Boolean operations on multi-region solids for mesh ... - Springer Link
Engineering with Computers (2012) 28:225–239 DOI 10.1007/s00366-011-0228-8
ORIGINAL ARTICLE
Boolean operations on multi-region solids for mesh generation Andre´ Maue´s Brabo Pereira • Marcos Chataignier de Arruda Antoˆnio Carlos de O. Miranda • William Wagner M. Lira • Luiz Fernando Martha
•
Received: 17 February 2010 / Accepted: 10 June 2011 / Published online: 28 June 2011 Springer-Verlag London Limited 2011
Abstract An algorithm for Boolean operations on nonmanifold models is proposed to allow the treatment of solids with multiple regions (internal interfaces) and degenerate portions (shells and wires), in the context of mesh generation. In a solid modeler, one of the most powerful tools to create three-dimensional objects with any level of geometric complexity is the Boolean set operators. They are intuitive and popular ways to combine solids, based on the operations applied to point sets. To assure that the resulting objects have the same dimension as the original objects, without loose or dangling parts, a regularization process is usually applied after a Boolean operation. In practice, the regularization is performed classifying the topological elements and removing internal or lowerdimensional structures. However, in many engineering applications, the adopted geometric model may contain idealized internal parts, as in the case of multi-region models, or lower-dimensional parts, as in the case of solids
that contain dangling slabs that are represented as zerothickness surfaces or wireframes in the model. Therefore, the aim of this work is the development of a generic algorithm that allows the application of the Boolean set operations in a geometric modeling environment applied to finite and boundary element mesh generation. This environment adopts a non-manifold boundary representation that considers an undefined number of topological entities (group concept), and works with objects of different dimensions and with objects not necessarily plane or polyhedral (parametric curved surfaces). Numerical examples are presented to illustrate the proposed methodology. Keywords Geometric modeling Non-manifold solids Boolean operations Boundary representation Mesh generation Degenerate portions
1 Introduction A. M. B. Pereira (&) Departamento de Engenharia Civil, Universidade Federal Fluminense, Rua Passo da Patria 156, Niteroi 24210-240, Brazil e-mail: [email protected]; [email protected] A. M. B. Pereira M. C. de Arruda L. F. Martha Departamento de Engenharia Civil, Pontifı´cia Universidade Cato´lica do Rio de Janeiro, Av. Marqueˆs de Sa˜o Vicente 225, Rio de Janeiro 22453-900, Brazil e-mail: [email protected] A. C. O. Miranda Departamento de Engenharia Civil e Ambiental, Universidade de Brasilia, Campus Universitario Darcy Ribeiro, Brasilia 70910-900, Brazil W. W. M. Lira Centro de Tecnologia, Universidade Federal de Alagoas, Campus A. C. Simo˜es, Maceio´ 57072-970, Brazil
The creation of geometric models in the context of mesh generation is an essential task when dealing with the numerical simulation of real engineering objects (Fig. 1). Numerical simulation techniques as the finite element method (FEM) or the boundary element method (BEM) require discrete models (i.e., domain meshes or boundary meshes, respectively) that are based on the geometric description of the model’s domain. In addition, the construction of the discrete models usually involves combinations of several solids that can still contain other objects of lower dimensions (such as shells and wires), resulting in complex multi-region solids. A natural manner to construct multi-region solids for mesh generation is through Boolean set operators (union, intersection and difference), which are essential tools in any solid modeling system.
123
226
Fig. 1 Real engineering objects containing multiple regions (internal interfaces) and degenerate portions (shells and wires)
Constructive solid geometry (CSG) and boundary representations (B-Rep) are traditional mathematical modeling schemes for representing the geometric shape of homogeneous objects, referred to as two-manifold solids [1]. However, these traditional schemes do not handle heterogeneous objects (multi-region solids) or models with degenerated portions. Non-manifold representations are strategies to deal with this class of problem. A modeler with a non-manifold environment allows Boolean operations to generate realistic FEM and BEM discrete models. In this representation, solids with multiple regions (internal interfaces) and degenerate portions (shells and wires) may be represented in a consistent way. These are situations that occur, e.g., in the problem illustrated in Fig. 1, which contains several materials, internal boundaries, fault surfaces, tunnel support surfaces and anchor bolts. Nevertheless, the implementation of a system that provides a non-manifold environment is a very complex task. 1.1 State of the art and literature review As already mentioned, two different strategies have been traditionally proposed to model homogeneous solid objects. In the first strategy, one aims to represent a solid through an explicit description of its boundary, which is the so-called boundary representation (B-rep) [2]. It is based on data structures that describe the adjacency relationships of the vertices, edges and faces of a solid. The other possibility is the CSG approach [3], which consists of representing a solid as a result of a sequence of set operations (union, intersection, and difference) performed on simple primitive solids. B-rep and CSG have complementary advantages and disadvantages [4], and much work has been carried out to convert from CSG to B-rep [5] and from B-rep to CSG [6].
123
Engineering with Computers (2012) 28:225–239
B-rep modeling is a natural way of constructing a solid that is defined by its bounding surfaces. Although complex objects can be described, in principle, by their defining surfaces, in many cases this is not a natural way of construction. In practice, modeling systems based on B-rep are usually capable of performing set operations, which enables one to use CSG-like operations to define solids, while retaining the full boundary representation. A topologically complete set operations algorithm for planar polyhedra two-manifold objects was described by Ma¨ntyla¨ [7], introducing the vertex neighborhood classifier for Boolean operations. Gardan and Perrin [8] proposed an interesting method for reducing 3D Boolean operations to a 2D problem in manifold environments. However, traditional B-rep and CSG techniques apply when one just needs to look at an object as inducing a three-part space decomposition: its interior, its exterior and its boundary. For many interesting applications, such as modeling of heterogeneous objects [9–13], this is not enough. In addition, in many engineering applications, the adopted geometric model may contain idealized lowerdimension parts, as in the case of finite element and boundary element systems [14]. These types of systems need the support of a unified and simultaneous representation of wires (e.g., beam elements), shells (e.g., concrete slabs that are represented as zero-thickness surfaces) and solid parts (e.g., foundation blocks). In these situations, one would like to have not only a representation of each part of the model, but also a description about the way these parts are connected to each other. Techniques that models heterogeneous objects and objects with lower-dimension parts are called non-manifold modeling techniques [1, 15, 16], because these objects are not necessarily two-manifolds embedded in threedimensional space. Traditional modelers (based on traditional B-rep and CSG modeling techniques) could only represent two-manifold objects, i.e., objects in which every point on the boundary has a neighborhood that is homeomorphic to an open two-dimensional disk [1]. Since the late 1980s, the advantages of non-manifold geometric modeling have been recognized [17] and several representation schemes for non-manifold objects have been developed [18–22]. The mathematical definition of nonmanifold objects has also been presented. Masuda [16] defined non-manifold geometric models as cell complexes that are subsets of 3D Euclidian space, and provided extended Euler operators for this type of model. The concept of cell complexes non-manifold geometric model is suitable, because the model may contain wires, shells and solid parts, or combinations of them. Arbab [23] presented an alternative CSG-like formalism based on open sets (called s-sets) for assemblies, i.e., objects with several solid
Engineering with Computers (2012) 28:225–239
parts or with internal boundaries, and formalized Boolean operations for heterogeneous objects (s-sets). One strategy for modeling non-manifold objects is the direct manipulation of cells. Rossignac and O’Connor [19] have proposed the concept of geometric complex (GC), in which a non-manifold object is decomposed in disjoint portions called cells, each one homogeneous in dimension and satisfying the condition that the intersection of the boundaries of any two cells is necessarily equal to the union of other cells of the decomposition. Cavalcanti et al. [24] generalized this concept and presented an approach based on spatial subdivision for non-manifold modeling. In this approach, any object is represented by a subdivision of the entire Euclidian space (including the external infinite region). This representation is called complete geometric complex (CGC) and was implemented using the radial edge data structure proposed by Weiler [20]. An incremental approach for CGC creation considers the inclusion of a patch of surface at a time, with the corresponding updating of the data structure after each insertion. Similarly, curve segment and vertices are inserted incrementally. The insertion of a new cell (surface patch, curve segment, or vertex) is a fundamental operation executed by the spatial subdivision scheme and may result in the refinement of new and existing cells due to geometric intersection. New solid regions (which are also cells in the decomposition) may result from surface patch insertions, as well as new surface patches may result from curve insertions. Cavalcanti et al. [24] have chosen CGC as the mathematical model to describe a spatial subdivision. This approach provides a basis for representing complex objects in two steps: first, a subdivision phase and, second, a phase with selection of active cells. The resulting object is called an selective complete geometric complex (SCGC). For instance, in a finite element analysis of a solid with a cavity, the infinite external region and the cavity would be inactive cells. Another strategy for creation of non-manifold models is through Boolean operations. Gursoz et al. [17] proposed a scheme for implementing Boolean operations based on the classification of six types of intersection pairs: vertex– vertex, vertex–edge, edge–edge, vertex–face, edge–face, and face–face. All of the intersections take place between pairs of elements that belong to two different objects. After having obtained a merged object, any Boolean operation is achieved by selecting the appropriate point set elements in the object. A more general process was introduced by Rossignac and Requicha [25]. They introduced a new class of objects, called constructive non-regularized geometry (CNRG) objects, and a family of operators that act on these objects. A CNRG tree is a rooted directed acyclic graph and represents an object. The leaves are the CNRG primitives.
