Complete swept volume generation, Part I: Swept volume of a piecewise C-continuous cutter at five-axis milling via Gauss map

In this paper, we present a methodology to generate swept volume of prevailing cutting tools undergoing multi-axis motion and it is proved to be robust and amenable for practical purposes with the help of a series of tests. The exact and complete SV, which is closed from the tool bottom to the top of the shaft, is generated by stitching up envelope profiles calculated by Gauss map.The novel approach finds the swept volume boundary for five-axis milling by extending the basic idea behind Gauss map. It takes piecewise C¹-continuous tool shape into account. At first, the tool shape is transformed from Euclidean space into Tool map (T-Map) on the unit sphere and the velocity vector of a cutter is transformed into Contact map (C-Map) using Gauss map. Then, closed intersection curve is found between T-Map and C-Map on the Gaussian sphere. At last, the inverse Gauss map is exploited to get envelope profile in Euclidean space from the closed curve in the range. To demonstrate its validity, a cutting simulation kernel for five-axis machining has been implemented and applied to mold and die machining.     

Boundary of the volume swept by a free-form solid in screw motion

The swept volume of a moving solid is a powerful computational and visualization concept. It provides an excellent aid for path and accessibility planning in robotics and for simulating various manufacturing operations. It has proven difficult to evaluate the boundary of the volume swept by a solid bounded by trimmed parametric surfaces undergoing an arbitrary analytic motion. Hence, prior solutions use one or several of the following simplifications: (1) approximate the volume by the union of a finite set of solid instances sampled along the motion; (2) approximate the curved solid by a polyhedron; and (3) approximate the motion by a sequence of simpler motions. The approach proposed here is based on the third type of simplification: it uses a polyscrew (continuous, piecewise-helical) approximation of the motion. This approach leads to a simple algorithm that generates candidate faces, computes the two-cells of their arrangement, and uses a new point-in-sweep test to select the correct cells whose union forms the boundary of the swept volume.     

Complete swept volume generation — Part II: NC simulation of self-penetration via comprehensive analysis of envelope profiles

In this paper the swept volume with self-penetration (or self-intersection) of the cutter is presented. The complete swept volume (SV), which describes the side and bottom shape of a milling cutter undergoing self-penetration, is generated by using the Gauss map method proposed in the authors’ previous paper [Lee SW, Nestler A. Complete swept volume generation—part I: swept volume of a piecewise C¹-continuous cutter at five-axis milling via Gauss map. Computer-Aided Design 2011; 43(4): 427–41]. Based on the Gauss map method, the comprehensive analysis of envelope profiles of the tool is accomplished. Through the analysis the necessary condition of the self-penetration of the cutter at five-axis movement is identified. After having classified movement types of the milling cutter in an in-depth manner, the topologically consistent boundary of SV is generated by trimming the invalid facets interior to the SV. To demonstrate the validity of the proposed method, a cutting simulation kernel for five-axis machining has been implemented and applied to cavity machining examples such as intake ports of automobile engines and so forth where the self-penetration occurs. The proposed method is proved to be robust and amenable for the practical purpose of the NC simulation.     

Hypertexturing complex volume objects

Published online: 15 March 2002     

Interactive modeling of implicit surfaces using a direct visualization approach with signed distance functions

Modeling appealing virtual scenes is an elaborate and time-consuming task, requiring not only training and experience, but also powerful modeling tools providing the desired functionality to the user. In this paper, we describe a modeling approach using signed distance functions as an underlying representation for objects, handling both conventional and complex surface manipulations. Scenes defined by signed distance functions can be stored compactly and rendered directly in real-time using sphere tracing. Hence, we are capable of providing an interactive application with immediate visual feedback for the artist, which is a crucial factor for the modeling process. Moreover, dealing with underlying mathematical operations is not necessary on the user level. We show that fundamental aspects of traditional modeling can be directly transferred to this novel kind of environment, resulting in an intuitive application behavior, and describe modeling operations which naturally benefit from implicit representations. We show modeling examples where signed distance functions are superior to explicit representations, but discuss the limitations of this approach as well.     

基于最佳圆弧样条逼近的快速距离曲面计算

距离曲面是一种常用的隐式曲面,它在几何造型和计算机动画中具有重要的应用价值,但以往往在对距离曲面进行多边形化时速较慢,为了提高点到曲线最近距离计算的效率,提出了一种基于最佳圆弧样条逼近的快速线骨架距离曲面计算方法,该算法对于一条任意的二维NURBS曲线,在用户给定的误差范围内,先用最少量的圆弧样条来逼近给定的曲线,从而把点到NURBS曲线最近距离的计算问题转化为点到圆弧样条最近距离的计算问题,由于在对曲面进行多边形化时,需要大量的点到曲线最近距离的计算,而该处可以将点到圆弧样条最近距离很少的计算量来解析求得,故该算法效率很高,该实验表明,算法简单实用,具有很大的应用价值。     

The differential equation algorithm for general deformed swept volumes

The differential equation approach for characterizing swept volume boundaries is extended to include objects experiencing deformation.For deformed swept volume,it is found that the structure and algorithm of sweep-envelope differential equation(SEDE)are similar between the deformed and the rigid swept volumes.The efficiency of SEDE approach for deformed swept volume is proved with an example.     

