In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state pr...In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.展开更多
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 ...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.展开更多
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 Schmi...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.展开更多
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 parameteri...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.展开更多
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...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.展开更多
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 th...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.展开更多
The characteristic of Quaternary codes is analyzed. The rule of distinguishing triangle direction is given out. An algorithm of neighbor finding by decomposing the Quaternary code from back to front is presented in th...The characteristic of Quaternary codes is analyzed. The rule of distinguishing triangle direction is given out. An algorithm of neighbor finding by decomposing the Quaternary code from back to front is presented in this paper. The contrastive analysis of time complexity between this algorithm and Bartholdi＇s algorithm is approached. The result illustrates that the average consumed time of this algorithm is about 23.66% of Bartholdi＇s algorithm.展开更多
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 iden...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.展开更多
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 alg...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.展开更多
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. T...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.展开更多
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 formulat...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.展开更多
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 algorithm is the linear combination of th...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 algorithm 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.展开更多
The continuous P_(3)and discontinuous P_(2)finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations.By putting two diagonal lines into every quadrilateral of a qua...The continuous P_(3)and discontinuous P_(2)finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations.By putting two diagonal lines into every quadrilateral of a quadrilateral mesh,we get a subquadrilateral triangular mesh.Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity.Numerical examples show an advantage of such a method over the Taylor-Hood P_(3)-P_(2)method,where the latter deteriorates when the viscosity becomes small.展开更多
基金supported by the NSFC Grant No.11872210 and Grant No.MCMS-I-0120G01Chi-Wang Shu:Research is supported by the AFOSR Grant FA9550-20-1-0055 and the NSF Grant DMS-2010107Jianxian Qiu:Research is supported by the NSFC Grant No.12071392.
文摘In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.
文摘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.
基金Project supported by the National Natural Science Foundation of China (No. 10371110) and the National Basic Research Program (973)of China (No. 2004CB318000)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 60673006 and 60533060)the Program for New Century Excellent Talents in University (No. NCET-05-0275), Chinathe IDeA Network of Biomedical Research Excellence Grant (No. 5P20RR01647206) from National Institutes of Health (NIH), USA
文摘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.
基金supported by the National Natural Science Foundation of China (No. 60773179)the National Basic Research Program (973) of China (No. 2004CB318000)
文摘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.
基金Project(60772089) supported by the National Natural Science Foundation of ChinaProject(20080440939) supported by the China Postdoctoral Science Foundation
文摘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.
基金Supported by the Natural Science Foundation of China （No. 40771169 No.40471108 No.40701152）.
文摘The characteristic of Quaternary codes is analyzed. The rule of distinguishing triangle direction is given out. An algorithm of neighbor finding by decomposing the Quaternary code from back to front is presented in this paper. The contrastive analysis of time complexity between this algorithm and Bartholdi＇s algorithm is approached. The result illustrates that the average consumed time of this algorithm is about 23.66% of Bartholdi＇s algorithm.
文摘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.
基金supported by the National Natural Science Foundation of China (60472083 60872141)
文摘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.
文摘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.
文摘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.
文摘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 algorithm 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.
文摘The continuous P_(3)and discontinuous P_(2)finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations.By putting two diagonal lines into every quadrilateral of a quadrilateral mesh,we get a subquadrilateral triangular mesh.Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity.Numerical examples show an advantage of such a method over the Taylor-Hood P_(3)-P_(2)method,where the latter deteriorates when the viscosity becomes small.