227
Internal nodes represent intermediate CNRG objects obtained by applying certain operations to their children. The operators are: aggregation, unification, sum, product, subtraction, complementation, interior, closure, boundary and regularization. CNRG operations provide a modeling process suitable for user interaction. When followed by a unification operator, which creates an object having only one component, they behave exactly as the corresponding point-set operators. Without unification, however, they return an aggregate (or assembly) of pair-wise disjoint components. Cavalcanti et al. [24] advocated that space subdivisions corresponding to CNRG objects can be implicitly represented by the corresponding tree, in the way as CSG trees can be used to represent B-rep solids. In this case, the CNRG operators are used only as a modeling process, as these operators provide a convenient way to define topological and Boolean operations for SCGC objects. Indeed, an SCGC can be seen as a CNRG whose components are its active cells. This allows one to use CNRG operators to build and modify SCGCs. In this context, these authors used the CNRG operators to create heterogeneous objects represented by SCGCs. They proposed a scheme to automatically determine the material to be attributed to each region based on region naming and a ‘‘dominance factor’’, which establishes the material that predominates in a combination operation. In the last decade, some attention has been given to the use of Boolean set operation for the creation of nonmanifold objects in the context of feature-based modeling and modeling of heterogeneous objects, including objects with functionally graded materials. Magalha˜es and Mesquita [9] proposed a solution for handling inner boundaries in the context of mesh generation for electromagnetic problems, assuring the design of manufacturable models. Their work is based on the s-set CSG-like formalism proposed by Arbab [23], and the implementation adopts the radial edge data structure of Weiler [20]. The study of Gardan et al. [26] describes the use of Boolean operations of feature-based models. As mentioned by these authors, feature-based modeling and Boolean operations are closely related areas, as such operations are involved in feature instantiation or modification. Boolean operators may also be used to combine two feature-based models. In both situations, the modeling process is based on the following sequence of statements: calculate the intersections, combine, classify and extract entities (faces, edges and vertices) of the operands. Following an approach similar to the use of a ‘‘dominance factor’’ [24], Sun and Hu [10] proposed reasoning Boolean operations for modeling heterogeneous objects. The reasoning Boolean operation algorithm consists of a merging operation and an extraction operation, and is
123
228
executed according to the material-dominant information extracted from the designed heterogeneous object database. In their work, they perform several geometric operations prior to the material classification. A very comprehensive work in this area is the one written by Qian and Dutta [11]. These authors proposed the direct face neighborhood alteration method to handle the two main tasks involved in the constructive operations of heterogeneous objects with functionally graded material: geometric boundary evaluation and material region forming. In a sense, the methodology devised by these authors generalizes and extends the use of region naming and material-dominant information in heterogeneous object modeling. They define mathematically three operations that manipulate material volumes: additive, subtractive and partition operations. These operations take into account both geometric properties and combined material behavior of each volume. A similar approach is adopted by Kou et al. [12]. However, the main difference is that these authors also consider smooth transitions of material properties throughout the complex geometry of the heterogeneous object. With the aid of cell adjacency information and attribute-based reasoning, complex, smooth and versatile material distributions can be defined upon the intricate geometries. Their work represents versatile material variations/distributions using different heterogeneous tree (HFT) structures. Keiser et al. [27] present the combination of curved primitives through the application of Boolean operations using the concept of boundary evaluation and exact arithmetic, and show some very expressive results. Numerical inaccuracies are a major problem in any system that manages simultaneously geometric and topological information. Since these systems must use floatingpoint numbers to save coordinates of points, parametric or implicit equations of curves and surfaces, and to perform translations, rotations and other kinds of transformations applied to primitives, they are prone to errors due to the lack of precision in some operations. Many solutions have already been proposed to deal with this problem, based on diverse concepts such as the use of model-based or machine tolerances [28], interval arithmetic and the exact computation paradigm [29]. To date, however, no system has been completely able to solve all kinds of problems related to numerical inaccuracies in environments where curved primitives are allowed. In summary, the two main strategies used to model nonmanifold objects are: direct cell manipulation and reasoning Boolean operations. The former naturally handles cells with different dimensions, such as solid regions, surface patches and wireframes. The latter is used to model heterogeneous objects with inner boundaries and with no
123
Engineering with Computers (2012) 28:225–239
lower-dimension or non-manufacturable part. The idea of this work relies exactly on how to tackle this deficiency: to consider idealized lower-dimension parts (dangling faces or wires) in non-manifold Boolean operations, in the context of mesh generation. 1.2 Main contributions The originality of the present work consists in the development of an algorithm to perform Boolean operations in nonmanifold environments containing curved primitives of any type and dimension. These operations handle specific problems of non-manifold geometric modeling to generate meshbased models. The main idea is to extend the methodology proposed by Cavalcanti et al. [24] based on the classification rules introduced by Ma¨ntyla¨ [7]. One key aspect of the proposed algorithm is that it explicitly maintains the two-manifold concept of boundary of topological entities. This concept allows applying the algorithm to any kind of object, with any shape or dimension, including dangling and loose parts. Another important aspect is the concept of group of entities, which allows performing Boolean operations not just on two homogeneous solids, but also on two sets of entities of different types and dimensions. Common algorithms for Boolean operations in three dimensions would only consider solids as possible input and output data. The proposed algorithm works with entities of different dimensions, since it treats separately each type of topological entity. Lower-dimension entities may be of interest when one does not want to discard internal parts of solids, dangling or loose. These parts might represent restrictions to solid finite element mesh generation inside a region (e.g., cracks in a fracture analysis) or interfaces between layers of different materials, with distinct properties. Another example occurs in geological subsurface objects where a set of faults crosses one or more horizons of rock layers. These faults are usually represented by surfaces that penetrate into the layers. Some faults might not cross an entire layer, becoming a dangling surface in its interior. In addition, tunneling design considers anchor bolts that are usually modeled as wires in the discrete model (Fig. 1). It should be observed that lower-dimension parts may result from Boolean operations between two (homogeneous or heterogeneous) objects. However, when dealing with manufacturable objects (which do not contain degenerated entities), these parts are detected by the non-manifold representation and treated in accordance with the target Boolean operation. In many situations, these parts are removed. The main difference between those situations and the methodology proposed in this work is that a lowerdimension cell may be an idealized feature of the object that should be preserved in many operations.
Engineering with Computers (2012) 28:225–239
Although the proposed algorithm does not involve any specific procedure to treat inaccuracy problems, it is important to keep in every stage consistency between geometry and topology, based on appropriate model-based or machine tolerances [28]. It is also important to point out that the algorithm proposed in the present paper has no intention to be the only possibility to deal with the combination of non-manifold solids through the application of Boolean operations. It is a satisfactory alternative to construct discrete models for these types of solids in a consistent manner. Finally, the main contributions may be summarized in a classification of entities that should be removed in Boolean operations of non-manifold and heterogeneous objects (see Sect. 3), which is not related to any specific data structure. Rather, the proposed classification may be adopted by any non-manifold modeler. 1.3 Outline of this paper In Sect. 2, the main definitions necessary to introduce the proposed algorithm are presented, which are the concepts of group, topological refinement of surfaces and curves, and boundary. The algorithm for applying Boolean set operations on B-Rep non-manifold solids is described in Sect. 3, with the basic procedures according to the desired operation. In Sect. 4, some examples of the application of the algorithm are shown. Finally, the conclusions with a summary of the main difficulties found during the development of the algorithm are discussed in Sect. 5.
2 Basic definitions There are three important issues to be considered before the presentation of the proposed algorithm. First, the idea of group of entities is discussed. Next, the topological refinement of surface and curves is defined, since it reflects directly on the applicability conditions of the Boolean operations. Finally, the concept of boundary of topological entities is introduced. 2.1 Group of entities The first issue to be considered is the definition of group of entities. In this context, a group is defined as a set of topological entities forming a sub-set of the studied model. Groups may contain vertices, edges, surface patches (faces) and regions. A group can contain several entities with different dimensions, including dangling entities or entities completely loose in space. For each topological entity, there must be a way of accessing its adjacency information, usually available in
229
any model data structure. For instance, the radial-edge data structure [20] is an example of a data representation scheme that may provide a sufficient set of adjacency relationships in a non-manifold environment. The Boolean operations will be applied to two groups of entities at a time. The two groups must be under the same topological framework. Each group can possess any number of topological entities. These entities must be refined, as described in the next subsection, to meet the applicability conditions of the Boolean operations. 2.2 Topological refinement of surfaces and curves The conditions of applicability of the algorithm are established to guarantee an essentially topological treatment of the problem. The pre-processing of the input data is made to guarantee that vertices, edges and faces do not intersect but only touch each other along their boundaries. This means that any type of intersections between points, curves and surface patches must be calculated, before applying the classification phase of the algorithm, by subdividing the respective entities into two or more entities of the same type and creating the topological elements (vertices, edges and faces) that will satisfy the non-intersection properties presented as follows: • • • •
A face can only intersect another face along the boundary of at least one of the two faces. Edges cannot intersect each other unless at their end points. Edges cannot intersect faces unless at the edges’ end points. Vertices must be spatially distinct.
This refinement process is described in more details in the work of Cavalcanti et al. [24]. Regions belonging to the same group cannot intersect each other, unless along their boundaries; and regions belonging to different groups may intersect (i.e., share the same space). However, no region is created or modified at the pre-processing stage. There are no restrictions concerning the geometry of the model. If all the conditions commented in this subsection are satisfied, the algorithm will not require additional topological information. 2.3 Boundary of topological entities The third issue for the application of the proposed algorithm is regarding the concept of boundary. The boundary of a topological entity is considered here as all entities hierarchically inferior to it (entities of lower dimensions) that define its contour and that can be obtained by adjacency relationships through its data structure. The boundary of an entity can contain several disconnect portions, but
123
230
each portion must form a connected and closed set of entities of lower dimensions. For example, the edges and vertices of the external loop of a face, forming a closed and alternated cycle of edges and vertices, constitute a part of the face boundary. The same apply to the edges and vertices that form an internal loop of the same face. However, a wire edge loose in the interior of a face constituting an internal loop formed only by this one edge is not considered as part of the face boundary. The same argument can be used for the faces belonging to an internal shell of a solid. If they form a closed set of faces, constituting a smaller region inside the main region, they are considered as part of its boundary. But an internal shell constituted by one or more faces without forming a closed region is not considered as part of the solid’s boundary. This particular definition of the boundary concept does not imply that these internal loose or dangling structures cannot be used as input data for the algorithm of Boolean operations. It only means that when the boundaries of entities are referred in some parts of the algorithm, it does not include such structures.
