Advances in the Treatment of Trimmed CAD Models due to Isogeometric Analysis
113th World Congress on Computational Mechanics (WCCM XIII) and 2nd Pan American Congress on Computational Mechanics(PANACM II), July 22-27, 2018, New York City, NY, USA
ADVANCES IN THE TREATMENT OF TRIMMED CAD MODELS DUE TOISOGEOMETRIC ANALYSISBENJAMIN MARUSSIG ∗ , † ∗ † Graz University of TechnologyInstitute of MechanicsKopernikusgasse 24/IV, 8010 Graz, Austria
Key words:
Isogeometric Analysis, Trimming, SSI operation, CAGD
Abstract.
Trimming is a core technique in geometric modeling. Unfortunately, the result-ing objects do not take the requirements of numerical simulations into account and yield vari-ous problems. This paper outlines principal issues of trimmed models and highlights differentanalysis-suitable strategies to address them. It is discussed that these concepts not only provideimportant computational tools for isogeometric analysis, but can also improve the treatment oftrimmed models in a design context.
Isogeometric analysis (IGA) aims to bridge the gap between computer aided geometric de-sign (CAGD) and analysis by using CAGD technologies for numerical simulations. Since theintroduction of IGA in 2005, it has been demonstrated that the synthesis of these disciplinesallows not only an improved interaction, but yields many computational advantages . Nowa-days, IGA is widely recognized as a powerful alternative to the conventional analysis methodol-ogy. In the following, the attention is drawn to a somewhat different aspect, namely (potential)benefits for CAGD due to developments made in IGA. It is focused on challenges concerningrobustness and interoperability. In particular, the treatment of trimmed models is addressed,because these representations play a central role in engineering design and the integration ofdesign and analysis . First, the evolution of trimmed CAD models is presented in order tooutline the related problems. Based on that, corresponding advances of IGA are discussed.1 enjamin Marussig The problem of computing surface-to-surface intersections (SSI) is closely related to trim-ming and thus, it is discussed at the beginning of this section. Then, the formulation of solidmodels defined by trimmed surfaces is presented and finally related robustness issues and therole of trimmed models with respect to the exchange of CAD data are discussed.
Computing intersections of surfaces is a crucial task in various types of modeling processes.First of all, it is the core ingredient for Boolean operations which are the most important func-tions in creating CAD objects . In general, the intersection of two parametric surfaces S ( u, v ) = ( x ( u, v ) , y ( u, v ) , z ( u, v )) (1) S ( s, t ) = ( x ( s, t ) , y ( s, t ) , z ( s, t )) (2)leads to a system of three nonlinear equations . These equations represent the three coordinatedifferences of the surfaces, S and S , with the four unknown surface parameters u, v, s, t . Inmost cases, the solution describes a curve, but intersection points, subsurfaces, or empty setsmay occur as well.Efficiently providing all features of these solutions is the purpose of SSI operations . Thedevelopment of a good SSI procedure is a very challenging task due to the fact that the oper-ation has to be accurate , efficient , and robust . These attributes are indeed quite contradictoryand the definition of an adequate balance between them depends strongly on the applicationcontext. Early solid modeling systems employed analytic methods to compute exact parametricdescriptions of intersections between linear and quadratic surfaces . Unfortunately, the alge-braic complexity of an intersection increases rapidly with the degree of S and S , which hasbeen thoroughly discussed by Sederberg and co-workers in the 1980s. This makes analyticapproaches impractical; a fact often illustrated by the algebraic degree of an intersection of twogeneral bicubic surfaces which is 324.Hence, alternative SSI schemes are needed. These concepts can be broadly classified aslattice evaluation schemes , subdivision methods , and marching methods . The formerreduces the dimensionality of the problem by computing intersections of a number of isocurvesof S with S and vice versa. The second strategy uses approximations of the actual surfaces,often defined by a set of piecewise linear elements, and computes the related intersections withrespect to the simplified objects. Finally, marching methods define an intersection curve bystepping piecewise along the curve. This requires detection of appropriate starting points, de-termination of point sequences along the intersection that emit from the starting points, andproper sorting and merging of these individual sequences. Marching methods are by far themost widely used schemes due to their generality and ease of implementation . However, eachintersection strategy has its advantages and drawbacks, hence SSI algorithms usually use hybridconcepts that combine different features of these approaches .2 enjamin Marussig S ( u, v ) S ( s, t ) ˆ C (a) Model space uv C t (b) Parameter space of S ( u, v ) st C t (c) Parameter space of S ( s, t ) Figure 1: Independent curve interpolation of an ordered point set to obtain approximations of the intersection of twopatches S ( u, v ) and S ( s, t ) . The set of sampling points depends on the SSI algorithm applied. The subsequentinterpolation of these points is performed in (a) the model space and the parameter space of (b) S ( u, v ) and (c) S ( s, t ) leading to independent curves ˆ C , C t , and C t . Irrespective of the scheme applied, the initial result of a SSI operation is usually a set ofsampling points that represent the intersection . An approximate intersection curve in modelspace ˆ C is subsequently obtained by some curve-fitting