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CG Reprentation of Wood Aging with Distortion, Cracking and Erosion
Abstract
Materials expod to the elements change in appearance becau of aging. Becau wood is an organic substance, cracks and the surface erosion occur easily. To produce realistic computer graphic images, we need simulate the aging phenome-non also. Here, we propo a visual simulation of the distortion, cracking, and erosion of wood. In this method, wood is reprented by a tetrahedral mesh. By tting mi-physical variables at each vertex in this mesh, a visual simulation of wood aging can be accomplished. The surface of the wood is defined by values assigned to the superficial tetrahedral mesh vertices. Changes in the surface are achieved by value changes. The effectiveness of this method is demonstrated by ap-plications on a plank and shapes such as a bunny and an armadillo statue.
Key words : Visual Simulation, Wood, Aging, Crack, Erosion.
1 Introduction
Natural factors such as light, wind, water, and biological activity influence expod objects. The expression of phenomena such as weathering, erosion, corrosion, rust, and dirt can be conducted using computer graphics (CG). Wood ud to make buildings and tools can develop cracks, become stresd by weathering, and attacked by blight. Figure 1 shows an example of aged wood. The caus of wood deterioration are complex and are not completely understood.
Visual simulations of material changes have been re-ported recently. Dory [4] reprented a surface as a -ries of layers, and simulated the patina. Dory [3] also reprented a surface as a slab structure to simulate weathered stone. The two studies mainly investigated changes on the surface of the materials. Recently, more basic distortions and changes on the surface of the object have been reported. Paquette [7] simulated paint cracking and peeling. Yin [12] propod a method for simulating phenomena like color variations involved in the weather-ing of wood. However, the studies investigated changes involving superficial changes of the materials, not major deformation or internal cracks.
Some studies involved visual simulation of cracks in an object. O'Brien [6] ud a finite element model to gener-ate cracks in the object with a tetrahedral mesh and simu-lated animation of the brittle fracture. Cracks initiate and propagate in the object by dividing into multiple tetrahe-dra. Gobro
n [5] propod a 3D surface cellular automata method. First, this method t a simple mi-physical variable on the surface, and then let the crack grow ac-cording to this variable. This method can simulate
Xin Yin Iwate University yinxin@cis.iwate-u.ac.jp
Tadahiro Fujimoto Iwate University
fujimoto@cis.iwate-u.ac.jp医学hr
Norishige Chiba Iwate University nchiba@cis.iwate-u.ac.jp
Figure 1:
Example of aged wood.
主题酒吧Figure 2: Visual simulation process. Crack surface and Eroded surface are computed individually, and then combined into Aged surface with cracks .
superficial cracks of many materials. In this paper, we ef-ficiently simulate crack surfaces in a 3D object using a mi-physical method.
The visual simulation process of wood changes was divided into two stages, as shown in Figure 2. One simu-lates distortions leading to cracks in the wood. The other simulates wood surface erosion. The wood is reprented with a tetrahedral mesh. To simulate the distortions and cracks, a tetrahedral mesh compod of big tetrahedra was ud to decrea the computing cost, which is explained in detail in ction 4. To simulate erosion, a superficial tetrahedral mesh compod of sma
ll tetrahedra was ud, as explained in ction 5. Finally, aged wood is obtained from combining the curved surface that shows the crack with the curved surface that shows erosion.
2 Background Knowledge
Many studies have been conducted in the field of wood rearch (Asano [1], Takahashi [9], Yaga [11]) that supply an introduction to the knowledge of wood.
The cells of a tree trunk are arranged in the direction of the trunk axis and in the radial direction. Therefore, the study of wood requires the consideration of the three main axes of the wood. A tree trunk form in a coordinate sys-tem is shown in Figure 3, indicating the three standard directions: trunk axis or fiber direction L, radial direction R that pass through the tree's core, and tangent direction T along the annual ring. The ction perpendicular to the trunk axis is designated the cross ction. The ction parallel to the trunk axis that pass through the tree's core is called the radial ction. The ction that follows the circumference of the trunk is called the tangential c-tion. 2.1 Distortion and Crack of Wood
The moisture content of the wood varies greatly according to the surrounding temperature and humidity conditions. Cracks occur during the wood drying process. Wood shrinkage shows remarkab
le anisotropy that is influenced by internal components and organization of the wood. Shrinkage in the tangential direction is largest, approxi-mately 3.5-15%, while shrinkage in the radial direction is 2.4-11%, and in the fiber direction is 0.1-0.9% (Takahashi [9]). Shrinkage differs considerably depending on tree species, the specific gravity, and other factors. In general, the anisotropy of the tangential and radial directions is larger in low specific gravity material.
