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A Survey of Partitional and Hierarchical Clustering Algorithms
Published in Charu C. Aggarwal, Chandan K. Reddy, Data Clustering, 2018
Chandan K. Reddy, Bhanukiran Vinzamuri
In a weighted graph, a minimum spanning tree is an acyclic subgraph that covers all the vertices with the minimum edge weights. Prim’s and Kruskal’s algorithms [9] are used for finding the minimum spanning tree (MST) in a weighted graph. In a Euclidean minimum spanning tree (EMST), the data points represent the vertices and the edge weights are computed using the Euclidean distance between two data points. Each edge in an EMST represents the shortest distance between those two points. Using this EMST a divisive clustering method can be developed which removes the largest weighted edge to get two clusterings and subsequently removes the next largest edge to get three clusterings and so on. This process of removing edges from an EMST gives rise to an effective divisive clustering method. The major advantage of this method is that it is able to detect clusters with nonspherical shapes effectively.
Conclusions
Published in Christopher M. Gold, Spatial Context: An Introduction to Fundamental Computer Algorithms for Spatial Analysis, 2018
We then move on to other boundary types, based on our Cosmic Model. As the galaxies move, under the influence of gravity, the voids tend to grow and boundaries form between them. These ‘slabs’ may be several galaxies thick, with the gravitational zones of influence of the exterior ones filling the voids on each side, forming what we think of as polyhedra or polygons. These exterior points form a chain (in 2D) defining the extent of the empty voids on each side: the interior points, in this model, have less influence but form a type of cluster, with multiple holes, and various types of cluster analysis may be used – in particular, for our purposes, analysis of the Voronoi cell shape to determine cluster boundaries (the same thing as void boundaries, if you think about it – just looking at them from the other side). A particularly useful form of cluster analysis uses the Euclidean Minimum Spanning Tree, a subset of the Delaunay triangulation – based on the connectivity of adjacent points: the dual is the context, again.
Basic Data Structures
Published in Charles J. Alpert, Dinesh P. Mehta, Sachin S. Sapatnekar, Handbook of Algorithms for Physical Design Automation, 2008
Given a set of n points in a plane, a spanning tree is a set of edges that connects all n points and contains no cycles. When each edge is weighted using some distance metric, the minimum spanning tree is a spanning tree whose sum of edge weights is minimum. If Euclidean distance (L2)is used, it is called the Euclidean minimum spanning tree; if rectilinear distance (L1) is used, it is called the rectilinear minimum spanning tree (RMST). The RMST is often used as a starting point for constructing a Steiner tree, which is used extensively in global routing (see Chapter 24).
Predicting jet ignitability using a PDF transport model
Published in Numerical Heat Transfer, Part A: Applications, 2022
More recently Cumber [26, 27], applied a composition PDF transport model to the natural gas jet of Birch et al. [28], using the Curl model [29], modified Curl (MC) model [30], interaction by exchange with the mean (IEM) model [31], a variant of the Langevin model [27], and the Euclidean minimum spanning tree (EMST) model [32]. Birch et al. [28], natural gas (95% methane) jet is used in Cumber [26, 27], for validation purposes as a wide range of measured data exist characterizing the jet, such as the PDF, mean mixture fraction, mixture fraction variance, as well as higher mixture fraction moments such as the skewness and kurtosis. From Cumber [26, 27], it is clear that the mean mixture fraction and mixture fraction variance predicted fields are insensitive to the micro-mixing model implemented. For higher mixture fraction moments and the PDF there is sensitivity to the micro-mixing model implemented with the MC model and a variant of the Langevin model being more accurate than the IEM model and the EMST model in this application.
A Fully Consistent Hybrid Les/Rans Conditional Transported Pdf Method for Non-premixed Reacting Flows
Published in Combustion Science and Technology, 2021
Federica Ferraro, Yipeng Ge, Michael Pfitzner, Matthew J. Cleary
In PDF methods the weakest part of the formulation is represented by the closure of the molecular mixing term, i.e., the conditional expectation of the scalar dissipation term in the PDF transport equation. A micro-mixing model needs to have some fundamental characteristics to physically reproduce the effects of the molecular diffusion. A detailed discussion can be found, e.g., in Chapter 8 of (Fox 2003) or Chapter 12 of (Pope 2000). Several studies have attempted to address the micro-mixing issue by proposing different closure approaches. The simplest model is the Interaction by Exchange with the Mean (IEM) (Villermaux and Devillon 1972) where particle compositions linearly relax toward the local mean value. Janicka, Kolbe, and Kollmann (1979) have formulated a modified Curl model (Curl 1963), where pairs of particles are mixed with a random level of mixing. Nooren et al. (1997) have analyzed different micro-mixing models and proposed a modification of the Curl model for particles of different weights. Subramaniam and Pope (1998) have formulated the Euclidean Minimum Spanning Tree (EMST) model, which enforces interactions between neighboring particles in composition space, improving the accuracy of the PDF method. Klimenko and Pope (2003) have developed the Multiple Mapping Conditioning (MMC) approach, which, in the stochastic formulation, provides the micro-mixing closure. This approach, based on the generalized mapping closure, fulfills the requirement of local mixing both in physical and composition space.
Consistent submodel coupling in hybrid particle/finite volume algorithms for zone-adaptive modelling of turbulent reactive flows
Published in Combustion Theory and Modelling, 2022
Tianwei Yang, Yu Yin, Hua Zhou, Yi Mo, Yuxuan Chen, Zhuyin Ren
The TPDF method is an approach where the evolution of a joint probability density function (PDF) of composition in the composition and physical space is computed, in which the highly nonlinear reaction source term appears in a closed form. For convenience, the Lagrangian Monte Carlo particle algorithm is the current mainstream approach for implementing the TPDF method, in which the joint PDF of composition is represented by a large number of computational particles. Each computational particle has its composition , position , and mass , and particles evolve in physical and composition space according to the following stochastic differential equations (SDEs) recast from the transport equation of the joint PDF of composition where the superscript ‘*’ denotes either an individual particle property or a value of a field evaluated at the particle’s location, is an independent Wiener increment, represents the rate of change due to the process of micro-mixing and is closed by a mixing model, such as the interaction by exchange with the mean (IEM) [28], the modified Curl (MC) [29–31], and the Euclidean minimum spanning tree (EMST) models [32–34]. For example, in the IEM model, particle compositions relax toward the local mean composition on the time scale [35,36], and the micro-mixing term is modelled by The SDEs (6) and (7) are solved using an operator-splitting scheme, in which the processes of transport in physical space, mixing in compositional space, and chemical reaction are solved in separate subsequential substeps.