Wednesdays at 4 PM in Eckhart 202.
We introduce a multiscale finite element method for computing flow and transport in strongly heterogeneous porous media which contain many spatial scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all the small scale features. This is accomplished by constructing the multiscale finite element base functions that incorporate local microstructures of the differential operator. Our method is applicable to general multiple-scale problems without restrictive assumptions on scale separation and periodicity. Convergence of our method has been established in the case of periodic oscillatory structures. The rate of convergence is shown to be independent of the small scales of the solution. We demonstrate the accuracy and robustness of our method through extensive numerical experiments, which include two-phase flows with strongly shear random permeability, wave propagation through heterogeneous media, and convection enhanced diffusion. Steady conduction through fiber composites and flows through random media with normal and fractal porosity distributions will also be considered. Parallel implementation and performance of the method will be addressed.
Moving mesh methods often work very well in practice, but our understanding of why they do is very incomplete. I will present some advances in they theory of such methods. We start with a curious method gotten by restricting Miller's MFE method so that mesh movement is the only flexibility that we have for solving a diffusion problem. For special case of the one-dimensional heat equation we see that mesh movement alone can model diffusion. Then we study moving mesh Galerkin method for advection-diffusion equations to obtain some error estimates which are independent of the bound of advection, i.e., mesh movement can also model advection. The techniques used here are related to characteristic-based methods. New symmetric error estimates are presented in this setting. If time allows, I will briefly talk about some results on moving mesh mixed method.
Computations of Marangoni-B\'enard convection are usually performed in two- or three-dimensional domains with rigid boundaries. Free surface deformations can result in qualitatively different behaviour. Bifurcations that arise in two-dimensional domains with a deformable free surface can be computed using the finite-element method, by combining an orthogonal mapping of the physical domain with extended system techniques for locating singularities. The fluid is assumed to be Newtonian, conform to the Boussinesq approximation and to have a surface tension that varies linearly with temperature. Contact angles other than 90 degrees will be shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner. The loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to, and less than, 90 degrees. Even small deviations from 90 degree contact angles result in surprisingly large changes in the critical Marangoni numbers at which bifurcation occurs.
Computations of slowly moving shocks by shock capturing schemes are often contaminated by oscillations that appear like a long wavy tail attached to the moving shock front. These oscillations are undesirable not only for aesthetic reasons but also, for example, because they inhibit convergence of transient solutions to their steady limit. They are generated even by first order schemes but become more pronounced in higher order schemes due to their lower dissipation. For first order schemes, these disturbances may be understood in the context of viscous shock profiles and the relevant theories. The results also bear on higher order shock-capturing schemes, since they all reduce to first order near shock fronts in order to maintain monotonicity. The phenomenon will be discussed and results will shown.
Radiation transport (energy or particles) is an important effect in simulations of fires, stars, and nuclear reactors and weapons. The method of discrete ordinates is commonly used to solve the Boltzmann transport equation. This leads to very large-scale computations since 3 dimensions are added to every spatial grid point -- 2 angular and 1 energy. Numerically, these techniques raise issues for gridding and discretization strategies, iterative vs direct solvers, and methods for accelerating convergence. Parallel computation poses additional challenges because of directional and long-range effects. In this talk I'll highlight some of the interesting questions that arise in discrete ordinates transport from a numerical and computer science perspective. I'll discuss in some detail a new method we're pursuing at Sandia, to develop parallel direct solvers for radiation on unstructured grids, including some unsolved problems we have. And I'll show some performance results for how these computations scale on the large ASCI machines at the DOE labs and what implications this has for future trends in supercomputer architectures.
In this talk, I will discuss the simulation of shallow water systems (bays, estuaries, etc.). These systems are environmentally sensitive and economically important. Consequently there has been a significant effort in the numerical approximation of these problems for a number of years. I will describe some of the standard models and also discuss some of their plusses and minuses. In particular, I will focus on a finite element model called ADCIRC (Advanced Circulation Model). I will also describe a new finite element approach that we have developed, and compare the models. Also of interest is the implementation of these models on distributed memory computers; I will discuss a parallel implementation of the ADCIRC model. The second part of the talk will focus on the coupling of these hydrodynamic flow models with water quality codes for modeling transport of contaminants, sediments, etc., through the domain.
Crinoids (echinoderms related to starfish and sea urchins) use an array of fine hair-like structures (tube feet, 50 Ám dia) to catch particles from the water. The only force driving fluid through this biological filter is the ambient currents. The tube feet are borne in a regular array on a branch of the animal's arms (called a pinnule). Re for the tube feet is about 1, for the pinnule about 100. The animals always feed with the tube feet on the downstream side of the pinnule; the tube feet are curved in the upstream direction; the tube foot arrays on adjacent pinnules are always separated by a gap of about 200 Ám. Using physical models 15x life size in a high viscosity fluid at in vivo Reynolds numbers, flow through an array of tube feet was quantified. The results imply unique mechanisms for moving fluids through feeding arrays at low to intermediate Reynolds numbers and suggest that at least some of these behaviors function to maximize flow through the tube foot array. These results are generalizable to all crinoids (which have a 500 million year fossil record) and possibly to a wide variety of passive suspension feeding animals; the mechanisms may also prove useful in the design of miniturized chemical sensors.
Molecular recognition and other biological reactions frequently occur at the surface regions of proteins. Proteins fold to adopt specific shapes to provide such surface regions with the necessary physicochemical microenvironment. Analysis of protein surface regions is becoming increasingly important for understanding protein functions, such as ligand binding and drug design. We describe a novel computational approach for rapid and analytical treatment of protein shapes. This method is based on weighted Delaunay simplicial complex and Voronoi diagram, as well as alpha shapes from computational geometry and computational topology. We describe how this method is applied to the molecular electrostatics problems by solving boundary integral version of the linearized Poisson-Boltzmann equation. A method for automated surface pockets and binding sites identification, as well as their analytical metric characterization will also be discussed. As an example of applications in drug discovery, we present a strategy for identifying target proteins for linked-fragment drug design method such as SAR by NMR. New findings about the relationship between protein packing and stability (thermophile vs. mesophile), as well as enzyme solvation changes will be discussed. Finally, we describe a new set of molecular descriptors based on alpha shapes and its validation for combinatorial drug discovery and chemical diversity analysis. These descriptors are advantageous for identifying novel drug leads.
1. Dynamical Systems and Neural Systems I: Death of Periodicity; 2. Dynamical Systems and Neural Systems II: Mapping Parameter Spaces; 3. Computing Periodic Orbits of Vector Fields
An efficient and accurate new numerical method for solving general moving interface problems such as dendritic solidification is presented. The method combines a re-initialized level set approach and a general velocity extension with an explicit unconditionally stable time stepping scheme. It merges moving interfaces naturally with large time steps, and builds an adaptive tree mesh to achieve almost optimal efficiency. An efficient new re-initializing technique based on trees enhances the method. Analysis of a linear one-dimensional model problem suggests an unexpected convergence criterion which is verified by two-dimensional numerical results. Further results show that complex merging interfaces are computed accurately and efficiently.
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