Point Membership Classification Research Articles

BeCOTS: Bent Corner-Operated Tran-Similar Maps and Lattices

A field (2-parameter family of similarities) S(x,y) is Tran-Similar (TS) if there exist similarities U and V such that S(x,y)=Ux⋅Vy and U⋅V=V⋅U. The recently proposed COTS map, M, is defined in terms of a planar TS field, S, as M(x,y)=S(x,y)⋅A and is Corner-Operated (CO), i.e., controlled by the images A, B, C, and D of the four corners of the unit-square in parameter space. The CO property enables intuitive design of planar warps. The TS property has several benefits: (1) The COTS map distributes distortion evenly; (2) It may be used to create pleasing patterns in which consecutive instances along any given row or column are related by the same similarity; and (3) Point-Membership Classification and Integral-Property Computation may be performed in constant time. However, the COTS map is only defined when the four corners are coplanar. To remove this restriction, we propose the Bent COTS (BeCOTS) generalization of the COTS map to three dimensions. It is CO (operated by its four non-coplanar corners) and TS (inheriting the properties of COTS mentioned above). We discuss potential benefits of BeCOTS for supporting the design and parameterization of highly-complex, bent lattices, for accelerating their processing, and for reducing the cost of manufacturing a class of large architectural structures.

Exact Representations and Geometric Queries for Lattice Structures with Quador Beams

Objects with architected internal structure often consist of a lattice of beams that are interconnected at nodes. Compared to the conventional cylindrical and conical beams, quador-beams are bounded by quadrics-of-revolution. We decompose the lattice into an assembly of hubs, wherein a hub is the union of a node (ball) with the half-beams incident upon it. In this paper, we propose (1) a hybrid representation of a quador-hub (hub with quador half-beams) consisting of exact CSG + CST + Boundary models, (2) a compact data structure and associated operators to work with the hybrid representation, and (3) algorithms to perform fundamental geometric queries on the hub. We provide the operators to query the half-beams of the hub, to inquire the trimming planes of a surface, or to traverse the faces, edges and vertices of the hub. We suggest efficient algorithms, that take advantage of the hybrid representation, for geometric queries such as point membership classification, planar slicing, surface meshing, and area/volume computation. We outline a possible approach for computing exact answers to all tests that are required for computing the exact topology of the proposed representation of the hub. Finally, we show that the simplified geometry of quador hubs leads to smaller data sizes and improved performance in a commercial modeler.

Integrating CAD and numerical analysis: ‘Dirty geometry’ handling using the Finite Cell Method

This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with “dirty geometries”. FCM, being a fictitious domain approach based on higher order finite elements, embeds the physical model into a fictitious domain, which can be discretized without having to take into account the boundary of the physical domain. The true geometry is captured by a precise numerical integration of elements cut by the boundary. Thus, an effective Point Membership Classification algorithm that determines the inside–outside state of an integration point with respect to the physical domain is a core operation in FCM. To treat also “dirty geometries”, i.e. imprecise or flawed geometric models, a combination of a segment-triangle intersection algorithm and a flood fill algorithm being insensitive to most CAD model flaws is proposed to identify the affiliation of the integration points. The present method thus allows direct computations on geometrically and topologically flawed models. The potential and merit for practical applications of the proposed method is demonstrated by several numerical examples.

GPU-accelerated generation and rendering of multi-level voxel representations of solid models

Solid models traditionally use boundary-representation (B-rep) to define and model their geometry. However, performing modeling operations such as Boolean operations or computing point membership classification with B-rep is computationally intensive, since B-reps do not have volumetric information. Voxelized representations, on the other hand, can be extended to include volumetric information of solid models. However, in order to use voxelized representations for solid modeling, efficient methods for voxelizing a B-rep solid model needs to be developed. In this paper, we present GPU-accelerated methods for creating and rendering a multi-level voxelization of a solid model that can be used for modeling operations. We describe two GPU-accelerated algorithms; one for creating a multi-level voxelization given a B-rep of a solid model and another for ray casting to render the multi-level voxelization of the solid model. We describe compact and flat data structures that can be used to store the multi-level voxelization data and can be efficiently retrieved in parallel by GPU-algorithms for rendering and modeling operations. The GPU-accelerated multi-level voxelization method can generate models with an effective voxel count of up to 8 billion voxels. In addition, the GPU voxelization algorithm is more than 40x faster than the CPU implementation in generating the voxelization. Finally, we outline a few applications for the voxel representation, which include fast point-membership classification, volume computation, and collision detection.

