机械CADChapter 4 Modelling of Solids

Chapter 4 Modeling of Solids4.1 IntroductionSo far we have represented objects with wireframe and surface modeling. Another method of geometric modeling is to use vertices, edges and surfaces to define a solid. Solid modeling systems allow users to create, store and manipulate unambiguous models of physical solid objects, In recent years, solid modeling has become the prevalent and favorable tool for applications in design and manufacturing.Some of the major advantages of solid modeling are listed here.(1) Visualization of components in 3D space or in realistic surroundings can be made easier.(2) Solid models can be used as data input to other systems, like FE-systems, integration between solid model and FE-model provides great possibility to cut short the product development process.(3) Components can be machined straight from the files created by these systems.A solid model contains both the geometry data and topological data. Topological data describe the connectivity and associability of the object entities. Solid model generation is often not unique, that is, there are often several different ways to create a solid model. CAD users need to generate solid models which can make the computer storage small and suited for later utilize.Boundary representation (B-rep) and Constructive Solid Geometry (CSG) are two most popular schemes for solid modeling. B-reps are based on the topological notion that an object is bounded by faces. CSG is based on that an object can be divided intoa set of primitives. Table 4.1 shows some widely used CAD systems.4.2 Solid RepresentationMost geometric objects we see every day are solids. Solids can be very simple like a cube or very complex like a piston engine. To be processed by computers, solids must have some representations that can describe the geometry and characteristics completely. In fact, a good representation should address the following issues.①Domain: While no representation can describe all possible solids, a representation should be able to represent a useful set of geometric objects. Domain should give a useful set of physical objects that can be represented.②Unambiguity: A solid should be represented without any doubt. An unambiguous representation is usually referred to as a complete one. Figure 4.1 shows an example of ambiguous solids.③Uniqueness: There is only one way to represent a particular solid. If a representation is unique, then it is easy to determine if two solids are identical since one can just compare their representations.④Accuracy: A representation is said accurately if no approximation is required.⑤Validity: A representation should not create any invalid or impossible solids. More precisely, a representation will not represent an object that does not correspond to a solid.⑥Closure; Solids will be transformed and used with other operations such as union and intersection, "Closure" means that transforming a valid solid always yields a valid solid.⑦Compactness and efficiency: A good representation should be compact enough for saving space and allow for efficient algorithms to determine desired physical characteristics.There are different methods to create models, such as half-space, boundary representation (B-rep), constructive solid geometry (CSG), sweeping, analytic solid modeling, cell decomposition. Following concepts are important in solid modeling: solid primitives, Boolean operations, geometry closure and regularized set operations.(1) PrimitivesSolid primitives are frequently used in solid modeling. CAD/CAM systems provide various primitives such as block, cylinder, cone, and sphere. Figure 4.2 shows an example of these solid entities.(2) Boolean operationsUnion, difference, complement and intersection are essential Boolean operations. Figure 4.3 shows Boolean operations used in the solid modeling. Boolean operations are intuitive to use and easy to understand, and they provide for rapid manipulation of large amounts of data. Because of this, Boolean operations are also used in many non-CSG systems.(3) Interior, exterior and closureWe need the concepts of interior, exterior and closure to understand regularized Boolean operators. Intuitively, the interior of a solid consists of all points lying inside of the solid; the closure consists of all interior points and all points on the surface of solids; and the exterior of a solid is the set of all points that do not belong to the closure.A open ball with centre (a, b, c) and radius r consists of all points that satisfy the following relation,(x-a)2+(y-b)2+(z-c)2<r2A point P is an interior point of a solid S if there exists a radius r such that the open ball with centre P and radius r is contained in the solid S. The set of all interior points of solid S is the interior of S, written as int (S). Based on this definition, the interior of an open ball is the open ball itself.On the other hand, a point Q is an exterior point of a solid S if there exists a radius r such that the open ball with centre Q and radius r does not intersect S. The set of all exterior point of solid S is the exterior of solid S, written as ext(S).Those points that are neither in the interior nor in the exterior of a solid S constitutes the boundary of solid S, written as b(S). Therefore, the union of interior, exterior and boundary of a solid is the whole space.The closure of a solid S is defined to be the union of the interior and the boundary of S, written as closure (S). Or, equivalently, the closure of solid S contains all points that are not in the exterior of S.