Wood strength depends on the direction of the cut. Strength weakens in the order of fiber, radial, and tangen-tial direction. In general, when wood dries, the vertical direction of the generated crack surface is in the direction of the tangent. That is, the wood crack occurs along the radial ction of the wood.
2.2 Deterioration of Wood
In general, wood deterioration occurs by microbial, water, heat, light, radiation, air, chemical, and mechanical action. Deterioration caud by sunlight, wind, and rain is called weathering. Through weathering, wood first discolors, the soft organization of the surface decompos, and hard portions of the wood are expod. Then, small cracks ap-pear that eventually extend across the entire surface, causing fractures at the end of the wood pieces.
3 Tetrahedral Mesh
Many studies utilize a tetrahedral mesh to reprent an object (Schöberl [8], Tsuji [10]). In a visual simulation, tetrahedron size uniformity influences the results. To minimize this influence, a tetrahedral mesh with a high degree of uniformity should be employed. In nature, crystal structures with a high degree of uniformity are formed by cloly packed spheres in 3D space. Therefore, we constructed a tetrahedral mesh consisting of a cloly packed spherical structure.
Figure 4 shows the 3D space created by cloly packed spheres. In the first layer (broken line in figure), six spheres are arranged around a single sphere. A sphere in the cond layer (solid line in figure) has three adjacent spheres in the first layer. A sphere in the third layer is lo-cated in a position different from tho in the first and cond layers. The structure generated in this manner is called a cubic cloly packed structure (Tsuji [10]). If ad-jacent sphere cores are connected by a line, the 3D space is asmbled into a tetrahedron and octahedron. If one octahedron is divided into four tetrahedrons, the space is reprented by a tetrahedral mesh.
Figure 3: Three main axes L, R and T, and three main ctions Cross ction, Radial ction and Tangent c-tion.
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An object is then placed in this space. The tetrahedra inside the object are called the internal tetrahedra. Tetra-hedra adjacent to the surface of the object are called su-perficial tetrahedra. We ud this total tetrahedral mesh to simulate distortion of and cracks in wood. The superficial tetrahedral mesh is ud to simulate erosion.
4 Visual Simulation of Distortion and Cracks
Wood contain branches. To simulate distortion and cracks, the wood is divided into three regions. Figure 5(c) shows the ction that pass through the trunk axis and branch axis. Part µ1 is the region of the trunk, while part µ2 is the region of the branch. Part µ3 is the region between the trunk and branch (shadowed area in figure).
4.1 Movement Tendency and Crack Tendency
Wood is an orthogonal anisotropic material. The shrink-age, mechanical characteristics, and strength of wood are dependent on the three main axes of wood. The character of wood around a knot is especially complex. This study avoids complex computation. Two variables, called movement tendency and crack tendency, are propod. The movement tendency is a vector that reprents the positional change of each vertex in the wood during dis-tortion. The movement tendency of each vertex in the in-terior of the wood depends on the shrinkage. When wood dries and shrinks, the shrinkage in the tangential, radial, and fiber directions occurs in a ratio of 10:5:0.5 (Yaga [11]). Therefore, in part µ1 and part µ2, the movement tendency in the coordinate system compod of the tan-gent, radial, and fiber directions is t to p = [10S, 5S, 0.5S], as shown in Figure 5(b). S is a parameter that indi-cates the size of the distortion. A large value of S indi-cates a large distortion. Becau shrinkage is inverly
proportional to humidity in wood (Takahashi [9]), we t the value of S inverly proportional to humidity in wood. Since humidity near the surface of object decreas fast, S value in the area near the surface of object is bigger than the interior of the object. Moreover, becau shrinkage in the tangential direction is largest and there is no tangential direction on truck core and branch core, the distortion in the area around the trunk core and branch core is small, so the S values of the area around the trunk axis and branch axis are smaller than the values of the other areas. For part µ3, p1 reprents the movement tendency according to the three main axes of the trunk; p2 reprents the movement tendency according to the three main axes of the branch. Thus, the movement tendency p3 of a vertex in part µ3 is obtained by:
2)1(13p p p αα−+= (1) where α is a coefficient in the range of 0 to 1. As shown in Figure 5(c), this coefficient is d2/(d1+d2). Here, d1 is the distance from part µ1, and d2 is the distance from part µ2. To obtain a more natural and realistic result, noi is added to the movement tendency determined by expres-sion (1).
