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.

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.

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 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.

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.

Enhanced optimal delaunay triangulation methods with connectivity regularization

In this paper,we study the underlying properties of optimal Delaunay triangulations(ODT)and propose enhanced ODT methods combined with connectivity regularization.Based on optimizing node positions and Delaunay triangulation iteratively,ODT methods are very effective in mesh improvement.This paper demonstrates that the energy function minimized by ODT is nonconvex and unsmooth,thus,ODT methods suffer the problem of falling into a local minimum inevitably.Unlike general ways that minimize the ODT energy function in terms of mathematics directly,we take an outflanking strategy combining ODT methods with connectivity regularization for this issue.Connectivity regularization reduces the number of irregular nodes by basic topological operations,which can be regarded as a perturbation to help ODT methods jump out of a poor local minimum.Although the enhanced ODT methods cannot guarantee to obtain a global minimum,it starts a new viewpoint of minimizing ODT energy which uses topological operations but mathematical methods.And in terms of practical effect,several experimental results illustrate the enhanced ODT methods are capable of improving the mesh furtherly compared to general ODT methods.

Tool path planning based on conformal parameterization for meshes

The similarity property of conformal parameterization makes it able to locally preserve the shapes between a surface and its parameter domain, as opposed to common parameterization methods. A parametric tool path planning method is proposed in this paper through such parameterization of triangular meshes which is furthermore based on the geodesic on meshes. The parameterization has the properties of local similarity and free boundary which are exploited to simplify the formulas for computing path parameters, which play a fundamentally important role in tool path planning, and keep the path boundary-conformed and smooth. Experimental results are given to illustrate the effectiveness of the proposed methods, as well as the error analysis.

NATURAL SUPERCONVERGENT POINTS OF EQUILATERAL TRIANGULAR FINITE ELEMENTS - A NUMERICAL EXAMPLE

A numerical test case demonstrates that the Lobatto and the Gauss points are not natural superconvergent points of the cubic and the quartic finite elements under equilateral triangular mesh for the Poisson equation.

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper,we propose a high-order accurate discontinuous Galerkin(DG)method for the compressible Euler equations under gravitationalfields on un-structured meshes.The scheme preserves a general hydrostatic equilibrium state and provably guarantees the positivity of density and pressure at the same time.Compar-ing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes,the extension to triangular meshes is conceptually plausible but highly nontrivial.Wefirst introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells,which plays an important role in the well-balanced and positivity-preserving properties.One main challenge is that the well-balanced schemes may not have the weak positivity property.In order to achieve the well-balanced and positivity-preserving properties simultaneously while maintaining high-order accuracy,we carefully design DG spa-tial discretization with well-balanced numericalfluxes and suitable source term ap-proximation.For the ideal gas,we prove that the resulting well-balanced scheme,cou-pled with strong stability preserving time discretizations,satisfies a weak positivity property.A simple existing limiter can be applied to enforce the positivity-preserving property,without losing high-order accuracy and conservation.Extensive one-and two-dimensional numerical examples demonstrate the desired properties of the pro-posed scheme,as well as its high resolution and robustness.

A clipping algorithm for real-scene 3D models

The development of unmanned aerial vehicle(UAV)oblique photogrammetric technology provides a good foundation for the rapid construction of large-scale and high-definition real-scene 3D models.However,due to the limitations of the modeling process,irrelevant feature data cannot be eliminated in the modeling stage.The built models contain irrelevant features and model distortions caused by errors.At present,most existing clipping algorithms cannot effectively clip real-scene 3D models that are organized as a whole or with levels of detail(LODs).Therefore,this paper proposes a novel algorithm for clipping real-scene 3D models from any perspective based on clipping boundary lines thatfit the surfaces of the models.The results of the clipping experiments for 3D models constructed with oblique UAV images show that this algorithm can effectively clip any part of the 3D models,that the clipping results of each level model closelyfit the corresponding clipping boundary lines,and that the accuracy of the clipping results is very high.Additionally,the time complexity of the algorithm is O(n^(2)).In conclusion,the algorithm proposed in this paper provides correct and effective clipping results for real-scene 3D models with LODs that are constructed with photogrammetric or 3D laser scanning data.

Automatic Mesh Generation on a Regular Background Grid

This paper presents an automatic mesh generation procedure on a 2D domain based on a regular background grid. The idea is to devise a robust mesh generation scheme with equal emphasis on quality and efficiency. Instead of using a traditional regular rectangular grid, a mesh of equilateral triangles is employed to ensure triangular element of the best quality will be preserved in the interior of the domain. As for the boundary, it is to be generated by a node/segment insertion process. Nodes are inserted into the background mesh one by one following the sequence of the domain boundary. The local structure of the mesh is modified based on the Delaunay criterion with the introduction of each node. Those boundary segments, which are not produced in the phase of node insertion, will be recovered through a systematic element swap process. Two theorems will be presented and proved to set up the theoretical basic of the boundary recovery part. Examples will be presented to demonstrate the robustness and the quality of the mesh generated by the proposed technique.