(4) Regularized set operationsWhen set operations are used in geometry modeling, unwanted geometry may begot. In Fig. 4.4, two cubes touch each other and their intersection is a rectangle shown on the right. A rectangle is not a three-dimensional object and hence not a solid.To eliminate these lower dimensional branches, the three set operations are regularized as follows. The idea is simple.Compute the result as usual and lower dimensional components may be generated. Compute the interior of the result. This step removes all the lower dimensional components.Compute the closure of the result obtained in the above step. This adds the boundary back.Let op* be the regularized operator and let A and B be two solids. Then, A∪* B , A ∩*B and A-* B can be defined mathematically based on the above description.A∪* B - c1osure(int(the set union of A and B))A∩*B - c1osure(int(the set intersection of A and B))A-* B - c1osure (int (the set difference of A and B))Based on this definition, the intersection of the two cubes shown in Fig. 4.4 is empty. The intersection of these two cubes is a rectangle, which is a two-dimensional object and has no interior. Hence, after taking interior (i.g. , int()), we get an empty set, whose closure is also empty. Consequently, the intersection is empty.4.3 Boundary RepresentationBoundary models have a hierarchical format. As mentioned earlier, B-reps describe objects bounded by a set of faces. Each face has its underlying closed (continuous) and oriented surface as shown in Fig. 4.5 and the face is bounded by edges. Each edge is bounded by vertices. A boundary model of an object is comprised of faces, edges and vertices.The surface of a solid consists of a set of well-organized faces, each of which is a piece of some surface (e. g., a surface patch). Faces may share vertices and edges that are curve segments. Therefore, a B-rep is an extension to the wireframe model by adding face information to the later.The orientation of each face is important. Normally, a face is surrounded by a set of vertices. Using the right-handed rule, the ordering of these vertices for describing a particular face must guarantee that the normal vector of that face is pointing to the exterior of the sol-id. Normally, the order is counter clockwise. Therefore, by inspecting normal vectors one can immediately tell the inside and outside of a solid under B-rep. This orientation must be done for all faces.The database of a boundary model contains both topological and geometric data. Topological data provide the relationships among vertices, edges and faces similar to that used in a wireframe model. In addition to connectivity, topological information also includes orientation of edges and faces. Geometric information is usually equations of the edges and faces. Topology is created by performing Euler operations and geometry is created by per-form Euclidean calculations. Both the topological and geometry validity should be checked to avoid nonsense object. Euler operators also provide designers with drafting functionality.Boolean operators are often used to create and edit models. Since B-rep requiresexplicit representation of the boundary of the solid, boundaries must be evaluated after the operation. For CSG model, the Boolean operation is simply an addition to the CSG tree.The main advantage of B-rep is its ability to construct solids that are difficult to build u-sing primitives. The main disadvantage is that it requires large amounts of storage because it stores the boundary explicitly.4.3.1 Euler FormulaObjects that are often encountered in engineering are polyhedral or curved objects. Figure 4.6 shows examples of polyhedral.Eu1er (or Euler-Poincare) law is that a polyhedron is topologically valid if they satisfy the following equation,F-E+V-L=2(B-G) (4.1) Where F, E, V, L, B and G are the number of faces, edges, vertices, faces’ inner loop bodies and genus (handles or through holes) respectively. For a simple polyhedron, we have: F-E+V=2.Euler law can be used to check the object’s validity, the system commands (Euler operators) are based on Euler law and thus ensure the validity simultaneously. Euler law given above applies to closed polyhedral objects. For open polyhedron, we have: P-E+V-L=B-G (4.2)If objects have curved surface, such as cylinders and spheres, Euler