CPU–GPU Parallel Framework for Real‐Time Interactive Cutting of Adaptive Octree‐Based Deformable Objects

A software framework taking advantage of parallel processing capabilities of CPUs and GPUs is designed for the real‐time interactive cutting simulation of deformable objects. Deformable objects are modelled as voxels connected by links. The voxels are embedded in an octree mesh used for deformation. Cutting is performed by disconnecting links swept by the cutting tool and then adaptively refining octree elements near the cutting tool trajectory. A surface mesh used for visual display is reconstructed from disconnected links using the dual contour method. Spatial hashing of the octree mesh and topology‐aware interpolation of distance field are used for collision. Our framework uses a novel GPU implementation for inter‐object collision and object self collision, while tool‐object collision, cutting and deformation are assigned to CPU, using multiple threads whenever possible. A novel method that splits cutting operations into four independent tasks running in parallel is designed. Our framework also performs data transfers between CPU and GPU simultaneously with other tasks to reduce their impact on performances. Simulation tests show that when compared to three‐threaded CPU implementations, our GPU accelerated collision is 53–160% faster; and the overall simulation frame rate is 47–98% faster.     

Robust interactive cutting based on an adaptive octree simulation mesh

We present an adaptive octree based approach for interactive cutting of deformable objects. Our technique relies on efficient refine- and node split-operations. These are sufficient to robustly represent cuts in the mechanical simulation mesh. A high-resolution surface embedded into the octree is employed to represent a cut visually. Model modification is performed in the rest state of the object, which is accomplished by back-transformation of the blade geometry. This results in an improved robustness of our approach. Further, an efficient update of the correspondences between simulation elements and surface vertices is proposed. The robustness and efficiency of our approach is underlined in test examples as well as by integrating it into a prototype surgical simulator.     

Multiresolution analysis on irregular surface meshes

Wavelet-based methods have proven their efficiency for visualization at different levels of detail, progressive transmission, and compression of large data sets. The required core of all wavelet-based methods is a hierarchy of meshes that satisfies subdivision-connectivity. This hierarchy has to be the result of a subdivision process starting from a base mesh. Examples include quadtree uniform 2D meshes, octree uniform 3D meshes, or 4-to-1 split triangular meshes. In particular, the necessity of subdivision-connectivity prevents the application of wavelet-based methods on irregular triangular meshes. In this paper, a “wavelet-like” decomposition is introduced that works on piecewise constant data sets over irregular triangular surface meshes. The decomposition/reconstruction algorithms are based on an extension of wavelet-theory allowing hierarchical meshes without property. Among others, this approach has the following features: it allows exact reconstruction of the data set, even for nonregular triangulations, and it extends previous results on Haar-wavelets over 4-to-1 split triangulations     

Refinement and coarsening of surface meshes

This paper presents an adaptation scheme for surface meshes. Both refinement and coarsening tools are based upon local retriangulation. They can maintain the geometric features of the given surface mesh and its quality as well. A mesh gradation tool to smooth out large size differences between neighboring (in space) mesh faces and a procedure to detect and resolve self-intersections in the mesh are also presented. Both are driven by an octree structure and make use of the presented refinement tool.     

基于变分隐式曲面的网格融合

三维物体融合利用三维模型之间的剪贴操作从两个或多个现有的几何模型中光滑融合出新的几何模型.作为一种新的几何造型方法,它正受到越来越多的关注.提出一种基于变分隐式曲面的网格融合新方法.首先利用平面截面切出网格物体的待融合边界,然后通过构造插值待融合网格物体边界的变分隐式曲面并对其进行多边形化,得到待融合网格物体间的过渡曲面,最后通过剪切掉过渡曲面的多余部分及拓扑合并操作以实现过渡网格曲面与原始网格间的光滑融合与现有的直接连接待融合网格物体边界以实现网格融合的算法相比,该方法不仅突破了对待融合物体的拓扑限制,允许多个物体同时进行融合,而且算法计算快速、鲁棒,使用方便,展示出良好的应用前景.     

Adaptive coding of generic 3D triangular meshes based on octree decomposition

In this paper, we present an adaptive-coding method for generic triangular meshes including both regular and irregular meshes. Though it is also based on iterative octree decomposition of the object space for the original mesh, as some prior arts, it has novelties in the following two aspects. First, it mathematically models the occupancy codes containing only a single–“1” bit for accurate initialization of the arithmetic coder at each octree level. Second, it adaptively prioritizes the bits in an occupancy code using a local surface smoothness measure that is based on triangle areas and therefore mitigates the effect of non-uniform vertex sampling over the surface. As a result, the proposed 3D mesh coder yields outstanding coding performance for both regular and irregular meshes and especially for the latter, as demonstrated by the experiments.     