Direct immersogeometric fluid flow analysis using B-rep CAD models

We present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche's method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the density and distribution of the surface quadrature points are crucial for the weak enforcement of Dirichlet boundary conditions and for obtaining accurate flow solutions. Also, with sufficient levels of surface quadrature element refinement, the quadrature error near the trim curves becomes insignificant. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor directly represented using B-rep.

Representation and analysis of additively manufactured parts

Representations of solid models were initially formulated partially in response to the need to support automation for numerically controlled machining processes. The assumed equivalence between shape, topology, and material properties of manufactured components and their computer representations led to the practice of modeling and simulating the behavior of physical parts before manufacture. In particular, representations of shape and material properties are treated in distinct nominal models for most unit manufacturing processes. Additively manufactured parts usually exhibit deviations from their nominal geometry in the form of stair-stepping artifacts and topological irregularities in the vicinity of small features. Furthermore, structural properties of additively manufactured parts have experimentally been shown to be dependent on the build orientation defining the cross sections where material is accumulated. Therefore geometric models of additively manufactured parts cannot be decoupled from the manufacturing process plan.In this paper we show that as-manufactured shapes may be represented in terms of the convolution operation to capture the additive deposition of material, measure the conformance to nominal geometry in terms of overlap volume, and model uncertainties involved in material flow and process control. We then demonstrate a novel interoperable approach to physical analysis on as-manufactured part geometry represented as a collection of machine-specific cross sections augmented with boundary conditions defined on the nominal geometry. The analysis only relies on fundamental queries of point membership classification and distance to boundary and therefore does not involve the overhead of model preparation required in approaches such as finite element analysis. Results are shown for non-trivial geometries to validate the proposed approach.

A comparison of sampling strategies for computing general sweeps

Sweeping moving objects has become one of the basic geometric operations used in engineering design, analysis and physical simulation. Despite its relevance, computing the boundary of the set swept by a non-polyhedral moving object is largely an open problem due to well-known theoretical and computational difficulties of the envelopes. We have recently introduced a generic point membership classification (PMC) test for general solid sweeping. Importantly, this PMC test provides complete geometric information about the set swept by the moving object, including the ability to compute the self-intersections of the sweep itself. In this paper, we compare two recursive strategies for sampling points of the space in which the object moves, and show that the sampling based on a fast marching cubes algorithm possesses the best combination of features in terms of performance and accuracy for the boundary evaluation of general sweeps. Furthermore, we show that the PMC test can be used as the foundation of a generic sweep boundary evaluator in conjunction with efficient space sampling strategies for solids of arbitrary complexity undergoing affine motions.

Classifying points for sweeping solids

Many diverse engineering problems can be modeled with solid sweeping in a conceptually simple and intuitive way, and sweeps are considered to be one of the basic representation schemes in solid modeling. However, many properties of sweeps as well as their “informational completeness” are not well understood, which is the primary reason why computational support for solid sweeping remains scarce. We propose a generic point membership classification (PMC) for sweeping solids of arbitrary complexity moving according to one parameter affine motions. The only restrictive assumption that we make in this paper is that the initial and final configurations of the moving object do not intersect during the sweep. Our PMC test is defined in terms of inverted trajectory tests against the original geometric representation of the generator object, which implies that this test can be implemented in any geometric representation that supports curve-solid intersections. Importantly, this PMC test provides complete geometric information about the set swept by 3-dimensional objects in general motions. At the same time, it establishes the foundations for developing a new generation of computational tools for sweep boundary evaluation and trimming, as well as a number of practical applications such as shape synthesis, contact analysis and path planning.

The HybridTree: Mixing skeletal implicit surfaces, triangle meshes, and point sets in a free-form modeling system

In this paper, we present a hybrid modeling framework for creating complex 3D objects incrementally. Our system relies on an extended CSG tree that assembles skeletal implicit primitives, triangle meshes and point set models in a coherent fashion: we call this structure the HybridTree. Editing operations are performed by exploiting the complementary abilities of implicit and polygonal mesh surface representations in a complete transparent way for the user. Implicit surfaces are powerful for combining shapes with Boolean and blending operations, while triangle meshes are well-suited for local deformations such as FFD and fast visualization. Our system can handle point sampled geometry through a mesh surface reconstruction algorithm. The HybridTree may be evaluated through four kinds of queries, depending on the implicit or explicit formulation is required: field function and gradient at a given point in space, point membership classification, and polygonization. Every kind of query is achieved automatically in a specific and optimized fashion for every node of the HybridTree.