law is still valid. Topologically, one can always stretch curved edges and faces so that they become flat without changing the relationships among them.4.3.2 Euler OperatorsOnce a polyhedron model is available, one might want to edit it by adding or deleting vertices, edges and faces to create a new polyhedron. These operations are called Euler operators.Recall from the discussion of the Euler formula that the following holds for all polyhedra:V-E+F-(L-F)-2(B-G) =0 (4.3) Based on this relation, some Euler operators have been selected for editing a polyhedron so that the Euler formula is always satisfied. There are two groups of such operators: the Make group and the Kill group. Operators starting with M and K are operators of the Make and Kill groups, respectively.Euler operators are written as Mxyz and Kxyz for operations in the Make and Kill groups, respectively, where x, y and z are elements of the model (e.g., a vertex, edge, face, loop, shell and genus). For example, MEV means adding an edge and a vertex while KEV means deleting an edge and a vertex. The user is not free to construct faces, edges or vertexes. For example, there are no such operators such as ME, MV or MF as they violate Euler law.Every topologically valid polyhedron can be constructed from an initial polyhedron by a finite sequence of Euler operators. Therefore, Euler operators are powerful operations.4.3.2.1 The Make Group of Euler OperatorsThe Make group consists of four operators for adding some elements into theexisting model creating a new one, and a Make-Kill operator for adding and deleting some elements at the same time. The operators are as shown in Table 4. 2.Table 4.2 shows the change of values of V, E, F, L. B and C. Note that adding a face produces a loop, the outer loop of that face. Therefore, when F is increased, L should also be increased. This new loop and the new face will cancel each other in the sub expression of L-F. Please verify that none of these operators would cause the Euler formula to fail.MBFV is often used to begin constructing the boundary models, so it could be through as the first vertex of the model. Table 4.3 illustrates the way of using Euler operators to construct a tetrahedron, in seven steps or seven Euler operators a tetrahedron is created.MBG simply makes a body with a hole. After this, one can add vertices, edges, faces, and loops. There must be loops, because the new hole penetrates at least one face.MEKL makes an edge and at the same time kills a loop. A commonly used MEKL is adding an edge connecting the outer loop and the inner loop of a face. In this case, the number of edges E is increased by 1and the number of loops L is decreased by 1 since that loop is killed.Higher level Euler operators can be developed, such as MCUBE (create a cube), MCYL(create a cylinder).4.3.2.2 The Kill Group of Euler OperatorsThe Kill group just performs the opposite of what the Make group does. In fact, replacing the M and K in all Make operators with K and M, respectively, would get the operators of the Kill group. Therefore, the Kill group consists of the five operators as shown in Table 4.4.With these operators, one can start with a tetrahedron and reduce it to nothing. These operators are the opposites of the Make operators. The advantages of Euler operators are that they ensure creating valid topology and reasonable is simple. Euler operators offer a mechanism to check the topological validity of the models. However the models are not unique because the boundary of any object can be divided into faces, edges and vertices in many ways. Euler operators are difficult for designers to use and usually not available to users at interface, only exist internal to software. Earlier user interfaces used commands such as "make edge/face" or "kill edge/face", although such interfaces are not convenient. Object-oriented (feature-based) user interfaces are more acceptable by users. For example, command "create hole" is better than "subtract cy-linder".4.3.3 The Winged-edge Data StructureThe winged data structure is useful data structure in B-rep. In this structure, all the adjacency relations of each edge are described explicitly. The winged-edge data structure uses edges to keep track almost everything.For each edge, the following information is important: vertices of this edge, its left and right faces, the predecessor and successor of this edge when traversing its left face, the predecessor and successor of this edge when traversing its right face.Each entry in the edge table contains above information mentioned earlier: edgename, start vertex and end vertex, left face and right face, the predecessor and successor edges for left and right faces. Note that clockwise ordering (viewing from outside of the polyhedron) is used for traverse.Figure 4.9 shows the information for the entry of edge a. The four edges b, c, d and e 1are the wings of edge a an d hence edge a is “winged”.Figure 