3 Proposed algorithm for Boolean operations This section presents the proposed algorithm for Boolean operations on non-manifold solids. The algorithm consists of two stages. The first stage is the pre-processing stage, which consists of the selection of the groups and their refinement. The second and main stage consists of the classification of the topological entities followed by the application of the desired Boolean operation over such entities. An optional additional stage could be a regularization procedure, which consists in eliminating from the result of a Boolean operation all the entities of lower dimensions that remained loose or dangling. The proposed classification phase of the algorithm is based on the ideas introduced by Ma¨ntyla¨ [7] for Boolean operations in B-Rep models. However, just the central idea of that algorithm was used, since it is only applicable to polyhedral two-manifold solids. In addition, Ma¨ntyla¨’s algorithm works only with two solids at a time, while the proposed algorithm works with the concept of group defined previously. The main steps of the proposed algorithm for Boolean operations in non-manifold solids are summarized in Fig. 2, for two target groups named A and B. The first step of the pre-processing stage consists in storing, in some appropriate lists, the topological entities (vertices, edges, faces and regions) belonging to each group, along with all the topologically inferior entities that form their boundaries or that remain dangling or loose in their interiors of regions (or faces). Every topological entity
123
Engineering with Computers (2012) 28:225–239
Fig. 2 Algorithm for Boolean operations in non-manifold solids
must be stored only once, i.e., the lists of topological entities of each group cannot have duplicated entities. However, a topological entity may belong to different groups. There must be separate lists to store entities in different groups and with different dimensions (e.g., a list containing the vertices of group A and another list containing the vertices of group B). The second step corresponds to the refinement phase. All intersections between surfaces and curves must be computed. Vertices created by the intersection of two edges of different groups are classified as belonging to both groups, as well as edges resulting from the intersection of two faces and faces created by the overlapping of two faces. The first step of the main stage consists in classifying the entities of each group that were stored in their respective lists. The entities from one group are classified against the entities from the other group. The following notation is adopted: AinB, AoutB and INTERS to indicate, respectively, an entity of group A that is located inside any region of group B, an entity of group A that is located outside all regions of group B, and an entity belonging to both groups. Procedures for point location (point in region and point on face) are necessary for the classification phase. The descriptions of some algorithms for point location can be found in works by De Berg et al. [30] and Preparata and Shamos [31]. Classifying a vertex means testing it against all the regions from the other group (the group that does not contain this vertex). Classifying an edge means checking the classification already given to its vertices. If at least one
Engineering with Computers (2012) 28:225–239
of the vertices is inside any region of the other group or outside all regions of the other group, the edge will be classified in the same way as the vertex. If both vertices are classified as INTERS, then any point belonging to the edge, except its end points, must be tested against the regions of the other group. The classification of the edge will be the same given to the point. Classifying a face (surface patch) means checking the classification already given to the vertices belonging to its outer loop. If at least one of the vertices is inside any region of the other group or outside all the regions of the other group, the face will be classified in the same way as the vertex. If all the vertices are classified as INTERS, then any point belonging to the interior of the face must be tested against the regions of the other group. The classification of the face will be the same given to the point. In essence, applying Boolean operations over the groups’ entities means removing vertices, edges and faces that are not part of the result, according to their
231
classification. Therefore, the second step of the main stage consists in removing undesirable faces, edges and vertices, for a target Boolean operation, which constitutes one of the main contributions of this paper. This step is summarized in the tables illustrated in Fig. 3. The order in which the removals occur is important: first the undesirable faces are removed, then undesirable edges and, lastly, undesirable vertices. Entities with different dimensions are separated, and the tables in Fig. 3 show which of these entities should be removed, according to the Boolean operation that is currently applied, with the assumption that group A is dominant in relation to group B or equal. Some of the terms used in Fig. 3 require more detailed explanation: •
numRegA refers to the number of regions of group A bounded by the face. Thus, numRegA = 0 means a dangling face; numRegA = 1 means an external boundary of a region; and numRegA = 2 means an internal
Fig. 3 Boolean operations algorithm: entities that should be removed
123
232
•
•
• • • • •
Engineering with Computers (2012) 28:225–239
boundary of a region. Therefore, it is important to point out here, the importance of the boundary concept previously described. nA refers to the direction of the outward normal vector of the face of group A, which is used to check the relative angle between the normal vectors of faces of different groups [11]. dangOrLoose means an edge that is located in the interior of a face, dangling or loose, belonging to a removed face. loose means a vertex that is loose in the interior of a face, belonging to removed face. numFaceA means the number of faces of group A bounded by the edge. numEdgeA means the number of edges of group A incident to the vertex. insideA means that the face is inside any region of group A. outsideA means that the face is outside all regions of group A.
In the boxes of Fig. 3 that have a horizontal dashed line, when at least one of the sentences is true, it means that the analyzed entities should be removed. In the specific case of difference operations, the faces of the list BinA that do not bound new regions, formed after the conclusion of the Boolean operation, should be removed. An example of Boolean operations (union, intersection and difference with dominance features) between two different groups (called G1 and G2) is presented in Fig. 4, to illustrate the very simple use of the algorithm proposed in Fig. 3. An optional stage may be a regularization of the results of a Boolean operation. Regularization means to eliminate from the result all the entities of lower dimensions, in such a way that only filling volumes are allowed, i.e., the only entities that should remain after the process of regularization are the faces belonging to the boundary of a given region, the edges of the boundaries of these faces and the vertices at the ends of these edges. Therefore, loose or dangling edges and faces and loose vertices that might eventually appear as a result of a Boolean operation must be removed if the user requires the regularization process. Regularized Boolean operations are important in applications that intend to avoid a subset of the nonmanifold results, which can be obtained by applying Boolean operations on manifold objects (e.g., homogeneous solid objects). They are also important in processes where only filling volumes are relevant or meaningful, as in industrial processes for constructing mechanical parts. In these processes, entities of lower dimensions, such as
123
faces or edges, are not representable because they have null width.
4 Practical applications In this section, some results obtained by implementing the proposed algorithm in a geometric modeler system are presented [32, 33]. It is important to emphasize that the modeler should provide tools for surface intersection and for point location (point in region and point on surface). First, some manifold outputs are shown to demonstrate the
Fig. 4 Application of the proposed algorithm into two groups of entities and the respective results for union, intersection and difference in Boolean operations
Engineering with Computers (2012) 28:225–239
validity of the algorithm in a manifold environment. Then, some non-manifold results are presented, generated by applying Boolean operations over manifold solids and over topological entities with different dimensions (e.g., faces and solids). Finally, a complex example is presented to show the capability of the proposed algorithm to generate complex engineering models. The meshes are generated using well-established algorithms such as [33–36] for surface mesh generation and [37] volume mesh generation. 4.1 Manifold outputs Applying Boolean operations over simple primitives as cylinders, boxes and spheres may lead to more complex solids of innumerous shapes. It is possible to notice some of these solids in Fig. 5. Figure 6 shows all the stages of the creation of a pile and the generation of a mesh of tetrahedrons for the solid that represents it. First, a union operation between two boxes and a cylinder is performed. Then, a new cylinder is inserted in the model and subtracted from the solid generated by the union operation. The last step is the generation of a finite element mesh of tetrahedrons for the solid generated after the difference operation. It is important to emphasize that each solid that will serve as input data for a
233
Boolean operation must be recognized as a closed region and that all the intersections between surface patches must be calculated before the application of any Boolean operation. An example of Boolean operations applied to two solids bounded by generic curved surfaces is shown in Fig. 7. The original solids (a ‘‘twisted cube’’ and a ‘‘bottle’’) with all the intersections between surfaces already computed are presented in Fig. 7a. Figure 7b shows the intersection between these two solids. The difference operation between the ‘‘twisted cube’’ and the ‘‘bottle’’ is shown in Fig. 7c. 4.2 Non-manifold outputs Sometimes, even when the input data are representable in a manifold environment, the output data may not be so. That is the case of the results shown in Fig. 8. In the first example, some dangling faces are part of the result (faces that are not part of the boundary of any solid). In the second example, there are two solids touching each other through a common edge. This edge is adjacent to four different faces. An example of the application of Boolean operations over entities of different dimensions is presented in Fig. 9.
Fig. 5 Examples of manifold solids obtained by applying Boolean operations over simpler primitives
123
234
Engineering with Computers (2012) 28:225–239
Fig. 6 Creation of a pile: a primitive solids, b refinement, c union operation, d insertion of a new cylinder, e difference between the solid and the new cylinder and f tetrahedron mesh for the resulting solid (pile)
This means that two groups of entities were selected, one containing only faces and the other containing a solid. 4.3 Complex engineering (geomechanics) model An appropriated set of Boolean operations over non-manifold solids plays an important role in the creation of complex discrete (finite element or boundary element) models, such as the model generation of a tunnel system crossing geological faults and with several horizons. It is shown in Fig. 10 the steps necessary to create a tunnel model with four geological layers and a fault using
123
the proposed Boolean operation algorithm. Figure 10a depicts the original groups used to generate the model: group G1 contains three geological horizons (layer’s interface), group G2 contains the fault, group G3 contains the model bounding box limited by the ground surface, and group G4 contains the tunnel. Figure 10b–f show preparations or results of Boolean operation applied consecutively on the groups to obtain the final model, and Fig. 10g shows a detailed view of the tunnel crossing the fault and a horizon after the application of the Boolean operations. The second operand G3 of the difference operations shown in Fig. 10c is the complement of group G3.
