Trimmed NURBS Surfaces Research Articles

Triangulation Method of 3D Scattered Data Points Based on CAD Model

A new method of triangulation for large scale scattered 3D points is proposed. This method is based on the available CAD model, with the thought of DC(divide and conquer).Alignment between the data and the CAD model, registration which establishes correspondence between the data points and those on the CAD trimmed NURBS surface entities;2D-Delaunay triangulation, performed on the corresponding points in the parametric domains(u, v)of each entity, application of the connectivity structure to the 3D data points for each mesh patch; Elimination of redundant triangles of each 3D mesh patch and stitching of patches together. Unlike many other methods, it is not constrained by certain types of measurement distribution or object shape. The experimental results testify that the approach is feasible and efficient.

Optimized GPU evaluation of arbitrary degree NURBS curves and surfaces

This paper presents a new unified and optimized method for evaluating and displaying trimmed NURBS surfaces using the Graphics Processing Unit (GPU). Trimmed NURBS surfaces, the de facto standard in commercial mechanical CAD modeling packages, are currently being tessellated into triangles before being sent to the graphics card for display since there is no native hardware support for NURBS. Other GPU-based NURBS evaluation and display methods either approximated the NURBS patches with lower degree patches or relied on specific hard-coded programs for evaluating NURBS surfaces of different degrees. Our method uses a unified GPU fragment program to evaluate the surface point coordinates of any arbitrary degree NURBS patch directly, from the control points and knot vectors stored as textures in graphics memory. This evaluated surface is trimmed during display using a dynamically generated trim-texture calculated via alpha blending. The display also incorporates dynamic Level of Detail (LOD) for real-time interaction at different resolutions of the NURBS surfaces. Different data representations and access patterns are compared for efficiency and the optimized evaluation method is chosen. Our GPU evaluation and rendering speeds are more than 40 times faster than evaluation using the CPU.

트림 NURBS 곡면의 T-스플라인 유한요소해석

In this present work, spline FEA for the trimmed NURBS surface of the 2D linear elasticity problem is presented. The main benefit of the proposed method is that no additional efforts for modeling of trimmed NURBS surfaces are needed and the information of the trimming curves and trimmed surfaces exported from the CAD system can be directly used for analysis. For this, trimmed elements are searched by using NURBS projection scheme. The integration of the trimmed elements is performed by using the NURBS-enhanced integration scheme. The formulation of constructing stiffness matrix of trimmed elements is presented. In this formulation, the information of the trimming curve is used for calculating the Jacobian as well as for obtaining integration points. The robustness and effectiveness of the proposed method are investigated through various numerical examples.

A reconstruction method using geometric subdivision and NURBS interpolation

Subdivision surfaces (SubD, NURSS) construction is a popular method to present and animate sculpture shapes for entertainment, and this is preferred due to advantage of polygon mesh. But subdivision surfaces have limited familiarity and portability in modern CAD, unlike NURBS-based entities. NURBS-based sculpting using dense point cloud data obtained through geometric subdivision has been so far a less discussed topic. The current work discusses the new methodology of using geometric subdivision and NURBS interpolation with an aim to add a new dimension to reverse engineering. This has distinct advantages like accurate shape building using scanned data, manufacturing ability of complex shapes, faster and accurate shape representation with high quality surfaces, model portability, and a better control on object shape and better patch-planning. Major milestones of the work are bridging the geometric subdivision and NURBS reconstruction on subdivided data, NURBS patch and topology planning (straight and PDPA scan), construction of trimmed NURBS surfaces and, finally, writing IGES of resultant patches. The work is a high-level automation solution for complex constructions. It is applied on human head-scan data and results are displayed.

Efficient rendering of deformable objects for real‐time applications

AbstractDeformable objects can be used to model soft objects such as clothing, human faces and animal characters. They are important as they can improve the realism of the applications. However, most existing hardware accelerators cannot render deformable objects directly. A tessellation process is often used to convert a deformable object into polygons so that the hardware graphics accelerator may render them. Unfortunately, this tessellation process is computationally very expensive. While the object is deforming, the tessellation process needs to be performed repeatedly to convert the deforming objects into polygons. As a result, deformable objects are seldom used in real‐time applications such as virtual environments and computer games. Since trimmed NURBS surfaces are often used to represent deformable objects, in this paper we present an efficient method for incremental rendering of deformable trimmed NURBS surfaces. A trimmed NURBS surface typically deforms through the deformation of the trimmed NURBS surface and/or the trimming curve. Our method handles both trimmed surface deformation as well as trimming curve deformation. Experimental results show that our method performs significantly faster than the method used in OpenGL and can be used in real‐time applications, such as computer games. Copyright © 2006 John Wiley & Sons, Ltd.