4.10 is a tetrahedron with four vertices A, B, C and D, six edges a, b, c, d, e and f, and four faces 1,2,3 (back) and 4 (bottom), whose edges are listed in Table 4.5. The winged-edge data structure requires two more tables, the vertex table and the face table. The vertex table has one entry for each vertex, which contains an edge, that is incident to this vertex. The face table has one entry for each face, which contains an edge i.e. one of this face's boundary edges.With this data structure, one can easily answer the question; which vertices, edges, faces are adjacent to each face, edge, or vertex. However, it may take longer time to answer other adjacency queries. Note also that once the numbers of vertices, edges and faces are known, the size of all three tables are fixed and will not change.If some faces of a solid have holes, the above form of winged-edge data structure does not work. There are two ways for resolving this problem:(1) For a face with inner loops, the outer boundary is ordered clockwise, while its inner loops, if any, are ordered counter clockwise as shown in Fig. 4.11.(2) Alternatively, a special auxiliary edge can be us ed to join each hole’s boundary to the outer boundary, as shown in Fig. 4. 12. Since each edge will have the same face on both of its sides, it can be easily identified.4.4 Constructive Solid GeometryA physical object can be divided into a set of primitives and can be combined by Boolean operations. CSG model is based on this principle.CSG is a solid modeling method that combines simple solid shapes called primitives to build more complex models, using the Boolean operators.The database of a CSG model, similar to B-rep, stores its topology and geometry. Topology is created via the regularized Boolean operations that combine primitives. Checking the used primitives can ensure the validity. The geometry stored in the database of a CSG includes configuration parameters of its primitives and rigid motion.A CSG model requires minimum storage to store solid definitions (graphs/trees) , as it does not store explicitly the faces, edges and vertices. It evaluates them whenever they are needed, e. g. generation of line drawings. This makes it slow to retrieve the model because it has to build a boundary. Due to the same reason, it is also slow to generate wireframes.4.4.1 CSG PrimitivesThe standard CSG primitives consist of the block (i.e. cube), triangular prism, sphere, cylinder, cone and torus as shown in Fig. 4. 13. All these primitives have their own local coordinate systems. These six primitives must be instantiated by the user to be used in his/her design. Moreover, the instantiated primitive may require transformations such as scaling translation and rotation to be positioned at the desired place.1. Block. A block is defined by its width, height and depth. The signs of W, H andD can be negative.2. Cylinder. This primitive is defined by its radius (or diameter) R and length H. the length H is usually taken along the direction of the z L axis. H can be positive or negative.3. Cone. The indata to define a cone is its base radius R. top radius (for truncated cone), and height H.4. Sphere. The indata is its radius.5. Wedge. A right-angled wedge is defined by its height H, width W. and base depthD.6. Torus. This primitive is generated by the revolution of a circle about an axis. It can be defined by the radius of its body R1, and the radius of the centre line of the torus body R2.4.4.2 Boolean OperatorsA solid can be considered as the result of applying Boolean operators to a set of instantiated and transformed CSG primitives. However, simple set operators may create problems as discussed in regularized Boolean operators. Let us just use set operations on following example.Let us construct a bracket with a hole, as shown on the right-most of Fig. 4.14. We start with two instantiations of blocks and one instantiation of a cylinder (the left-most figure). These three instantiations are then rotated or/and transformed to their desiredpositions. The final product is obtained by subtracting the cylinder from block 1, and then computing the union with block 2.Please note that the design of the above solid is not unique. For example, one may compute the union of block l and block 2, and then compute the difference.4.4.3 CGS TreeTrees and graphs are used for the CSG representation as in winged-edge structure usedin B-rep.A tree is defined as a data structure accessed from the root node. Each node is either a leaf or an internal node. An internal node has one or more child nodes and is called the parent of its child nodes. Figure 4.15 shows a tree. A binary tree is defined as a tree with at most two children for each node.A CSG tree is defined as an inverted ordered binary tree whose leafs are primitives and interior nodes are regularized set operations.The procedure to build the bracket with Boolean operators can be written as: union(diff(trans(block 1), trans(cylinder)), trans(block2)) , where union(A,B) and diff(A,B) are the union and difference of A and B, and trans( ) indicates appropriate transformations This procedure can be converted to a CSG tree of the design, as shown in Fig. 4. 