A new method of quality improvement for quadrilateral mesh based on small polygon reconnection

In this paper, a new method of topological cleanup for quadrilateral mesh is presented. The method first selects a patch of mesh around an irregular node. It then seeks the best connection of the selected patch according to its irregular valence using a new topological operation： small polygon reconnection （SPR）. By replacing the original patch with an optimal one that has less irregular valence, mesh quality can be improved. Three applications based on the proposed approach are enumerated： （1） improving the quality of a quadrilateral mesh, （2） converting a triangular mesh to a quadrilateral one, and （3） adapting a triangle generator to a quadrilateral one. The presented method is highly effective in all three applications.

Accuracy assessment of mine walls＇ surface models derived from terrestrial laser scanning

The monitoring of highwall slopes at open-pit mines is an important task to ensure safe mining. For this reason, several techniques such as total station, radar, terrestrial Light Detection and Ranging （LIDAR） can be employed for surface measurement. The objective of this study is to investigate mesh algorithms, which can be used to interpolate 3D models of pit walls. Experiments were carried out at Coc Sau open-pit mine at Quang Ninh province of Vietnam, and at experimental mine of Akademia Gtrniczo-Hutnicza University of Science and Technology in Cracow, Poland. First, 3D point cloud data for the study area was acquired by using terrestrial LIDAR, then was used to generate mesh surfaces using three algorithms--Delaunay 2.5D XY Plane, Delaunay 2.5D Best Fitting Plane, and Mesh from Points. After that, the results were rectified and optimized. Subsequently, the optimized meshes were used for generation of non-uniform rational basis spline （NURBS） surfaces. Then, the NURBS surface accuracy was assessed. The results showed that the average distance between surface and point cloud was within range of 5.6-5.8 mm with deviation of 6.2-6.8 mm, depending on the used mesh. Additionally, the quality of surfaces depends on the quality of input data set and the algorithm used to generate mesh network, and the accuracy of computed NURBS surfaces fitting into pointset was 4-5 times lower than that of optimized mesh fitting. However, the accuracy of the final product allows determining displacements on the level of centimeters.

3D Engineering Model Retrieval Based on Enhanced Shape Distributions

To reuse and share the valuable knowledge embedded in repositories of engineering models for accelerating the design process, improving product quality, and reducing costs, it is crucial to devise search engines capable of matching 3D models efficiently and effectively. In this paper, an enhanced shape distributions-based technique of using geometrical and topological information to search 3D engineering models represented by polygonal meshes was presented. A simplification method of polygonal meshes was used to simplify engineering model as the pretreatment for generation of sample points. The method of sampling points was improved and a pair of functions that was more sensitive to shape was employed to construct a 2D shape distribution. Experiments were conducted to evaluate the proposed algorithm utilizing the Engineering Shape Benchmark (ESB) database. The experiential results suggest that the search effectiveness is significantly improved by enforcing the simplification and enhanced shape distributions to engineering model retrieval.

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and analyze.Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise constant.Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.

A unified framework for isotropic meshing based on narrowband Euclidean distance transformation

In this paper, we propose a simpleyet-effective method for isotropic meshing relying on Euclidean distance transformation based centroidal Voronoi tessellation(CVT). Our approach improves the performance and robustness of computing CVT on curved domains while simultaneously providing highquality output meshes. While conventional extrinsic methods compute CVTs in the entire volume bounded by the input model, we restrict the computation to a 3D shell of user-controlled thickness. Taking voxels which contain surface samples as sites, we compute the exact Euclidean distance transform on the GPU. Our algorithm is parallel and memory-efficient,and can construct the shell space for resolutions up to 20483 at interactive speed. The 3D centroidal Voronoi tessellation and restricted Voronoi diagrams are also computed efficiently on the GPU. Since the shell space can bridge holes and gaps smaller than a certain tolerance, and tolerate non-manifold edges and degenerate triangles, our algorithm can handle models with such defects, which typically cause conventional remeshing methods to fail. Our method can process implicit surfaces, polyhedral surfaces, and point clouds in a unified framework. Computational results show that our GPU-based isotropic meshing algorithm produces results comparable to state-ofthe-art techniques, but is significantly faster than conventional CPU-based implementations.

