Adaptive and Feature-Preserving Subdivision for High-Quality Tetrahedral Meshes

We present an adaptive subdivision scheme for unstructured tetrahedral meshes inspired by the       -subdivision scheme for triangular meshes. Existing tetrahedral subdivision schemes do not support adaptive refinement and have traditionally been driven by the need to generate smooth three-dimensional deformations of solids. These schemes use edge bisections to subdivide tetrahedra, which generates octahedra in addition to tetrahedra. To split octahedra into tetrahedra one routinely chooses a direction for the diagonals for the subdivision step. We propose a new topology-based refinement operator that generates only tetrahedra and supports adaptive refinement. Our tetrahedral subdivision algorithm is motivated by the need to have one representation for the modeling, the simulation and the visualization and so to bridge the gap between CAD and CAE. Our subdivision algorithm design emphasizes on geometric quality of the tetrahedral meshes, local and adaptive refinement operations, and preservation of sharp geometric features on the boundary and in the interior of the physical domain.     

Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods

In this paper we present an algorithm to refine space–time finite element meshes as needed for the numerical solution of parabolic initial boundary value problems. The approach is based on a decomposition of the space–time cylinder into finite elements, which also allows a rather general and flexible discretization in time. This also includes adaptive finite element meshes which move in time. For the handling of three-dimensional spatial domains, and therefore of a four-dimensional space–time cylinder, we describe a refinement strategy to decompose pentatopes into smaller ones. For the discretization of the initial boundary value problem we use an interior penalty Galerkin approach in space, and an upwind technique in time. A numerical example for the transient heat equation confirms the order of convergence as expected from the theory. First numerical results for the transient Navier–Stokes equations and for an adaptive mesh moving in time underline the applicability and flexibility of the presented approach.     

面向三角网格的自适应细分

细分曲面存在的一个问题是随着细分次数的增多,网格的面片数迅速增长,巨大的数据量使得细分后的模难以进行其它处理。针对这个问题,该文利用控制点的局部信息提出了一种基于Loop模式的自适应细分算法,利用该算法可避免在相对光滑处再细分,与正常细分相比,既大大减少了数据量,提高了模型的处理速度,又达到了对模型进行细分的目的。     

Hybrid Discretization Methods with Adaptive Yield Surface Detection for Bingham Pipe Flows

We devise a hybrid low-order method for Bingham pipe flows, where the velocity is discretized by means of one unknown per mesh face and one unknown per mesh cell which can be eliminated locally by static condensation. The main advantages are local conservativity and the possibility to use polygonal/polyhedral meshes. We exploit this feature in the context of adaptive mesh refinement to capture the yield surface by means of local mesh refinement and possible coarsening. We consider the augmented Lagrangian method to solve iteratively the variational inequalities resulting from the discrete Bingham problem, using piecewise constant fields for the auxiliary variable and the associated Lagrange multiplier. Numerical results are presented in pipes with circular and eccentric annulus cross-section for different Bingham numbers.     

Implicit Hierarchical Quad‐Dominant Meshes

We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.     

Interpolatory √3-Subdivision

We present a new interpolatory subdivision scheme for triangle meshes. Instead of splitting each edge and performing a 1-to-4 split for every triangle we compute a new vertex for every triangle and retriangulate the old and the new vertices. Using this refinement operator the number of triangles only triples in each step. New vertices are computed with a Butterfly like scheme. In order to obtain overall smooth surfaces special rules are necessary in the neighborhood of extraordinary vertices. The scheme is suitable for adaptive refinement by using an easy forward strategy. No temporary triangles are produced here which allows simpler data structures and makes the scheme easy to implement.     

Fractality of refined triangular grids and space-filling curves

In this paper we explain the fractal geometry of refined and derefined triangular and tetrahedral meshes by means of the application of iterated function systems (IFS). These meshes feature a remarkable amplifying invariance under changes of scale. The applications of IFS families are shown equivalent to the use of adaptive strategies that combine the refinement procedure with the derefinement procedure. In addition, space-filling curves (SFC) are used to assign a binary code for any 2D triangular refined mesh. SFC are also shown as useful for the problem of automatic domain decomposition.     

