Regular Triangular Mesh Research Articles

Construction of a modified butterfly subdivision scheme with [formula omitted]-smoothness and fourth-order accuracy

This article presents a modified butterfly subdivision scheme with improved smoothness over regular triangular meshes. The proposed technique is an approximating scheme with a tension parameter. It achieves fourth-order accuracy and generates C2 limit surfaces for a suitable range of the parameter while maintaining the same support of the original butterfly scheme. To validate the theoretical results, some numerical examples are provided.

A computation design method for architectural artifacts adapted from traditional Kagome bamboo basketry techniques

This study is a digital form-finding and manual fabrication experiment in woven architectural design, with one traditional weaving style, Kagome, used to scale the craft up into an architectural-scale bamboo woven artifact. Kagome is a trihexagonal pattern employed in traditional bamboo basketry as a triaxial weaving system, resulting in an object with a self-bracing capacity without the use of fasteners owing to its interlacing lattices. Although existing studies and tools have addressed triaxial weaving design and representation, the current consideration of the advantages of weaving with bamboo is insufficient. To address this research gap, this study develops a computational design method based on studies on bamboo basketry. This allows for the representation and exploration of design geometries using combinations of regular triangular meshes for the fabrication of Kagome woven bamboo artifacts. A full-scale mock-up was fabricated to evaluate the effectiveness of the method. The mock-up demonstrated the self-bracing properties of Kagome, but there were discrepancies between the mock-up and the design. Factors affecting bamboo weaving on an architectural scale have been identified within this study to inform future research on woven bamboo structures.

Dr. KID: Direct Remeshing and K-Set Isometric Decomposition for Scalable Physicalization of Organic Shapes.

Dr. KID is an algorithm that uses isometric decomposition for the physicalization of potato-shaped organic models in a puzzle fashion. The algorithm begins with creating a simple, regular triangular surface mesh of organic shapes, followed by iterative K-means clustering and remeshing. For clustering, we need similarity between triangles (segments) which is defined as a distance function. The distance function maps each triangle's shape to a single point in the virtual 3D space. Thus, the distance between the triangles indicates their degree of dissimilarity. K-means clustering uses this distance and sorts segments into k classes. After this, remeshing is applied to minimize the distance between triangles within the same cluster by making their shapes identical. Clustering and remeshing are repeated until the distance between triangles in the same cluster reaches an acceptable threshold. We adopt a curvature-aware strategy to determine the surface thickness and finalize puzzle pieces for 3D printing. Identical hinges and holes are created for assembling the puzzle components. For smoother outcomes, we use triangle subdivision along with curvature-aware clustering, generating curved triangular patches for 3D printing. Our algorithm was evaluated using various models, and the 3D-printed results were analyzed. Findings indicate that our algorithm performs reliably on target organic shapes with minimal loss of input geometry.

Numerical convergence of the family of flux-continuous schemes with variable quadrature (qu1,qu2) for single phase flow in porous media

Finite-volume schemes, which honor pressure and flux-continuity conditions, is developed using double quadrature (qu1,qu2), referred as double family scheme. The scheme is applicable to solve the elliptic pressure equation used in reservoir simulation. Schemes are applicable on both regular cartesian and unstructured triangular meshes. The scheme is defined over a control-volume distributed formulation. The scheme can be applied to both diagonal and full permeability tensor elliptic pressure equation with discontinuous coefficients. The scheme removes the first order errors, which are introduced by standard reservoir simulation schemes when applied to full tensor flow. The scheme is quantified with help of a quadrature rule. When the scheme is applied to highly heterogeneous and anisotropic porous media it does not honor maximum principle resulting in unstable solution with oscillatory behavior. The numerical solution is termed non-monotonicity for high anisotropy ratios with results showing oscillations in the numerical pressure solution. In this paper a double (qu1,qu2) quadrature flux continuous schemes is presented, which with specific choice of quadrature (qu1,qu2) helps in improved stability of the numerical solutions. Numerical convergence of the scheme is also demonstrated with help of a number of numerical test cases and schemes impact on monotonicity behavior is also demonstrated with numerical examples.

