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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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TYPE SAND Report YEAR 2007
OSTIDOI

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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TYPE SAND Report YEAR 2005
OSTIDOI

An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics

Computer Methods in Applied Mechanics and Engineering

Trask, Nathaniel A.; You, Huaiqian; Yu, Yue; Parks, Michael L.

We present a meshfree quadrature rule for compactly supported nonlocal integro-differential equations (IDEs) with radial kernels. We apply this rule to develop a meshfree discretization of a peridynamic solid mechanics model that requires no background mesh. Existing discretizations of peridynamic models have been shown to exhibit a lack of asymptotic compatibility to the corresponding linearly elastic local solution. By posing the quadrature rule as an equality constrained least squares problem, we obtain asymptotically compatible convergence by introducing polynomial reproduction constraints. Our approach naturally handles traction-free conditions, surface effects, and damage modeling for both static and dynamic problems. We demonstrate high-order convergence to the local theory by comparing to manufactured solutions and to cases with crack singularities for which an analytic solution is available. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff–Winkler experiments.

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TYPE Journal Article YEAR 2019
ScopusOSTIDOI
Results 1–25 of 116
Results 1–25 of 116