained through extracting iso-contour of distance field. The space of adjacent contours is a. The tool paths for material deposition will move along these iso-contours; (3) The volume ratios of primary materials are calculated using Eq. (5) and the primary materials are mixed by powder mixer according to the volume ratios.
3. Boolean operations of distance field In current constructive solid geometry (CSG) modeling systems, regularized boolean operations are used to create complex objects from simple shapes (cube, sphere and cylinder etc.). Similarly, boolean operations of distance field need to be defined in order to create and manipulate complex heterogeneous solid models. The boolean operations on object models depend on the operations that are defined for manipulating the geometry models as well as the material models.
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Fig. 3. Consistent-feature objects: (a) Sphere of point feature, (b) Cylinder of line feature, (c) Box of axis feature and (d) Surface of arbitrary curve feature.
Fig. 4. (a–b) Single-feature, (c) Multiple-feature (the weight of point feature is 0.3 and that of line feature is 0.7) and (d) Multiple-feature (the weight of point feature is 0.7 and that of line feature is 0.3).
Fig. 5. (a) Laser-based LM process and (b) Iso-contours of multi-color distance field.
For the purpose of convenient observation, the boolean operations are described in 2D example. Boolean operations of distance field are performed for two simple models: T1 (A, m1) (Fig. 6 a), T2 (B, m2) (Fig. 6b). According to the position between A and B, it can be divided into two cases (as discussed in Sections 3.1 and 3.2, respectively). In the two cases, firstly, a bounding box C which encloses the two models (A and B) is obtained. The bounding box C is then uniformly subdivided into sub-squares and the distance fields (A and B) are calculated, respectively. Note that the distance fields of A and B must be calculated at the same resolution, or the boolean operations between A and B will be a troublesome work. Thus, in order to satisfy the precision require-
ment, the bounding box C has to be subdivided to the prescribed size. 3.1. Case I In Fig. 6c, A \ B = B, the signed distance fields of A and B are calculated, denoted as dA and dB, respectively. The geometry model can be gained through the following boolean operations. Geometric intersection operation:
dA\B ¼ dA \ dB ¼ dB : Geometric union operation:
Fig. 6. (a) Model A, (b) Model B, (c) A \ B = B and (d) A \ B = D.
ð10Þ
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dA[B ¼ dA [ dB ¼ dA :
ð11Þ
(1) If p is inside of D (dA(i,j,k) > 0 and dB(i,j,k) > 0)
Geometric difference operation:
dA\Bði;j;kÞ ¼ minðdAði;j;kÞ ; dBði;j;kÞ Þ;
dA=B ¼ dA =dB :
dA=Bði;j;kÞ ¼ dBði;j;kÞ ;
ð12Þ
Obviously the above boolean operations (intersection and union) need not compute a new distance field. In order to obtain the difference result (Eq. (12)), grid point p (i, j, k) inside of C is divided into three types.
dA[Bði;j;kÞ ¼ minðdðp; L3Þ; dðp; L4ÞÞ;
dB=Aði;j;kÞ ¼ dAði;j;kÞ , where, dðp; L3Þ and dðp; L4Þ denote the distance value from p to L3 and from p to L4, respectively. (2) If p is inside of A and outside of B (dA(i,j,k) > 0 and dB(i,j,k) < 0)
dA\Bði;j;kÞ ¼ dBði;j;kÞ ;
dA[Bði;j;kÞ ¼ minðdðp; L3Þ; dðp; L4ÞÞ;
dA=Bði;j;kÞ ¼ minðdAði;j;kÞ ; jdBði;j;kÞ jÞ;
(1) If p is inside of B (dB(i,j,k) > 0)
dB=Aði;j;kÞ ¼ dAði;j;kÞ :
(3) If p is inside of B and outside of A (dA(i,j,k) < 0 and dB(i,j,k) > 0)
dA=Bði;j;kÞ ¼ dBði;j;kÞ : (2) If p is outside of B and inside of A (dA(i,j,k) > 0 and dB(i,j,k) < 0)
dA=Bði;j;kÞ ¼ minðdAði;j;kÞ ; jdBði;j;kÞ jÞ; where min(dA(i,j,k),jdB(i,j,k)j) denotes the smaller value between dA(i,j,k) and jdB(i,j,k)j. (3) If p is outside of A (dA(i,j,k) < 0)
dA=Bði;j;kÞ ¼ dAði;j;kÞ : All above distance values of grid points are combined together and a new distance field of difference operation is formed in the Fig. 7a. The material boolean operator () is consistently applicable to modeling operations such as material intersection, union and differm1 in Fig. 7b, ence. The material difference operation of A and B is dA=B where m1denotes its material composition. The material intersecm1 m2 in Fig. 7c, where m1 m2 denotes tion operation of A and B is dA\B the grading varying material component. The material union operm1 m2 m1 m2 m1 m1 m2 in Fig. 7d, where dA[B ¼ dA=B þ dA\B . Of ation of A and B is dA[B course, the new geometry model may redistribute new material feature to model new material composition according to user’s requirement.