An approach to integrating shape and biomedical attributes in vascular models

Computational models have been used widely in tissue engineering research and have proven to be powerful tools for bio-mechanical analysis (i.e., blood flow, growth models, drug delivery, etc). This paper focuses on developing higher-fidelity models for vascular structures and blood vessels that integrate computational shape representations with biomedical properties and features. Previous work in computer-aided vascular modeling comes from two communities. For those in biomedical imaging, the goal of past research has been to develop image understanding techniques for the interpretation of x-ray, magnetic resonance imaging (MRI), or other radiological data. These representations are predominantly discrete shape models that are not tied to physiological properties. The other corpus of existing work comes from those interested in developing physiological models for vascular growth and behavior based on bio-medical attributes. These models usually either have a highly simplified shape representation, or lack one entirely. Further, neither of these representations are suitable for the kind of interactive modeling required by tissue engineering applications.This paper aims to bridge these two approaches and develop a set of mathematical tools and algorithms for feature-based representation and computer-aided modeling of vascular trees for use in computer-aided tissue engineering applications. The paper offers a multi-scale representation based on swept volumes and a feature-based representation that can attribute the geometric representation with information about blood flow, pressure, and other biomedical properties. The paper shows how the resulting representation can be used as part of an overall approach for designing and visualizing vascular scaffolds. As a real-world example, we show how this computational model can be used to develop a tissue scaffold for liver tissue engineering. Such scaffolds may prove useful in a number of biomedical applications, including the growth of replacement tissue grafts and in vitro study of the pharmacological affects of new drugs on tissue cultures.     

A high quality interpolation method for colocated polyhedral/polygonal control volume methods

A high quality and efficient interpolation method for polyhedral/polygonal control volume simulation data is presented. The proposed method utilizes a non-ambiguous and efficient mesh decomposition technique. A pseudo-Laplacian is used to solve an optimization problem to approximate the variation between discrete data points in a linear fashion. The interpolation method guarantees continuous interpolation data throughout the control volume mesh topology and faithfully reproduces the input control volume data. The interpolation connectivity is structured to mimic the interpolation methods utilized by the control volume discretization. The method only requires the geometry of the input data to perform interpolations. This allows key interpolation data to be calculated once and stored for efficient interpolations. The benefits of the proposed algorithm are highlighted by an interpolation test case which demonstrates the benefits of the current method compared to a popular interpolation method currently used in industry. Since the proposed method is designed to augment an existing mesh data structure it can be used to update existing control volume software.     

一种基于凸包近似的快速体积计算方法

体积是物体的基本几何属性,在许多应用场合需要频繁地被计算。目前基本上通过重构物体曲面而间接求取体积,增加了许多不必要的工作。提出一种快速求取点云模型体积的方法,使用增量式算法计算点云的凸包用来近似物体,将凸包分解成上下两个三角网格面,使用正投影法分别求取它们的投影体积,它们两者之差即是所求模型体积。实验表明该算法实现简单,可快速地求解处理具有任何几何和拓扑复杂性的点云模型。     

Generating cutter swept envelopes in five-axis milling by two-parameter families of spheres

This paper presents an efficient parametric approach of determining the shape of the envelope surface by a generalized cutter that follows five-axis tool path during NC machining. In this approach the cutter is modeled as a canal surface. By considering the tool motions the cutter is decomposed into a set of characteristic and great circles which are generated by two-parameter families of spheres. The center of a sphere from these families is described by two parameters which represent the spine curve and the tool path, and the radius of the sphere is described by one parameter representing the spine curve. Considering the relationship between characteristic and great circles the grazing points on the tool surface are identified. Analytically it is proven for the NC cutter geometries that any point on the envelope surface is located at the intersection of the characteristic and great circles. Then based on the proofs a closed-form solution for computing the grazing points generated by a surface of revolution is presented. The presented methodology is reduced to a simpler parametric form when the NC cutters are described by pipe surfaces.     

Finite volume approximation of stiff problems on two-dimensional curvilinear domain

The aim of this article is to apply a novel finite volume method to approximate a stiff problem for a two-dimensional curvilinear domain. The stiffness is caused by the existence of a small parameter in the equation which introduces a boundary layer along parts of the curvilinear boundary. Incorporating in the finite volume space the boundary layer correctors, the boundary layer singularities are absorbed. Hence, we propose a second order scheme for curvilinear domains using uniform meshes thus avoiding the costly refinement of mesh in the boundary layers.     