Displacement-Correlated XFEM for Simulating Brittle Fracture

We present a remeshing-free brittle fracture simulation method under the assumption of quasi-static linear elastic fracture mechanics (LEFM). To achieve this, we devise two algorithms. First, we develop an approximate volumetric simulation, based on the extended Finite Element Method (XFEM), to initialize and propagate Lagrangian crack-fronts. We model the geometry of fracture explicitly as a surface mesh, which allows us to generate high-resolution crack surfaces that are decoupled from the resolution of the deformation mesh. Our second contribution is a mesh cutting algorithm, which produces fragments of the input mesh using the fracture surface. We do this by directly operating on the half-edge data structures of two surface meshes, which enables us to cut general surface meshes including those of concave polyhedra and meshes with abutting concave polygons. Since we avoid triangulation for cutting, the connectivity of the resulting fragments is identical to the (uncut) input mesh except at edges introduced by the cut. We evaluate our simulation and cutting algorithms and show that they outperform state-of-the-art approaches both qualitatively and quantitatively.     

Hexahedral Meshing With Varying Element Sizes

Hexahedral (or Hex‐) meshes are preferred in a number of scientific and engineering simulations and analyses due to their desired numerical properties. Recent state‐of‐the‐art techniques can generate high‐quality hex‐meshes. However, they typically produce hex‐meshes with uniform element sizes and thus may fail to preserve small‐scale features on the boundary surface. In this work, we present a new framework that enables users to generate hex‐meshes with varying element sizes so that small features will be filled with smaller and denser elements, while the transition from smaller elements to larger ones is smooth, compared to the octree‐based approach. This is achieved by first detecting regions of interest (ROIs) of small‐scale features. These ROIs are then magnified using the as‐rigid‐as‐possible deformation with either an automatically determined or a user‐specified scale factor. A hex‐mesh is then generated from the deformed mesh using existing approaches that produce hex‐meshes with uniform‐sized elements. This initial hex‐mesh is then mapped back to the original volume before magnification to adjust the element sizes in those ROIs. We have applied this framework to a variety of man‐made and natural models to demonstrate its effectiveness.     

从表面重构的体数据实现三维网格剖分

针对有限元分析中网格最优化问题,本文提出一种改进的生成四面体网格的自组织算法。该算法首先应用几何方法将三角形表面模型重新构造成规定大小的分类体数据,同时由该表面模型建立平衡八叉树,计算用以控制网格尺寸的三维数组;然后将体数据转换成邻域内不同等值面的形态一致的边界指示数组;结合改进的自组织算法和相关三维数据的插值函数,达到生成四面体网格的目的。实验对比表明,该方法能够生成更高比例的优质四面体,同时很好地保证了边界的一致。在对封闭的三维表面网格进行有限元建模时,本文算法为其提供了一种有效、可靠的途径。     

Incremental reconstruction of sharp edges on mesh surfaces

Limited by the regular grids in computing, many modelling approaches (e.g. field-based methods) sample 3D shape insensitive to sharp features therefore exhibit aliasing errors, by which a lot of sharp edges and corners are lost on the reconstructed surface. An incremental approach for recovering sharp edges on an insensitive sampled triangular mesh is presented in this paper, so that shape approximation errors are greatly reduced. Either chamfered or blended sharp edges on an input triangular mesh could be successfully reconstructed by the signals inherent in the mesh. As a non-iterative method, our approach could be finished in a very short time comparing to those diffusion-based sharp-feature reproducers. The region embedding sharp features is first identified through normal variations. The positions of vertices in the sharp-feature embedded region are then predicted progressively from outer to the inner of sharp regions so that sharp edges could be recovered in the sense of region shrinking.     

网格图形编辑的样条方法

提出基于样条的网格图形编辑方法,首先在网格表面附近构建近似的样条曲面,同时预计算网格顶点在样条上的对应点处局部标架下的坐标表示,并作为不变量在变形中进行保持;然后编辑样条的位置和形状,利用局部标架和细节坐标重建变形后的网格,同时进行网格光滑和网格细分,改善变形效果,以实现复杂模型简单快捷的编辑/变形.方法在保细节的同时允许对网格在多个尺度下编辑.实验结果表明,融合了样条的三角网格方法较传统的样条编辑方法可避免产生过多的控制点,大大地简化了操作.     