素菜做法大全The crack tendency is a normal vector of the crack sur-face generated during wood distortion. Wood generates cracks along the radial ction as described in ction 2.1. Therefore, similar to f shown in Figure 5(a), the crack tendency in part µ1 and part µ2 is t according to the tangent direction. In
part µ3, we u f1 to reprent the crack tendency according to the three main axes of the trunk; f2 reprents the crack tendency according to the three main axes of the branch. Then, the crack tendency f3 of a vertex in part µ3 is obtained by:
2)1(13f f f αα−+= (2) where α is the same as in expression (1). Noi is added to
Figure 4: Construction of the cubic cloly packed structure. The spheres in the first layer are arranged as broken line and the spheres in the cond layer are ar-ranged as solid line. A sphere in the third layer is located in a position different from tho in the first and cond layers.
First layer
Second layer
(a) (b)
(c)
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the crack tendency obtained by expression (2), which provides the crack tendency of each internal vertex of the wood.
First, the object is distorted according to the movement tendency. Then, cracks are generated where they can oc-cur easily. After a crack is generated, the stress around the crack decreas. Therefore, the crack tendency around the crack surface also decreas. After that, object distortion occurs again and new crack surfaces appear in the tetra-hedral mesh according to new crack tendency values. By repeating this process, the distortion and crack surface of the object develops. Finally, the crack tendency of each vertex in the wood is minimized and the wood shape and cracks stabilize.
4.2 Distortion of Wood
天猫营销策略
The movement of internal wood vertices is influenced by the movement tendency.
The vertex in the tetrahedral mesh is divided into two types: a vertex that has been moved (the fixed vertex), and a vertex that has not been moved (the free vertex). Figure 6 shows an algorithm that mo
ves free vertex D, which becomes fixed vertex D1. Vertices A, B, and C are all fixed vertices adjacent to free vertex D. Vector P is a movement tendency of vertex D, obtained by the method described in ction 4.1. The direction of vector P is from free vertex D to fixed vertex D1. Vector P1 is a projection of vector P according to line DA. Vector P2 is a projec-tion of vector P according to line DB. Vector P3 is a pro-jection of vector P according to line DC. The vector Pt of vertex D is obtained by (P1+P2+P3)/n, where n is the number of fixed vertices adjacent to D vertex (n=3 in this ca). Free vertex D is moved according to vector Pt, and it becomes fixed vertex D1.
The first fixed vertex is the vertex nearest to center of gravity of the object in a tetrahedral mesh. The free vertex adjacent to this fixed vertex moves according to the movement tendency before becoming a fixed vertex. Next, all of the free vertices adjacent to the fixed vertex move according to the movement tendency before becoming
fixed vertices. As this process repeats, all of the vertices of a tetrahedral mesh become fixed vertices. The result is a distorted object.
Becau of the anisotropy of wood, the amount of shrinkage along the tetrahedral edge varies during the drying process. Cracks appear to occur easily where shrinkage is most extreme.
4.3 Cracks in Wood
Cracks in wood occur according to the crack tendency of the tetrahedral mesh. A crack surface is generated verti-cally in the direction of the crack tendency. The magni-tude of the crack tendency of the vertex around the crack surface decreas after the crack is generated. If the mag-nitude of the crack tendency is greater than zero, we say a crack tendency exists. Two positions exist where a new crack can occur: a position of extreme distortion, and the border of a crack that has already been generated.
O’Brien [6] ud a tetrahedron division method, which generated a realistic crack shape. This tetrahedron divi-sion method is applied here.