Engineering with Computers (2012) 28:225–239
235
Fig. 7 a Refinement, b intersection and c difference between two solids bounded by curved surfaces
5 Conclusions An algorithm for Boolean operations on non-manifold models is proposed to allow the treatment of solids with multiple regions (internal interfaces) and degenerate portions (shells and wires), in the context of mesh generation. The algorithm was implemented in a pre-existing geometric modeler for finite element solid mesh generation [32, 33]. This modeler works with a hybrid data representation in which every model, besides being represented by the modeler’s internal data structure, can have another representation given by the CGC library [24], whenever the model needs to be topologically tied up. In general, there are two main modeling strategies for the creation of non-manifold solids. The first is analogous to the B-rep approach of the two-manifold modeling of
homogeneous solids. This strategy is the referred to here as direct cell manipulation [19, 20, 22, 24], in which a cell can be a topological entity of any dimension, such as a solid region, a surface patch or a wire curve. The second approach, analogous to the CSG approach of the twomanifold modeling, is called reasoning Boolean operations [9–12, 16, 17, 23, 25, 26]. This strategy, in general, is used to model manufacturable heterogeneous objects with inner boundaries, but with no lower-dimension portions. This works proposes an algorithm for the creation of discrete models, in the context of mesh generation, whose highlevel interface are Boolean operations. These operations apply to two objects that are organized as groups of selected entities of different dimensions. The main contribution of this study is that the proposed methodology handles specifically lower-dimension entities that may be
123
236
Engineering with Computers (2012) 28:225–239
Fig. 8 Non-manifold results obtained by applying Boolean operations over manifold solids
Fig. 9 Boolean operations applied over entities of different dimensions
of interest in idealized models with dangling or loose parts. These lower-dimension entities might represent restrictions to solid finite element mesh generation inside a region, interfaces between layers of different materials or anchor bolts in a tunnel design. To achieve the level of generality desired, some initial conditions had to be imposed to guarantee an essentially topological treatment of the problem. These initial conditions implicate pre-processing the input data to assure that all topological entities just touch each other (even if not along their boundaries). This pre-processing stage is a
123
refinement of surfaces and curves of the groups (operands of the Boolean operators). In this refinement, the original regions of the groups are preserved because they are used in the classification phase of the proposed algorithm. After refinement, the classification phase relies on region point location, which is the only geometric information that is necessary to apply the algorithm. The remaining information is essentially topological adjacency relationship. The pre-processing stage is a refinement of surfaces and curves of the groups (operands of the Boolean operators). In this refinement, the original regions of the groups are
Engineering with Computers (2012) 28:225–239
237
Fig. 10 Creation of a tunnel model crossing a fault and with three horizons: a original groups, b preparation and c results of the differences G1 ¼ G1 G3 and G2 ¼ G2 G3, d union G1 = G1UG2 (G1 = G2), e union G1 = G1UG3 (G1 [ G3), f difference G1 - G4, g detailed view of tunnel intersections
preserved because they are used in the classification phase of the proposed algorithm. After refinement, the classification phase relies on region point location, which is the only geometric information that is necessary to apply the algorithm. The remaining information is essentially topological. The classification phase of the algorithm is summarized in three tables (Fig. 3), which constitute another contribution of this work. These tables indicate the topological
entities that should be removed according to the target Boolean operation (union, intersection, or difference). A dominance factor of a group in relation to the other is used in an entity selection. The selection also takes into account the case in which the groups have equal dominance factor. Usually, this case occurs in modeling of homogenous material solids. In this case, a regularization procedure might be necessary to eliminate undesirable dangling and loose parts after the removal of entities for a specific
123
238
Boolean operation. Finally, after removing entities, a detection of new regions of the resulting non-manifold object is performed. The algorithm is very simple and intuitive, and its implementation in any B-Rep-based non-manifold system does not demand much effort. However, the system must incorporate non-linear primitive intersection, region detection and point location algorithms to assure the correct functioning of the Boolean operations algorithm. Several tests were performed, using manifold and nonmanifold input data and leading to manifold and nonmanifold output data, yet not, respectively. The algorithm proved to be quite efficient for relatively low-complexity models. An efficiency analysis is necessary for highly complex models, i.e., models with a significant number of high-degree curved primitives, a considerable number of surface intersections, a considerable number of closed regions, too many topological vertices of one group located inside the regions of the other group, and the presence of surface overlapping. It is important to mention that the algorithm depends and relies on the geometric capabilities of the modeler, since it should treat the geometric intersection problems, as well as region detection and point location procedures in a reasonable way. Acknowledgments The authors would like to thank the Computer Graphics Technology Group (Tecgraf) and Pontifical Catholic University of Rio de Janeiro (PUC-Rio) for the financial support and for providing the necessary space and resources used during the development of this work.
References 1. Hoffmann CM (1989) Geometric and solid modeling: an introduction. Morgan Kaufmann, San Fransisco 2. Ma¨ntyla¨ M (1988) An introduction to solid modeling Computer. Science Press, Rockville 3. Requicha A (1977) Constructive solid geometry. Technical Memo 25, University of Rochester, Production Automation Project 4. Miller JR (1989) Architectural issues in solid modelers. IEEE Comput Graph Appl 9:72–87 5. Requicha AG, Voelcker HB (1985) Boolean operations in solid modeling: boundary evaluation and merging algorithms. Proc IEEE 73:30–44 6. Shapiro V, Vossler DL (1993) Separation for boundary to CSG conversion. ACM Trans Graph 12:33–55 7. Ma¨ntyla¨ M (1986) Boolean operations of 2-manifolds through vertex neighborhood classification. ACM Trans Graph 5:1–29 8. Gardan Y, Perrin E (1995) An algorithm for reducing 3D Boolean operations to a 2D problem: concepts and results. Comput Aided Des 28:277–287 9. Magalha˜es ALCDC, Mesquita RC (2000) Exploring inner boundaries in solid modelers applied to electromagnetic problems. IEEE Trans Magn 36:1682–1686 10. Sun W, Hu X (2002) Reasoning Boolean operation based modeling for heterogeneous objects. Comput Aided Des 34:481–488
123
Engineering with Computers (2012) 28:225–239 11. Qian X, Dutta D (2003) Heterogeneous object modeling through direct face neighborhood alteration. Comput Graph 27:943–961 12. Kou XY, Tan ST, Sze WS (2006) Modeling complex heterogeneous objects with non-manifold heterogeneous cells. Comput Aided Des 38:457–474 13. Sen D, Srikanth TK (2008) Efficient computation of volume fractions for multi-material cell complexes in a grid by slicing. Comput Geosci 34:754–782 14. Colyer B et al (1997) Project MIDAS: magnet integrated design and analysis system. IEEE Trans Magn 33:1143–1148 15. Cracker GA, Reinke WF (1991) An editable nonmanifold boundary representation. IEEE Comput Graph Appl 11:39–51 16. Masuda H (1993) Topological operators and Boolean operations for complex-based non-manifold geometric models. Comput Aided Des 25:119–129 17. Gursoz EL, Choi Y, Prinz FB (1991) Boolean set operations on non-manifold boundary representation objects. Comput Aided Des 23:33–39 18. Gursoz EL, Choi Y, Prinz FB (1990) Vertex-based representation of non-manifold boundaries. In: Wozny M, Turner J, Preiss K (eds) Geometric modeling for product engineering. NorthHolland, Amsterdam, pp 107–130 19. Rossignac JR, O’Connor MA (1989) A dimensional-independent model for pointsets with internal structures and incomplete boundaries. Geometric modeling for product engineering. NorthHolland, Amsterdam, pp 145–180 20. Weiler K (1986) Topological structures for geometric modeling. Dissertation, Rensselaer Polytechnic Institute, Troy, NY 21. Weiler K (1988) The radial-edge structure: a topological representation for non-manifold geometric boundary representation. Geometric modeling for CAD applications. North Holland, Amsterdam, pp 3–36 22. Lienhardt P (1991) Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput Aided Des 23:59–82 23. Arbab F (1990) Set models and Boolean operations for solids and assemblies. IEEE Comput Graph Appl 10:76–86. doi: 10.1109/38.62698 24. Cavalcanti PR, Carvalho PCP, Martha LF (1997) Non-manifold modeling: an approach based on spatial subdivision. Comput Aided Des 29:209–220 25. Rossignac JR, Requicha AG (1991) Constructive non-regularized geometry. Comput Aided Des 23:21–32 26. Gardan Y, Minich C, Perrin E (2003) Boolean operations on feature-based models. J WSCG (Winter School of Computer Graphics) 27. Keyser JC, Culver T, Foskey M, Krishnan S, Manocha D (2004) ESOLID—a system for exact boundary evaluation. Comput Aided Des 36:175–193 28. Shewchuk JR (1997) Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput Geometry 18:305–363 29. Keyser JC (2000) Exact boundary evaluation for curved solids. Dissertation, University of North Carolina 30. De Berg M, Van Kreveld M, Overmars M, Schwarzkopf O (1997) Computational geometry: algorithms and applications. Springer, Berlin 31. Preparata FP, Shamos MI (1990) Computational geometry: an introduction. Springer, New York 32. Coelho LCG, Gattass M, Figueiredo LH (2000) Intersecting and trimming parametric meshes on finite-element shells. Int J Numer Methods Eng 47:777–800 33. Lira WWM, Coelho LCG, Cavalcanti PR, Martha LF (2002) A modeling methodology for finite element mesh generation of multi-region models with parametric surfaces. Comput Graph 26:907–918
Engineering with Computers (2012) 28:225–239 34. Miranda ACO, Martha LF (2002) Mesh generation on highcurvature surfaces based on a background quadtree structure. In: Proceedings of 11th international meshing roundtable, Ithaca, pp 333–341 35. Lira WWM, Coelho LCG, Martha LF (2002) Multiple intersections of finite-element surface meshes. In: Proceedings of 11th international meshing roundtable, Ithaca, pp 355–363
239 36. Miranda ACO, Martha LF, Wawrzynek PA, Ingraffea AR (2009) Surface mesh regeneration considering curvatures. Eng Comput 25:207–219. doi:10.1007/s00366-008-0119-9 37. Cavalcante-Neto JB, Wawrzynek PA, Carvalho MTM, Martha LF, Ingraffea AR (2001) An algorithm for three-dimensional mesh generation for arbitrary regions with cracks. Eng Comput 17:75–91
123
ORIGINAL ARTICLE