QUANTITATIVE APPROACH TO INFLUENCES OF HIGH SEDIMENT LADEN INUNDATION FLOWS ON THE MORPHOLOGICAL VARIATION OF FLOODPLAINS

This article mainly aims at developing an integrated 2-D numerical simulation model on inundation, sediment transportation and the morphological variations of floodplains due to high sediment-laden inundation flows. Due to the complexity of inner and outer boundaries and the arbitrary structures within the computational domain of floodplains, an unstructured Finite-Volume Method （FVM） based on an irregular polygon mesh was worked out so that the influences of complex boundaries can be integrated into the simulation. A case study was conducted in the Lower Yellow River Basin, in which a dike-break at the Huayuankou Hydrological Station was assumed to happen when a flood scale of 1982 was suffered in the region. The simulated spatial distribution of sediment deposition and erosion can be used to reasonably explain the natural phenomena of ＂suspended river＂ of the lower part of the Yellow River. It is concluded that the inundation process of water is similar to a variable-river-bed condition during the simulation because the sediment deposition and erosion are modified by new values at the end of each time step. The mass and momentum conservation were strictly followed during the simulation. Therefore, the prediction of floodplain evolutions by the integrated simulation model, proposed in this study, can be adequately and accurately given if the real condition of an floodplain can be obtained in detail.

PDE-BASED MEDIAL AXIS EXTRACTION AND SHAPE MANIPULATION OF ARBITRARY MESHES

Shape skeletonization （i.e., medial axis extraction） is powerful in many visual computing applications, such as pattern recognition, object segmentation, registration, and animation. In this paper, the authors expand the use of diffusion equations combined with distance field information to approximate medial axes of arbitrary 3D differential properties. It offers an alternative solids represented by polygonal meshes based on their but natural way for medial axis extraction for commonly used 3D polygonal models. By solving the PDE along time axis, this system can not only quickly extract diffusion-based medial axes of input meshes, but also allow users to visualize the extraction process at each time step. In addition, the proposed model provides users a set of manipulation toolkits to sculpt extracted medial axes, then use diffusion-based techniques to recover corresponding deformed shapes according to the original input datasets. This skeleton-based shape manipulation offers a fast and easy way for animation and deformation of complicated mesh objects.

A VARIATIONAL APPROACH FOR DETECTING FEATURE LINES ON MESHES

Feature lines are fundamental shape descriptors and have been extensively applied to computer graphics, computer-aided design, image processing, and non-photorealistic renderingi This paper introduces a unified variational framework for detecting generic feature lines on polygonal meshes. The classic Mumford-Shah model is extended to surfaces. Using F-convergence method and discrete differential geometry, we discretize the proposed variational model to sequential coupled sparse linear systems. Through quadratic polyno- mials fitting, we develop a method for extracting valleys of functions defined on surfaces. Our approach provides flexible and intuitive control over the detecting procedure, and is easy to implement. Several measure functions are devised for different types of feature lines, and we apply our approach to various polygonal meshes ranging from synthetic to measured models. The experiments demonstrate both the effectiveness of our algorithms and the visual quality of results.

A Weak Galerkin Mixed Finite Element Method for Second Order Elliptic Equations on 2D Curved Domains

This article concerns the weak Galerkin mixed finite element method(WGMFEM)for second order elliptic equations on 2D domains with curved boundary.The Neumann boundary condition is considered since it becomes the essential boundary condition in this case.It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretizationwith polynomial order a>1.The purpose of this paper is two-fold.First,we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains,which exhibits an O(h^(1/2))convergence for all a≥1.It is a little surprising to see that even the lowest-order WG-MFEM(a=1)experiences a loss of accuracy.This is different from known results for the finite element method(FEM)or the mixed FEM,and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain.Second,we propose a remedy to bring the approximation rate back to optimal by employing two techniques.One is a specially designed boundary correction technique.The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes,which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements.Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all a.Numerical results further confirm this conclusion.