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A finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl., 1(4)(1980)217–224 (in Chinese)], which is the so-called nine point scheme on structured quadrilateral meshes. The scheme has both cell-centered unknowns and vertex unknowns which are usually expressed as a linear weighted interpolation of the cell-centered unknowns. The critical factor to obtain the optimal accuracy for the scheme is the reconstruction of vertex unknowns. However, when the mesh deformation is severe or the diffusion tensor is discontinuous, the accuracy of the scheme is not satisfactory, and the author hope to improve this scheme.

The authors propose an explicit weighted vertex interpolation algorithm which allows arbitrary diffusion tensors and does not depend on the location of discontinuity. Both the derivation of the scheme and that of vertex reconstruction algorithm satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The vertex interpolation algorithm can be easily extended to 3 D case.

Numerical results show that it maintain optimal convergence rates for the solution and flux on 2 D and 3 D meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous.

This paper proposes a linearity preserving and explicit weighted vertex interpolation algorithm for cell-centered finite volume approximations of diffusion equations on general grids. The proposed finite volume scheme with the new interpolation algorithm allows arbitrary continuous or discontinuous diffusion tensors; the final scheme is applicable to arbitrary polygonal grids, which may have concave cells or degenerate ones with hanging nodes. The final scheme has second-order convergence rate for the approximate solution and higher than first-order accuracy for the flux on 2 D and 3 D meshes. The explicit weighted interpolation algorithm is easy to implement in three dimensions in case that the diffusion tensor is continuous or discontinuous.

This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository.

The nonlinear two-point flux approximation is used to discretize the diffusion flux and has a fixed stencil. The cell-vertex unknowns are applied to define the auxiliary unknowns and can be interpolated by the cell-centered unknowns. The approximation of convection flux is based on the second-order upwind method with a slope limiter.

Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

A new positivity-preserving nonlinear finite volume scheme is proposed to simulate the far-field model used in the geological radioactive waste repository. Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

This paper aims to describe a recently proposed algorithm in terrain‐based cooperative UAV mapping of the unknown complex obstacle in a stationary environment where the complex obstacles are represented as curved in nature. It also aims to use an extended Kalman filter (EKF) to estimate the fused position of the UAVs and to apply the 2‐D splinegon technique to build the map of the complex shaped obstacles. The path of the UAVs are dictated by the Dubins path planning algorithm. The focus is to achieve a guaranteed performance of sensor based mapping of the uncertain environments using multiple UAVs.

An extended Kalman filter is used to estimate the position of the UAVs, and the 2‐D splinegon technique is used to build the map of the complex obstacle where the path of the UAVs are dictated by the Dubins path planning algorithm.

The guaranteed performance is quantified by explicit bounds of the position estimate of the multiple UAVs for mapping of the complex obstacles using 2‐D splinegon technique. This is a newly proposed algorithm, the most efficient and a robust way in terrain based mapping of the complex obstacles. The proposed method can provide mathematically provable and performance guarantees that are achievable in practice.

The paper describes the main contribution in mapping the complex shaped curvilinear objects using the 2‐D splinegon technique. This is a new approach where the fused EKF estimated positions are used with the limited number of sensors' measurements in building the map of the complex obstacles.

Results are given which establish a computational foundation for simplicial approximation and design centering of a convex body. A simplicial polyhedron is used to approximate the convex body and the “design center”, i.e. the point inside the body furthest in some norm from its exterior, is approximated by the point in the polyhedron furthest from its exterior. A point representation of the polyhedron is used, so that there is no necessity for computing or storing the faces of the approximation. Since in N space there can be factorially more faces than points, we are able to achieve significant efficiencies in both operation count and storage requirements, compared to previously reported methods. We give results for the 2 norm and the max norm, and demonstrate that our new method is operable in the nonconvex case, and can handle a mixed basis of faces and points as well.

A control‐volume based method for the numerical calculation of axisymmetric incompressible fluid flow and heat transfer is presented. The proposed method extends the staggered grid approach to unstructured triangular meshes. The velocities are stored at the vertices and the edges of a triangle, pressure and temperature are stored at the vertices. Accordingly, velocities are interpolated in a quadratic way, pressure and temperature linearly. The accuracy of the proposed method is examined for a number of different testproblems. Compared to a linear interpolation scheme implemented in the same code, more accurate solutions and smaller computation times are obtained for the proposed quadratic scheme. The method was designed for and is about to be applied to the numerical simulation of crystal growth.

