Samuel Scott Collis

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

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.

Unstructured Discontinuous Galerkin For Seismic Inversion

This abstract explores the potential advantages of discontinuous Galerkin (DG) methods for the time-domain inversion of media parameters within the earths interior. In particular, DG methods enable local polynomial refinement to better capture localized geological features within an area of interest while also allowing the use of unstructured meshes that can accurately capture discontinuous material interfaces. This abstract describes our initial findings when using DG methods combined with Runge-Kutta time integration and adjoint-based optimization algorithms for full-waveform inversion. Our initial results suggest that DG methods allow great flexibility in matching the media characteristics (faults, ocean bottom and salt structures) while also providing higher fidelity representations in target regions. Time-domain inversion using discontinuous Galerkin on unstructured meshes and with local polynomial refinement is shown to better capture localized geological features and accurately capture discontinuous-material interfaces. These approaches provide the ability to surgically refine representations in order to improve predicted models for specific geological features. Our future work will entail automated extensions to directly incorporate local refinement and adaptive unstructured meshes within the inversion process.

A "Gaussian Explosion" Seismic Energy Source.

A point seismic energy source, with spatial support given by a three-dimensional (3D) Dirac delta function, is commonly utilized in seismic wave propagation theory and numerical modeling. However, there is increasing interest in spatially-extended sources, perhaps because they constitute more realistic representations of physical seismic sources. Aldridge (2002) summarizes an extensive body of literature dating back to Jeffreys (1931) and Sharpe (1942) treating the classic “pressurized spherical cavity” source. More recently, Blair (2007, 2010) has re-examined and improved Heelan’s (1953) elastic radiation solution for a finite-length explosive column. A common aspect of these sources is that they are mathematically characterized as time-varying boundary conditions applied to the surface of a medium. In this investigation, we adopt the alternative point of view that a spatially-extended source may be considered a body source of seismic waves, and is thus represented by inhomogeneous terms in the governing system of partial differential equations. In effect, the source energy is instantaneously “deposited” within a portion of the medium supporting wave propagation. This may be a reasonable approximation in certain seismic source scenarios, as with large chemical or nuclear explosions.