Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing

We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.     

An automatic 3D mesh generation method for domains with multiple materials

This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form non-manifold surfaces. In earlier papers, we developed an octree-based isocontouring method to construct unstructured 3D meshes for a single material (homogeneous) domain with manifold boundary. In this paper, we introduce the notion of a material change edge and use it to identify the interface between two or several different materials. A novel method to calculate the minimizer point for a cell shared by more than two materials is provided, which forms a non-manifold node on the boundary. We then mesh all the material regions simultaneously and automatically while conforming to their boundaries directly from volumetric data. Both material change edges and interior edges are analyzed to construct tetrahedral meshes, and interior grid points are analyzed for proper hexahedral mesh construction. Finally, edge-contraction and smoothing methods are used to improve the quality of tetrahedral meshes, and a combination of pillowing, geometric flow and optimization techniques is used for hexahedral mesh quality improvement. The shrink set of pillowing schemes is defined automatically as the boundary of each material region. Several application results of our multi-material mesh generation method are also provided.     

Guaranteed-quality all-quadrilateral mesh generation with feature preservation

In this paper, a quadtree-based mesh generation method is described to create guaranteed-quality, geometry-adapted all-quadrilateral (all-quad) meshes with feature preservation for arbitrary planar domains. Given point cloud, our method generates all-quad meshes with these points as vertices and all the angles are within [45°, 135°]. For given planar curves, quadtree-based spatial decomposition is governed by the curvature of the boundaries and narrow regions. 2-refinement templates are chosen for local mesh refinement without creating any hanging nodes. A buffer zone is created by removing elements around the boundary. To guarantee the mesh quality, the angles facing the boundary are improved via template implementation, and two buffer layers are inserted in the buffer zone. It is proved that all the elements of the final mesh are quads with angles between 45° ± ε and 135° ± ε (ε ≤ 5°) with the exception of badly shaped elements that may be required by the sharp angles in the input geometry. We also prove that the scaled Jacobians defined by two edge vectors are in the range of [sin(45° ? ε), sin90°], or [0.64, 1.0]. Furthermore, sharp features and narrow regions are detected and preserved automatically. Boundary layer meshes are generated by splitting elements of the second buffer layer. We have applied our algorithm to a set of complicated geometries, including the Lake Superior map and the air foil with multiple components.     

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.     

A computational approach to joint line detection on triangular meshes

In this paper, we present formulae for evaluating differential quantities at vertices of triangular meshes that may approximate potential piecewise smooth surfaces with discontinuous normals or discontinuous curvatures at the joint lines. We also define the C ¹ and C ² discontinuity measures for surface meshes using changing rates of one-sided curvatures or changing rates of curvatures across mesh edges. The curvatures are computed discretely as of local interpolating surfaces that lie within a tolerance to the mesh. Together with proper estimation of local shape parameters, the obtained discontinuity measures own properties like sensitivity to salient joint lines and being scale invariant. A simple algorithm is finally developed for detection of C ¹ or C ² discontinuity joint lines on triangular meshes with even highly non-uniform triangulations. Several examples are provided to demonstrate the effectiveness of the proposed method.     

Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization

Gradient-based aerodynamic shape optimization using computational fluid dynamics (CFD), and time dependent problems in aeroelasticity, that is, coupled calculations between computational structural mechanics (CSM) and CFD, require repeated deformations of the CFD mesh.An interpolation scheme, based on radial basis functions (RBF), is devised in order to propagate the deformations from the boundaries to the interior of the CFD mesh. This method can lower the computational costs due to the deformation of the mesh, in comparison with the usual Laplace smoothing. Moreover, the algorithm is independent of the mesh connectivities. Therefore, structured and unstructured meshes are equally treated as well as hybrid meshes.The application of this interpolation scheme in problems of aerodynamic shape optimization is also carefully investigated. When the optimization is executed by a gradient-based algorithm the cost function is differentiated with respect to the design parameters in order to obtain the gradient. The gradient is most efficiently and accurately calculated by solving a certain adjoint equation derived from the discretized flow equations. The calculation of the gradient, which is detailed in this presentation, involves the Jacobian matrix of the mesh deformation.Finally, we present the results of an optimization of the ONERA M6 wing at transonic speed using the interpolation algorithm. The results are used for comparison with another technique of mesh deformation. The quality of the mesh obtained by the new algorithm, and the interpolation error, are analyzed with respect to the parameters of the interpolation scheme: the type of RBF, the RBF’s shape parameter, and the sets of control points.     

