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At least 109 records · Page 6

MGLab: An Interactive Multigrid Environment

MGLab is a set of Matlab functions that defines an interactive environment for experimenting with multigrid algorithms. The package solves two-dimensional elliptic partial differential equations discretized using either finite differences or finite volumes, depending on the problem. Built-in problems include the Poisson equation, the Helmholtz equation, a convection-diffusion problem, and a discontinuous coefficient problem. A number of parameters controlling the multigrid V-cycle can be set using a point-and-click mechanism. The menu-based user interface also allows a choice of several Krylov subspace methods, including CG, GMRES(k), and Bi-CGSTAB, which can be used either as stand-alone solvers or as multigrid acceleration schemes. The package exploits Matlab's visualization and sparse matrix features and has been structured to be easily extensible.

Bordner, James↗

Annotations on the virtual element method for second-order elliptic problems

This document contains working annotations on the Virtual Element Method (VEM) for the approximate solution of diffusion problems with variable coefficients. To read this document you are assumed to have familiarity with concepts from the numerical discretization of Partial Differential Equations (PDEs) and, in particular, the Finite Element Method (FEM). This document is not an introduction to the FEM, for which many textbooks (also free on the internet) are available. Eventually, this document is intended to evolve into a tutorial introduction to the VEM (but this is really a long-term goal).

97 MATHEMATICS AND COMPUTING↗

Iterative solutions for one-dimensional diffusion with time varying surface composition and composition-dependent diffusion coefficient

Solutions are given for one-dimensional diffusion problems with a time varying surface composition and also a composition dependent diffusion coefficient. The most general solution does not require special mathematical functions to fit the variation in surface composition or D(C). In another solution, a series expansion may be used to fit the time dependent surface concentration. These solutions make use of iterative calculations that converge rapidly and are highly stable. Computer times are much shorter than that required for finite difference calculations and can efficiently make use of interactive graphics terminals. Existing gas carburization data were used to provide an illustration of an iterative approach with a time varying carbon composition at the free surface.

Chow, M.↗

Single-, two-, and three-phase binary-alloy systems

Series of three computer programs solves one-dimensional transient diffusion problems in single-and multiphase binary-alloy systems. Accurate understanding of diffusion process in binary-alloy system is important for development of metal matrix composites, some protective coatings, and thin-film technology.

Tenney, D. R.↗

The Diffuse Interstellar Bands: Solving a Century Old Problem

The Diffuse Interstellar Bands (DIBs) are a set of apporoximately 500 absorption bands that are seen in the spectra of reddened stars (i.e., stars obscured by the presence of interstellar clouds in their line of sight). The first DIBs were detected in the visible over a century ago. Diffuse Interstellar Bands are now detected from the near ultraviolet to the near infrared in the spectra of reddened stars spanning a variety of interstellar environments in our local, and in other galaxies. Although DIB carriers are a significant part of the interstellar chemical inventory as they account for a noticeable fraction of the interstellar extinction, the nature of their carriers is still unknown over a century after the detection of the first bands. DIB carriers are stable and ubiquitous in a broad variety of interstellar environments and play a unique role in interstellar physics and chemistry. It has long been realized that the solving of the DIB problem requires a strong synergy between astronomical observations, laboratory astrophysics and astrochemistry, quantum chemistry calculations and astrophysical modeling of line-of-sights. In this review, we'll present and discuss the current state of this perplexing problem. We'll review the progress and the failures that have been encountered in the long quest for the identification of the carriers of these ubiquitous interstellar bands.

Molecules in Space↗

A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems

We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share “nonlocal” interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. Finally, this work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.

42 ENGINEERING↗

MULTI-PRED: A Software Module for Predictive Modeling of Coupled Multi-Physics Systems: User's Manual

This User’s Manual describes the code module MULTI-PRED, written in FORTRAN which implements the methodology for “predictive modeling of coupled multi-physics systems (PM-CMPS)” formulated by Cacuci (2014). This methodology fully takes into account the coupling terms between the systems but requires only the computational resources that would be needed to perform predictive modeling on each system separately. The PM-CMPS methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution based on a priori known mean values and uncertainties characterizing the experimental and computational parameters and results of interest responses, called for the multi-physics models under consideration. This “maximum entropy” a priori distribution is combined, using Bayes’ theorem, with the “likelihood” provided by the multi-physics simulation models to obtain a formal posterior distribution. Subsequently, the posterior distribution thus obtained is evaluated using the saddle-point method to obtain analytical expressions for the optimally predicted values for the multi-physics models parameters and responses along with corresponding reduced uncertainties. Noteworthy, the predictive modeling methodology for the coupled systems is constructed such that the systems can be considered sequentially rather than simultaneously, while preserving exactly the same results as if the systems were treated simultaneously. Consequently, very large coupled systems, which could perhaps exceed available computational resources if treated simultaneously, can be treated with the PM-CMPS methodology presented in this work sequentially and without any loss of generality or information, requiring just the resources that would be needed if the systems were treated sequentially. Three illustrative demonstration problems are also provided. The first problem presents the application of the PM-CMPS methodology to a simple particle diffusion problem which admits a closed-form analytical solution which facilitates a rapid understanding of this methodology and its predicted results. The second demonstration problem presents the application of the PM-CMPS methodology to the problem of inverse prediction, from detector responses in the presence of counting uncertainties, of the thickness of a homogeneous slab of material containing uniformly distributed gamma-emitting sources, for optically thin and thick slabs. This problem highlights the essential role played by the relative uncertainties (or, conversely, accuracies) of measured and computed responses. The third demonstration problem presents the application of the PM-CMPS methodology to the F-area cooling towers at the Savannah River National Lab. This problem demonstrates that the PM-CMPS methodology reduces the predicted response uncertainties not only at locations where measurements are available, but also at locations where measurements are not available.