dA\Bði;j;kÞ ¼ dAði;j;kÞ ;
dA[Bði;j;kÞ ¼ minðdðp; L3Þ; dðp; L4ÞÞ;
dA=Bði;j;kÞ ¼ minðdðp; L1Þ;
ðp; L3ÞÞ; dB=Aði;j;kÞ ¼ minðdBði;j;kÞ ; jdAði;j;kÞ jÞ:
(4) If p is outside of A and B (dA(i,j,k) < 0 and dB(i,j,k) < 0)
dA\Bði;j;kÞ ¼ minðdðp;L1Þ;dðp;L3ÞÞ;dA[Bði;j;kÞ ¼ minðdðp;L3Þ;dðp;L4ÞÞ; dA=Bði;j;kÞ ¼ minðdðp;L1Þ;dðp;L3ÞÞ;dB=Aði;j;kÞ ¼ minðdðp;L2Þ;dðp;L4ÞÞ: All above distance values of grid points are combined together and the results are shown in Fig. 8. Then the material information is incorporated into the above m1 and geometry model. The material difference operations are dA=B m2 dB=A between A and B in Fig. 9a-b, where m1 and m2denote their material compositions. The material intersection operation of A m1 m2 in Fig. 9c, where, m1 m2 denotes the grading varyand B is dA\B ing material component and the boundary L1 and L2 are material m1 m2 in Fig. features. The material union operation of A and B is dA[B m1 m2 m1 m2 9d, which comprise three parts: dA=B , dB=A and dA\B , where, the material feature of intersection section is boundary L2. 4. Engineering applications
3.2. Case II 4.1. Example 1 In Fig. 6d, A \ B = D, the two models are subdivided into L1, L2, L3 and L4. The signed distance fields of A and B are calculated, defined as dA and dB, respectively. The geometry model can be gained through the following boolean operations. Geometric intersection operation:
dA\B ¼ dA \ dB ¼ dD :
For a homogeneous gear there is always a possibility that the heat from the outer surface transmits to the shaft causing adverse effects on its function and efficiency as discussed in [20]. To prevent heat from being transferred to the shaft and improve the
ð13Þ
Geometric union operation:
dA[B ¼ dA [ dB :
ð14Þ
Geometric difference operation:
dA=B ¼ dA =dB ;
ð15Þ
dB=A ¼ dB =dA :
ð16Þ
In order to obtain the above operation results (Eqs. (13)–(16)), grid point p (i,j,k) inside of C is divided into four types.
Fig. 8. Geometric boolean operations: (a)dA[B, (b) dA\B, (c) dA/B and (d) dB/A.
m
m m2
1 1 Fig. 7. Material boolean operations: (a) dA/B, (b) dA=B , (c) dA\B
m m2
1 and (d) dA[B
.
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m
m
m m2
1 2 1 Fig. 9. Material boolean operations: (a) dA=B , (b) dB=A , (c) dA\B
m m2
1 and (d) dA[B
.
performance of the gear, the gear can be made of FGM with ceramic at the outer surface for its high hardness and fine wearing resistance properties, metal at the inner surface for its fine mechanical properties. The point cloud (Fig. 10b) can be gained through scanning the surface of spur gear (Fig. 10a) using coordinate measuring machine Leitz PMM866. The signed distance field (Fig. 10c) can be obtained by the above geometry modeling method (in Section 2.1). The material feature is the gear axis and the material ratio is shown in Fig. 11a. Thus, the inconsistent-feature object needs to compute the material field (Fig. 11b) from a query point inside the object to gear axis. Then, the layers (Fig. 11c) can be gained by slicing the material field model. The gap between two adjacent slices defines the layer thickness.
