Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

Polycube splines

This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains, except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.     

Smoothing of bicubic parametric surfaces

Bicubic parametric surfaces are often used to represent complex shapes in systems for computer-aided design and manufacture. Such as surface can be defined by a topologically rectangular mesh of cubic parametric splines, a curve which is an approximate mathematical model of the linear elastic beam.Smoothing a bicubic parametric surface can be done by smoothing the curve net that defines it. This paper describes a method for moving datapoints in a curve net to new ‘smoother’ positions. Different techniques to analyse the result of the smoothing are also discussed.     

形状模型在混合建模环境中的映射研究

混合建模技术是ＣＡＤ研究继承参数化特征建模技术之后的又一研究热点,线框,曲面和实体模型３种模型间的转换是混合建模技术的核心,本文介绍了：１）实体模型到线框和曲面模型的映射;（２）线框模型到实体模型的映射,３）曲面框型到实体模型的映射,为混合建模技术做了一些有益的探讨,并将模型映射算法成功地应用于ＩＧＥＳ和ＳＴＥＰ的前后置处理器的设计中,取得了满意的结果。     

A new quadratic and biquadratic algorithm for curve and surface estimation

A new deterministic quadratic parametric algorithm is introduced for curve estimation. A parametric biquadratic algorithm for surface estimation, based on the one for curve estimation is also presented. Our algorithm does not assume that the surface to be estimated, based on a given set of data in the three-dimensional space, has a continuous first derivative, nor does it assume that the data satisfy the assumption of stationarity or the intrinsic hypothesis. The grid formed by the given data does not have to be equidistant; in other words the distance between neighboring points in the two-dimensional domain does not have to be the same. Also since the algorithm leads to parametric equations for the patches of the surface, the estimating surface does not need to be a function. Appropriate parameters are introduced in the blending functions of the parametric equations to produce tension. The algorithm does not require inversion of matrices and is faster than splines and kriging. The estimated surface passes through the given data points. Error analysis based on estimating surfaces of known functions from a sample of data and then comparing to their value, are made. A comparison with biqubic natural splines based on data generated from known functions is also given.     

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

We provide a simple method that extracts an isosurface that is manifold and intersection‐free from a function over an arbitrary octree. Our method samples the function dual to minimal edges, faces, and cells, and we show how to position those samples to reconstruct sharp and thin features of the surface. Moreover, we describe an error metric designed to guide octree expansion such that flat regions of the function are tiled with fewer polygons than curved regions to create an adaptive polygonalization of the isosurface. We then show how to improve the quality of the triangulation by moving dual vertices to the isosurface and provide a topological test that guarantees we maintain the topology of the surface. While we describe our algorithm in terms of extracting surfaces from volumetric functions, we also show that our algorithm extends to generating manifold level sets of co‐dimension 1 of functions of arbitrary dimension.     

Manifold splines with a single extraordinary point

This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g−2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.     

Topological Zone Organization of Scalar Volume Data

We present a new method for preprocessing and organizing discrete scalar volume data of any dimension on external storage. We describe our implementation of a visual navigation system using our method. The techniques have important applications for out-of-core visualization of volume data sets and image understanding. The applications include extracting isosurfaces in a manner that helps reduce both I/O and disk seek time, a priori topologically correct isosurface simplification (prior to extraction), and producing a visual atlas of all topologically distinct objects in the data set. The preprocessing algorithm computes regions of space that we call topological zone components, so that any isosurface component (contour) is completely contained in a zone component and all contours contained in a zone component are topologically equivalent. The algorithm also constructs a criticality tree that is related to the recently studied contour tree. However, unlike the contour tree, the zones and the criticality tree hierarchically organize the data set. We demonstrate that the techniques work on both irregularly and regularly gridded data, and can be extended to data sets with nonunique values, by the mathematical analysis we call Digital Morse Theory (DMT), so that perturbation of the data set is not required. We present the results of our initial experiments with three dimensional volume data (CT) and describe future extensions of our DMT organizing technology.     

Integration of design and manufacturing for structural shape optimization

This paper presents an integrated design and manufacturing approach that supports shape optimization of structural components. The approach starts from a primitive concept stage, where boundary and loading conditions of the structural component are given to the designer. Topology optimization is conducted for an initial structural layout. The discretized structural layout is smoothed using parametric B-Spline surfaces. The B-Spline surfaces are imported into a CAD system to construct parametric solid models for shape optimization. Virtual manufacturing (VM) techniques are employed to ensure that the optimized shape can be manufactured at a reasonable cost. The solid freeform fabrication (SFF) system fabricates physical prototypes of the structure for design verification. Finally, a computer numerical control (CNC) machine is employed to fabricate functional parts as well as mold or die for mass production of the structural component. The main contribution of the paper is incorporating manufacturing into the design process, where manufacturing cost is considered for design. In addition, the overall design process starts from a primitive stage and ends with functional parts. A 3D tracked vehicle roadarm is employed throughout this paper to illustrate the overall design process and various techniques involved.     

B样条曲面间G1连续条件及局部格式构造问题

研究了内部单节点张量积B样条曲面间G1连续的条件.通过选择特殊类型的拼接函数,打破了公共边界必须是整体多项式曲线的限制,给出了以内部单节点双四次B样条曲面为工具、使用局部格式构造G1连续曲面的算法.最后给出了计算实例.     