Boolean operations on multi-region solids for mesh generation Andre´ Maue´s Brabo Pereira • Marcos Chataignier de Arruda Antoˆnio Carlos de O. Miranda • William Wagner M. Lira • Luiz Fernando Martha
•
Received: 17 February 2010 / Accepted: 10 June 2011 / Published online: 28 June 2011 Springer-Verlag London Limited 2011
Abstract An algorithm for Boolean operations on nonmanifold models is proposed to allow the treatment of solids with multiple regions (internal interfaces) and degenerate portions (shells and wires), in the context of mesh generation. In a solid modeler, one of the most powerful tools to create three-dimensional objects with any level of geometric complexity is the Boolean set operators. They are intuitive and popular ways to combine solids, based on the operations applied to point sets. To assure that the resulting objects have the same dimension as the original objects, without loose or dangling parts, a regularization process is usually applied after a Boolean operation. In practice, the regularization is performed classifying the topological elements and removing internal or lowerdimensional structures. However, in many engineering applications, the adopted geometric model may contain idealized internal parts, as in the case of multi-region models, or lower-dimensional parts, as in the case of solids
that contain dangling slabs that are represented as zerothickness surfaces or wireframes in the model. Therefore, the aim of this work is the development of a generic algorithm that allows the application of the Boolean set operations in a geometric modeling environment applied to finite and boundary element mesh generation. This environment adopts a non-manifold boundary representation that considers an undefined number of topological entities (group concept), and works with objects of different dimensions and with objects not necessarily plane or polyhedral (parametric curved surfaces). Numerical examples are presented to illustrate the proposed methodology. Keywords Geometric modeling Non-manifold solids Boolean operations Boundary representation Mesh generation Degenerate portions
1 Introduction A. M. B. Pereira (&) Departamento de Engenharia Civil, Universidade Federal Fluminense, Rua Passo da Patria 156, Niteroi 24210-240, Brazil e-mail: [email protected]; [email protected] A. M. B. Pereira M. C. de Arruda L. F. Martha Departamento de Engenharia Civil, Pontifı´cia Universidade Cato´lica do Rio de Janeiro, Av. Marqueˆs de Sa˜o Vicente 225, Rio de Janeiro 22453-900, Brazil e-mail: [email protected] A. C. O. Miranda Departamento de Engenharia Civil e Ambiental, Universidade de Brasilia, Campus Universitario Darcy Ribeiro, Brasilia 70910-900, Brazil W. W. M. Lira Centro de Tecnologia, Universidade Federal de Alagoas, Campus A. C. Simo˜es, Maceio´ 57072-970, Brazil
The creation of geometric models in the context of mesh generation is an essential task when dealing with the numerical simulation of real engineering objects (Fig. 1). Numerical simulation techniques as the finite element method (FEM) or the boundary element method (BEM) require discrete models (i.e., domain meshes or boundary meshes, respectively) that are based on the geometric description of the model’s domain. In addition, the construction of the discrete models usually involves combinations of several solids that can still contain other objects of lower dimensions (such as shells and wires), resulting in complex multi-region solids. A natural manner to construct multi-region solids for mesh generation is through Boolean set operators (union, intersection and difference), which are essential tools in any solid modeling system.
123
226
Fig. 1 Real engineering objects containing multiple regions (internal interfaces) and degenerate portions (shells and wires)
Constructive solid geometry (CSG) and boundary representations (B-Rep) are traditional mathematical modeling schemes for representing the geometric shape of homogeneous objects, referred to as two-manifold solids [1]. However, these traditional schemes do not handle heterogeneous objects (multi-region solids) or models with degenerated portions. Non-manifold representations are strategies to deal with this class of problem. A modeler with a non-manifold environment allows Boolean operations to generate realistic FEM and BEM discrete models. In this representation, solids with multiple regions (internal interfaces) and degenerate portions (shells and wires) may be represented in a consistent way. These are situations that occur, e.g., in the problem illustrated in Fig. 1, which contains several materials, internal boundaries, fault surfaces, tunnel support surfaces and anchor bolts. Nevertheless, the implementation of a system that provides a non-manifold environment is a very complex task. 1.1 State of the art and literature review As already mentioned, two different strategies have been traditionally proposed to model homogeneous solid objects. In the first strategy, one aims to represent a solid through an explicit description of its boundary, which is the so-called boundary representation (B-rep) [2]. It is based on data structures that describe the adjacency relationships of the vertices, edges and faces of a solid. The other possibility is the CSG approach [3], which consists of representing a solid as a result of a sequence of set operations (union, intersection, and difference) performed on simple primitive solids. B-rep and CSG have complementary advantages and disadvantages [4], and much work has been carried out to convert from CSG to B-rep [5] and from B-rep to CSG [6].
123
Engineering with Computers (2012) 28:225–239
B-rep modeling is a natural way of constructing a solid that is defined by its bounding surfaces. Although complex objects can be described, in principle, by their defining surfaces, in many cases this is not a natural way of construction. In practice, modeling systems based on B-rep are usually capable of performing set operations, which enables one to use CSG-like operations to define solids, while retaining the full boundary representation. A topologically complete set operations algorithm for planar polyhedra two-manifold objects was described by Ma¨ntyla¨ [7], introducing the vertex neighborhood classifier for Boolean operations. Gardan and Perrin [8] proposed an interesting method for reducing 3D Boolean operations to a 2D problem in manifold environments. However, traditional B-rep and CSG techniques apply when one just needs to look at an object as inducing a three-part space decomposition: its interior, its exterior and its boundary. For many interesting applications, such as modeling of heterogeneous objects [9–13], this is not enough. In addition, in many engineering applications, the adopted geometric model may contain idealized lowerdimension parts, as in the case of finite element and boundary element systems [14]. These types of systems need the support of a unified and simultaneous representation of wires (e.g., beam elements), shells (e.g., concrete slabs that are represented as zero-thickness surfaces) and solid parts (e.g., foundation blocks). In these situations, one would like to have not only a representation of each part of the model, but also a description about the way these parts are connected to each other. Techniques that models heterogeneous objects and objects with lower-dimension parts are called non-manifold modeling techniques [1, 15, 16], because these objects are not necessarily two-manifolds embedded in threedimensional space. Traditional modelers (based on traditional B-rep and CSG modeling techniques) could only represent two-manifold objects, i.e., objects in which every point on the boundary has a neighborhood that is homeomorphic to an open two-dimensional disk [1]. Since the late 1980s, the advantages of non-manifold geometric modeling have been recognized [17] and several representation schemes for non-manifold objects have been developed [18–22]. The mathematical definition of nonmanifold objects has also been presented. Masuda [16] defined non-manifold geometric models as cell complexes that are subsets of 3D Euclidian space, and provided extended Euler operators for this type of model. The concept of cell complexes non-manifold geometric model is suitable, because the model may contain wires, shells and solid parts, or combinations of them. Arbab [23] presented an alternative CSG-like formalism based on open sets (called s-sets) for assemblies, i.e., objects with several solid
Engineering with Computers (2012) 28:225–239
parts or with internal boundaries, and formalized Boolean operations for heterogeneous objects (s-sets). One strategy for modeling non-manifold objects is the direct manipulation of cells. Rossignac and O’Connor [19] have proposed the concept of geometric complex (GC), in which a non-manifold object is decomposed in disjoint portions called cells, each one homogeneous in dimension and satisfying the condition that the intersection of the boundaries of any two cells is necessarily equal to the union of other cells of the decomposition. Cavalcanti et al. [24] generalized this concept and presented an approach based on spatial subdivision for non-manifold modeling. In this approach, any object is represented by a subdivision of the entire Euclidian space (including the external infinite region). This representation is called complete geometric complex (CGC) and was implemented using the radial edge data structure proposed by Weiler [20]. An incremental approach for CGC creation considers the inclusion of a patch of surface at a time, with the corresponding updating of the data structure after each insertion. Similarly, curve segment and vertices are inserted incrementally. The insertion of a new cell (surface patch, curve segment, or vertex) is a fundamental operation executed by the spatial subdivision scheme and may result in the refinement of new and existing cells due to geometric intersection. New solid regions (which are also cells in the decomposition) may result from surface patch insertions, as well as new surface patches may result from curve insertions. Cavalcanti et al. [24] have chosen CGC as the mathematical model to describe a spatial subdivision. This approach provides a basis for representing complex objects in two steps: first, a subdivision phase and, second, a phase with selection of active cells. The resulting object is called an selective complete geometric complex (SCGC). For instance, in a finite element analysis of a solid with a cavity, the infinite external region and the cavity would be inactive cells. Another strategy for creation of non-manifold models is through Boolean operations. Gursoz et al. [17] proposed a scheme for implementing Boolean operations based on the classification of six types of intersection pairs: vertex– vertex, vertex–edge, edge–edge, vertex–face, edge–face, and face–face. All of the intersections take place between pairs of elements that belong to two different objects. After having obtained a merged object, any Boolean operation is achieved by selecting the appropriate point set elements in the object. A more general process was introduced by Rossignac and Requicha [25]. They introduced a new class of objects, called constructive non-regularized geometry (CNRG) objects, and a family of operators that act on these objects. A CNRG tree is a rooted directed acyclic graph and represents an object. The leaves are the CNRG primitives.
