Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes

A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulation.This work is a continuation of our investigation of the SFWG method for the biharmonic equation.The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular grids.This new method also keeps the formulation that is symmetric,positive definite,and stabilizer-free.Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)norm.Superconvergence of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation polynomial.The postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal order.Numerical examples are tested to verify the theor ies.

A Novel VOF-Type Volume-Tracking Method for Free-Surface Flows Based on Unstructured Triangular Mesh

A novel VOF-type volume-tracking method for two-dimensional free-surface flows based on the unstructured triangular mesh is presented. Owing to the inherent merit of the unstructured triangular mesh in fitting curved boundaries, this method can handle the free-surface problems with complex geometries accurately and directly, without introducing any complicated boundary treatment or artificial diffusion. The method solves the volume transport equation geometrically through the Modified Lagrangian-Eulerian Re-map （MLER） method, which is applied to advective fluid volumes. Moreover, the PLIC method is adopted to give a second-order reconstructed interface approximation. To validate this method, two advection tests were performed for the establishment of the accuracy and convergence rate of the solutions. Numerical results for these complex tests provide convincing evidence for the excellent solution quality and fidelity of the method.

Algorithm of Detecting and Filling Small Holes in Triangular Mesh Surface

In some scattered point cloud triangular mesh restoration algorithm, small triangular mesh holes problem will often affect the quality of the model. For small holes at the details, this paper propose a method for identifying and extracting hollow edge,and use a triangle growth way based on boundary edge angle to fill the empty void. First, according the relationship of the point, side and face of the triangle mesh model to identify the hole, then extracting the holes boundary edge and classifying it. Finally, using a triangle growth method based on holes boundary edge angle to fill each small holes separated from the boundary. Compared with other algorithm of filling holes, this method is high efficiency for small holes of smooth surface,and itimprovesthe quality of the triangular mesh model.

Improving the Quality of Conforming Triangular Meshes by Topological Clean up and DSI

A new method as a post-processing step is presented to improve the shape quality of triangular meshes, which uses a topological clean up procedure and discrete smoothing interpolate (DSI) algorithm together. This method can improve the angle distribution of mesh element. while keeping the resulting meshes conform to the predefined constraints which are inputted as a PSLG.

Fast view prediction for stereo images based on Delaunay triangular mesh model

The view prediction is an important step in stereo/multiview video coding, wherein, disparity estil mation （DE） is a key and difficult operation. DE algorithms usually require enormous computing power. A fast DE algorithm based on Delaunay triangulation （DT） is proposed. First, a flexible and content adaptive DT mesh is established on a target frame by an iterative split-merge algorithm. Second, DE on DT nodes are performed in a three-stage algorithm, which gives the majority of nodes a good estimate of the disparity vectors （DV）, by removing unreliable nodes due to occlusion, and forcing the minority of ＇problematic nodes＇ to be searched again, within their umbrella-shaped polygon, to the best. Finally, the target view is predicted by using affine transformation. Experimental results show that the proposed algorithm can give a satisfactory DE with less computational cost.

Curvatures estimation on triangular mesh

Curvatures are important geometric attributes of surfaces. There are many applications that require as a first step the accurate estimation of curvatures at arbitrary vertices on a triangulated surface. Chen and Schmitt (1992) and Taubin (1995) presented two simple methods to estimate principal curvatures. They used circular arcs to approximate the normal curvature. We find this may cause large error in some cases. In this paper, we describe a more accurate method to estimate the normal curvature, and present a novel algorithm to estimate principal curvatures by simplifying the Chen and Schmitt’s method. Some comparison results are also shown in this paper.