The vertex in the tetrahedron where a new crack will occur is called the crack vertex. Figure 7 shows the gen-eration of a crack from the crack vertex in a tetrahedron. The tetrahedron shown in Figure 7(a) has one crack ver-tex. Vertex A is a crack vertex. Vector f is the crack ten-dency of vertex A. In this ca, crack surface AEF is per-pendicular to f and is generated by including vertex A. Vertices E and F are border vertices of the crack surface, and are in a position where a new crack is likely to occur. If a crack tendency of the F and E vertices exists, they become new crack vertices. The tetrah
edron shown in Figure 7(b) has two crack vertices. Vertices A and B are crack vertices. Vector f is a crack tendency of vertex A and vector f’ is a crack tendency of vertex B. Crack sur-face ABE is perpendicular to f +f’ and is generated by in-cluding vertices A and B. If the crack tendency of vertex E exists, vertex E becomes a new crack vertex. The tetra-hedron shown in Figure 7(c) has three crack vertices. Vertices A, B, and C are crack vertices. Becau vertices
A, B, and C are clo to one another, the crack tendencies
C Figure 6: Update of the position of a vertex. Vertex
D is moved to position D1 according to the average vector Pt of projection vectors P1, P2 and P3 obtained from origi-
nal movement tendency P.
(a)
(b)
(c)
Figure 7: Cracks growing in tetrahedral mesh. Figures (a), (b) and (c) show the three cas where one, two and three crack tendencies are given in one tetrahedron.
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of the three vertices are very similar. In this ca, the crack surface is plane ABC.
As described by O'Brien [6], a discontinuous tetrahe-dral mesh is generated becau of new edge vertices (such as vertices E and F in Figure 7(a)) that exist after forma-tion of a crack surface. For example, as shown in Figure 7(a), a polyhedron such as that defined by ABDEF, which is not a tetrahedron, can be created. Therefore, if the polyhedron and discontinuous tetrahedron are divided into small tetrahedra, the tetrahedral mesh becomes continu-ous. Through this process, the generated crack surface becomes the surface of a tetrahedron in the tetrahedral mesh. When the tetrahedron is distorted, the crack surface is also distorted.
Problems can ari when the crack surface is generated by the above-mentioned method. First, as shown in Figure 8(a), when one edge is broken more than once, the crack surface may become discontinuous. Plane AEF is a crack surface contain vertex A. Plane HIJ is a crack surface contain vertex H. Becau the directions of the crack ten-dency of vertex A and vertex H are different, the gener-ated crack surface becomes discontinuous on edge BC. To solve this problem, the positions of the vertices J and E are adjusted to their mean position, resulting in the crack surface including vertex A become plane AMF and the crack surface including vertex H become plane HIM. This restores continuity in the crack surface. A cond problem occurs when the crack surface is near a vertex of the tet-rahedron. As shown in Figure 8(a), the distance between vertex F and vertex C may
be very small, making the tet-rahedron in the new generated mesh to be thin. This problem can be avoided by tting a minimum distance between a crack surface and a tetrahedral vertex. A third problem aris when the crack surface grows in a wrong direction, as shown in Figure 8(b). AB is an old crack. A new crack should grow along BC. However, a crack may also grow along BD. A crack along BD is considered ill
growth and is called opposite growth. It is necessary to specify the scope within a new crack can develop to pre-vent opposite growth. The shadowed area in Figure 8(b) reprents the scope of new crack growth.
5 Visual Simulation of Erosion
Yin [12] ud voxel data for a wood erosion visual simu-lation. Here, data from the superficial tetrahedron was ud to express detail on the surface of the object, for simulated erosion. The vertices of the surface tetrahedron are renewed by decreasing the value of superficial tetra-hedral vertices, resulting in exposure of vertices from the interior of the object, thus simulating erosion.
5.1 Erosion Data Structure
什么分明成语
The vertex of superficial tetrahedron obtained in ction 3 is called a surface control vertex. As shown in Figure 9(a), the triangle is a reprentation of the object. Vertices A
and B are two vertices of the superficial tetrahedron. Such surface control vertices can be classified as minus or plus
vertex. Vertex B, which is outside the surface, is a minus
vertex; vertex A, which is inside the surface, is a plus
vertex. Simulation of surface erosion is possible by
changing the values of the two types of vertices.
First, the value of the surface control vertices is deter-mined from the triangular surface area of the original ob-ject. As shown in Figure 9(a), vertex C is at the interc-tion of line AB and the surface triangle. Vertex D is in the
middle of line AB. When distance |AC| is smaller than distance |AD|, the value of the plus vertex A is t to 1-0.5|CD|/|AD| and the value of the minus vertex B is t to 0. However, when distance |AC| is larger than distance |AD|, the value of the plus vertex A is t to 1 and value
(a) (b) (a)
(b) Figure 9: Update of eroded surfaces. (a) Initial values of tetrahedral mesh vertices A and B are t according to the location of interction C of an original surface triangle with line AB. (b) If the decread value of vertex A is lower than 0.5, vertex F becomes a plus vertex. (c) An updated eroded surface is determined from the updated values of vertices.
