Boundary-conformed toolpath generation for trimmed free-form surfaces

In this paper, we adopt a 2D reparameterization procedure to regenerate boundary-conformed toolpaths. Three methods for the 2D reparameterization of trimmed boundaries in parametric space are examined and compared. They are the Coons method, the Laplace method, and a newly developed boundary-blending method. These three methods represent three different approaches to 2D surface parameterization, namely, the algebraic interpolation approach, the partial differential equation approach, and a geometric offsetting approach, respectively. Complete algorithms for surface reparameterization and toolpath generation are developed and implemented. The results showed that the Coons method is relatively simple yet might cause anomalies when the complexities of the boundary are high. The Laplace method is robust but takes relatively more computational time and also has the problem of uneven distribution of iso-parametrics. For the newly developed boundary-blending method, both the computational efficiency and parameterization robustness are quite good, in addition, it alleviates the uneven distribution problem appeared in the Laplace results.     

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.     

Gauss map computation for free-form surfaces

The Gauss map of a smooth doubly-curved surface characterizes the range of variation of the surface normal as an area on the unit sphere. An algorithm to approximate the Gauss map boundary to any desired accuracy is presented, in the context of a tensor-product polynomial surface patch, r(u,v) for (u,v)[0,1]×[0,1]. Boundary segments of the Gauss map correspond to variations of the normal along the patch boundary or the parabolic lines (loci of vanishing Gaussian curvature) on the surface. To compute the latter, points of vanishing Gaussian curvature are identified with the zero-set of a bivariate polynomial, expressed in the numerically-stable Bernstein basis—the subdivision and variation-diminishing properties then govern an adaptive quadtree decomposition of the (u,v) parameter domain that captures the zero-set of this polynomial to any desired accuracy. Loci on the unit sphere corresponding to the patch boundaries and parabolic lines are trimmed at their mutual or self-intersection points (if any), and the resulting segments are arranged in a graph structure with the segment end-points as nodes. By appropriate traversal of this graph, the Gauss map boundary segments may then be identified in proper order, and extraneous segments (lying in the Gauss map interior) are discarded. The symmetrization of the Gauss map (by identification of antipodal points) and its stereographic projection onto a plane are also discussed.     

Isogeometric topology optimization using trimmed spline surfaces

In the present work, a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed. In the proposed approach, the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by CAD systems is employed for structural response analysis and sensitivity calculation in the topology optimization. The outer and inner boundaries of design models are represented by a spline surface and trimming curves. Design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves. New sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified. The creation of new inner fronts during optimization is allowed for the topological flexibility. An inner front merging algorithm is also presented. The proposed spline based topology optimization is used to solve some benchmarking problems. Design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization. Design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one. It is also shown that post-processing effort for converting to CAD model is eliminated by using the same spline information in numerical analysis and design optimization.     

Correct resolution rendering of trimmed spline surfaces

Current strategies for real-time rendering of trimmed spline surfaces re-approximate the data, pre-process extensively or introduce visual artifacts. This paper presents a new approach to rendering trimmed spline surfaces that guarantees visual accuracy efficiently, even under interactive adjustment of trim curves and spline surfaces. The technique achieves robustness and speed by discretizing at a near-minimal correct resolution based on a tight, low-cost estimate of adaptive domain griding. The algorithm is highly parallel, with each trim curve writing itself into a slim lookup table. Each surface fragment then makes its trim decision robustly by comparing its parameters against the sorted table entries. Adding the table-and-test to the rendering pass of a modern graphics pipeline achieves anti-aliased sub-pixel accuracy at high render-speed, while using little additional memory and fragment shader effort, even during interactive trim manipulation.     

Localization and comparison of two free-form surfaces

Comparison of two free-form surfaces based on discrete data points is of paramount importance for reverse engineering. It can be used to assess the accuracy of the reconstructed surfaces and to quantify the difference between two such surfaces. The entire process involves three main steps: data acquisition, 3D feature localization and quantitative comparison. This paper presents models and algorithms for 3D feature localization and quantitative comparison. Complex free-form surfaces are represented by bicubic parametric spline surfaces using discrete points. A simple yet effective pseudoinverse algorithm was developed and implemented for localization. It consists of two iterative operations, namely, constructing a pseudo transformation matrix and point matching. A computing algorithm was developed to compare two such surfaces using optimization techniques. Since this approach does not involve solving non-linear equations for the parameters of positions and orientations, it is fast and robust. The algorithm was implemented and tested with several examples. It is effective and can be used in industry for sculptured surface comparison.     

Iso-level tool path planning for free-form surfaces

The aim of tool path planning is to maximize the efficiency against some given precision criteria. In practice, scallop height should be kept constant to avoid unnecessary cutting, while the tool path should be smooth enough to maintain a high feed rate. However, iso-scallop and smoothness often conflict with each other. Existing methods smooth iso-scallop paths one-by-one, which make the final tool path far from being globally optimal. This paper proposes a new framework for tool path optimization. It views a family of iso-level curves of a scalar function defined over the surface as tool path so that desired tool path can be generated by finding the function that minimizes certain energy functional and different objectives can be considered simultaneously. We use the framework to plan globally optimal tool path with respect to iso-scallop and smoothness. The energy functionals for planning iso-scallop, smoothness, and optimal tool path are respectively derived, and the path topology is studied too. Experimental results are given to show effectiveness of the proposed methods.     

