Towards and FVE-FAC Method for Determining Thermocapillary Effects on Weld Pool Shape

SEE ParentDocumentRecord|Ntt=19970006857 "Seventh Copper Mountain Conference on Multigrid Methods"; p. 147-166; Part 1; NASA-CP-3339Several practical materials processes, e.g., welding, float-zone purification, and Czochralski crystal growth, involve a pool of molten metal with a free surface, with strong temperature gradients along the surface. In some cases, the resulting thermocapillary flow is vigorous enough to convect heat toward the edges of the pool, increasing the driving force in a sort of positive feedback. In this work we examine this mechanism and its effect on the solid-liquid interface through a model problem: a half space of pure substance with concentrated axisymmetric surface heating, where surface tension is strong enough to keep the liquid free surface flat. The numerical method proposed for this problem utilizes a finite volume element (FVE) discretization in cylindrical coordinates. Because of the axisymmetric nature of the model problem, the control volumes used are torroidal prisms, formed by taking a polygonal cross-section in the (r, z) plane and sweeping it completely around the z-axis. Conservation of energy (in the solid), and conservation of energy, momentum, and mass (in the liquid) are enforced globally by integrating these quantities and enforcing conservation over each control volume. Judicious application of the Divergence Theorem and Stokes' Theorem, combined with a Crank-Nicolson time-stepping scheme leads to an implicit algebraic system to be solved at each time step. It is known that near the boundary of the pool, that is, near the solid-liquid interface, the full conduction-convection solution will require extremely fine length scales to resolve the physical behavior of the system. Furthermore, this boundary moves as a function of time. Accordingly, we develop the foundation of an adaptive refinement scheme based on the principles of Fast Adaptive Composite Grid methods (FAC). Implementation of the method and numerical results will appear in a later report.N00014-92-WR-24009Approved for public release; distribution is unlimited

An Ensemble Framework for Detecting Community Changes in Dynamic Networks

Dynamic networks, especially those representing social networks, undergo constant evolution of their community structure over time. Nodes can migrate between different communities, communities can split into multiple new communities, communities can merge together, etc. In order to represent dynamic networks with evolving communities it is essential to use a dynamic model rather than a static one. Here we use a dynamic stochastic block model where the underlying block model is different at different times. In order to represent the structural changes expressed by this dynamic model the network will be split into discrete time segments and a clustering algorithm will assign block memberships for each segment. In this paper we show that using an ensemble of clustering assignments accommodates for the variance in scalable clustering algorithms and produces superior results in terms of pairwise-precision and pairwise-recall. We also demonstrate that the dynamic clustering produced by the ensemble can be visualized as a flowchart which encapsulates the community evolution succinctly.Comment: 6 pages, under submission to HPEC Graph Challeng

Multilevel Aggregation Methods for Small-World Graphs with Application to Random-Walk Ranking

We describe multilevel aggregation in the specific context of using Markov chains to rank the nodes of graphs. More generally, aggregation is a graph coarsening technique that has a wide range of possible uses regarding information retrieval applications. Aggregation successfully generates efficient multilevel methods for solving nonsingular linear systems and various eigenproblems from discretized partial differential equations, which tend to involve mesh-like graphs. Our primary goal is to extend the applicability of aggregation to similar problems on small-world graphs, with a secondary goal of developing these methods for eventual applicability towards many other tasks such as using the information in the hierarchies for node clustering or pattern recognition. The nature of small-world graphs makes it difficult for many coarsening approaches to obtain useful hierarchies that have complexity on the order of the number of edges in the original graph while retaining the relevant properties of the original graph. Here, for a set of synthetic graphs with the small-world property, we show how multilevel hierarchies formed with non-overlapping strength-based aggregation have optimal or near optimal complexity. We also provide an example of how these hierarchies are employed to accelerate convergence of methods that calculate the stationary probability vector of large, sparse, irreducible, slowly-mixing Markov chains on such small-world graphs. The stationary probability vector of a Markov chain allows one to rank the nodes in a graph based on the likelihood that a long random walk visits each node. These ranking approaches have a wide range of applications including information retrieval and web ranking, performance modeling of computer and communication systems, analysis of social networks, dependability and security analysis, and analysis of biological systems

DFTS on irregular grids : the anterpolated DFT

In many instances the discrete Fourier transform (DFT) is desired for a data set that occurs on an irregular grid. Commonly the data are interpolated to a regular grid, and a fast Fourier transform (FFT) is then applied. A drawback to this approach is that typically the data have unknown smoothness properties, so that the error in the interpolation is unknown. An alternative method is presented, based upon multilevel integration techniques introduced by A. Brandt. In this approach, the kernel, e(-iwt), is interpolated to the irregular grid, rather than interpolating the data to the regular grid. This may be accomplished by pre-multiplying the data by the adjoint of the interpolation matrix (a process dubbed anterpolation), producing a new regular-grid function, and then applying a standard FFT to the new function. Since the kernel is C infinity the operation may be carried out to any preselected accuracy. A simple optimization problem can be solved to select the problem parameters in an efficient way. If the requirements of accuracy are not strict, or if a small bandwidth is of interest, the method can be used in place of an FFT even when the data are regularly spaced.http://archive.org/details/dftsonirregularg00hensN

The Table of Analytical Discrete Fourier Transforms

While most people rely on numerical methods (most notably the fast Fourier transform) for computing discrete Fourier transforms (DFTs), there is still an occasional need to have analytical DFTs close at hand. Such a table of analytical DFTs is provided in this paper, along with comments and observations, in the belief that it will serve as a useful resource or teaching aid for Fourier practioners

Wavelets and Multigrid

The last few years have seen a remarkable amount of activity and interest in the field of wavelet theory and multiresolution analysis. With this heightened level of interest, researchers in diverse fields have begun to consider wavelet-based methods. The work presented in this paper was done in an exploratory spirit, by investigating the very suggestive similarities between multiresolution analysis and multigrid methods. The results are preliminary and only point to several avenues of future work

Resume of Van Emden Henson, 1992-02

Naval Postgraduate School Faculty Resum

Assistant Professor Van Emden Henson (archived)

Webpage describing Prof. Van Emden Henson, his role in the Applied Mathematics Department, and professional works and status

Multigrid Methods For Nonlinear Problems: An Overview

Since their early application to elliptic partial differential equations, multigrid methods have been applied successfully to a large and growing class of problems, from elasticity and computational fluid dynamics to geodetics and molecular structures. Classical multigrid begins with a two-grid process. First, iterative relaxation is applied, whose effect is to smooth the error. Then a coarse-grid correction is applied, in which the smooth error is determined on a coarser grid. This error is interpolated to the fine grid and used to correct the fine-grid approximation. Applying this method recursively to solve the coarse-grid problem leads to multigrid. The coarse-gri

A Multilevel Cost-Space Approach To Solving The Balanced Long Transportation Problem

INTRODUCTION The transportation problem is the simplest of network flow problems. It is posed on a bipartite graph, consisting of a set of M supply nodes, a set of N demand nodes, and a set of arcs connecting them. Each supply node S i has a fixed amount s i of a commodity which it can provide. Each demand node D j has a fixed requirement d j for that commodity, and for each arc (i; j) connecting supply node S i to demand node D j there is an associated cost per unit flow c ij . When the total supply equals the total demand the problem is balanced. When M !! N , the problem is referred to as a long transportation problem. Denoting the flow on arc (i; j) by x ij , the transportation prob