On divergence- and gradient-preserving coarse-graining for finite volume primitive equation ocean models

We consider the problem of coarse-graining in the context of finite-volume fluid models. If a variable is defined on a high-resolution grid it may be coarse-grained so that it is defined on a grid of lower resolution. In general this will cause some information about the variable to be lost. In particular, horizontal divergences, gradients or other operators calculated on the coarse grid after projecting may differ from those calculated on the fine grid. In some cases we are able to choose averaging weights for coarse-graining such that the coarse-grid operators will give a result approximating that of the corresponding fine-grid operators applied on the fine grid. In this work we derive general conditions on the averaging weights that allow the divergence and gradient to be preserved. These conditions are applied to a regular triangular mesh with B-grid variable placement in which the fine-grid resolution is some integer multiple N of the coarse-grid resolution. For this case we find particular values for the averaging weights that preserve the divergence, and a set of different averaging weights that preserve the gradient. We observe that the vertical component of the curl is also preserved by the same coarse-graining that preserves divergence. These coarse-grainings are applied to data from FESOM2 simulations and we demonstrate that using the particular coarse-grainings derived herein gives an overall reduction in L1 error when compared with other methods.

Steady seepage analysis in soil-rock-mixture slope using the numerical manifold method

The NMM (numerical manifold method) was widely adopted to deal with many types of engineering problems involving moving boundaries, such as multiple crack propagation in rock masses. With regular triangular mathematical meshes, the NMM is further developed for two typical steady seepage problems, such as the confined seepage analysis and the unconfined seepage analysis. For unconfined seepage analysis, procedures to determine free surface location are introduced, and incorporated into the NMM code. Several classic numerical examples with analytical or reference solutions, and two challenge examples including a soil-rock-mixture (SRM) dam and a SRM slope are adopted to confirm capability and accuracy of the proposed numerical model for steady seepage problems. The results show that the exit gradient and free surface for steady seepage problems can be accurately calculated with the proposed numerical model. In addition, the existence of rock blocks have an impact to the distribution pattern of predicted contours of water head. More importantly, the proposed numerical model is able to calculate locations of free surface in SRM dam and SRM slope, which is very important in stability analysis of SRM dam and slope.

Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct

A finite-difference spatial discretization scheme that preserves the port-Hamiltonian structure of infinite dimensional systems governed by the wave equation is proposed. The scheme is based on the use of staggered grids for the discretization of different variables of the system. The discretization is given in 2D for rectilinear and regular triangular meshes. The proposed method is completed with the midpoint rule for time integration and numerical results are provided, including considerations for interconnection and closed loop behaviors and isotropy comparison between the proposed meshes.

차분진화알고리즘을 이용한 네트워크 돔의 정삼각형 메쉬 모델링에 관한 연구

This paper aimed at modeling a triangular mesh for network dome by using differential evolution algorithm. For this purpose, an optimization process to find a regular triangular mesh on the curved surface was proposed and a program for this was developed. The objective function consisted of the standard deviations of the area and the boundary length of each triangle, and the design variables were regulated by the upper and lower boundary conditions which were calculated depending on the nodal connectivity. Plate model and triangular network dome model, which initially had consisted of irregular triangular meshes, were selected as target examples. In the case of the plate model its final mesh corresponded with the regular triangular mesh exactly. In network dome model, its result was very approached to the minimum standard deviation of the area and theboundary length of each triangle, too. Additionally, the more rise-span ratio of the dome decreased, the more the standard deviation of the mesh modeling decreased.

BIORTHOGONAL WAVELETS WITH SIX-FOLD AXIAL SYMMETRY FOR HEXAGONAL DATA AND TRIANGLE SURFACE MULTIRESOLUTION PROCESSING

This paper discusses the construction of highly symmetric compactly supported wavelets for hexagonal data/image and triangle surface multiresolution processing. Recently, hexagonal image processing has attracted attention. Compared with the conventional square lattice, the hexagonal lattice has several advantages, including that it has higher symmetry. It is desirable that the filter banks for hexagonal data also have high symmetry which is pertinent to the symmetric structure of the hexagonal lattice. The high symmetry of filter banks and wavelets not only leads to simpler algorithms and efficient computations, it also has the potential application for the texture segmentation of hexagonal data. While in the field of computer-aided geometric design (CAGD), when the filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms for regular vertices have high symmetry, which make it possible to design the corresponding multiresolution algorithms for extraordinary vertices. In this paper we study the construction of six-fold axial symmetric biorthogonal filter banks and the associated wavelets, with both the dyadic and [Formula: see text]-refinements. The constructed filter banks have the desirable symmetry for hexagonal data processing. By associating the outputs (after one-level multiresolution decomposition) appropriately with the nodes of the regular triangular mesh with which the input data is associated (sampled), we represent multiresolution analysis and synthesis algorithms as templates. The six-fold axial symmetric filter banks constructed in this paper result in algorithm templates with desirable symmetry for triangle surface processing.