一种基于数据差异的逻辑卷快速同步方法

在数据保护系统中,经常需要对源卷和目的卷进行同步使彼此数据保护一致。提出了一种新的基于数据块的同步方法。该方法充分利用了源卷和目的卷的位图信息,在计算数据块的特征值时引入了动态算法。实验表明该方法显著地减少了同步数据量。     

Field modeling with sampled distances

Traditional mesh-based approaches to the modeling and analysis of physical fields within geometric models require some form of topological reconstruction and conversion in the mesh generation process. Such manipulations tend to be tedious and error-prone manual processes that are not easily automated. We show that most field problems may be solved directly by using approximate distance fields computed from designed or sampled geometric data, thus avoiding many of the difficult reconstruction and meshing problems. With distances we can model fields that satisfy boundary conditions while approximating the governing differential equations to arbitrary precision. Because the method is based on sampling, it provides natural control for multi-resolution both in geometric detail of the domain and in accuracy of the computed physical field. We demonstrate the field modeling capability with several heat transfer applications, including a typical transient problem and a ‘scan and solve’ approach to the simulation of a physical field in a real-world artifact.     

Fast and intuitive metamorphosis of 3D polyhedral models using SMCC mesh merging scheme

A very fast and intuitive approach to generate the metamorphosis of two genus 0 3D polyhedral models is presented. There are two levels of correspondence specified by animators to control morphs. The higher level requires the animators to specify scatter features to decompose the input models into several corresponding patches. The lower level optionally allows the animators to specify extra features on each corresponding patch for finer correspondence control. Once these two levels of correspondence are established, the proposed schemes automatically and efficiently establish a complete one-to-one correspondence between two models. We propose a novel technique called SMCC (Structures of Minimal Contour Coverage) to efficiently and robustly merge corresponding embeddings. The SMCC scheme can compute merging in linear time. The performance of the proposed methods is comparable to or better than state-of-the-art 3D polyhedral metamorphosis. We demonstrate several examples of aesthetically pleasing morphs, which can be created very quickly and intuitively.     

Perspective silhouette of a general swept volume

We present an efficient and robust algorithm for computing the perspective silhouette of the boundary of a general swept volume. We also construct the topology of connected components of the silhouette. At each instant t, a three-dimensional object moving along a trajectory touches the envelope surface of its swept volume along a characteristic curve K^t. The same instance of the moving object has a silhouette curve L^t on its own boundary. The intersection K^t∩L^t contributes to the silhouette of the general swept volume. We reformulate this problem as a system of two polynomial equations in three variables. The connected components of the resulting silhouette curves are constructed by detecting the instances where the two curves K^t and L^t intersect each other tangentially on the surface of the moving object. We also consider a general case where the eye position changes while moving along a predefined path. The problem is reformulated as a system of two polynomial equations in four variables, where the zero-set is a two-manifold. By analyzing the topology of the zero-set, we achieve an efficient algorithm for generating a continuous animation of perspective silhouettes of a general swept volume.     

Fast update of sliced voxel workpiece models using partitioned swept volumes of three-axis linear tool paths

This paper presents a new and effective method to achieve fast machining simulation via the frame-sliced voxel representation (FSV-rep) workpiece modeling environment. The FSV-rep workpiece...     

脑血管体绘制的快速表意式增强

利用3维可视化技术重构脑血管模型,获取脑血管及其相关组织的立体结构,对于辅助诊断脑血管疾病具有重要意义.鉴于脑血管位置的特殊性、形态的复杂性及灰度信息的多变性,要求重构技术能够清晰还原其空间结构.基于CUDA(computed unified device architecture)的光线投射体绘制,引入深度和轮廓宽度等因子,采用基于曲率的轮廓增强、基于深度的边界增强以及基于立体显示和颜色融合的深度线索提示等表意式技术,实时重构高质量3维血管模型,生动展现脑血管的3维结构信息,如深度、梯度、观察方向等.实验结果表明,本文方法的有效性,在精确显示脑血管结构的同时能够增强体绘制效果.     

Material side tracing and curve refinement for pencil-cut machining of complex polyhedral models

In this paper, a new Material-Side-Tracing method and a pencil-cut curve refinement technique are proposed for 3-axis pencil-cut path generation. Pencil-cut machining has been used to remove remaining material at highly curved regions or corners after the finishing process. Procedures of evaluating and extracting valid pencil-cut points are developed by taking practical cases into account. With the strategy of using material-side information in the tracing process, smooth and clean pencil-cut curves can be generated even if the actual adjacent pencil-cut curves are very close. A technique of pencil-cut curve refinement is presented to overcome the limitation due to the discrete CL-net intervals, and the smooth pencil-cut paths are made complete at sharp corners. Computer implementation and practical examples are also presented in this paper. The proposed techniques can be used in the CAD/CAM systems to generate pencil-cut paths for machining complex polyhedral models.