A survey of interactive mesh‐cutting techniques and a new method for implementing generalized interactive mesh cutting using virtual tools‡

In our experience, mesh‐cutting methods can be distinguished by how their solutions address the following major issues: definition of the cut path, primitive removal and re‐meshing, number of new primitives created, when re‐meshing is performed, and representation of the cutting tool. Many researches have developed schemes for interactive mesh cutting with the goals of reducing the number of new primitives created, creating new primitives with good aspect ratios, avoiding a disconnected mesh structure between primitives in the cut path, and representing the path traversed by the tool as accurately as possible. The goal of this paper is to explain how, by using a very simple framework, one can build a generalized cutting scheme. This method allows for any arbitrary cut to be made within a virtual object, and can simulate cutting surface, layered surface or tetrahedral objects using a virtual scalpel, scissors, or loop cautery tool. This method has been implemented in a real‐time, haptic‐rate surgical simulation system allowing arbitrary cuts to be made on high‐resolution patient‐specific models. Published in 2002 by John Wiley & Sons, Ltd.     

基于边长的三维形状插值

形状插值在计算机图形学和几何处理中是一个极其重要而基础的问题,在计算机动画等领域有 着广泛应用。注意到在平面三角网格和三维四面体网格插值问题中,对边长平方插值等价于对回拉度量进行插 值,因此具有等距扭曲和共形扭曲同时有界的良好性质。通过将其推广至曲面三角网格,提出了一种完全基于 边长的曲面三角网格插值算法。给定边长,在重建网格阶段,使用牛顿法对边长误差能量进行优化。并且给出 了其海森矩阵的解析正定化形式,从而避免了高代价的特征值分解步骤。注意到四面体网格的边长平方插值结 果具有极低曲率,意味着只需少许修改即可将其压平从而嵌入三维空间。因此提出先将曲面三角网格四面体化, 再从四面体网格的插值结果提取表面。然后将这表面作为初始化用于边长误差能量的牛顿迭代,从而使得收敛 结果更加接近全局最优。在一系列三角网格上进行了实验,结果说明了本文方法比之前方法的边长误差更小, 且得到的结果还是有界扭曲的。     

Diffusion equations over arbitrary triangulated surfaces for filtering and texture applications

In computer graphics, triangular mesh representations of surfaces have become very popular. Compared with parametric and implicit forms of surfaces, triangular mesh surfaces have many advantages, such as easy to render, convenient to store and the ability to model geometric objects with arbitrary topology. In this paper, we are interested in data processing over triangular mesh surfaces through PDEs (partial differential equations). We study several diffusion equations over triangular mesh surfaces, and present corresponding numerical schemes to solve them. Our methods work for triangular mesh surfaces with arbitrary geometry (the angles of each triangle are arbitrary) and topology (open meshes or closed meshes of arbitrary genus). Besides the flexibility, our methods are efficient due to the implicit/semi-implicit time discretization. We finally apply our methods to several filtering and texture applications such as image processing, texture generating and regularization of harmonic maps over triangular mesh surfaces. The results demonstrate the flexibility and effectiveness of our methods.     

Deformable mesh for virtual shape sculpting

A novel, interactive virtual sculpting framework based upon a deformable mesh model generated by a self-organizing feature map (SOFM) is described in this paper. The three-dimensional lattice of the SOFM maintains the relative connectivity of neighbouring nodes in the hexahedral mesh as it transforms from the initial reference geometry into the desired shape. Material and dynamic properties are incorporated into the deformable mesh by treating surface and internal nodes as point masses connected by a network of springs. The initial SOFM mesh can be either retrieved from a library of primitive shapes, or created by automatically adapting the 3D mesh to fit selected surface points. Once the initial mesh has been generated, the designer reshapes the virtual object by introducing external forces to the nodal mesh. The process of virtual sculpting is analogous to hand moulding of clay in the physical world where the material mass remains constant. During sculpting, the dynamically changing mesh can be easily rendered in VRML for visualization in a virtual reality environment. The deformable mesh generator and shape-sculpting system are illustrated by reshaping solid meshes created from scanned human heads.     