An Algorithm of Neighbor Finding on Sphere Triangular Meshes with Quaternary Code

第四级的代码的特征被分析。区分三角方向的规则被给出。邻居由从回到前面分解第四级的代码发现的一个算法在这篇论文被介绍。在这个算法和 Bartholdi 之间的时间复杂性的对比的分析是算法被接近。结果说明一般水准消费了这个算法的时间，这是大约 23.66% Bartholdi 的算法。

Method of triangular mesh modeling in numerical control machining simulation

In the process of numerical control machining simulation,the workpiece surface is usually described with the uniform triangular mesh model.To alleviate the contradiction between the simulation speed and accuracy in this model,two improved methods,i.e.,the local refinement triangular mesh modeling method and the adaptive triangular mesh modeling method were presented.The simulation results show that when the final shape of the workpiece is known and its mathematic representation is simple,the local refinement triangular mesh modeling method is preferred;when the final shape of the workpiece is unknown and its mathematic description is complicated,the adaptive triangular mesh modeling method is more suitable.The experimental results show that both methods are more targeted and practical and can meet the requirements of real-time and precision in simulation.

PARAMETERIZATION OF TRIANGULAR MESHES

A new algorithm called the weighted least square discrete parameterization (WLSDP) is presented for the parameterization of triangular meshes over a convex planar region. This atgorithm is the linear combination of the discrete Conformal mapping (DCM) and the discrete Authalic mapping (DAM). It provides the good properties of both DCM and DAM, such as robustness and low distortion. By adjusting the scaling factor q embedded in the WLSDP, satisfactory parameterizations in different special applications can be achieved.

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

In this paper,a new multi-resolution weighted essentially non-oscillatory(MR-WENO)limiter for high-order local discontinuous Galerkin(LDG)method is designed for solving Navier-Stokes equations on triangular meshes.This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes.Such new limiter uses information of the LDG solution essentially only within the troubled cell itself,to build a sequence of hierarchical L^(2)projection polynomials from zeroth degree to the highest degree of the LDGmethod.As an example,a third-order LDGmethod with associated same orderMR-WENO limiter has been developed in this paper,which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities.The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one.This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell.This MR-WENO limiter is very simple to construct,and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes.Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes.Some classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

A Coercivity Result of Quadratic Finite Volume Element Schemes over Triangular Meshes

In this work,we study the coercivity of a family of quadratic finite volume element(FVE)schemes over triangular meshes for solving elliptic boundary value problems.The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as H^(1) and L^(2) error estimates.The novelty of this paper is that,each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts:the first part is the element stiffness matrix of the standard quadratic finite element method(FEM),the second part is the difference between the FVE and FEM on the element boundary,while the third part can be expressed as the tensor product of two vectors.As a result,we reach a sufficient condition to guarantee the existence,uniqueness and coercivity result of the FVE solution on general triangular meshes.Moreover,based on this sufficient condition,some minimum angle conditions with simple,analytic and computable expressions are obtained.By comparison,the existing minimum angle conditions were obtained numerically from a computer program.Theoretical findings are conformed with the numerical results.

A NUMERICAL MODEL OF FLOW FIELD AND WATER QUALITY FOR TIDAL REGIONS BASED ON TRIANGULAR MESHES

In tidal areas, natural land boundary is complex and underwater topographyvaries acutely due to influence of upstream runoff and outer tide. The simulation and forecast ofwater current and mass transport play an important role in practical engineering. According to thesituation of irregular natural boundaries in tidal region, unsturctured triangular grid arrangementis applied to suit for complex conditions. A finite difference method with alternating directionalimplicit scheme for triangular grid is established in this paper. The model has been applied incalculation of flow and concentration fields for Nantong reach of the Yangtze River. It is satisfiedthat the calculated values are in agreement with observed data.

A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO(SWENO)scheme.The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism.The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENOscheme efficiently.Besides,a scaling technique is used to circument the growth of condition numbers as mesh refined.However,weak oscillations still appear when the SWENO scheme deals with complex low density equations.In order to guarantee the maximum-principle-preserving(MPP)property,we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy.A novel procedure is designed to prove this property theoretically.Finally,numerical examples for one-and two-dimensional problems are presented to verify the good performance,maximum principle preserving,essentially non-oscillation and high resolution of the proposed scheme.