On increasing the developability of a trimmed NURBS surface

Developable surfaces are desired in designing products manufactured from planar sheets. Trimmed non-uniform rational B-spline (NURBS) surface patches are widely adopted to represent 3D products in CAD/CAM. This paper presents a new method to increase the developability of an arbitrarily trimmed NURBS surface patch. With this tool, designers can first create and modify the shape of a product without thinking about the developable constraint. When the design is finished, our approach is applied to increase the developability of the designed surface patches. Our method is an optimisation-based approach. After defining a function to identify the developability of a surface patch, the objective function for increasing the developability is derived. During the optimisation, the positions and weights of the free control points are adjusted. When increasing the developability of a given surface patch, its deformation is also minimised and the singular points are avoided. G0 continuity is reserved on the boundary curves during the optimisetion, and the method to reserve G1 continuity across the boundaries is also discussed in this paper. Compared to other existing methods, our approach solves the problem in a novel way that is close to the design convention, and we are dealing with the developability problem of an arbitrarily trimmed NURBS patch.

Constrained Scaling of Catmull-Clark Subdivision Surfaces

AbstractA method to scale a Catmull-Clark subdivision surface while holding the shape and size of specific features (sub-structures) unchanged is presented. The concept of a previous approach for trimmed NURBS surfaces, fix-and-stretch [20], is followed here, i.e., the new surface is formed by fixing selected regions of the given subdivision surface that contain the features, and scaling and stretching the remaining part. But the fixing process and the stretching process are performed differently because of the topology complexity of subdivision surfaces. The resulting surface is again a Catmull-Clark subdivision surface and the method ensures that the resulting surface reflects the shape and curvature distribution of the original surface. The contributions of the paper also include efficient strain energy computation techniques and energy optimization techniques for regions around extra-ordinary points. The new method can handle shapes that can not be represented by trimmed NURBS surfaces and, consequent...

Characterization of Trimmed NURBS Surface Boundaries Using Topological Criteria

This paper presents an approach based on topology for the determination of characteristics and properties of curves used in the trimming of NURBS surfaces. Through discrete subdivision and topological criteria, a method is presented to determine characteristics of the boundary; such as whether the set of trimming curves forms a set of closed loops, whether trimming curves contain singularities or self intersections, and whether the boundary is simply connected. A surface mesh partitionning the parameter space is used, formed of isoparametric lines in both parametric directions. Topological properties of the cells of this mesh and their intersections with the trimming curves allow to localize the boundary. Topological treatment of this localization allows to define the interior and exterior of the face, and to refine the boundary localization. Singularities and self intersections of the boundary as well as voids in the face are investigated through the study of topological properties of neighbors. As an application, an algorithm for point localization is presented that very rapidly allows to determine whether a given point in parameter space lies inside, on the boundary or outside of the trimmed surface.

A surface based approach to recognition of geometric features for quality freeform surface machining

The paper presents a surface-based approach for geometric feature recognition for the purpose of automating the process planning of freeform surface machining. The proposed approach consists of the following four steps for recognition of the geometric features: conversion and preprocessing of the surface geometry data, subdivision of NURBS surface, reconstruction of surface orientation areas, and recognition of geometric features. The proposed scheme assumes that the input geometry data form is based on an IGES CAD model and the surface model can be represented in the form of trimmed NURBS surfaces. The connectivity relations of the product surfaces are analyzed and each surface is subdivided into orientation regions based on the surface normal vector over a certain point density grid, and then all the connected regions with the same orientation type are grouped into surface orientation areas. After that, the geometric feature will be recognized through the analysis of area connectivity and relationship. The paper describes the developed algorithms on surface orientation region subdivision, surface orientation area reconstruction, and geometric feature recognition. The verified feasibility study of the developed method is also presented.