Presents an interpolatory subdivision scheme to generate adaptively refined quadrilateral meshes which approximate a smooth surface of arbitrary topology. The described method differs significantly from classical mesh generation techniques based on spline surfaces or implicit representations since no explicit description of the limit surface is used. Instead, simple affine combinations are applied to compute new vertices if a face of the net is split. These rules are designed to guarantee asymptotic smoothness, i.e. the sequence of refined nets converges to a smooth limit surface. Subdivision techniques are useful mainly in applications where a given quadrilateral net is a coarse approximation of a surface and points on a refined grid have to be estimated. To evaluate the proposed approach, shows examples for FE‐computations on surfaces generated by this algorithm.

The purpose of this paper is to propose an automatic interpolation algorithm for robot spraying trajectories based on cubic Non-Uniform Rational B-Splines (NURBS) curves, to solve the problem of sparse and incomplete trajectory points of the head and heel of the shoe sole when extracting robot motion trajectories using structured-light 3D cameras and to ensure the robot joints move smoothly, so as to achieve a good effect of automatic spraying of the shoe sole with a 7-degree-of-freedom (DOF) robot.

Firstly, the original shoe sole edge trajectory position points acquired by the 3D camera are fitted with NURBS curves. Then, the velocity constraint at the local maximum of the trajectory curvature is used as the reference for curve segmentation and S-shaped acceleration and deceleration planning. Immediately, real-time interpolation is performed in the time domain to obtain the position and orientation of each point of the robot motion trajectory. Finally, the inverse kinematics of the anthropomorphic motion of the 7-DOF robot arm is used to obtain the joint motion trajectory.

The simulation and experiment prove that the shoe sole spraying trajectory is complete, the spraying effect is good and the robot joint movement is smooth, which show that the algorithm is feasible.

This study is of good practical value for improving the quality of automated shoe sole spraying, and it has wide applicability for different shoe sole shapes.

To retrieve the geometric information contained in CT images, a surface reconstruction method, which is based on the similarity between the corresponding contours of adjacent sections, is presented in this paper. The correspondence of the contours of adjacent sections is determined by incorporating the topological rules and overlaps of the convex hulls of the contours. Then, the similar vertices of the corresponding contours are matched using a two‐phase strategy, consisting of overall matching followed by local matching. Dissimilar portions are extracted to construct the triangulable spatial dissimilar polygons. Finally, triangular meshes interpolating the contours are obtained by triangulating the dissimilar polygons and similar portions separately. The reconstructed surface models can be used in rapid prototyping as well as visualization. Experimental results demonstrate the validity of the method in reconstructing the surface from severe dissimilar contours.

In the newly released ASTM standard specification for additive manufacturing file (AMF) format – version 1.1 – Hermite curve-based interpolation is used to refine input triangles to generate denser mesh with smoother geometry. This paper aims to study the problems of constructing smooth geometry based on Hermite interpolation on curves and proposes a solution to overcome these problems.

A formulation using triangular Bézier patch is proposed to generate smooth geometry from input polygonal models. Different configurations on the boundary curves in the formulation are analyzed to further enrich this formulation.

The study shows that the formulation given in the AMF format (version 1.1) can lead to the problems of inconsistent normals and undefined end-tangents.

The scheme has requirements on the input normals of a model, only C0 interpolation can be generated on those cases with less-proper input.

To overcome the problems of smooth geometry generation in the AMF format, the authors propose an enriched scheme for computing smooth geometry by using triangular Bézier patch. For the configurations with less-proper input, the authors adopt the Boolean sum and the Nielson’s point-opposite edge interpolation for triangular Coons patch to generate the smooth geometry as a C0 interpolant.

In GIS applications for a realistic representation of a terrain a great number of triangles are needed that ultimately increases the data size. For online GIS interactive programs it has become highly essential to reduce the number of triangles in order to save more storing space. Therefore, there is need to visualize terrains at different levels of detail, for example, a region of high interest should be in higher resolution than a region of low or no interest. Wavelet technology provides an efficient approach to achieve this. Using this technology, one can decompose a terrain data into hierarchy. On the other hand, the reduction of the number of triangles in subsequent levels should not be too small; otherwise leading to poor representation of terrain.

This paper proposes a new computational code (please see Appendix for the flow chart and pseudo code) for triangulated irregular network (TIN) using Delaunay triangulation methods. The algorithms have proved to be efficient tools in numerical methods such as finite element method and image processing. Further, second generation wavelet techniques popularly known as “lifting schemes” have been applied to compress the TIN data.

A new interpolation wavelet filter for TIN has been applied in two steps, namely splitting and elevation. In the splitting step, a triangle has been divided into several sub‐triangles and the elevation step has been used to “modify” the point values (point coordinates for geometry) after the splitting. Then, this data set is compressed at the desired locations by using second generation wavelets.

A new algorithm for second generation wavelet compression has been proposed for TIN data compression. The quality of geographical surface representation after using proposed technique is compared with the original terrain. The results show that this method can be used for significant reduction of data set.