Figure 1.25: Some of the limitations of Tate’s algorithm [60]: (a) asymmetry of a detail producing the same global parameters; (b) different loops with same properties;
(c) rotation problem; (d) no distinction between protrusion and depression.
general problem, because a B-Rep model always describes surfaces of revolution with two or more pieces, which breaks the object symmetry (see Chapter4).
Compared to the other approaches, Tate’s method is the one that uses effectively a B-Rep model, as an infinite point set model. As discussed in Section 1.1, B-Rep models are used in a design process as basic digital models and widely subjected to shape transformation processes. Therefore, using a B-Rep model that directly matches exactly its boundary, avoids problems of referring to its faceted representation. It is a significant advantage compared to other approaches.
Conclusion 35 applicable in the engineering context.
In a PDP context, B-Rep CAD models are more useful and more important than the others but they are infinite point set representation and the main group of re- searches uses a discretized version of these models as input. This does not provide a better answer to the engineering needs. Tate’s work is an entirely new approach that directly takes a B-Rep CAD model as input. With the description of loop properties, the candidate symmetry informations are created but this method has many limita- tions that restrict the results’ validity to special cases. This is inherently due to the use of global parameters attached to areas of the object boundary that cannot reflect precisely their spatial configuration. Finally, after many years of research, symme- try detection on B-Rep CAD models is still an issue for its use in a PDP and other engineering applications.
Chapter 2
Principle of the symmetry analysis approach and hypotheses
This chapter gives a overview of the symmetry analysis approach and sets the objectives of the present work. The symmetry analysis is devoted to engineering applications with CAD based models to process shapes close to real objects. The proposed approach is a semi global one since an object boundary is decomposed into patches and curves defining the boundary of these patches. The objects are described in STEP format, hence they are decomposed into patches and their symmetry properties are not readily available.
Surfaces and curves bounding objects stands for infinite point sets and they form the basis of the approach. In this case, maximal surfaces and curves generation is mandatory before the symmetry analysis and performed with the help of hypergraphs, which form the main datastructure of the algorithm. Then, candidate symmetry properties are extracted from each entity of the hypergraphs to initiate a divide phase of a divide and conquer process. The conquer phase consists in propagation processes producing global as well as local symmetry information about the object analyzed.
2.1 Principles of the symmetry analysis approach
First of all, it has to be recalled that reflexive symmetry is, by definition, a point based property (see eq. 1.1and eq. 1.2), which has been exploited essentially for dis- crete (point sets) and piecewise linear (mesh based or faceted models) representations of objects. Consequently, the algorithms currently available have a rather high com- plexity (polynomial of high degree) and have to be applied to a large amount of entities due to the discrete representation of the objects processed. In addition, many of them produce an approximate symmetry information because of these discrete representa- tions. Effectively, these representations cannot represent all the details of a physical object like engineering components. Axisymmetry cannot be extracted from discrete representations and the location of symmetry planes stays approximative as a function of a shape discretization, which is often not acceptable for engineering applications.
Indeed, engineering applications require symmetry properties evaluation at the level of accuracy of manufactured components to be useful in a PDP. It is the purpose of the proposed approach to address this issue.
Consequently, the purpose is to set up an approach of higher level such that infinite point sets rather than finite ones can be processed, allowing for a precise description of objects while enabling a much faster processing. In that sense the proposed approach is a semi global approach since an object boundary is decomposed into patches and curves defining the boundary of these patches. There are two categories of infinite point sets: curves defining the boundary of patches and patches defining a subset of the object boundary. The goal of the approach is to analyze an object symmetry without limitation of discrete representations: the resolution of point sets and the chordal deviation between a mesh and the precise model of this object. As a result, the input model can contain all the details necessary to get very close to real objects if needed. Regarding the applications of symmetry properties in a PDP, the proposed approach forms a basis to add symmetry information to several of its steps, e.g.
restructuring a modeling tree to incorporate the object symmetries since a design process rarely expresses all the object symmetries; exploiting symmetry properties to simplify an object for finite element simulations, to structure an assembly process, to improve the trajectory planning of manufacturing processes, to compare objects and characterize similarities in databases, etc. Whatever, the step considered, symmetry properties being intrinsic to an object, they must be independent from the boundary decomposition of this object. Indeed, this decomposition is the result of modeling and modification processes in a PDP whereas it must be intrinsic to the object symmetries.
It is part of the principle of the proposed approach to set up an intrinsic framework to analyze and exploit the symmetry properties of an object.