CFL Conditions for Finite Integration Methods on Triangular Meshes

In this paper, the CFL conditions of several 2-D finite integration (FI) methods on regular triangular meshes are derived. A piecewise uniform element allows a time step that is twice as large as that for the Whitney element.

Analysis of optimal bivariate symmetric refinable Hermite interpolants

Multivariate refinable Hermite interpolants with high smoothness and small support are of interest in CAGD and numerical algorithms. In this article, we are particularly interested in analyzing some univariate and bivariate symmetric refinable Hermite interpolants, which have some desirable properties such as short support, optimal smoothness and spline property. We shall study the projection method for multivariate refinable function vectors and discuss some properties of multivariate spline refinable function vectors. Here a compactly supported multivariate spline function on $\mathbb R^s$ just means a function of piecewise polynomials supporting on a finite number of polygonal partition subdomains of $\mathbb R^s$. We shall discuss spline refinable function vectors by investigating the structure of the eigenvalues and eigenvectors of the transition operator. To illustrate the results in this paper, we shall analyze the optimal smoothness and spline properties of some univariate and bivariate refinable Hermite interpolants. For the regular triangular mesh, we obtain a bivariate $C^2$ symmetric dyadic refinable Hermite interpolant of order $2$ whose mask is supported inside $[-1,1]^2$.

DIAGRAMMATIC TOOLS FOR GENERATING BIORTHOGONAL MULTIRESOLUTIONS

Elsewhere we have introduced a construction to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work presents the construction for regular meshes using diagrams (stencils, masks) and interactions between diagrams to replace matrices and matrix multiplication. Regular triangular meshes with butterfly subdivision and a variant of Loop subdivision due to Litke, et al. are used as examples.

Optimal $C^2$ two-dimensional interpolatory ternary subdivision schemes with two-ring stencils

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Holder smoothness exponent of its basis function cannot exceed log 3 11(≈ 2.18266), where the critical Holder smoothness exponent of a function f: R 2 R is defined to be v ∞ (f):= sup{v: f ∈ Lipv}. On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Holder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound log 3 11. Consequently, we obtain optimal smoothest C 2 interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the l p -norm joint spectral radius.

Complete Multiple Description Mesh-Based Video Coding Scheme and Its Performance

This paper proposes a multiple description (MD) mesh-based motion coding method, which generates two descriptions for mesh-based motion by subsampling the nodes of a right-angled triangular mesh and dividing them into two groups. Motion vectors associated with the mesh nodes in each group are transmitted over distinct channels. With the nodes in each group, two other regular triangular meshes besides the original one can be constructed, and three different prediction images can be reconstructed according to descriptions available. The proposed MD mesh-based motion coding method is then combined with the pairwise correlating transform (PCT), and a complete MD video coding scheme is proposed. Further measures are taken to reduce the mismatch between the encoder and decoder that occurs when only one description is received and the decoder reconstruction is different from the encoder. The performance of the proposed scheme is evaluated using computer simulations, and the results show, compared to Reibman's MD transform coding (MDTC) method, the proposed scheme achieves better redundancy rate distortion (RRD) performance. In packet loss scenario, the proposed scheme outperforms the MDTC method.