Parallel adaptive simulations of dynamic fracture events

Finite element simulations of dynamic fracture problems usually require very fine discretizations in the vicinity of the propagating stress waves and advancing crack fronts, while coarser meshes can be used in the remainder of the domain. This need for a constantly evolving discretization poses several challenges, especially when the simulation is performed on a parallel computing platform. To address this issue, we present a parallel computational framework developed specifically for unstructured meshes. This framework allows dynamic adaptive refinement and coarsening of finite element meshes and also performs load balancing between processors. We demonstrate the capability of this framework, called ParFUM, using two-dimensional structural dynamic problems involving the propagation of elastodynamic waves and the spontaneous initiation and propagation of cracks through a domain discretized with triangular finite elements.     

Efficient view-dependent refinement of 3D meshes using sqrt{3}-subdivision

In this paper we introduce an efficient view-dependent refinement technique well suited to adaptive visualization of 3D triangle meshes on a graphic terminal. Our main goal is the design of fast and robust, smooth surface reconstruction from coarse meshes. We demonstrate that the sqrt{3}-subdivision operator recently introduced by Kobbelt offers many benefits, including view-dependent refinement, removal of polygonal aspect and a highly tunable level of detail adaptation. In particular, we propose a new data structure that requires neither edges nor hierarchies for efficient and reversible view-dependent refinement. Results on various 3D meshes illustrate the relevance of the technique.     

Context‐Based Coding of Adaptive Multiresolution Meshes

Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression scheme for these adaptive structures that exploits the parent–child relationships inherent to the mesh hierarchy. We use the rules that correspond to the adaptive refinement scheme and store bits only where some freedom of choice is left, leading to compact codes that are free of redundancy. Moreover, we extend the coder to sequences of meshes with varying refinement. The connectivity compression ratio of our method exceeds that of state‐of‐the‐art coders by a factor of 2–7. For efficient compression of vertex positions we adapt popular wavelet‐based coding schemes to the adaptive triangular and quadrangular cases to demonstrate the compatibility with our method. Akin to state‐of‐the‐art coders, we use a zerotree to encode the resulting coefficients. Using improved context modelling we enhanced the zerotree compression, cutting the overall geometry data rate by 7% below those of the successful Progressive Geometry Compression. More importantly, by exploiting the existing refinement structure we achieve compression factors that are four times greater than those of coders which can handle irregular meshes.     

An anisotropic unstructured triangular adaptive mesh algorithm based on error and error gradient information

In this paper, an algorithm based on unstructured triangular meshes using standard refinement patterns for anisotropic adaptive meshes is presented. It consists of three main actions: anisotropic refinement, solution-weighted smoothing and patch unrefinement. Moreover, a hierarchical mesh formulation is used. The main idea is to use the error and error gradient on each mesh element to locally control the anisotropy of the mesh. The proposed algorithm is tested on interpolation and boundary-value problems with a discontinuous solution.     

Localized Delaunay Refinement for Sampling and Meshing

The technique of Delaunay refinement has been recognized as a versatile tool to generate Delaunay meshes of a variety of geometries. Despite its usefulness, it suffers from one lacuna that limits its application. It does not scale well with the mesh size. As the sample point set grows, the Delaunay triangulation starts stressing the available memory space which ultimately stalls any effective progress. A natural solution to the problem is to maintain the point set in clusters and run the refinement on each individual cluster. However, this needs a careful point insertion strategy and a balanced coordination among the neighboring clusters to ensure consistency across individual meshes. We design an octtree based localized Delaunay refinement method for meshing surfaces in three dimensions which meets these goals. We prove that the algorithm terminates and provide guarantees about structural properties of the output mesh. Experimental results show that the method can avoid memory thrashing while computing large meshes and thus scales much better than the standard Delaunay refinement method.     

Parallel quadrilateral subdomain generation for finite element analysis

In this article the Sub-domain Generation Method (SGM) previously developed for finite element meshes composed of triangular elements is generalised to account for meshes composed of arbitrary quadrilateral elements. A neural network of enhanced accuracy was developed to predict the number of quadrilaterals generated within each coarse quadrilateral element after refinement using an adaptive re-meshing procedure. Partitioning of the meshes was undertaken on the coarse mesh using a genetic algorithm to optimise the partitions considering both load balancing and interprocessor communication of the subsequent finite element analysis. A series of examples of increasing refinement were decomposed by considering a single coarse mesh of a fixed number of elements.     