3D finite element meshing from imaging data

This paper describes an algorithm to extract adaptive and quality 3D meshes directly from volumetric imaging data. The extracted tetrahedral and hexahedral meshes are extensively used in the finite element method (FEM). A top-down octree subdivision coupled with a dual contouring method is used to rapidly extract adaptive 3D finite element meshes with correct topology from volumetric imaging data. The edge contraction and smoothing methods are used to improve mesh quality. The main contribution is extending the dual contouring method to crack-free interval volume 3D meshing with boundary feature sensitive adaptation. Compared to other tetrahedral extraction methods from imaging data, our method generates adaptive and quality 3D meshes without introducing any hanging nodes. The algorithm has been successfully applied to constructing quality meshes for finite element calculations.     

Key-components: detection of salient regions on 3D meshes

In this paper, we present a method to detect stable components on 3D meshes. A component is a salient region on the mesh which contains discriminative local features. Our goal is to represent a 3D mesh with a set of regions, which we called key-components, that characterize the represented object and therefore, they could be used for effective matching and recognition. As key-components are features in coarse scales, they are less sensitive to mesh deformations such as noise. In addition, the number of key-components is low compared to other local representations such as keypoints, allowing us to use them in efficient subsequent tasks. A desirable characteristic of a decomposition is that the components should be repeatable regardless shape transformations. We show in the experiments that the key-components are repeatable and robust under several transformations using the SHREC’2010 feature detection benchmark. In addition, we discover the connection between the theory of saliency of visual parts from the cognitive science and the results obtained with our technique.     

Constrained mesh optimization on boundary

This paper presents a new mesh optimization approach aiming to improve the mesh quality on the boundary. The existing mesh untangling and smoothing algorithms (Vachal et al. in J Comput Phys 196: 627–644, 2004; Knupp in J Numer Methods Eng 48: 1165–1185, 2002), which have been proved to work well to interior mesh optimization, are enhanced by adding constrains of surface and curve shape functions that approximate the boundary geometry from the finite element mesh. The enhanced constrained optimization guarantees that the boundary nodes to be optimized always move on the approximated boundary. A dual-grid hexahedral meshing method is used to generate sample meshes for testing the proposed mesh optimization approach. As complementary treatments to the mesh optimization, appropriate mesh topology modifications, including buffering element insertion and local mesh refinement, are performed in order to eliminate concave and distorted elements on the boundary. Finally, the optimization results of some examples are given to demonstrate the effectivity of the proposed approach.     

LayTracks3D: A new approach for meshing general solids using medial axis transform

This paper presents an extension of the all-quad meshing algorithm called LayTracks to generate high quality hex-dominant meshes of general solids. LayTracks3D uses the mapping between the Medial Axis (MA) and the boundary of the 3D domain to decompose complex 3D domains into simpler domains called Tracks. Tracks in 3D have no branches and are symmetric, non-intersecting, orthogonal to the boundary, and the shortest path from the MA to the boundary. These properties of tracks result in desired meshes with near cube shape elements at the boundary, structured mesh along the boundary normal with any irregular nodes restricted to the MA, and sharp boundary feature preservation. The algorithm has been tested on a few industrial CAD models and hex-dominant meshes are shown in the Results section. Work is underway to extend LayTracks3D to generate all-hex meshes.     

Towards flattenable mesh surfaces

In many industries, products are constructed by assembled surface patches in ℜ³, where each patch is expected to have an isometric map to a corresponding region in ℜ². The widely investigated developable surfaces in differential geometry show this property. However, the method to model a piecewise-linear surface with this characteristic is still under research. To distinguish from the continuous developable surface, we name them as flattenable mesh surfaces since a polygonal mesh has the isometric mapping property if it can be flattened into a two-dimensional sheet without stretching. In this paper, a novel flattenable mesh surface (Flattenable Laplacian mesh) is introduced and the relevant modelling tool is formulated. Moreover, for a given triangular mesh which is almost flattenable, a local perturbation approach is developed to improve its flattenability. The interference between the meshes under process and their nearby objects has been prevented in this local flattenable perturbation. Both the computations of Flattenable Laplacian meshes and the flattenable perturbation are based on the constrained optimization technology.     

基于二维轮廓序列的膝关节三维重建

提出了一个基于二维轮廓序列的四面体网格生成方法,用于医学图像三维几何模型重构.该方法首先对各选定的断层图像提取目标轮廓并做分支匹配等处理,然后生成各轮廓内部平面域的三角网格,最后在相邻断层之间根据三角网格连接四面体单元.该方法被应用于人体膝关节虚拟手术系统的三维几何建模,得到的膝部股骨模型包含494个节点和2 046个四面体单元,膝部脂肪模型包含2 854个节点和14011个四面体单元,这些模型被成功地应用于膝关节手术仿真,从而证明了该三维模型重建方法的可行性和有效性.     