NURBS surface fitting using orthogonal coordinate transform for rapid product development

Constructing a CAD model from a physical model plays a key role in some rapid product development processes. Presented in the paper is a method of fitting NURBS surfaces for rotational freeform shapes: (1) cloud-of-points data (COP-data) representing a rotational freeform shape are transformed into an orthogonal coordinate system, (2) a single-valued B-spline surface is fitted to the transformed data, and (3) the resulting B-spline surface is converted to a 3D NURBS surface by applying a symbolic product operation with a quadratic NURBS base-geometry. Compared to the existing ‘direct’ fitting methods, the proposed method has some distinctive advantages: it provides a natural means to parameterization, enables to recover exact NURBS geometry when the COP-data represent a true surface-of-revolution, and allows an easy point-membership classification for NURBS-bounded solid objects.

WELL-FORMED SET REPRESENTATIONS OF SOLIDS

All point Membership Classification (PMC) algorithms on solids constructed using regularized set operations require representing and computing neighborhoods of a point with respect to the represented solid. Such computations involve some of the more sensitive and difficult algorithms in solid modeling. We establish the necessary and sufficient conditions under which set-theoretic constructions are well-formed so that PMC amounts to evaluating a ternary logic expression and does not require any neighborhood computations. We demonstrate well-formedness of any monotone set expression whose primitives form a simple arrangement in Ed, and of common representations for polyhedral objects constructed using 'decreasing convex hull' algorithms. Well-formed representations exist for every solid but not for every fixed collection of primitives. We also give a necessary and sufficient test for existence of such representations. Identifying and eliminating neighborhood computations whenever possible should lead to simpler, more efficient, and robust geometric algorithms that are also more suitable for hardware implementation. Finally, well-formed set representations can be trivially translated into representations of solids by real-valued implicit functions.

Quasinonparametric surfaces

The concept of nonparametric surfaces is extended to the definition of quasinonparametric surfaces. These surfaces project one to one onto the plane z = 0 and are built up over the area between two curves that are single-valued, either in Cartesian or in polar coordinates. In the Cartesian case, a subset of nonrational B-spline surfaces is used. The scheme in polar coordinates exhibits all the positive properties of the B-spline tensor-product approach, and yields a piecewise rational Bézier surface. The main interest of this representation is that there exists a very simple point membership classification algorithm for the volume defined between two quasinonparametric patches. Therefore, such a volume can be incorporated as a primitive into a CSG solid-modelling system.

Single-valued surfaces in spherical coordinates

A method is given for the construction of piecewise surfaces that can be parameterized as single-valued surfaces in spherical coordinates. The main advantage of this representation is that, contrary to general NURBS-bounded primitives, there exists a very simple Point Membership Classification algorithm for the volume defined between two of these surfaces. Therefore, such a volume can be incorporated as a primitive into a CSG solid-modelling system. The chief ingredients for the construction of such surfaces are single-valued spline curves in polar coordinates and single-valued curves on coordinate cones, which are combined following a B-spline scheme. The resulting surfaces are composed of rational Bézier patches belonging to a certain subset, and which join with specified continuity conditions. These spherical spline surfaces exhibit all the positive properties of the B-spline tensor-product approach, including local control or convex hull.

Single-valued tubular patches

A method is given for the construction of tubular surface patches. These surfaces are built from a 3D directrix curve that is single-valued either in Cartesian or in cylindrical coordinates. Control points for defining the patch lie on certain directions the distribution of which is controlled by two knot vectors. The surface is evaluated following a B-spline scheme. The distance between the surface and the directrix is a bivariate single-valued expression of two variables: the parameter corresponding to the directrix and an angular parameter around the directrix. The main advantage of this representation is that there exists a simple Point Membership Classification algorithm for the volume defined between two tubular patches. Therefore, such volumes can be incorporated as nodes in a CSG context.

THE EXTENDED CONVEX DIFFERENCES TREE (ECDT) REPRESENTATION FOR N-DIMENSIONAL POLYHEDRA

We introduce the Extended Convex Differences Tree (ECDT) representation for n-dimensional polyhedra. The ECDT is simultaneously a representation and an efficiency scheme. A set is represented by a tree. Every node holds a convex bound to the set which it represents (not necessarily the convex hull). The union of the sets represented recursively by the children is the set difference between the parent's convex bound and the set the parent represents. The fact that a node holds a bound to its set is useful for avoidance of unnecessary computations. This bound is convex, permitting efficient algorithms. The ECDT uses convex differences and is therefore able lo look at concave areas as being convex. The ECDT can be viewed as an extension to the Convex Differences Tree scheme, without its drawbacks, or as a restricted form of GSG. We show how Boolean operations are performed directly on the ECDT and how a CSG tree is converted to ECDT form. Various geometric operations on the ECDT are detailed, including point membership classification, slicing by a hyper-plane, and boundary evaluation.