Tate’s approach [58][60], is among the closest to the proposed one. The input model is of type B-Rep and it can contain a combination of planes, cylinders, cones, spheres and tori with possible spline surfaces. This is an advantage but the symmetry detection does not rely on an intrinsic decomposition, hence ambiguous parameters and heuristics have been used and reduce the efficiency of this approach. Here, the purpose is to preserve the intrinsic framework throughout the symmetry analysis pro- cess to obtain a reliable and robust process. According to the contribution of different models into a PDP as addressed at chapter 1, the B-Rep NURBS is used by many famous CAD software to generate digital models often regarded as reference ones by companies. B-Rep NURBS models can contain many shape details as defined by engineers and technicians and can be combined with CSG operators.
However, patch boundaries of these models are often resulting from intersection computations, hence they are approximated and their symmetry properties can be perturbed and difficult to obtain robustly like faceted representations are approxi- mations of smooth objects through chordal deviation. Here, the approach aims at favoring the use of intrinsic parameters all through the symmetry analysis. To this end, the intrinsic parameters of surfaces should be used as much as possible to derive symmetry properties and addressing the geometry of intersection curves is avoided to preserve the robustness of the analysis.
Principles of the symmetry analysis approach 39 If the proposed approach could be integrated in industrial CAD platforms di- rectly or in tight connection with them, this would be very helpful for the design and product development processes. Modification suggestions could be easily exploited in the original CAD software because it would be connected to the modeling tree of objects. Because of the commercial protection, these modeling tree and B-Rep datas- tructures are internal, hence difficult to access and hardly generic. Even if some of these datastructures are partly accessible, it holds in one software environment and can be transposed to another one only with a fair amount of software development effort.
Another possible integration focuses on shape transformations often taking place between product views in a PDP. It can be associated with all CAD softwares and all simulation platforms when B-Rep NURBS datastructures rely on standard formats.
In this context, object shapes can be generated from an original industrial CAD software and may contain numerous details, which justify modification requirements, e.g. simplifications for finite element analyses, for digital review visualization, etc.
In the present approach, the STEP format [26][71] has been selected because it is an ISO standard where an object boundary is described with surfaces and curves forming infinite sets of points and analytic surfaces can be explicitly available too, i.e. planes, cylinders, cones, etc. The topology of volume boundaries is explicit and can be used to robustly extend symmetry properties attached to patches and their boundaries.
As far as symmetry analysis is concerned, both categories of integration are part of the current approach since symmetry is an intrinsic shape property, which relies on datastructures and processes rather independent from CAD modeling issues. Both categories of integration will be observed here.
Analytic surfaces are widespread in mechanical engineering applications, which entails the description of a wide range of components while benefiting well known symmetry properties to initiate a high level approach for symmetry analysis. Conse- quently, the main idea of the approach is to analyze object symmetries from the level of patches, seen as infinite point sets, and to extend progressively these point sets to neighboring ones. This can be described as a divide-and-conquer process:
• Every surface patch is described by its intrinsic parameters and its location in space. This is independent of their parametric or implicit equations that could be used and it concentrates on the embedding of each patch in IR3 where the symmetry properties take place;
• Every surface patch has self symmetry properties that can be combined with the symmetry constraints of its adjacent surfaces to take into account its boundary as patch boundary of a B-Rep model: this produces Candidate Symmetry Planes (CSPs). It is the result of a division phase;
• Then, a propagation process can extend the validity of each CSP over the largest
possible area of a model boundary to structure the global symmetry properties of the object: it is the conquer phase.
Apart from symmetry analysis, asymmetry is also very interesting and is more useful for design and PDPs since many components are not globally symmetric, rather symmetry exists only at the level of a boundary subset and the loss of symmetry can be used as a means to evaluate shape transformations that could be useful for simplification purposes. The symmetry analysis proposed in the current approach intend to address this issue since the propagation process helps identify and structure areas where symmetry properties are valid.
Finally, the objectives of the approach can be summarized as the algorithms to answer the following questions:
1. Is a B-Rep NURBS model symmetric with respect to some symmetry planes or symmetry axes? If so, where are located these symmetry planes and symmetry axes?
2. If a B-Rep NURBS model has no global symmetry property, doest it benefit local symmetry planes or axes? Where are located these symmetry planes or axes and what are their extents of validity over the model boundary?
3. How does these symmetry properties can be obtained at various steps of the design and PDPs? Into which extent they can be obtained with a process in- trinsic to the object shape and how robust is this process under a wide diversity of shapes?