technique such as point interpolationor least-squares approximation. Thus, ˆ C does not lie on either of the intersecting surfaces ingeneral. Furthermore, the sampling points are mapped into the parameter spaces of S and S ,where they are again used as input for a curve-fitting procedure. This yields the main resultof the SSI process, namely trimming curves C t in the two-dimensional parametric domains.These C t are usually represented by spline curves. They are essential because they allow thedefinition of arbitrarily shaped partitions within a tensor product surface, which enables propervisualization of intersecting surfaces and the application of Boolean operations. Every C t canbe mapped into model space, but the resulting image ˜ C t will not coincide with ˆ C . In short, SSIoperations yield various independent approximations of the actual intersection (see Figure 1),rather than an unambiguous solution. It is emphasized that there is no direct mapping betweenthese different approximations and that the sampling point data for their construction is usuallydiscarded once the curves are computed. There are various approaches for representing geometric objects . The most popularone in engineering design is the boundary representation (B-Rep) and the benefits of storingan object’s shape by means of its boundary were already elaborated in the seminal work ofBraid . B-Rep solid modeling utilizes SSI schemes to create arbitrarily defined free-formgeometric entities. The corresponding algorithms, however, require more than the computation3 enjamin Marussig of intersection curves. Essential attributes of geometric modeling operators are : • the determination of the geometric surface descriptions, • the determination of the topological descriptions, and • the guarantee that the geometry corresponds unambiguously to the topology.Topological data is not metrical, but addresses connectivity and dimensional continuity of amodel . Its determination requires the classification of the neighborhood of various entities(faces, edges, and vertices) involved in the intersections . In CAGD, the term solid model em-phasizes that a representation contains the descriptions of an object’s shape, i.e., the geometry,as well as its structure, i.e., the topology; it does not refer to the dimension of the object defined.The idea of a trimmed model appeared already in 1974 and was proposed by Pierre Bézier .However, the approach was presented with little theoretical support and it took some time todevelop a rigorous way to represent trimmed free-form solid models. The first formulationsupporting Boolean operations and free-form geometry was presented by Farouki as well asCasale and Bobrow in the late 1980s. In general, the connectivity between intersectingsurfaces is established by assigning the approximate intersection curves (which do not coin-cide) to a single topological entity. Further, Boolean operations define the relation of the faces,edges, and vertices of a model. Various data structures for B-Reps have been proposed to find acompromise between storage requirements and response to topological questions. The crucialdiscrepancy, which still exits, is that solid modeling is concerned with the use of unambiguous representations, but SSI schemes introduce approximations and do not provide a unique repre-sentation of an intersection. In other words, all these modeling approaches have to deal withimprecise data and thus, fail to guaranty exact topological consistency . Thus, the robustnessof a trimmed B-Rep becomes a crucial factor. Several robustness issues arise in case of imprecise geometric operations. As a matter offact, numerical output from simple geometric operations can already be quite inaccurate - evenfor linear elements . For SSI schemes, ill-conditioned intersection problems are particularlytroublesome. Such cases occur when intersections are tangential or surfaces overlap, for in-stance. Since geometrical decisions are based on approximate data and arithmetic operationsof limited precision, there is an interval of uncertainty in which the numerical data cannot yieldfurther information and the fact that SSI operations do not provide a unique intersection curvemakes the situation even more delicate.The most common strategy to address robustness issues is the use of tolerances . Theyshall assess the quality of geometrical operations and may be adaptively defined or dynam-ically updated . Alternative approaches employ interval arithmetic or exact arithmetic , but4 enjamin Marussig these concepts have certain drawbacks (especially with respect to efficiency) and hence, toler-ance based approaches are usually preferred. Unfortunately, tolerances cannot guarantee robustalgorithms since they do not deal with the inherent problem of limited-precision arithmetic.Overall, the formulation of robust solid models with trimmed patches is still an open issue.This is particularly true when a model shall be transferred from a CAD system to another soft-ware tool. Since there is no canonical representation of trimmed solid models, different systemsmay employ different data structures and robustness checks. Consequently, data exchange in-volves a translation process which can lead to misinterpretation. This makes the treatment oftrimmed solid models a key aspect for the interoperability of design and analysis. Since the introduction of IGA, more and more scientists in the field of computational me-chanics have become aware of the advantages and deficiencies of design models and variousanalysis-suitable approaches dealing with CAD-related challenges have been proposed. Here,we highlight advances made in the context of local refinement of multivariate splines, whichare important to derive watertight models, and the treatment of trimmed geometries.