On the meshing of trimmed 3D surfaces

A novel approach for generating quadrilateral meshes on trimmed three-dimensional surfaces is proposed. The parametric plane to Cartesian space mapping technique is extensively employed in this approach. Newly defined ‘separators’, are created on a given surface and nodes are generated on them. The relationship between nodes and separators, which is invariant in both the parametric plane and Cartesian space, is maintained for the ease of triangulation. Trimmed surfaces are discretized and the resulting meshes are presented to validate the proposed algorithm.     

5-axis flank milling free-form surfaces considering constraints

A new tool path generation method of flank milling considering constraints is proposed for ball-end cutters in this paper. It will not only reduce the machining error range but also meet the following two constraints: (a) The ball end of the milling tool is tangential to the constraint surface; (b) There is no overcut and the minimum error is zero, which is called nonnegative-error constraint. The two constraints are very useful in some situations of engineering applications, such as flank milling impeller blades. Based on the proposed method, two types of cutter will be used to generate tool paths for the same designed surface and constraint surface. The effectiveness and accuracy of the proposed method will be finally proved with some examples.     

SURFED — an interactive editor for free-form surfaces

An editor is described for creating and modifying free-form surfaces. A modular system was developed in order to provide the researchers with a facility for communicating design ideas and new mathematical forms through an interactive graphical interface to a computer-based model. To achieve this it was necessary to invent a new graphical construct called a ‘spider’ for inputting three-dimensional parameters. This experimental system has the essential features of a large-scale implementation, with the capability of utilizing many new surface forms that have not yet been tried in actual applications.     

A spherical representation for recognition of free-form surfaces

Introduces a new surface representation for recognizing curved objects. The authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. The authors define local regularity constraints that the mesh must satisfy. The authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is pose-invariant is stored at every node of the mesh. The authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. The authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation. The authors present results on real range images which show that this approach to modelling and recognizing 3D objects has three main advantages: (1) it is applicable to complex curved surfaces that cannot be handled by conventional techniques; (2) it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search; and (3) even though it is based on a spherical mapping, the approach can handle occlusions and partial views     

Improvement of free-form surfaces for product styling applications

There are a number of ways of describing free-form surfaces within geometric modelling systems. For product styling and related activities, there is a need to ensure that surface quality is good and that patches join together smoothly. Additional parameters can be introduced to allow surfaces to be modified. This raises the question of whether these can be chosen automatically, and this in turn requires measures of what is a ‘fair’ surface. Measures based on surface curvature are discussed and applied to adjust surface patches presented in terms of point meshes.     

Semi-automatic system for defining free-form curves and surfaces

Among curves and surfaces defined by parametric polynomials, the cases dealt with here are those which only have to comply with the requirement to run through a certain number of points previously located in space. The process can be totally automatic, but the results are liable to be altered by arbitrary decisions.     

A tolerant approach to reconstruct topology from unorganized trimmed surfaces

Presented in this paper is a topology reconstruction algorithm from a set of unorganized trimmed surfaces. Error-prone small geometric elements are handled to give proper topological information. It gives complete topology to topologically complete models, and it is also tolerant to incomplete models. The proposed algorithm is vertex-based in that clues for topological information are searched from the set of vertices first, not from that of edges.     

Hermite approximation for free-form deformation of curves and surfaces

Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications, we need an explicit expression of the object, rather than a collection of sampled points. If both object and deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree, something undesirable in real applications. To solve this problem, we transform each curve or surface composing the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients. Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending and twisting, by computing a polynomial approximant of the deformation.     

A vision-aided reverse engineering approach to reconstructing free-form surfaces

A vision-aided reverse engineering approach (VAREA) has been developed to reconstruct least-square free-form surface models from physical models, with a coordinate measurement machine (CMM) equipped with a touch-triggered probe and a computer vision system. The VAREA integrates computer vision, surface data digitization and surface modelling into a single process. Two main steps are applied in this innovative approach. The initial vision-driven surface triangulation process (IVSTP) generates a triangular patch by using stereo image detection and a constrained Delaunay triangulation method. The adaptive model-based digitizing process is then used to refine the surface reconstruction and to control accuracy to within user-specified tolerances. As a result, a least-squares bicubic B-spline surface model with the controlled accuracy of digitization can be obtained for further application in product design and manufacturing processes. More than 85% reduction has been achieved in the time required to construct free-form surfaces using this approach, as compared with traditional manual methods with CMM. Therefore, product design lead time can be significantly reduced.     

Design and manufacture of free-form surfaces by cross-sectional approach

Conventional methods for manual design and analysis are not necessarily the best methods for computer implementation, lofting techniques, or computer aided design to produce a satisfactory result for the design of a wide range of shapes and the general form of the B-spline curve with the defining polygon adequately fulfils most essential requirements. Fourth-order curves seemed to be best suited for most cases, as they provide a compromise between smoothness and ‘localness’, yet curve orders of varying degrees still have some merit in certain circumstances.The most attractive feature realized with this method is the simplicity of design and the determination of the main input elements. Surface interpolation curves are shown and the production of a mould for a plastic handle indicated.