(c)
儿童素描画
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of the minus vertex B is t to 0.5|CD|/|BD|. Several ad-jacent surface control vertices exist around one surface control vertex. Therefore, the value of one surface control vertex is computed as described above. Then, the values average is determined, which is the original value of the vertex before erosion.
It is possible to convert the surface vertices onto the surface triangles according to the value of the surface control vertex using the method of Doi [2]. This method generates an equivalent value surface from the value of the surface vertex. Concrete coordinates of the surface triangle are obtained by:
2
11v v c
v t −−=
(3)
)(121Cor Cor t Cor Cor −+= (4) v 1 is the value of the plus vertex A and v 2 is the value of the minus vertex B; c is the value of equal value surface (0.5 in this ca). Cor is the coordinate of new triangular vertex E. As shown as Figure 9(c), Cor 1 is the coordinate of plus vertex A and Cor 2 is coordinate of minus vertex B. t is the computed coefficient that reprents the relation among the coordinates of vertex A, B, and E. Such con-version has an error margin between the new and original surfaces. The error margin can be minimized by using a smaller tetrahedron, which increas computational accu-racy.
5.2 Renewal of Surface Control Vertex
The value reduction method is the same as propod by Yin [12]. Value changes are computed from the distribu-tion of water on the surface. Where water collects and the density of wood is low, values
decrea rapidly. As the value decreas, the surface control vertex and superficial tetrahedron are renewed, and new surface of the object is got, which is the eroded surface.
The renewal algorithm of the surface control vertex is explained by the two-dimensional space shown in Figure 9(b). Vertex A is a former plus vertex. A becomes the minus vertex when the value of vertex A decreas to 0.5 or lower. An internal vertex (vertex F in Figure 9(b)) ad-jacent to vertex A becomes a plus vertex on the surface. However, if a plus vertex is not adjacent to former minus vertex B, meaning B is a vertex outside of the object, ver-tex B is removed from the minus vertex group. Thus, the object becomes smaller by renewing a plus vertex and a minus vertex on the surface. This is the process of object erosion.
5.3 Smoothing the Surface
The eroded surface is generated from the data of the plus vertices and the minus vertices by the method described in ction 5.1. However, tetrahedral meshes influence the
牛筋面是什么做的generated surface, and the surface is rough. For better reprentation, a smoothing computation of the surface is needed. Many smoothing surface algorithms exist, in-cluding the simple one ud here. The coordinates of a given vertex are adjusted by averaging the coordinates of the surroundin
g vertices, which results in smoothing of the surface. Becau tetrahedral mesh with small tetrahe-dra is ud to simulate erosion, the distance between ver-tices on the generated surface is small. Therefore, such a smoothness method does not adverly affect surface simulations of the object.
6 Combination of Crack Surfaces and Eroded Surfaces
Becau wood is compod of fibers, there are fibers in the cracks that develop in wood.
We performed calculations on the crack surface and eroded surface parately as introduction above. The crack is applied on the eroded surface by combining the crack surface and the eroded surface. A part of the crack surface cross out of the eroded surface, as shown in Figure 10(a). Thus, the part of the crack surface outside the eroded surface must be removed. First, the interc-tion of the crack surface and the eroded surface is com-puted. The area of the eroded surface near the cross line is removed as shown in Figure 10(b). The hole reprenting the crack on the eroded surface is generated. Then, as shown in Figure 10(c), this hole is filled by a new curved surface that is displayed as a gray color indicating shadow. To reprent the fibers in the crack, thin curved surfaces are drawn using the same color of the wood. The long di-rection on the thin curved surfaces is identical to the di-rection of the fibers.
7 Implementation
This visual simulation was run on a system with an Intel PIII 1.6Ghz CPU, 512 MB memory, and 32MB ATI Rage 128 Ultra display adapter. The wood texture was gener-ated by the method of Yin [12].茨威格的简介
Crack surface
Eroded surface
Crack hole
(a)
(b)
(c)
Figure 10: Combination of crack surfaces and eroded surfaces. (a) A crack surface intercts an ero
ded surface.(b) A crack hole is generated. (c) The crack hole is cov-ered by new surfaces.
New surfaces