shank interface [21], and which generates stress concentration near such interface. Furthermore, this stress concentration could lead to the undesired functional loss of tool. Thus, the FGM tool is designed to improve the stress concentration. Fig. 12 shows a heterogeneous boring cutter tool. The FGM graded region is inserted between shank and the tip in order to enforce the material composition continuity. The object is partitioned into a cemented carbide cutter and a shank that is manufactured from FGM of steel and cemented carbide. Assume that the material composition of the cutter tool is varied with explicit function along the x-direction. The modeling of inconsistent-feature material field is gained through computing the distance from the query point to the left face feature. The two kinds of distribution are as follows: (1) Linear material distribution (Fig. 13a)
8 0 6 x 6 40; > < 1; Steel : 1 ðx 40Þ=24; 40 < x < 64; > : 0; 64 6 x 6 76: 8 0 6 x 6 40; > < 0; Cemented carbide : ðx 40Þ=24; 40 < x < 64; > : 1; 64 6 x 6 76: (2) Parabolic material distribution (Fig. 13b)
4.2. Example 2 As is widely known, conventional tools used in metal or glass cutting machineries are manufactured by welding together tip and shank of different materials. As a result, there is the sharp discontinuity in the material composition distribution across the tip-
8 0 6 x 6 40; > < 1; Steel : 1 ððx 40Þ=24Þ2 ; 40 < x < 64; > : 0; 64 6 x 6 76:
Fig. 10. (a) Model of a spur gear, (b) Point cloud, (c) Geometric distance field and (d) Slices of (c).
Fig. 11. (a) Material distribution function, (b) Multi-color distance field model and (c) Slices of (b).
Fig. 12. Heterogeneous boring cutter tool.
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Fig. 13. (a) Material field modeling of linear material distribution and (b) Material field modeling of non-linear material distribution.
Fig. 14. (a) Material distribution of turbine blade, (b) Ceramic part, (c) FGM part (d1) (h1 + h2)-contour (d2–d4) inner surface (d5) Metal part (difference operations: d1/d2/d3/ d4) and (e) The union model of (b), (c) and (d5).
8 0 6 x 6 40; > < 0; 2 Cemented carbide : ððx 40Þ=24Þ ; 40 < x < 64; > : 1; 64 6 x 6 76: Because the material composition varies along the x-direction, the object is sliced with planes at different x-heights (planes perpendicular to the x direction and parallel to YZ plane) to obtain the layers. The composition ratios of each layer are gained through the above material distribution and the tool paths for material deposition are determined by the slicing geometric models. 4.3. Example 3 In aerospace applications, in order to reduce the mechanical stress and thermal stress at the same time, metals on the low temperature side and ceramic on the high temperature side are used. Moreover, in order to reduce the high stress concentration at the interfaces of metal and ceramics, a mixture of metal and ceramic with varying graded material are designed. The material composition of turbine blade is varying along the thickness direction of model surface. The material composition ratio and stress property are shown in Fig. 14a. The thickness of cera-
mic part is h1, then, the model of ceramic part (Fig. 14b) is gained through extracting the zero-contour and h1-contour from the signed distance field of the outer surface. For the FDM part (Fig. 14c, its thickness is h2), the geometric model is gained through extracting the h1-contour and (h1 + h2)-contour from the signed distance field of the outer surface. However the material composition can be directly obtained in Eq. (4) because of the consistentfeature material. Finally metal part (Fig. 14d5) is constructed via difference operations (in Section 3.1) of several distance fields: (h1 + h2)-contour (Fig. 14d1) and inner surface (Fig. 14d2–d4). The whole turbine blade (Fig. 14e) is obtained by the union operation of three models (Fig. 14b, c, and d5). 5. Conclusion A novel approach of modeling heterogeneous objects based on multi-color distance field has been developed in this study. Most material modeling applications may be reformulated in a terms of distance fields, which serve as convenient and intuitive parameters for specifying variation in material properties. In this method, the geometry modeling and material modeling are unified with multi-color distance field. The unified distance field can be also
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used to model much more complex heterogeneous object through the extended boolean operations of distance field. Finally, we list some issues that were briefly discussed or not discussed in this paper but need to be addressed in details forming the basis of out ongoing work. (1) Boolean operations for those models with different resolution will be a challenge work in the future research. (2) The precision of geometry modeling must be improved further on the basis of same resolution. (3) The optimization and simulation object’s performances by selecting material parameter is a valuable issue.
Acknowledgements The work described in this paper was supported by a Grant from the National Natural Science Foundation of China (Projects No. 50575177 and 50305027). The correlative members of the projects are hereby acknowledged.
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