An algorithm for adaptive mesh refinement inn dimensions

The author describes a fast algorithm for local adaptive mesh refinement inn dimensions based on simplex bisection. A ready-to-use implementation of the algorithm in C++ pseudocode is given. It is proven that the scheme satisfies all conditions one usually places on grid refinement in the context of finite-element calculations. Bisection refinement also offers an interesting additional feature over the usual, regular, refinement scheme: all linear finite-element basis functions of one generation are of disjoint support. In the way the scheme is presented here, all generated simplex meshes satisfy a ‘structural condition’ which is exploited to simplify bookkeeping of the neighbour graph. However, bisection refinement places certain restrictions on the initial, coarsest grid. For a simply connected domain, a precise and useful criterion for the applicability of the described refinement scheme is formulated and proven.     

An adaptive mesh refinement of quadrilateral finite element meshes based upon an a posteriori error estimation of quantities of interest: modal response

Modal analysis is commonly performed in a vehicle development process to assess dynamic responses of structure designs. This paper presents an adaptive quadrilateral refinement process for modal analysis of elastic shells based upon a posteriori error estimation in natural frequencies. The process provides engineers with an estimation of their modal analysis quality and an effective adaptive refinement tool for quadrilateral meshes. The effectiveness of the process is demonstrated on the eigenvalue analyses of two numerical examples, a shock tower cap and a roof structure. It shows that the solution error in the frequency of interest is effectively reduced through the adaptive refinement process, and the resulting frequency of interest converges to the solution of a very fine model.     

Weighted Error Estimates for Finite Element Solutions of Variational Inequalities

In this note the studies begun in Blum and Suttmeier (1999) on adaptive finite element discretisations for nonlinear problems described by variational inequalities are continued. Similar to the concept proposed, e.g., in Becker and Rannacher (1996) for variational equalities, weighted a posteriori estimates for controlling arbitrary functionals of the discretisation error are constructed by using a duality argument. Numerical results for the obstacle problem demonstrate the derived error bounds to be reliable and, used for an adaptive grid refinement strategy, to produce economical meshes. Received September 6, 1999; revised February 8, 2000     

解Hamilton-Jacobi方程的无振荡局部加密方法

０．引 言 近年来,Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程（简称Ｈ－Ｊ方程）的数学理论与数值逼近已引起人们越来越多的关注．Ｈ－Ｊ方程不仅在原有的领域例如控制论、微分几何等有非常重要的应用［８］,而且不断开拓新的应用领域,例如用于网格生成［５］以及流体界面的水平集方法计算 ［９,１２,１３,１５］等．由于 Ｈ－Ｊ方程解的导数会出现间断,导致解曲面（线）出现尖点或纽结等现象［７］,故如何做到既节省计算时间,又能在光滑区域高精度数值求解和较好地分辨间断是一个十分重要的问题．文卜］通过在每个坐标方向构造单变量的…     

Mesh generation and mesh refinement procedures for the analysis of concrete shells

In this paper, a mesh generation and mesh refinement procedure for adaptive finite element (FE) analyses of real-life surface structures are proposed. For mesh generation, the advancing front method is employed. FE meshes of curved structures are generated in the respective 2D parametric space of the structure. Thereafter, the 2D mesh is mapped onto the middle surface of the structure. For mesh refinement, two different modes, namely uniform and adaptive mesh refinement, are considered. Remeshing in the context of adaptive mesh refinement is controlled by the spatial distribution of the estimated error of the FE results. Depending on this distribution, remeshing may result in a partial increase and decrease, respectively, of the element size. In contrast to adaptive mesh refinement, uniform mesh refinement is characterized by a reduction of the element size in the entire domain. The different refinement strategies are applied to ultimate load analysis of a retrofitted cooling tower. The influence of the underlying FE discretization on the numerical results is investigated.     

基于几何特征和力学特性的自适应网格生成算法

为获得适合有限元分析的满意网格划分,提出了平面域的基于几何特征和力学特性相结合的自适应网络生成方法,实现了应力集中区的网格局部加密及平稳变密度的网格自动剖分,通过实例表明本方法实用性强、效果良好。