Spectral 3D mesh segmentation with a novel single segmentation field

We present an automatic mesh segmentation framework that achieves 3D segmentation in two stages, hierarchical spectral analysis and isoline-based boundary detection. During the hierarchical spectral analysis stage, a novel segmentation field is defined to capture a concavity-aware decomposition of eigenvectors from a concavity-aware Laplacian. Specifically, a sufficient number of eigenvectors is first adaptively selected and simultaneously partitioned into sub-eigenvectors through spectral clustering. Next, on the sub-eigenvectors level, we evaluate the confidence of identifying a spectral-sensitive mesh boundary for each sub-eigenvector by two joint measures, namely, inner variations and part oscillations. The selection and combination of sub-eigenvectors are thereby formulated as an optimization problem to generate a single segmentation field. In the isoline-based boundary detection stage, the segmentation boundaries are recognized by a divide-merge algorithm and a cut score, which respectively filters and measures desirable isolines from the concise single segmentation field. Experimental results on the Princeton Segmentation Benchmark and a number of other complex meshes demonstrate the effectiveness of the proposed method, which is comparable to recent state-of-the-art algorithms.     

A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids

A Cartesian cut-cell method which allows the solution of two- and three-dimensional viscous, compressible flow problems on arbitrarily refined graded meshes is presented. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative in terms of mass, momentum, and energy. For three-dimensional compressible flows, such a method has not been presented in the literature, yet. Since ghost cells can be arbitrarily positioned in space the proposed method is flexible in terms of shape and size of embedded boundaries. A key issue for Cartesian grid methods is the discretization at mesh interfaces and boundaries and the specification of boundary conditions. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh, which are used to formulate the surface flux. Expressions to impose boundary conditions and to compute the viscous terms on the boundary are derived. The overall discretization is shown to be second-order accurate in L¹. The accuracy of the method and the quality of the solutions are demonstrated in several two- and three-dimensional test cases of steady and unsteady flows.     

A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.     

从表面重构的体数据实现3维网格剖分

目的 针对有限元分析中网格最优化问题,提出一种改进的生成四面体网格的自组织算法。方法 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3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3维数据的插值函数,达到生成四面体网格的目的。结果 实验结果对比表明,该方法能够生成更高比例的优质四面体,增强了对扁平面体的抑制能力,同时很好地保证了边界的一致。结论 在对封闭的3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。     

Numerical modeling of interaction of a current with a circular cylinder near a rigid bed

The numerical modeling of 2D turbulent flow around a smooth horizontal circular cylinder near a rigid bed with gap ratio G/D = 0.3 at Reynolds number Re_D = 9500 is investigated. Ansys^® 10.0-FLOTRAN program package is used to solve the governing equations by FEM, and the performance of the standard k ? ε, standard k ? ω, and SST turbulence models are examined. A sensitivity study for the three turbulence models is carried out on three computational meshes with different densities near the cylinder surface. The computational velocity fields and the Strouhal numbers from the present simulations are compared with those obtained from the PIV measurement. It is found that the time-averaged velocity field of the flow in the proximity of the cylinder is closely affected by the mesh resolution near the cylinder surface, and the mesh refinement in radial direction improves the results of present simulations. The shedding of vortices in the cylinder wake is not predicted by k ? ε model on all the three meshes. The results for the time-averaged velocity field show that the numerical modeling using either of k ? ω and SST turbulence models on the finest mesh used on the cylinder surface is reasonably successful.     

Generating well-shaped d-dimensional Delaunay Meshes

A d-dimensional simplicial mesh is a Delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside. A mesh is well shaped if the maximum aspect ratio of all its simplices is bounded from above by a constant. It is a long-term open problem to generate well-shaped d-dimensional Delaunay meshes for a given polyhedral domain. In this paper, we present a refinement-based method that generates well-shaped d-dimensional Delaunay meshes for any PLC domain with no small input angles. Furthermore, we show that the generated well-shaped mesh has O(n) d-simplices, where n is the smallest number of d-simplices of any almost-good meshes for the same domain. Here a mesh is almost-good if each of its simplices has a bounded circumradius to the shortest edge length ratio.     

Embedding a family of disjoint 3D meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint 3D meshes into a crossed cube. Two major contributions of this paper are: (1) for n?4, a family of two disjoint 3D meshes of size 2×2×2^n-3 can be embedded in an n-D crossed cube with unit dilation and unit expansion, and (2) for n?6, a family of four disjoint 3D meshes of size 4×2×2^n-5 can be embedded in an n-D crossed cube with unit dilation and unit expansion. These results mean that a family of two or four 3D-mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends the results recently obtained by Fan and Jia [J. Fan, X. Jia, Embedding meshes into crossed cubes, Information Sciences 177(15) (2007) 3151-3160].     

Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises a B-spline collocation method on a uniform mesh, which leads to a tridiagonal linear system. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The analysis is done on a uniform mesh, which permits its extension to the case of adaptive meshes which may be used to improve the solution. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.