16.In fact, every solid constructed using the CSG technique has a corresponding CSG tree. The expression of the CSG tree is a representation of the final design.The total number of nodes in a CSG tree of a solid depends on the number of primitives the solid is decomposed to. If a solid has n primitives, then there are (n-1) Boolean operations for a total of (2n-1) nodes in its tree.Balanced trees are preferred in various applications. A balanced tree is defined as a tree whose left and right subtrees have almost an equal number of nodes, that is, the absolute value of the difference (n L-n R) is as small as possible, where (n L+ n R) = 2n-2. The general rule to create balanced trees is to start to build the model from an almost central position and branch out in two opposite directions.To transverse a tree means to pass through the tree and visit each of its nodes. There are mainly three methods to transverse a CSG tree: preorder, in order and postorder. To transverse a tree in preorder, we have he following recursive definition:①Visit the root;②Transverse the left subtree in preorder;③Transverse the right subtree in preorder.Then the reversed preorder is defined by reversing the above operations, that is, visit the right subtree, then the left subtree, and the rootThe order to transverse a tree in inorder is,①Transverse the left subtree in inorder;②Visit the root;③Transverse the right subtree in inorder.And the order to transverse a tree in postorder is,①Transverse the left subtree in postorder;②Transverse the right subtree in postorder.③Visit the rootTo get the reversed inorder and postorder, operations in respective method are reversed.4.5 Sweep GenerationThere are three types of sweep operation: linear, nonlinear and hybrid sweeps. Path in a linear sweep is linear or circular vector describe by a linear parametric equation. Path in a nonlinear sweep is a curve described by a higher-order equation. Hybrid sweep combines linear and/or nonlinear sweep via set operations.Many commands in CAD/CAM software are based on sweeping.4.6 Modeling of Analytic SolidA short introduction of analytic solid is given below. The definition of a parametric solid can be extended from curves and surfaces. Points in the interior or on the boundary of a solid are given by[][])w ,v ,u (z )w ,v ,u (y )w ,v ,u (x z y x )v ,u (P ==,w w w ,v v v ,u u u max min max min max min ≤≤≤≤≤≤ (4.4) As shown in Fig. 4.18 hyperpatch is a geometric entity, which defines a volume in 3Dspace. It is represented internally as a parametric tricubic solid.A tricubic hyperpatch can be given by∑∑∑====30i 30j 30k k j i ijk w v u C )w ,v ,u (P ,1w 0,1v 0,1u 0≤≤≤≤≤≤ (4.5)By substituting proper value of the parameters, we can get face surface, edge curve and corner vertices. For example, P (0,u,w) and P(1,u, w) are two faces, P(u, 0, 0) is an edge and P(0, 0, 0) is a vertex. The tricubic hyperpatch has the same disadvantages of cubic curve and bicubic surface, and it even requires the input of w v u /P 3∂∂∂∂.A tricubic Bezier Hyperpatch is defined by the following equation,∑∑∑====30i 30j 30k k j ijk )w (B )v (B )u (Bi P )w ,v ,u (P (4.6)4.7 Solid Manipulation(1) DisplayingThere are two methods to display a solid; wire display and shaded display. Shaded display provides realistic visual images. Shaded images can be generated from B-reps by visible-surface algorithms. Shading can be performed from CSG by means of ray-casting or z-buffer algorithms. More on shading will be discussed in Chapter 5.(2) Evaluating points, curves and surfaces on solidEvaluating points and curves on solids can be viewed as intersection problem. Points and curves are generated by finding curve/solid and surface/solid intersections. Evaluating surfaces on a solid can be regarded as extracting the underlying surfaces of the solid faces. These surfaces can be bonded by the proper solid edges oruser-defined boundaries.(3) SegmentationSegmentation of a solid is to split it into two or more valid subsolids. Each resulting subsolid should have its own topology and geometry, In a B-rep model, new vertices, edges and faces are created, In a CSG model, splitting the solid would also require splitting its CSG graph and tree.(4) Trimming and intersectionTrimming a solid is using trimming boundaries to split a solid and removing the portions outside these boundaries. Intersection can be performed by Boolean operations.(5) TransformationSolids can be translated, rotated by using homogeneous transformation.(6) EditingEditing a solid involves changing its existing topological and geometrical information. For CGS models, graphs and trees are used to edit the model. For B-rep model, a graph or tree can be created and modified.。