The lack of local refinement of conventional tensor product splines was one of the firstissues tackled by the IGA community. The topic emerged to an active area of research andseveral techniques have been developed, such as T-splines , LR-B-splines , hierarchi-cal B-splines , and truncated hierarchical B-splines . Some of these concepts were firstpresented in the context of CAGD (e.g., T-splines and hierarchical B-splines). However, theirapplication in an analysis setting has provided a huge impetus to their further enhancement. Infact, these concepts have become so technically mature that the question is no longer if localrefinement of multivariate spline is feasible, but what technique do you prefer.Besides the apparent computational benefits, these advances in local refinement techniquesalso offer new possibilities for the design community. Admittedly, these local refinement con-cepts are usually not incorporated in current CAD systems (yet), but a strong indicator for theimpact of IGA is a novel capability of the next version of the Standard for the Exchange ofProduct Model Data (STEP) – the most involved neutral exchange standard. That is, it will in-clude entities that facilitate a canonical representation of locally refined tensor product splines .To be precise, this feature affects the part “geometric and topological representations," whichfocuses on the definition of geometric models and represents a core component of STEP. Re-garding trimmed models, the ability of local refinement can be a powerful tool as well. Forinstance, effects of trimming may be localized or trimmed surfaces may even be joined as itis done during the conversion of trimmed B-Reps to watertight T-spline models .5 enjamin Marussig The term "non-watertight" is commonly used to stress that trimmed models have small gapsand overlaps between their intersecting surfaces. They occur due to the inevitable approxima-tions introduced by SSI operations as discussed in Section 2.1. Watertight representations, onthe other hand, possess unambiguously-defined edges and a direct link between adjacent el-ements. This link is missing in case of trimmed models and has to be established (or at leasttaken into account) in order to make them analysis-suitable. Current attempts for the integrationof trimmed geometries into IGA can be divided into global and local approaches . The formeraims to convert trimmed solid models to watertight ones in a pre-processing step (or even al-ready during the design stage), whereas the latter intends to enhance the simulation tool so thatit is able to cope with the models’ flaws. The basic idea of local approaches is that trimmed parameter spaces are used as backgroundparameterization for the simulation. Hence, there is a close relation to fictitious domain meth-ods and the corresponding challenges are indeed similar: First, the elements needed for theanalysis have to be detected . Second, special integration techniques for elements cut by atrimming curve have to be employed. Third, weak enforcement of boundary conditions orweak coupling of adjacent surfaces has to be addressed . Finally, stability issues of cutelements with small support should be taken into account . The main difference to fictitiousdomain methods is that an additional effort is required to associate the degrees of freedom ofadjacent patches, keeping in mind that their intersections have non-matching parameterizations,gaps, and overlaps. Usually, point inversion algorithms are utilized to establish a link be-tween adjacent surfaces. Alternatively, simulation methods that allow discontinuities betweenelements can be applied.The majority of the publications on IGA with trimmed geometries employs such local con-cepts. A possible reason could be that these approaches focus on analysis aspects and thus,may seem more feasible for researchers in the field of computational mechanics. On the otherhand, this also means that the number of subjects that may affect CAGD is relatively small. Theessential common ground is the problem of finding robust procedures and the use of tolerancesto achieve a proper model treatment. However, this does not mean that the task is trivial. Asoutlined in Section 2.3, the robust treatment of trimmed models is a really challenging issue inCAGD. Regarding IGA, an additional obstacle complicates the situation, that is, analysis soft-ware has to deal with extracted data. In other words, the input data provides only a reducedportion of the information that would be available in the initial CAD tool. Furthermore, thisportion may be altered due to the translation process that might be required for the exchange.This aspect could be improved when the exchange procedure is tailored to a specific CAD sys-tem using its native data format. Yet, this would require vendor interaction and the restrictionto a single software. Most importantly, this option is not very sustainable since a native format6 enjamin Marussig of a CAD system may become obsolete after a new software version is released.