High Order Cubic-Polynomial Interpolation Schemes on Triangular Meshes

The Cubic-Polynomial Interpolation scheme has been developed and applied to many practical simulations.However,it seems the existing Cubic-Polynomial Interpolation scheme are restricted to uniform rectangular meshes.Consequently,this scheme has some limitations to problems in irregular domains.This paper will extend the Cubic-Polynomial Interpolation scheme to triangular meshes by using some spline interpolation techniques.Numerical examples are provided to demonstrate the accuracy of the proposed schemes.

A spherical parameterization approach based on symmetry analysis of triangular meshes

We present an efficient spherical parameterization approach aimed at simultaneously reducing area and angle dis-tortions. We generate the final spherical mapping by independently establishing two hemisphere parameterizations. The essence of the approach is to reduce spherical parameterization to a planar problem using symmetry analysis of 3D meshes. Experiments and comparisons were undertaken with various non-trivial 3D models, which revealed that our approach is efficient and robust. In particular, our method produces almost isometric parameterizations for the objects close to the sphere.

Adaptive triangular mesh coarsening with centroidal Voronoi tessellations

We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on centroidal Voronoi tessellations (CVTs). The accuracy and good quality of the output triangular mesh are achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.

基于改进MeshCNN的机械零件加工特征识别方法

为了解决传统CAPP系统中特征识别效率低且智能化程度不高的问题,将原始的MeshCNN与Faster RCNN相结合,提出一种基于Mesh-Faster RCNN的机械零件加工特征识别方法,该方法通过将自定义的加工数据集作为神经网络的输入端,获得最优的神经网络模型;然后利用MBD技术对加工模进行标注,通过PMI的信息标注获取待加工特征,将其转化为三角网格数据;在此基础上结合三角网格数据处理的算法,将处理好的加工特征数据导入最优神经网络模型中,完成特征识别过程。最后以某型号船用柴油机部分关键件为例,验证了所提方法的可行性和有效性。

基于Transformer的三角形网格分类分割网络

三角形网格是一种重要的几何数据结构,能有效地表达三维模型的形状细节,但三角形网格面元素的分布并不规则,因此将现有的深度神经网络直接应用到网格上较为困难。针对三角形网格不规则的结构问题,直接将网格的面作为Token,提出一种将Transformer应用于三角形网格的深度神经网络。首先,将面的重心坐标或谱域特征作为位置信息,融合其内蕴特征作为输入特征,并对输入特征位置嵌入;其次,利用自注意力模块提取全局特征,利用面卷积模块提取局部特征,以增强网络局部特征的提取能力;最后,融合局部特征和全局特征构建应用于三角形网格上的分类和分割深度神经网络。在SHREC分类数据集和COSEG分割数据集上的实验结果表明,该方法准确率较高且可以有效地提升训练速度。

保留STL模型特征细节的3D打印自适应分层算法

针对传统的3D打印自适应分层算法存在计算过程复杂、无法消除阶梯效应影响、不能保留模型特征以及特征容易丢失等问题,提出了一种改进的保留STL模型特征细节的自适应分层算法。该算法首先判断模型当前层是否含有显著特征边;然后针对当前层含有显著特征边的情况,通过相邻2个三角网格法矢量的夹角来适应性确定层厚,对不含显著特征边的情况,利用三角网格法矢量与成型方向的夹角来自适应确定层厚,对于模型存在特征偏移的情况,则通过相邻切平面封闭曲线个数变化来适应性地确定层厚;最后将上述3种情况下自适应层厚的最小值作为当前分层的依据。研究结果表明:对于所采用的恐龙模型,所提算法不仅很好地保留了模型的细节特征,而且显著提升了模型成型质量;与基于三角面片法的矢量自适应分层算法和仅保留模型特征的自适应分层算法相比,所提算法的模型精度分别提升了28.6%和31.7%。该算法减小了阶梯效应以及模型特征丢失和偏移对打印质量的影响,提高了模型的打印精度。