Finite element mesh deformation with the skeleton-section template

To develop fast finite element (FE) adaptation methods for simulation-driven design optimization, we propose a radial basis functions (RBF) method with a skeleton-section template to globally and locally deform FE meshes of thin-walled beam structures.The skeleton-section template is automatically formulated from the input mesh and serves as a hierarchical parameterization for the FE meshes. With this hierarchical parameterization, both the global and the local geometries of a thin-walled beam can be processed in the same framework, which is of importance for designing engineering components. The curve skeleton of the mesh is constructed with Voronoi decomposition, while the cross-sections are extracted from the mesh based on the curve skeleton.The RBF method is employed to locally and globally deform the mesh model with the cross-sections and the skeleton, respectively. The RBF method solves the spatial deformation field given prescribed deformations at the cross-sections. At the local scale, the user modifies the cross-sections to deform a region of the surface mesh. At the global level, the skeleton is manipulated and its deformation is transferred to all cross-sections to induce the mesh deformation.In order to handle curved mesh models and attain flexible local deformations, the input mesh is embedded into its skeleton frame field using an anisotropic distance metric. In this way, even strip-like features along arbitrary directions can be created on the mesh model using only a few cross-sections as the deformation handles. In addition, form features can be rigidly preserved at both deformation levels.Numerical examples demonstrate that intuitive and qualified FE mesh deformations can be obtained with manipulation of the skeleton-section template.     

Feature-Sensitive Tetrahedral Mesh Generation with Guaranteed Quality

Tetrahedral meshes are being extensively used in finite element methods (FEM). This paper proposes an algorithm to generate feature-sensitive and high-quality tetrahedral meshes from an arbitrary surface mesh model. A top-down octree subdivision is conducted on the surface mesh and a set of tetrahedra are constructed using adaptive body-centered cubic (BCC) lattices. Special treatments are given to the tetrahedra near the surface such that the quality of the resulting tetrahedral mesh is provably guaranteed: the smallest dihedral angle is always greater than 5.71°. The meshes generated by our method are not only adaptive from the interior to the boundary, but also feature-sensitive on the surface with denser elements in high-curvature regions where geometric feature most likely reside. A variety of experimental results are presented to demonstrate the effectiveness and robustness of this algorithm.     

Multiresolution free-form deformation with subdivision surface of arbitrary topology

A new free-from deformation method is presented in this paper. Object deformation is controlled by a mesh of arbitrary topology, namely a control mesh. The subdivision surface determined by the control mesh spans an intermediate deformation space. The object is embedded into the space by the nearest point rule. When the shape of the control mesh is changed, the object embedded in the intermediate deformation space will be deformed accordingly. Since the subdivision surface has a multiresolution property, the proposed deformation method naturally has a multiresolution property. A technique for generating control meshes is also introduced in the paper. Compared with previous deformation methods with arbitrary topology control tools, the proposed method has the advantages of flexible control and computational efficiency.     

Adaptive surface‐deformable model with shape‐preserving spring

This paper presents a multi‐resolutional surface deformable model with physical property adjustment scheme and shape‐preserving springs to represent surface‐deformable objects efficiently and robustly. In order to reduce the computational complexity while ensuring the same global volumetric behaviour for the deformable object, we introduce a multi‐resolutional mass‐spring model that is locally refined using the modified‐butterfly subdivision scheme. For robust deformation, a shape‐preserving spring, which helps to restore the model to the original shape, is proposed to reduce the animation instability. Volume and shape preservation is indirectly achieved by restoring the model to the original shape without computing the actual volume and associated forces at every iteration. Most existing methods concentrate on visual realism of multi‐resolutional deformation and often neglect to maintain the dynamic behavioural integrity between detail levels. In order to preserve overall physical behaviour, we present a new scheme for adjusting physical properties between different levels of details. During the animation of deformable objects, the part of the object under external forces beyond a threshold or with large surface curvature variations is refined with a higher level of detail. The physical properties of nodes and springs in the locally refined area are adjusted in order to preserve the total mass and global behaviour of the object. The adequacy of the proposed scheme was analysed with tests using practical mesh examples. Experimental results demonstrate improved efficiency in object deformation and preservation of overall behaviour between different levels. Copyright © 2005 John Wiley & Sons, Ltd.     

A Shrink Wrapping Approach to Remeshing Polygonal Surfaces

Due to their simplicity and flexibility, polygonal meshes are about to become the standard representation for surface geometry in computer graphics applications. Some algorithms in the context of multiresolution representation and modeling can be performed much more efficiently and robustly if the underlying surface tesselations have the special subdivision connectivity. In this paper, we propose a new algorithm for converting a given unstructured triangle mesh into one having subdivision connectivity. The basic idea is to simulate the shrink wrapping process by adapting the deformable surface technique known from image processing. The resulting algorithm generates subdivision connectivity meshes whose base meshes only have a very small number of triangles. The iterative optimization process that distributes the mesh vertices over the given surface geometry guarantees low local distortion of the triangular faces. We show several examples and applications including the progressive transmission of subdivision surfaces.