A COMPUTATIONAL-RAM (C-RAM) ARCHITECTURE FOR REAL-TIME MESH-BASED VIDEO MOTION TRACKING PART 1: MOTION ESTIMATION

This paper presents a new Computational-RAM (C-RAM) architecture for real-time mesh-based video motion tracking. The motion tracking consists of two operations: mesh-based motion estimation and compensation. The proposed motion estimation architecture is presented in Part 1 and the proposed motion compensation architecture is presented in Part 2. The motion estimation architecture stores two frames and computes motion vectors for a regular triangular mesh structure as defined by MPEG-4 Part 2.1 The motion estimation architecture uses the block-matching algorithm (BMA) to estimate the vertical and horizontal motion vectors for each mesh node. Parallel and pipelined implementations have been used to overcome the huge computational requirements of the motion estimation process. The two frames are stored in embedded S-RAMs generated with Virage™ Memory Compiler. The proposed motion estimation architecture has been prototyped, simulated and synthesized using the TSMC 0.18 μm CMOS technology. At 100 MHz clock frequency, the proposed architecture processes one CIF video frame (i.e., 352×288 pixels) in 1.48 ms, which means it can process up to 675 frames per second. The core area of the proposed motion estimation architecture is 24.58 mm2 and it consumes 46.26 mW.

On application of two superconvergent recovery procedures to plate problems

AbstractIn this paper a study is performed on application of two recovery methods, i.e. superconvergent patch recovery (SPR) and the recovery by equilibrium of patches (REP), to plate problems. The two recovery methods have been recognized to give similar results in adaptive solutions of two dimensional stress problems. While the former applies a least square fit over a set of values at the so called superconvergent points, the latter does not need any knowledge of such points and thus has a wider application especially in non‐linear problems. The formulation of REP is extended to Reissner–Mindlin plate problems. The convergence rates of the recovered fields of the gradients obtained from application of the two methods are compared using series of regular triangular and rectangular meshes for thick and thin plate solution cases. Assumed strain formulation based elements, i.e. the elements formulated by mixed interpolation of tensorial components, as well as conventional from of elements based on selective integration schemes are employed for the study.In order to investigate the possibility of any improvement in the results by adding equilibrium constraints to SPR, as some authors suggest for simple two‐dimensional problems, some weighted forms of such conditions are designed and added to the formulation. Comprehensive study has been given first by varying the weight terms to obtain the best enhanced results and then using the optimal values to investigate the effects of the constraints on the rate of convergence. It is observed that despite of the cost of this approach, due to the coupling of the gradient terms, no significant improvement is achieved. Copyright © 2004 John Wiley & Sons, Ltd.

INTER-FRAME PREDICTION OF MEDICAL AND VIDEOPHONE SEQUENCES: A DEFORMABLE TRIANGLE-BASED APPROACH

Motion compensation using deformable triangle patches has been successfully used for low bit rate coding of videophone sequences. They were also shown to be particularly efficient for interframe coding of MRI sequences, for which the difference between image slices can be well modeled by locally affine deformations. Regular triangular meshes were used in previous works. In this paper, we present a quadtree decomposition algorithm to generate a triangle mesh for which smaller triangles are used in image areas where the motion or deformation is more complex. Grid points are recursively added to areas where the reduction in prediction error is more significant. Results show that using variable size triangular patches increases the PSNR of the motion-compensated image while reducing the number of grid points when compared to a regular triangular mesh.

The P1mod Element: A New Nonconforming Finite Element for Convection-Diffusion Problems

We consider a nonconforming streamline diffusion finite element method for solving convection-diffusion problems. The loss of the Galerkin orthogonality of the streamline diffusion method when applied to nonconforming finite element approximations results in an additional error term which cannot be estimated uniformly with respect to the perturbation parameter for the standard piecewise linear or rotated bilinear elements. Therefore, starting from the Crouzeix--Raviart element, we construct a modified nonconforming first order finite element space on shape regular triangular meshes satisfying a patch test of higher order. A rigorous error analysis of this P1mod element applied to a streamline diffusion discretization is given.The numerical tests show the robustness and the high accuracy of the new method.

A Novel Video Coding Scheme Using Delaunay Triangulation

In this paper, a novel triangle-based video coding scheme for very low bit-rate transmission is presented. In the scheme, regular triangular mesh as used in Brusewitz [6] and Nakaya [7] is replaced by adaptive DT mesh generated by a split–merge approach, and then a new method based on the umbrella matching algorithm (UMA) is developed to refine the coarse motion vectors of DT's vertices derived from a BMA. Computer simulations are carried out to evaluate the coding efficiency at 16 kbps. The results show that the quality of the decoded images is improved greatly compared to BMA or other triangle-based methods.

Multigrid solution of the Navier-Stokes equations on triangular meshes

A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.