227
Internal nodes represent intermediate CNRG objects obtained by applying certain operations to their children. The operators are: aggregation, unification, sum, product, subtraction, complementation, interior, closure, boundary and regularization. CNRG operations provide a modeling process suitable for user interaction. When followed by a unification operator, which creates an object having only one component, they behave exactly as the corresponding point-set operators. Without unification, however, they return an aggregate (or assembly) of pair-wise disjoint components. Cavalcanti et al. [24] advocated that space subdivisions corresponding to CNRG objects can be implicitly represented by the corresponding tree, in the way as CSG trees can be used to represent B-rep solids. In this case, the CNRG operators are used only as a modeling process, as these operators provide a convenient way to define topological and Boolean operations for SCGC objects. Indeed, an SCGC can be seen as a CNRG whose components are its active cells. This allows one to use CNRG operators to build and modify SCGCs. In this context, these authors used the CNRG operators to create heterogeneous objects represented by SCGCs. They proposed a scheme to automatically determine the material to be attributed to each region based on region naming and a ‘‘dominance factor’’, which establishes the material that predominates in a combination operation. In the last decade, some attention has been given to the use of Boolean set operation for the creation of nonmanifold objects in the context of feature-based modeling and modeling of heterogeneous objects, including objects with functionally graded materials. Magalha˜es and Mesquita [9] proposed a solution for handling inner boundaries in the context of mesh generation for electromagnetic problems, assuring the design of manufacturable models. Their work is based on the s-set CSG-like formalism proposed by Arbab [23], and the implementation adopts the radial edge data structure of Weiler [20]. The study of Gardan et al. [26] describes the use of Boolean operations of feature-based models. As mentioned by these authors, feature-based modeling and Boolean operations are closely related areas, as such operations are involved in feature instantiation or modification. Boolean operators may also be used to combine two feature-based models. In both situations, the modeling process is based on the following sequence of statements: calculate the intersections, combine, classify and extract entities (faces, edges and vertices) of the operands. Following an approach similar to the use of a ‘‘dominance factor’’ [24], Sun and Hu [10] proposed reasoning Boolean operations for modeling heterogeneous objects. The reasoning Boolean operation algorithm consists of a merging operation and an extraction operation, and is
123
228
executed according to the material-dominant information extracted from the designed heterogeneous object database. In their work, they perform several geometric operations prior to the material classification. A very comprehensive work in this area is the one written by Qian and Dutta [11]. These authors proposed the direct face neighborhood alteration method to handle the two main tasks involved in the constructive operations of heterogeneous objects with functionally graded material: geometric boundary evaluation and material region forming. In a sense, the methodology devised by these authors generalizes and extends the use of region naming and material-dominant information in heterogeneous object modeling. They define mathematically three operations that manipulate material volumes: additive, subtractive and partition operations. These operations take into account both geometric properties and combined material behavior of each volume. A similar approach is adopted by Kou et al. [12]. However, the main difference is that these authors also consider smooth transitions of material properties throughout the complex geometry of the heterogeneous object. With the aid of cell adjacency information and attribute-based reasoning, complex, smooth and versatile material distributions can be defined upon the intricate geometries. Their work represents versatile material variations/distributions using different heterogeneous tree (HFT) structures. Keiser et al. [27] present the combination of curved primitives through the application of Boolean operations using the concept of boundary evaluation and exact arithmetic, and show some very expressive results. Numerical inaccuracies are a major problem in any system that manages simultaneously geometric and topological information. Since these systems must use floatingpoint numbers to save coordinates of points, parametric or implicit equations of curves and surfaces, and to perform translations, rotations and other kinds of transformations applied to primitives, they are prone to errors due to the lack of precision in some operations. Many solutions have already been proposed to deal with this problem, based on diverse concepts such as the use of model-based or machine tolerances [28], interval arithmetic and the exact computation paradigm [29]. To date, however, no system has been completely able to solve all kinds of problems related to numerical inaccuracies in environments where curved primitives are allowed. In summary, the two main strategies used to model nonmanifold objects are: direct cell manipulation and reasoning Boolean operations. The former naturally handles cells with different dimensions, such as solid regions, surface patches and wireframes. The latter is used to model heterogeneous objects with inner boundaries and with no
123
Engineering with Computers (2012) 28:225–239
lower-dimension or non-manufacturable part. The idea of this work relies exactly on how to tackle this deficiency: to consider idealized lower-dimension parts (dangling faces or wires) in non-manifold Boolean operations, in the context of mesh generation. 1.2 Main contributions The originality of the present work consists in the development of an algorithm to perform Boolean operations in nonmanifold environments containing curved primitives of any type and dimension. These operations handle specific problems of non-manifold geometric modeling to generate meshbased models. The main idea is to extend the methodology proposed by Cavalcanti et al. [24] based on the classification rules introduced by Ma¨ntyla¨ [7]. One key aspect of the proposed algorithm is that it explicitly maintains the two-manifold concept of boundary of topological entities. This concept allows applying the algorithm to any kind of object, with any shape or dimension, including dangling and loose parts. Another important aspect is the concept of group of entities, which allows performing Boolean operations not just on two homogeneous solids, but also on two sets of entities of different types and dimensions. Common algorithms for Boolean operations in three dimensions would only consider solids as possible input and output data. The proposed algorithm works with entities of different dimensions, since it treats separately each type of topological entity. Lower-dimension entities may be of interest when one does not want to discard internal parts of solids, dangling or loose. These parts might represent restrictions to solid finite element mesh generation inside a region (e.g., cracks in a fracture analysis) or interfaces between layers of different materials, with distinct properties. Another example occurs in geological subsurface objects where a set of faults crosses one or more horizons of rock layers. These faults are usually represented by surfaces that penetrate into the layers. Some faults might not cross an entire layer, becoming a dangling surface in its interior. In addition, tunneling design considers anchor bolts that are usually modeled as wires in the discrete model (Fig. 1). It should be observed that lower-dimension parts may result from Boolean operations between two (homogeneous or heterogeneous) objects. However, when dealing with manufacturable objects (which do not contain degenerated entities), these parts are detected by the non-manifold representation and treated in accordance with the target Boolean operation. In many situations, these parts are removed. The main difference between those situations and the methodology proposed in this work is that a lowerdimension cell may be an idealized feature of the object that should be preserved in many operations.
Engineering with Computers (2012) 28:225–239
Although the proposed algorithm does not involve any specific procedure to treat inaccuracy problems, it is important to keep in every stage consistency between geometry and topology, based on appropriate model-based or machine tolerances [28]. It is also important to point out that the algorithm proposed in the present paper has no intention to be the only possibility to deal with the combination of non-manifold solids through the application of Boolean operations. It is a satisfactory alternative to construct discrete models for these types of solids in a consistent manner. Finally, the main contributions may be summarized in a classification of entities that should be removed in Boolean operations of non-manifold and heterogeneous objects (see Sect. 3), which is not related to any specific data structure. Rather, the proposed classification may be adopted by any non-manifold modeler. 1.3 Outline of this paper In Sect. 2, the main definitions necessary to introduce the proposed algorithm are presented, which are the concepts of group, topological refinement of surfaces and curves, and boundary. The algorithm for applying Boolean set operations on B-Rep non-manifold solids is described in Sect. 3, with the basic procedures according to the desired operation. In Sect. 4, some examples of the application of the algorithm are shown. Finally, the conclusions with a summary of the main difficulties found during the development of the algorithm are discussed in Sect. 5.
2 Basic definitions There are three important issues to be considered before the presentation of the proposed algorithm. First, the idea of group of entities is discussed. Next, the topological refinement of surface and curves is defined, since it reflects directly on the applicability conditions of the Boolean operations. Finally, the concept of boundary of topological entities is introduced. 2.1 Group of entities The first issue to be considered is the definition of group of entities. In this context, a group is defined as a set of topological entities forming a sub-set of the studied model. Groups may contain vertices, edges, surface patches (faces) and regions. A group can contain several entities with different dimensions, including dangling entities or entities completely loose in space. For each topological entity, there must be a way of accessing its adjacency information, usually available in
229
any model data structure. For instance, the radial-edge data structure [20] is an example of a data representation scheme that may provide a sufficient set of adjacency relationships in a non-manifold environment. The Boolean operations will be applied to two groups of entities at a time. The two groups must be under the same topological framework. Each group can possess any number of topological entities. These entities must be refined, as described in the next subsection, to meet the applicability conditions of the Boolean operations. 2.2 Topological refinement of surfaces and curves The conditions of applicability of the algorithm are established to guarantee an essentially topological treatment of the problem. The pre-processing of the input data is made to guarantee that vertices, edges and faces do not intersect but only touch each other along their boundaries. This means that any type of intersections between points, curves and surface patches must be calculated, before applying the classification phase of the algorithm, by subdividing the respective entities into two or more entities of the same type and creating the topological elements (vertices, edges and faces) that will satisfy the non-intersection properties presented as follows: • • • •
A face can only intersect another face along the boundary of at least one of the two faces. Edges cannot intersect each other unless at their end points. Edges cannot intersect faces unless at the edges’ end points. Vertices must be spatially distinct.
This refinement process is described in more details in the work of Cavalcanti et al. [24]. Regions belonging to the same group cannot intersect each other, unless along their boundaries; and regions belonging to different groups may intersect (i.e., share the same space). However, no region is created or modified at the pre-processing stage. There are no restrictions concerning the geometry of the model. If all the conditions commented in this subsection are satisfied, the algorithm will not require additional topological information. 2.3 Boundary of topological entities The third issue for the application of the proposed algorithm is regarding the concept of boundary. The boundary of a topological entity is considered here as all entities hierarchically inferior to it (entities of lower dimensions) that define its contour and that can be obtained by adjacency relationships through its data structure. The boundary of an entity can contain several disconnect portions, but
123
230
each portion must form a connected and closed set of entities of lower dimensions. For example, the edges and vertices of the external loop of a face, forming a closed and alternated cycle of edges and vertices, constitute a part of the face boundary. The same apply to the edges and vertices that form an internal loop of the same face. However, a wire edge loose in the interior of a face constituting an internal loop formed only by this one edge is not considered as part of the face boundary. The same argument can be used for the faces belonging to an internal shell of a solid. If they form a closed set of faces, constituting a smaller region inside the main region, they are considered as part of its boundary. But an internal shell constituted by one or more faces without forming a closed region is not considered as part of the solid’s boundary. This particular definition of the boundary concept does not imply that these internal loose or dangling structures cannot be used as input data for the algorithm of Boolean operations. It only means that when the boundaries of entities are referred in some parts of the algorithm, it does not include such structures.