Computing offsets of trimmed NURBS surfaces

Offsetting of trimmed NURBS surfaces is one of the widely used functionalities in the design and manufacture simulations of composite laminates. This paper presents an approach for the offsetting of a trimmed NURBS surface. The approach has been developed mainly to meet the stringent accuracy requirements in the simulation of composite laminate design and manufacturing processes. However, the approach is applicable for the offset of a general trimmed NURBS surface. Though the method is based on known techniques in literature, the practical approach and the treatment of the subject presented is unique and has not been reported earlier. The basic approach consists of offsetting the underlying surface, offsetting of all the trimming loops and the creation of offset trimmed surface using the offset surface and the offset trimming loops. This is a unified offset approach for trimmed surfaces where in the offset of underlying surface and the offset of trimming loops are obtained using the same approach. The approach has better error control and results in less number of control points. Further the approach can be extended to obtain offsets of a general B-Rep. The approach has been used in the creation of offset surfaces of various aircraft components.

Quadrilateral mesh generation on trimmed NURBS surfaces

An automatic mesh generation scheme with unstructured quadrilateral elements on trimmed NURBS surfaces has been developed. In this paper NURBS surface geometries in the IGES format have been employed to represent geometric models. For unstructured mesh generation with quadrilateral elements, a domain decomposition algorithm employing loop operators has been modified. As for the surface meshing, an indirect 2D approach is proposed in which both quasi-expanded planes and projection planes are employed. Sample meshes for complex models are presented to demonstrate the robustness of the algorithm.

Geometry based triangulation of multiple trimmed NURBS surfaces

NURBS surfaces, conforming to the geometrical contours of the part to be realised, play an important role in the design and simulation of the manufacturing process of many engineering applications, particularly in the area of composite laminates. A laminate can have an arbitrary topology, and its constituent layers invariably have different surface and contour characteristics. Consequently, the geometrical complexity to be dealt with, in the realm of computer aided design of composite laminates, is considerably enhanced. The individual layers constituting the laminate are represented in the form of B-Rep with constituent NURBS surfaces and curves. For the design and manufacturing simulation of a laminate, various operations like trimming, sewing and offsetting have to be carried out. To perform these operations directly on the B-Rep is complex and very tedious; a better approach is to break it down into precise piecewise planar representation. This paper presents a methodology for generating such a representation (triangulation). Though the method is based on known techniques in literature, the analytical approach and the treatment of the subject presented is unique and has not been reported before. A few important features of the proposed approach are: (1) simultaneous triangulation of multiple surfaces yielding a crack-free representation, (ii) tight control over the quality of triangulation leading to better results for both singly curved and multiply connected surfaces, and (iii) automatic elimination of spurious triangles. Also, the merging techniques of polygonised trimmed NURBS surfaces proposed in the paper is independent of B-Rep tolerance and has been carried out in the Euclidean space, unlike the existing works. The model assumes the surface to be no more than C 0 continuous. The methodology has been adopted to generate triangulation of various real life aircraft component surfaces (B-Reps). The methodology proposed is applicable for the triangulation of general NURBS B-Rep.

Constrained scaling of trimmed NURBS surfaces based on fix-and-stretch approach

A new method to scale a trimmed NURBS surface while holding the shape and size of specific features (trimming curves) unchanged is presented. The new method is fix-and-stretch based: the new surface is formed by fixing selected regions of the given trimmed NURBS surface that contain the trimming curves and stretching the remaining part of the surface to reach certain boundary conditions. The stretching process is performed using an optimization process to ensure that the resulting surface reflects the shape and curvature distribution of the scaled version of the given surface. The resulting surface maintains a NURBS representation and, hence, is compatible with most of the current data-exchange standards. The new approach is more robust than a previous, attach-and-deform based approach (Zhang P, Zhang C, Cheng F. Constrained shape scaling of trimmed NURBS surfaces. In: Proceedings of the 1999 ASME Design Theory and Methodology Conference. Las Vegas, Nevada, 1999) in that it can tolerate scaling factors of bigger values (up to 2 in some cases). The new approach also guarantees that the features remain exactly the same after scaling. Test results on several car parts with trimming curves and comparison with the previous approach are included. The quality of the resulting surfaces is examined using the highlight line model. The presented technique is important for integrating standard parts into a sculptured product.