Global approaches decompose trimmed model components into a set of regular surfaces orreplace them by other spline representations such as subdivision surfaces or T-splines. Similarto the developments regarding local refinement, some strategies may originate from CAGD. Forinstance, isogeometric analysis with subdivision surfaces and T-splines can be includedinto the class of global techniques. Reconstruction concepts proposed in the context of analysisusually aim to replace trimmed surfaces by a set of regular ones. This is done by means ofruled surfaces , Coons patches , triangular Bézier patches , or a reconstruction based onisocurves .Global approaches seek to resolve the core problem of trimmed solid models and hence,they are more related to CAGD than their local counterpart. Consequently, advances in thisresearch area are more likely to have an impact in the design community. T-splines are a primeexample in this regard. The introduction of T-splines in IGA has led to various enhancementssuch as analysis-suitable T-spline spaces that guarantee linear independent basis functions andit would be no exaggeration to say that IGA has been a driving force for the development T-splines in the past years. Approaches emerging from an analysis perspective can also be veryuseful for design applications. For instance, the reconstruction scheme introduced by Urick could be utilized to create watertight Boolean operations. This possibility is currently underinvestigation and a preliminary example is illustrated in Figure 2. Note that Figure 2(c) showsa single surface with a matching parameterization across the computed intersection. (a) Original surfaces (b) State-of-the-art Boolean operation (c) Watertight Boolean operationMatching parameterizationFigure 2: Comparison of conventional and watertight Boolean operations: (a) initial surfaces and their controlgrids colored in blue and red, respectively, (b) the outcome based on a conventional Boolean operation, and (c) theresult of the watertight counterpart. The control grids in (a) and (b) are identical, whereas in (c) the control pointsare updated to reflect the intersection which is defined as an isocurve of the watertight surface. enjamin Marussig In contrast to local approaches, it is hard to identify general ingredients associated to globalreconstruction procedures. Each global strategy requires a self-contained concept which be-comes more and more sophisticated with its capabilities. This is indeed a potential drawback,especially when new features are added later on. Once a global scheme can be successfullyapplied, that is, it leads to a watertight model, two fundamental questions have to be addressed:(i) the representation of unstructured meshes and (ii) the treatment of extraordinary points (EPs).These topics are indeed of great interest for CAGD. Current model data is usually based on astructured mesh setting, where all control points of a surface are arranged in a regular grid.When local refinement of tensor product surfaces is considered as well, a structured mesh ad-mits only interior points of valance 4 and T-junctions. But in case of smooth watertight modelspoints with any valence (e.g., 3, 4, 5, ...) can occur and the arising non-regular points are re-ferred to as EPs. These EPs also affect analysis properties and hence, their proper treatment isimportant for IGA . It is worth noting that IGA researchers are also included in recent attemptsseeking to include the capability of representing unstructured meshes in STEP.Looking at the overall scope of global schemes, it is fair to say that they do address coreissues of trimmed models. A compelling analysis-suitable approach could eventually resolvethe robustness and interoperability of trimmed models not only for analysis, but all downstreamapplications. On the other hand, they are more complex and their success will also depend ontheir acceptance in CAGD. A brief overview of the development of surface-to-surface intersection operations and theformulation of trimmed solid models is provided to indicate the potential problems related tothese popular computer aided geometric design (CAGD) representations. Strategies for isoge-metric analysis (IGA) with trimmed geometries are listed and divided into two categories: (i)local approaches aim to enhance the analysis process and (ii) global approaches try to converttrimmed objects to regular models before the simulation. It is argued that these advances alsobring new insights for CAGD, indicating the mutual benefits due to the interaction of the designand analysis communities. That IGA solutions lead to improvements for analysis as well asdesign has already been demonstrated by the evolution of local refinement concepts for mul-tivariate splines and the recent developments regarding the treatment of trimmed models areindeed on a similar trajectory.
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