3 Proposed algorithm for Boolean operations This section presents the proposed algorithm for Boolean operations on non-manifold solids. The algorithm consists of two stages. The first stage is the pre-processing stage, which consists of the selection of the groups and their refinement. The second and main stage consists of the classification of the topological entities followed by the application of the desired Boolean operation over such entities. An optional additional stage could be a regularization procedure, which consists in eliminating from the result of a Boolean operation all the entities of lower dimensions that remained loose or dangling. The proposed classification phase of the algorithm is based on the ideas introduced by Ma¨ntyla¨ [7] for Boolean operations in B-Rep models. However, just the central idea of that algorithm was used, since it is only applicable to polyhedral two-manifold solids. In addition, Ma¨ntyla¨’s algorithm works only with two solids at a time, while the proposed algorithm works with the concept of group defined previously. The main steps of the proposed algorithm for Boolean operations in non-manifold solids are summarized in Fig. 2, for two target groups named A and B. The first step of the pre-processing stage consists in storing, in some appropriate lists, the topological entities (vertices, edges, faces and regions) belonging to each group, along with all the topologically inferior entities that form their boundaries or that remain dangling or loose in their interiors of regions (or faces). Every topological entity
123
Engineering with Computers (2012) 28:225–239
Fig. 2 Algorithm for Boolean operations in non-manifold solids
must be stored only once, i.e., the lists of topological entities of each group cannot have duplicated entities. However, a topological entity may belong to different groups. There must be separate lists to store entities in different groups and with different dimensions (e.g., a list containing the vertices of group A and another list containing the vertices of group B). The second step corresponds to the refinement phase. All intersections between surfaces and curves must be computed. Vertices created by the intersection of two edges of different groups are classified as belonging to both groups, as well as edges resulting from the intersection of two faces and faces created by the overlapping of two faces. The first step of the main stage consists in classifying the entities of each group that were stored in their respective lists. The entities from one group are classified against the entities from the other group. The following notation is adopted: AinB, AoutB and INTERS to indicate, respectively, an entity of group A that is located inside any region of group B, an entity of group A that is located outside all regions of group B, and an entity belonging to both groups. Procedures for point location (point in region and point on face) are necessary for the classification phase. The descriptions of some algorithms for point location can be found in works by De Berg et al. [30] and Preparata and Shamos [31]. Classifying a vertex means testing it against all the regions from the other group (the group that does not contain this vertex). Classifying an edge means checking the classification already given to its vertices. If at least one
Engineering with Computers (2012) 28:225–239
of the vertices is inside any region of the other group or outside all regions of the other group, the edge will be classified in the same way as the vertex. If both vertices are classified as INTERS, then any point belonging to the edge, except its end points, must be tested against the regions of the other group. The classification of the edge will be the same given to the point. Classifying a face (surface patch) means checking the classification already given to the vertices belonging to its outer loop. If at least one of the vertices is inside any region of the other group or outside all the regions of the other group, the face will be classified in the same way as the vertex. If all the vertices are classified as INTERS, then any point belonging to the interior of the face must be tested against the regions of the other group. The classification of the face will be the same given to the point. In essence, applying Boolean operations over the groups’ entities means removing vertices, edges and faces that are not part of the result, according to their
231
classification. Therefore, the second step of the main stage consists in removing undesirable faces, edges and vertices, for a target Boolean operation, which constitutes one of the main contributions of this paper. This step is summarized in the tables illustrated in Fig. 3. The order in which the removals occur is important: first the undesirable faces are removed, then undesirable edges and, lastly, undesirable vertices. Entities with different dimensions are separated, and the tables in Fig. 3 show which of these entities should be removed, according to the Boolean operation that is currently applied, with the assumption that group A is dominant in relation to group B or equal. Some of the terms used in Fig. 3 require more detailed explanation: •
numRegA refers to the number of regions of group A bounded by the face. Thus, numRegA = 0 means a dangling face; numRegA = 1 means an external boundary of a region; and numRegA = 2 means an internal
Fig. 3 Boolean operations algorithm: entities that should be removed
123
232
•
•
• • • • •
Engineering with Computers (2012) 28:225–239
boundary of a region. Therefore, it is important to point out here, the importance of the boundary concept previously described. nA refers to the direction of the outward normal vector of the face of group A, which is used to check the relative angle between the normal vectors of faces of different groups [11]. dangOrLoose means an edge that is located in the interior of a face, dangling or loose, belonging to a removed face. loose means a vertex that is loose in the interior of a face, belonging to removed face. numFaceA means the number of faces of group A bounded by the edge. numEdgeA means the number of edges of group A incident to the vertex. insideA means that the face is inside any region of group A. outsideA means that the face is outside all regions of group A.
In the boxes of Fig. 3 that have a horizontal dashed line, when at least one of the sentences is true, it means that the analyzed entities should be removed. In the specific case of difference operations, the faces of the list BinA that do not bound new regions, formed after the conclusion of the Boolean operation, should be removed. An example of Boolean operations (union, intersection and difference with dominance features) between two different groups (called G1 and G2) is presented in Fig. 4, to illustrate the very simple use of the algorithm proposed in Fig. 3. An optional stage may be a regularization of the results of a Boolean operation. Regularization means to eliminate from the result all the entities of lower dimensions, in such a way that only filling volumes are allowed, i.e., the only entities that should remain after the process of regularization are the faces belonging to the boundary of a given region, the edges of the boundaries of these faces and the vertices at the ends of these edges. Therefore, loose or dangling edges and faces and loose vertices that might eventually appear as a result of a Boolean operation must be removed if the user requires the regularization process. Regularized Boolean operations are important in applications that intend to avoid a subset of the nonmanifold results, which can be obtained by applying Boolean operations on manifold objects (e.g., homogeneous solid objects). They are also important in processes where only filling volumes are relevant or meaningful, as in industrial processes for constructing mechanical parts. In these processes, entities of lower dimensions, such as
123
faces or edges, are not representable because they have null width.
4 Practical applications In this section, some results obtained by implementing the proposed algorithm in a geometric modeler system are presented [32, 33]. It is important to emphasize that the modeler should provide tools for surface intersection and for point location (point in region and point on surface). First, some manifold outputs are shown to demonstrate the
Fig. 4 Application of the proposed algorithm into two groups of entities and the respective results for union, intersection and difference in Boolean operations
Engineering with Computers (2012) 28:225–239
validity of the algorithm in a manifold environment. Then, some non-manifold results are presented, generated by applying Boolean operations over manifold solids and over topological entities with different dimensions (e.g., faces and solids). Finally, a complex example is presented to show the capability of the proposed algorithm to generate complex engineering models. The meshes are generated using well-established algorithms such as [33–36] for surface mesh generation and [37] volume mesh generation. 4.1 Manifold outputs Applying Boolean operations over simple primitives as cylinders, boxes and spheres may lead to more complex solids of innumerous shapes. It is possible to notice some of these solids in Fig. 5. Figure 6 shows all the stages of the creation of a pile and the generation of a mesh of tetrahedrons for the solid that represents it. First, a union operation between two boxes and a cylinder is performed. Then, a new cylinder is inserted in the model and subtracted from the solid generated by the union operation. The last step is the generation of a finite element mesh of tetrahedrons for the solid generated after the difference operation. It is important to emphasize that each solid that will serve as input data for a
233
Boolean operation must be recognized as a closed region and that all the intersections between surface patches must be calculated before the application of any Boolean operation. An example of Boolean operations applied to two solids bounded by generic curved surfaces is shown in Fig. 7. The original solids (a ‘‘twisted cube’’ and a ‘‘bottle’’) with all the intersections between surfaces already computed are presented in Fig. 7a. Figure 7b shows the intersection between these two solids. The difference operation between the ‘‘twisted cube’’ and the ‘‘bottle’’ is shown in Fig. 7c. 4.2 Non-manifold outputs Sometimes, even when the input data are representable in a manifold environment, the output data may not be so. That is the case of the results shown in Fig. 8. In the first example, some dangling faces are part of the result (faces that are not part of the boundary of any solid). In the second example, there are two solids touching each other through a common edge. This edge is adjacent to four different faces. An example of the application of Boolean operations over entities of different dimensions is presented in Fig. 9.
Fig. 5 Examples of manifold solids obtained by applying Boolean operations over simpler primitives
123
234
Engineering with Computers (2012) 28:225–239
Fig. 6 Creation of a pile: a primitive solids, b refinement, c union operation, d insertion of a new cylinder, e difference between the solid and the new cylinder and f tetrahedron mesh for the resulting solid (pile)
This means that two groups of entities were selected, one containing only faces and the other containing a solid. 4.3 Complex engineering (geomechanics) model An appropriated set of Boolean operations over non-manifold solids plays an important role in the creation of complex discrete (finite element or boundary element) models, such as the model generation of a tunnel system crossing geological faults and with several horizons. It is shown in Fig. 10 the steps necessary to create a tunnel model with four geological layers and a fault using
123
the proposed Boolean operation algorithm. Figure 10a depicts the original groups used to generate the model: group G1 contains three geological horizons (layer’s interface), group G2 contains the fault, group G3 contains the model bounding box limited by the ground surface, and group G4 contains the tunnel. Figure 10b–f show preparations or results of Boolean operation applied consecutively on the groups to obtain the final model, and Fig. 10g shows a detailed view of the tunnel crossing the fault and a horizon after the application of the Boolean operations. The second operand G3 of the difference operations shown in Fig. 10c is the complement of group G3.
