A Subdivision-based Geometric Multigrid Method

In this paper we introduce a smooth subdi- vision theory-based geometric multigrid method. While theory and efficiency of geometric multigrid methods rely on grid regularity, this requirement is often not directly fulfilled in applications where partial differential equations are defined on complex geometries. Instead of generating multigrid hierarchies with classical linear refinement, we here propose the use of smooth subdivision theory for automatic grid hierarchy regularization within a geometric multigrid solver. This subdivi- sion multigrid method is compared to the classical geometric multigrid method for two benchmark problems. Numerical tests show significant improvement factors for iteration numbers and solve times when comparing subdivision to classical multigrid. A second study fo- cusses on the regularizing effects of surface subdivision refinement, using the Poisson-Nernst-Planck equations. Subdivision multigrid is demonstrated to outperform classical multigrid.

A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution

A simplest way is introduced to generate a generalized algorithm of univariate and bivariate subdivision schemes. This generalized algorithm is based on the symbol of uniform B-splines subdivision schemes and probability generating function of Binomial distribution. We present a family of binary approximating subdivision schemes which has maximum continuity and less support size. Our proposed family members P4, P5, P6, and P7, have C7, C9, C11 and C13 continuities respectively. In fact, we use Binomial probability distribution to increase the continuity of uniform B-splines subdivision schemes up to more than double. We present the complete analysis of one family member of proposed schemes and give a visual performance to check smoothness graphically. In our analysis, we present continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils analysis of proposed family of subdivision schemes. We also present a survey of high continuity subdivision schemes. Comparison shows that our proposed family of subdivision schemes gives high continuity of subdivision schemes comparative to existing subdivision schemes. We have found that as complexity increases the continuity also increases. In the last, we give the general formula for tensor product surface subdivision schemes and also present the visual performance of proposed tensor product surface subdivision schemes.

Multi-Temporal Image Analysis for Fluvial Morphological Characterization with Application to Albanian Rivers

A procedure for the characterization of the temporal evolution of river morphology is presented. Wet and active river channels are obtained from the processing of imagery datasets. Information about channel widths and active channel surface subdivision in water, vegetation and gravel coverage classes are evaluated along with channel centerline lengths and sinuosity indices. The analysis is carried out on a series of optical remotely-sensed imagery acquired by different satellite missions during the time period between 1968 and 2017. Data from the CORONA, LANDSAT and Sentinel-2 missions were considered. Besides satellite imagery, a digital elevation model and aerial ortho-photos were also used. The procedure was applied to three, highly dynamic, Albanian rivers: Shkumbin, Seman and Vjosë, showing a high potential for application in contexts with limitations in ground data availability. The results of the procedure were assessed against reference data produced by means of expert interpretation of a reference set of river reaches. The results differ from reference values by just a few percentage points (<6%). The time evolution of hydromorphological parameters is well characterized, and the results support the design of future studies aimed at the understanding of the relations between climatic and anthropogenic controls and the response of river morphological trajectories. Moreover, the high spatial and temporal resolution of the Sentinel-2 mission motivates the development of an automatic monitoring system based on a rolling application of the defined procedure.

Geometric modeling of lattice structures for additive manufacturing

Purpose This paper aims to propose a consistent approach to geometric modeling of optimized lattice structures for additive manufacturing technologies. Design/methodology/approach The proposed method applies subdivision surfaces schemes to an automatically defined initial mesh model of an arbitrarily complex lattice structure. The approach has been developed for cubic cells. Considering different aspects, five subdivision schemes have been studied: Mid-Edge, an original scheme proposed by the authors, Doo–Sabin, Catmull–Clark and Bi-Quartic. A generalization to other types of cell has also been proposed. Findings The proposed approach allows to obtain consistent and smooth geometric models of optimized lattice structures, overcoming critical issues on complex models highlighted in literature, such as scalability, robustness and automation. Moreover, no sharp edge is obtained, and consequently, stress concentration is reduced, improving static and fatigue resistance of the whole structure. Originality/value An original and robust method for modeling optimized lattice structures was proposed, allowing to obtain mesh models suitable for additive manufacturing technologies. The method opens new perspectives in the development of specific computer-aided design tools for additive manufacturing, based on mesh modeling and surface subdivision. These approaches and slicing tools are suitable for parallel computation, therefore allowing the implementation of algorithms dedicated to graphics cards.

Efficient NC Tool Path Planning Based on the Subdivision of Sculptured Surface

In this paper, we have presented a method to generate efficient NC tool paths based on the surface subdivision. The main objective is to achieve high efficiency in the machining of sculptured surface. The NC machining efficiency can be improved by segmenting the whole surface into distinct areas according to the characters of sculptured surface and by using different size mills and different tool path planning methods to machine the areas. The iso-parametric method and large mills are used in the curvature changing little areas. While the iso-scallop method and small mills are used in curvatures changing large areas. This can make full use of tool path generation methods and mills, which improve the machining efficiency of sculpture effectively.

Sampling Strategy for Free-Form Surface Inspection Using Coordinate Measuring Machines

Due to the complexity and non-rotational symmetry of free-form surface, it is difficult to achieve accurate and efficient inspection method. In order to solve this problem, three types of sampling sequences are proposed to specify a set of measuring points of free-form surface. For comparing the results of different sampling strategies, the profile errors of free-form surface are calculated based on a quasi particle swarm optimization (QPSO) searching the transformation parameters to implement localization and surface subdivision method finding the closest points on the design model corresponding to measured points. In order to obtain effective sampling strategies, four design models are generated by non-uniform rational basis spline (NURBS) and parts are manufactured on two machining centers to obtain surfaces of different roughness and measured on CMMs by selecting different sampling methods and sample sizes. The profile errors of parts are calculated by the proposed method and CMMs software, respectively. The results show that randomized Hammersley sampling sequence and medium sample size are preferred for the profile error inspection of given parts if accuracy and time are all considered. The research provides a method for free-form surface accurate inspection while minimizing the sampling time and cost.