Contact treatment algorithm for the trimmed NURBS surface

As a kind of surface description, the trimmed non-uniform rational B-spline (NURBS) surface is widely employed in CAD/CAM. The trimmed NURBS surface has many advantages in describing tool surfaces because of its simplicity in data preparation and the small amount of surface data compared to that of other surfaces. Despite its inherent advantages, the trimmed NURBS surface is rarely employed in the analysis of forming problems due to the complexity in the contact treatment. In this study, a contact treatment based on a sheet normal vector in the trimmed NURBS surface model is presented. For the sake of importing surface data, an initial graphics exchange specification (IGES) format input module has also been developed. The computed results for deep drawings of a clover shaped cup and an L-shaped cup are compared with experiments and the results of the commercial stamping analysis code PAM-STAMP.

Practical Ray Tracing of Trimmed NURBS Surfaces

Abstract A system is presented for ray tracing trimmed NURBS surfaces. While approaches to components are drawn largely from existing literature, their combination within a single framework is novel. This paper also differs from prior work in that the details of an efficient implementation are fleshed out. Throughout, emphasis is placed on practical methods suitable to implementation in general ray-tracing programs.

Geometry-based triangulation of trimmed NURBS surfaces

An algorithm for obtaining a piecewise triangular approximation of a trimmed NURBS surface is presented. The algorithm is geometry based, i.e. the surface is subdivided into triangular facets based on its geometric characteristics and not on its parametrization. No assumption is made about the surface's parametrical representation; it does not have to be continuously differentiable, only C o continuity is assumed. The surface subdivision is performed in model space, however, the triangulation is carried out in parameter space using the parametric vertices of subdivision rectangles. Along with computing the triangulation, the method produces a compact database for browsing in the triangular irregular network, e.g. finding all neighbors of a given triangle.

A tessellation algorithm for the representation of trimmed nurbs surfaces with arbitrary trimming curves

In most cad systems, a trimmed parametric surface is defined by two things: the control information of the surface itself and the control information of its trimming curves, which are usually defined as parametric curves in the parameter space of the surface. Often, trimmed parametric surfaces cause problems in the context of data exchange between cad systems, surface evaluation/rendering, and grid/mesh generation. This paper describes a new approach for representing a trimmed surface. The approach is based on the idea of decomposing the valid part of the parameter space of a trimmed surface by a set of planar, ruled surfaces, which are defined in parameter space and whose union defines the entire valid part. A generalized Voronoï diagram is constructed for the set of trimming curves in parameter space. This diagram defines tiles around each trimming curve, which are further tessellated by utilizing a scan line technique for identifying and filling the interior of polygons. Here, the scan line technique is used to extract non-horizontal segments of the given trimming curves. Pairs of these segments are then linearly interpolated in horizontal parameter direction, thus defining a set of planar, ruled surfaces. The set of all these planar, ruled surfaces defines the entire valid part of a trimmed surface exactly.

Efficient rendering of trimmed nurbs surfaces

An algorithm for the interactive display of trimmed nurbs surfaces is presented. The algorithm converts the nurbs surfaces to Bézier surfaces, and nurbs trimming curves to Bézier curves. It tessellates each trimmed Bézier surface into triangles, and renders them using the triangle rendering capabilities common in current graphics systems. It makes use of tight bounds for the uniform tessellation of Bézier surfaces into cells and it traces the trimming curves to compute the trimmed regions of each cell. This operation is based on the tracing of trimming curves, intersection computation with the cells, and triangulation of the cells. The resulting technique also makes use of spatial and temporal coherence between successive frames for cell computation and triangulation. Polygonization anomalies such as cracks and angularities are avoided as well. The algorithm can display trimmed models described using thousands of Bézier surfaces at interactive frame rates on high end graphics systems.

Tessellating trimmed nurbs surfaces

An algorithm for obtaining a piecewise planar approximation of a trimmed nurbs surface is presented. Given a model space tolerance ε. the algorithm triangulates the parameter space domain of the trimmed surface such that the 3D planar approximation, obtained by mapping 2D triangles onto the surface, deviates from the trimmed surface by no more than ε. The number of triangles computed in parameter space depends on the bounds of the second derivatives. A detailed discussion of the algorithm and a practical error analysis of the tessellation are provided.