Engineering with Computers (2012) 28:225–239
235
Fig. 7 a Refinement, b intersection and c difference between two solids bounded by curved surfaces
5 Conclusions An algorithm for Boolean operations on non-manifold models is proposed to allow the treatment of solids with multiple regions (internal interfaces) and degenerate portions (shells and wires), in the context of mesh generation. The algorithm was implemented in a pre-existing geometric modeler for finite element solid mesh generation [32, 33]. This modeler works with a hybrid data representation in which every model, besides being represented by the modeler’s internal data structure, can have another representation given by the CGC library [24], whenever the model needs to be topologically tied up. In general, there are two main modeling strategies for the creation of non-manifold solids. The first is analogous to the B-rep approach of the two-manifold modeling of
homogeneous solids. This strategy is the referred to here as direct cell manipulation [19, 20, 22, 24], in which a cell can be a topological entity of any dimension, such as a solid region, a surface patch or a wire curve. The second approach, analogous to the CSG approach of the twomanifold modeling, is called reasoning Boolean operations [9–12, 16, 17, 23, 25, 26]. This strategy, in general, is used to model manufacturable heterogeneous objects with inner boundaries, but with no lower-dimension portions. This works proposes an algorithm for the creation of discrete models, in the context of mesh generation, whose highlevel interface are Boolean operations. These operations apply to two objects that are organized as groups of selected entities of different dimensions. The main contribution of this study is that the proposed methodology handles specifically lower-dimension entities that may be
123
236
Engineering with Computers (2012) 28:225–239
Fig. 8 Non-manifold results obtained by applying Boolean operations over manifold solids
Fig. 9 Boolean operations applied over entities of different dimensions
of interest in idealized models with dangling or loose parts. These lower-dimension entities might represent restrictions to solid finite element mesh generation inside a region, interfaces between layers of different materials or anchor bolts in a tunnel design. To achieve the level of generality desired, some initial conditions had to be imposed to guarantee an essentially topological treatment of the problem. These initial conditions implicate pre-processing the input data to assure that all topological entities just touch each other (even if not along their boundaries). This pre-processing stage is a
123
refinement of surfaces and curves of the groups (operands of the Boolean operators). In this refinement, the original regions of the groups are preserved because they are used in the classification phase of the proposed algorithm. After refinement, the classification phase relies on region point location, which is the only geometric information that is necessary to apply the algorithm. The remaining information is essentially topological adjacency relationship. The pre-processing stage is a refinement of surfaces and curves of the groups (operands of the Boolean operators). In this refinement, the original regions of the groups are
Engineering with Computers (2012) 28:225–239
237
Fig. 10 Creation of a tunnel model crossing a fault and with three horizons: a original groups, b preparation and c results of the differences G1 ¼ G1 G3 and G2 ¼ G2 G3, d union G1 = G1UG2 (G1 = G2), e union G1 = G1UG3 (G1 [ G3), f difference G1 - G4, g detailed view of tunnel intersections
preserved because they are used in the classification phase of the proposed algorithm. After refinement, the classification phase relies on region point location, which is the only geometric information that is necessary to apply the algorithm. The remaining information is essentially topological. The classification phase of the algorithm is summarized in three tables (Fig. 3), which constitute another contribution of this work. These tables indicate the topological
entities that should be removed according to the target Boolean operation (union, intersection, or difference). A dominance factor of a group in relation to the other is used in an entity selection. The selection also takes into account the case in which the groups have equal dominance factor. Usually, this case occurs in modeling of homogenous material solids. In this case, a regularization procedure might be necessary to eliminate undesirable dangling and loose parts after the removal of entities for a specific
123
238
Boolean operation. Finally, after removing entities, a detection of new regions of the resulting non-manifold object is performed. The algorithm is very simple and intuitive, and its implementation in any B-Rep-based non-manifold system does not demand much effort. However, the system must incorporate non-linear primitive intersection, region detection and point location algorithms to assure the correct functioning of the Boolean operations algorithm. Several tests were performed, using manifold and nonmanifold input data and leading to manifold and nonmanifold output data, yet not, respectively. The algorithm proved to be quite efficient for relatively low-complexity models. An efficiency analysis is necessary for highly complex models, i.e., models with a significant number of high-degree curved primitives, a considerable number of surface intersections, a considerable number of closed regions, too many topological vertices of one group located inside the regions of the other group, and the presence of surface overlapping. It is important to mention that the algorithm depends and relies on the geometric capabilities of the modeler, since it should treat the geometric intersection problems, as well as region detection and point location procedures in a reasonable way. Acknowledgments The authors would like to thank the Computer Graphics Technology Group (Tecgraf) and Pontifical Catholic University of Rio de Janeiro (PUC-Rio) for the financial support and for providing the necessary space and resources used during the development of this work.
References 1. Hoffmann CM (1989) Geometric and solid modeling: an introduction. Morgan Kaufmann, San Fransisco 2. Ma¨ntyla¨ M (1988) An introduction to solid modeling Computer. Science Press, Rockville 3. Requicha A (1977) Constructive solid geometry. Technical Memo 25, University of Rochester, Production Automation Project 4. Miller JR (1989) Architectural issues in solid modelers. IEEE Comput Graph Appl 9:72–87 5. Requicha AG, Voelcker HB (1985) Boolean operations in solid modeling: boundary evaluation and merging algorithms. Proc IEEE 73:30–44 6. Shapiro V, Vossler DL (1993) Separation for boundary to CSG conversion. ACM Trans Graph 12:33–55 7. Ma¨ntyla¨ M (1986) Boolean operations of 2-manifolds through vertex neighborhood classification. ACM Trans Graph 5:1–29 8. Gardan Y, Perrin E (1995) An algorithm for reducing 3D Boolean operations to a 2D problem: concepts and results. Comput Aided Des 28:277–287 9. Magalha˜es ALCDC, Mesquita RC (2000) Exploring inner boundaries in solid modelers applied to electromagnetic problems. IEEE Trans Magn 36:1682–1686 10. Sun W, Hu X (2002) Reasoning Boolean operation based modeling for heterogeneous objects. Comput Aided Des 34:481–488
123
Engineering with Computers (2012) 28:225–239 11. Qian X, Dutta D (2003) Heterogeneous object modeling through direct face neighborhood alteration. Comput Graph 27:943–961 12. Kou XY, Tan ST, Sze WS (2006) Modeling complex heterogeneous objects with non-manifold heterogeneous cells. Comput Aided Des 38:457–474 13. Sen D, Srikanth TK (2008) Efficient computation of volume fractions for multi-material cell complexes in a grid by slicing. Comput Geosci 34:754–782 14. Colyer B et al (1997) Project MIDAS: magnet integrated design and analysis system. IEEE Trans Magn 33:1143–1148 15. Cracker GA, Reinke WF (1991) An editable nonmanifold boundary representation. IEEE Comput Graph Appl 11:39–51 16. Masuda H (1993) Topological operators and Boolean operations for complex-based non-manifold geometric models. Comput Aided Des 25:119–129 17. Gursoz EL, Choi Y, Prinz FB (1991) Boolean set operations on non-manifold boundary representation objects. Comput Aided Des 23:33–39 18. Gursoz EL, Choi Y, Prinz FB (1990) Vertex-based representation of non-manifold boundaries. In: Wozny M, Turner J, Preiss K (eds) Geometric modeling for product engineering. NorthHolland, Amsterdam, pp 107–130 19. Rossignac JR, O’Connor MA (1989) A dimensional-independent model for pointsets with internal structures and incomplete boundaries. Geometric modeling for product engineering. NorthHolland, Amsterdam, pp 145–180 20. Weiler K (1986) Topological structures for geometric modeling. Dissertation, Rensselaer Polytechnic Institute, Troy, NY 21. Weiler K (1988) The radial-edge structure: a topological representation for non-manifold geometric boundary representation. Geometric modeling for CAD applications. North Holland, Amsterdam, pp 3–36 22. Lienhardt P (1991) Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput Aided Des 23:59–82 23. Arbab F (1990) Set models and Boolean operations for solids and assemblies. IEEE Comput Graph Appl 10:76–86. doi: 10.1109/38.62698 24. Cavalcanti PR, Carvalho PCP, Martha LF (1997) Non-manifold modeling: an approach based on spatial subdivision. Comput Aided Des 29:209–220 25. Rossignac JR, Requicha AG (1991) Constructive non-regularized geometry. Comput Aided Des 23:21–32 26. Gardan Y, Minich C, Perrin E (2003) Boolean operations on feature-based models. J WSCG (Winter School of Computer Graphics) 27. Keyser JC, Culver T, Foskey M, Krishnan S, Manocha D (2004) ESOLID—a system for exact boundary evaluation. Comput Aided Des 36:175–193 28. Shewchuk JR (1997) Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput Geometry 18:305–363 29. Keyser JC (2000) Exact boundary evaluation for curved solids. Dissertation, University of North Carolina 30. De Berg M, Van Kreveld M, Overmars M, Schwarzkopf O (1997) Computational geometry: algorithms and applications. Springer, Berlin 31. Preparata FP, Shamos MI (1990) Computational geometry: an introduction. Springer, New York 32. Coelho LCG, Gattass M, Figueiredo LH (2000) Intersecting and trimming parametric meshes on finite-element shells. Int J Numer Methods Eng 47:777–800 33. Lira WWM, Coelho LCG, Cavalcanti PR, Martha LF (2002) A modeling methodology for finite element mesh generation of multi-region models with parametric surfaces. Comput Graph 26:907–918
Engineering with Computers (2012) 28:225–239 34. Miranda ACO, Martha LF (2002) Mesh generation on highcurvature surfaces based on a background quadtree structure. In: Proceedings of 11th international meshing roundtable, Ithaca, pp 333–341 35. Lira WWM, Coelho LCG, Martha LF (2002) Multiple intersections of finite-element surface meshes. In: Proceedings of 11th international meshing roundtable, Ithaca, pp 355–363
239 36. Miranda ACO, Martha LF, Wawrzynek PA, Ingraffea AR (2009) Surface mesh regeneration considering curvatures. Eng Comput 25:207–219. doi:10.1007/s00366-008-0119-9 37. Cavalcante-Neto JB, Wawrzynek PA, Carvalho MTM, Martha LF, Ingraffea AR (2001) An algorithm for three-dimensional mesh generation for arbitrary regions with cracks. Eng Comput 17:75–91
123