97 MATHEMATICS AND COMPUTING↗

On upstream blocking in a viscous diffusive stratified flow

The effect of diffusion of specie upon the flow about a transverse flat plate moving horizontally in a viscous stratified medium is considered. Asymptotic expansions are used to define a parameter regime where a viscous-diffusive-buoyancy balance is dominant. The solution, expressed in terms of an inverse Fourier transform, is numerically integrated. The results show that, as in the non-diffusive problem, a region of closed streamlines exists ahead of the body. However, unlike the case where diffusion is neglected, the density field within this recirculating region is uniquely determined and found to be statically stable. It is also found that varying the relative amount of diffusion affects not only the density distribution, but the velocity profile as well, indicating a strong coupling between the vorticity and specie equation.

Koop, C. G.↗

Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction–diffusion systems

The generalized fractional reaction–diffusion equations which exist in the form of noninteger order partial differential equations have now found wide application for illustrating important and useful physical phenomena, such as subdiffusive and superdiffusive scenarios. The space fractional derivatives are defined in the Riesz sense on the intervals 0 < α < 1 and 1 < α ≤ 2. We propose robust numerical techniques, such as a spectral representation of the fractional Laplacian operator in conjunction with the exponential time differencing method, and present the equivalent relationship between the Riesz fractional derivative and fractional Laplacian operator. We apply these techniques to numerically solve a range of chaotic processes, such as the Chua’s equations, Rössler system, Lorenz and Lorenz-type systems. Simulation results revealed various complex and spatiotemporal chaos, spiral chaos, intermittent chaos and spots patterns in two-dimensional space fractional reaction–diffusion problems.

97 MATHEMATICS AND COMPUTING↗

Domain-decomposed preconditionings for transport operators

The performance was tested of five different interface preconditionings for domain decomposed convection diffusion problems, including a novel one known as the spectral probe, while varying mesh parameters, Reynolds number, ratio of subdomain diffusion coefficients, and domain aspect ratio. The preconditioners are representative of the range of practically computable possibilities that have appeared in the domain decomposition literature for the treatment of nonoverlapping subdomains. It is shown that through a large number of numerical examples that no single preconditioner can be considered uniformly superior or uniformly inferior to the rest, but that knowledge of particulars, including the shape and strength of the convection, is important in selecting among them in a given problem.

Chan, Tony F.↗

A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integro-differential operator consists of both local convection-diffusion and nonlocal diffusion operators. Here, our numerical scheme is based on the direct approximation of the nonlinear Feynman–Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman–Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas.

97 MATHEMATICS AND COMPUTING↗

Variational Principle for Mass Transport

A variation principle for mass transport in solids is derived that recasts transport coefficients as minima of local thermodynamic average quantities. The result is independent of diffusion mechanisms and applies to amorphous and crystalline systems. This unifies different computational approaches for diffusion and provides a framework for the creation of new approximation methods with error estimation. It gives a different physical interpretation of the Green function. Lastly, the variational principle quantifies the accuracy of competing approaches for a nontrivial diffusion problem.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Probing surface recombination velocities in semiconductors using two-photon microscopy

The determination of minority-carrier lifetimes and surface recombination velocities is essential for the development of semiconductor technologies such as solar cells. The recent development of two-photon time-resolved microscopy allows for better measurements of bulk and subsurface interfaces properties. Here, we analyze the diffusion problem related to this optical technique. Our three-dimensional treatment enables us to separate lifetime (recombination) from transport effects (diffusion) in the photoluminescence intensity. It also allows us to consider surface recombination occurring at a variety of geometries: a single plane (representing an isolated exposed or buried interface), a two parallel planes (representing two inequivalent interfaces), and a spherical surface (representing the enclosing surface of a grain boundary). We provide fully analytical results and scalings directly amenable to data fitting and apply those to experimental data collected on heteroepitaxial CdTe/ZnTe/Si.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Multigrid Reduction in Time for Nonlinear Parabolic Problems