Subdivision surfaces for CAD—an overview

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surface modelling are addressed. Several key topics, such as scheme construction, property analysis, parametric evaluation and subdivision surface fitting, are discussed. Some other important topics are also summarized for potential future research and development. Several examples are provided to highlight the modelling capability of subdivision surfaces for CAD applications.     

Function Representation of Solids Reconstructed from Scattered Surface Points and Contours

This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ≥ 0. The volume spline is based on use of the Green's function for interpolation of scalar function values of a chosen “carrier” solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross-sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.     

Reconstruction of volume data with quadratic super splines

We propose a new approach to reconstruct nondiscrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition. We discuss the smoothness and approximation properties of our model and compare to alternative piecewise polynomial constructions. We observe, as a nonstandard phenomenon, that the derivatives of our splines yield optimal approximation order for smooth data, while the theoretical error of the values is nearly optimal due to the averaging rules. Our approach enables efficient reconstruction and visualization of the data. As the piecewise polynomials are of the lowest possible total degree two, we can efficiently determine exact ray intersections with an isosurface for ray-casting. Moreover, the optimal approximation properties of the derivatives allow us to simply sample the necessary gradients directly from the polynomial pieces of the splines. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.     

2D-Manifold Boundary Surfaces Extraction from Heterogeneous Object on GPU

The conventional isosurface techniques are not competent for meshing a heterogeneous object because they assume that the object is homogeneous.Thus the visualization method taking the heterogeneity into account is desired.In this paper,we propose a novel algorithm to extract the boundary surfaces from a heterogeneous object in one pass,whose remarkable advantage is free of the number of materials contained.The heterogeneous object is first classified into a series of homogeneous material components.Then each component is enclosed with a 2D-manifold boundary surface extracted via our algorithm.The information important to the heterogeneous object is also provided,such as the interface between two materials,the intersection curve where three materials meet and the intersection point where four materials meet.To improve the performance,the algorithm is also designed and implemented on GPU.Experimental results demonstrate the effectiveness and efficiency of the proposed algorithm.     

On marching cubes

A characterization and classification of the isosurfaces of trilinear functions is presented. Based upon these results, a new algorithm for computing a triangular mesh approximation to isosurfaces for data given on a 3D rectilinear grid is presented. The original marching cubes algorithm is based upon linear interpolation along edges of the voxels. The asymptotic decider method is based upon bilinear interpolation on faces of the voxels. The algorithm of this paper carries this theme forward to using trilinear interpolation on the interior of voxels. The algorithm described here will produce a triangular mesh surface approximation to an isosurface which preserves the same connectivity/separation of vertices as given by the isosurface of trilinear interpolation.     

基于曲面模型的IGES前后置处理器的设计

符合ＩＧＥＳ标准的曲面模型的输入和输出是大多数ＣＡＤ系统必备的数据接口之一,本文设计了一种基于裁剪ＮＵＲＢＳ曲面片的ＩＧＥＳ曲面模型的前后置处理器,并 详细讨论了Ｂｒｅｐ表示的实体模型和曲面模型的相互转换,在曲央模型到实体模型转换中使用了基于ｏｐｅｎ ｅｄｇｅ结构的曲面拓扑信息重建技术。     

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。     

Feature-varying skeletonization

Current skeletonization algorithms strive to produce a single centered result which is homotopic and insensitive to surface noise. However, this traditional approach may not well capture the main parts of complex models, and may even produce poor results for applications such as animation. Instead, we approximate model topology through a target feature size ω, where undesired features smaller than ω are smoothed, and features larger than ω are retained into groups called bones. This relaxed feature-varying strategy allows applications to generate robust and meaningful results without requiring additional parameter tuning, even for damaged, noisy, complex, or high genus models.     

Volumetric T-spline construction using Boolean operations

In this paper, we present a novel algorithm for constructing a volumetric T-spline from B-reps inspired by constructive solid geometry Boolean operations. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into two groups represented, topologically, by either a cube or a torus. We perform two Boolean operations (union and difference) with the primitives and convert them into polycubes through parametric mapping. With these polycubes, octree subdivision is carried out to obtain a volumetric T-mesh, and sharp features detected from the input model are also preserved. An optimization is then performed to improve the quality of the volumetric T-spline. The obtained T-spline surface is C ² everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C ⁰ to G ¹. Finally, we extract trivariate Bézier elements from the volumetric T-spline and use them directly in isogeometric analysis.     

Polycube Simplification for Coarse Layouts of Surfaces and Volumes

Representing digital objects with structured meshes that embed a coarse block decomposition is a relevant problem in applications like computer animation, physically‐based simulation and Computer Aided Design (CAD). One of the key ingredients to produce coarse block structures is to achieve a good alignment between the mesh singularities (i.e., the corners of each block). In this paper we improve on the polycube‐based meshing pipeline to produce both surface and volumetric coarse block‐structured meshes of general shapes. To this aim we add a new step in the pipeline. Our goal is to optimize the positions of the polycube corners to produce as coarse as possible base complexes. We rely on re‐mapping the positions of the corners on an integer grid and then using integer numerical programming to reach the optimal. To the best of our knowledge this is the first attempt to solve the singularity misalignment problem directly in polycube space. Previous methods for polycube generation did not specifically address this issue. Our corner optimization strategy is efficient and requires a negligible extra running time for the meshing pipeline. In the paper we show that our optimized polycubes produce coarser block structured surface and volumetric meshes if compared with previous approaches. They also induce higher quality hexahedral meshes and are better suited for spline fitting because they reduce the number of splines necessary to cover the domain, thus improving both the efficiency and the overall level of smoothness throughout the volume.