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Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
subdivision: To subdivide an edge {v,w} $ \{ v, w\} $ of a graph G is to construct a new graph G’ from G by removing the edge {v,w} $ \{ v, w\} $ and introducing new vertices xi and new edges {v,x1}, $ \{ v, x_{1} \} , $ {xk,w} $ \{ x_{k} , w\} $ and {xi,xi+1} $ \{ x_{i} , x_{i + 1} \} $ for 1 ≤ i < k. A subdivision of a graph is a graph obtained by subdividing one or more edges of the graph.
When a tree model meets texture baking: an approach for quality-preserving lightweight visualization in virtual 3D scene construction
Published in International Journal of Digital Earth, 2023
Chen Zhang, Biao He, Renzhong Guo, Ding Ma
The subdivided surface obtained by the Catmull-Clark algorithm is shown in Figure 10. The mesh is much smoother than the low-poly mesh after a surface subdivision because the essence of surface subdivision is vertex interpolation. The operations inevitably result in an increase in the number of triangles. A comparison of the number of triangles among the low-poly mesh, high-poly mesh and subdivided surface is shown in Table 1. Although the number of triangles in the subdivided surface approximates or, in most cases, exceeds that in the high-poly mesh, the subdivided surface is still not as detailed as the high-poly model. For this reason, the subsequent employment of the baked textures is required and plays a critical role in reflecting model appearance.
A review on big data real-time stream processing and its scheduling techniques
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
Nicoleta Tantalaki, Stavros Souravlas, Manos Roumeliotis
Eskandari et al. [66] presented a hierarchical adaptive scheduling scheme called P-Scheduler. The P-Scheduler is also a topology-aware strategy that uses weights on the graph edges, which store the data transfer rates. These rate values are obtained by initially submitting the known topology to the cluster and running it on the default Storm scheduler. The number of nodes needed to run a topology is computed on the basis of the topology load. These nodes share the entire topology and their communications are determined. The algorithm assigns the highest communication pairs of the topology to the same worker (JVM) and the other workers are loaded with a subdivision of the entire topology. The topology is partitioned by METIS, a specialised software for graph partitioning. By this arrangement, the inter-process and inter-worker traffic is reduced. The scheduler is also a workload-aware strategy, in the sense that most of the Storm topologies have almost similar workload and generally, no load balancing issue exists. The authors implemented P-Scheduler in a homogeneous cluster with 8 worker nodes and tested it on three topologies. The results have shown that P-scheduler reduces the average tuple processing time be almost 50% as a result of reducing the inter-node and inter-worker communications.
Efficient preprocessing of complex geometries for CFD simulations
Published in International Journal of Computational Fluid Dynamics, 2019
Zaib Ali, James Tyacke, Rob Watson, Paul G. Tucker, Shahrokh Shahpar
The medial axis transform (MAT)-based algorithms for the domain decomposition have been presented in, for example, (Tam and Armstrong 1991; Price, Armstrong, and Sabin 1995a, 1995b; Sheehy, Armstrong, and Robinson 1995). Here the medial axis is generated using the Voronoi-based method. A subdivision is created resulting in one block for each medial vertex, medial edge and medial face. A midpoint subdivision is then used for meshing the blocks. An alternative has been presented by Rigby (2004), called the ‘TopMaker’ approach, which makes use of medial vertices and parts of medial axis to block the domain. Medial vertices are defined as the points which are equidistant from three locations from the domain boundary. Consequently, six types of medial edges and appropriate rules are defined for creating the blocks. Further enhancements have been included to produce a good quality mesh, however this technique has yet to be extended for 3D.