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A mathematical framework for multiscale science and engineering : the variational multiscale method and interscale transfer operators

Bochev, Pavel B.; Collis, Samuel S.; Jones, Reese E.; Lehoucq, Richard B.; Parks, Michael L.; Scovazzi, Guglielmo S.; Silling, Stewart A.; Templeton, Jeremy A.

This report is a collection of documents written as part of the Laboratory Directed Research and Development (LDRD) project A Mathematical Framework for Multiscale Science and Engineering: The Variational Multiscale Method and Interscale Transfer Operators. We present developments in two categories of multiscale mathematics and analysis. The first, continuum-to-continuum (CtC) multiscale, includes problems that allow application of the same continuum model at all scales with the primary barrier to simulation being computing resources. The second, atomistic-to-continuum (AtC) multiscale, represents applications where detailed physics at the atomistic or molecular level must be simulated to resolve the small scales, but the effect on and coupling to the continuum level is frequently unclear.

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Recycling Krylov subspaces for sequences of linear systems

SIAM Journal on Scientific Computing

Parks, Michael L.; De Sturler, Eric; Mackey, Greg; Johnson, Duane D.; Maiti, Spandan

Many problems in science and engineering require the solution of a long sequence of slowly changing linear systems. We propose and analyze two methods that significantly reduce the total number of matrix-vector products required to solve all systems. We consider the general case where both the matrix and right-hand side change, and we make no assumptions regarding the change in the right-hand sides. Furthermore, we consider general nonsingular matrices, and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix. Our methods work well under these general assumptions, and hence form a significant advancement with respect to related work in this area. We can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems. We consider two approaches that allow for the continuous improvement of the recycled subspace at low cost. We consider both Hermitian and non-Hermitian problems, and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling. We also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics, materials science, and computational physics. © 2006 Society for Industrial and Applied Mathematics.

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A reduced order model for the study of asymmetries in linear gas chromatography for homogeneous tubular columns

Romero, L.A.; Whiting, Joshua J.; Parks, Michael L.

In gas chromatography, a chemical sample separates into its constituent components as it travels along a long thin column. As the component chemicals exit the column they are detected and identified, allowing the chemical makeup of the sample to be determined. For correct identification of the component chemicals, the distribution of the concentration of each chemical along the length of the column must be nearly symmetric. The prediction and control of asymmetries in gas chromatography has been an active research area since the advent of the technique. In this paper, we develop from first principles a general model for isothermal linear chromatography. We use this model to develop closed-form expressions for terms related to the first, second, and third moments of the distribution of the concentration, which determines the velocity, diffusion rate, and asymmetry of the distribution. We show that for all practical experimental situations, only fronting peaks are predicted by this model, suggesting that a nonlinear chromatography model is required to predict tailing peaks. For situations where asymmetries arise, we analyze the rate at which the concentration distribution returns to a normal distribution. Numerical examples are also provided.

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Results 101–116 of 116
Results 101–116 of 116