The need for parallel-in-time is being driven by changes in computer architectures, where future speed-ups will be available through greater concurrency, but not faster clock speeds, which are stagnant.This leads to a bottleneck for sequential time marching schemes, because they lack parallelism in the time dimension. Multigrid Reduction in Time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficient is defined as achieving similar performance when compared to a corresponding linear problem. As our benchmark, we use the p-Laplacian, where p = 4 corresponds to a well-known nonlinear diffusion equation and p = 2 corresponds to our benchmark linear diffusion problem. When considering linear problems and implicit methods, the use of optimal spatial solvers such as spatial multigrid imply that the cost of one time step evaluation is fixed across temporal levels, which have a large variation in time step sizes. This is not the case for nonlinear problems, where the work required increases dramatically on coarser time grids, where relatively large time steps lead to worse conditioned nonlinear solves and increased nonlinear iteration counts per time step evaluation. This is the key difficulty explored by this paper. We show that by using a variety of strategies, most importantly, spatial coarsening and an alternate initial guess to the nonlinear time-step solver, we can reduce the work per time step evaluation over all temporal levels to a range similar with the corresponding linear problem. This allows for parallel scaling behavior comparable to the corresponding linear problem.

97 MATHEMATICS AND COMPUTING↗

Variance reduction in Monte Carlo analysis of rarefied gas diffusion.

The problem of rarefied diffusion between parallel walls is solved using the Monte Carlo method. The diffusing molecules are evaporated or emitted from one of the two parallel walls and diffuse through another molecular species. The Monte Carlo analysis treats the diffusing molecule as undergoing a Markov random walk, and the local macroscopic properties are found as the expected value of the random variable, the random walk payoff. By biasing the transition probabilities and changing the collision payoffs, the expected Markov walk payoff is retained but its variance is reduced so that the Monte Carlo result has a much smaller error.

Perlmutter, M.↗

The crack-contact and the free-end problem for a strip under residual stress

The plane problem for an infinite strip with two edge cracks under a given state of residual stress is considered. The residual stress is compressive near and at the surfaces and tensile in the interior of the strip. If the crack is deep enough to penetrate into the tensile zone, then the problem is one of crack-contact where the depth of the contact area is an unknown which depends on the crack depth and the residual stress profile. The problem has applications to the static fatigue of glass plates and is solved for three typical residual-stress profiles. In the limiting case of the crack's crossing the entire plate thickness, the problem becomes a stressfree end problem for a semiinfinite strip under a given residual-stress state away from the end. This is a typical stress diffusion problem in which decay behavior of the residual stress near and the nature of the normal displacement at the end of the semiinfinite strip are of special interest. For two typical residual-stress states the solution is obtained, and some numerical results are given.

Bakioglu, M.↗

The Crack-contact and the Free End Problem for a Strip Under Residual Stress

The plane problem for an infinite strip with two edge cracks under a given state of residual stress is considered. The residual stress is compressive near and at the surfaces and tensile in the interior of the strip. If the crack is deep enough to penetrate into the tensile zone, then the problem is one of crack-contact problem in which the depth of the contact area is an unknown which depends on the crack depth and the residual stress profile. The problem has applications to the static fatigue of glass plates and is solved for three typical residual stress profiles. In the limiting case of the crack crossing the entire plate thickness, the problem becomes a stress-free end problem for a semi-infinite strip under a given residual stress state away from the end. This is a typical stress diffusion problem in which decay behavior of the residual stress near and the nature of the normal displacement at the end of the semi-infinite strip are of special interest. For two typical residual stress states the solution is obtained, and some numerical results are given.

Bakioglu, M.↗

Scaling Structured Multigrid to 500K+ Cores Through Coarse-Grid Redistribution

The effcient solution of sparse, linear systems resulting from the discretization of partial differential equations is crucial to the performance of many physics-based simulations. The algorithmic optimality of multilevel approaches for common discretizations makes them a good candidate for an efficient parallel solver. Yet, modern architectures for high-performance computing systems continue to challenge the parallel scalability of multilevel solvers. While algebraic multigrid methods are robust for solving a variety of problems, the increasing importance of data locality and cost of data movement in modern architectures motivates the need to carefully exploit structure in the problem. Robust logically structured variational multigrid methods, such as Black Box Multigrid (BoxMG), maintain structure throughout the multigrid hierarchy. This avoids indirection and increased coarse- grid communication costs typical in parallel algebraic multigrid. Nevertheless, the parallel scalability of structured multigrid is challenged by coarse-grid problems where the overhead in communication dominates computation. In this paper, an algorithm is introduced for redistributing coarse-grid problems through incremental agglomeration. Guided by a predictive performance model, this algorithm provides robust redistribution decisions for structured multilevel solvers. A two-dimensional diffusion problem is used to demonstrate the significant gain in performance of this algorithm over the previous approach that used agglomeration to one processor. In addition, the parallel scalability of this approach

97 MATHEMATICS AND COMPUTING↗