Mini-Workshop: Numerical Upscaling for Flow Problems: Theory and Applications

The objective of this workshop was to bring together researchers working in multiscale simulations with emphasis on multigrid methods and multiscale finite element methods, aiming at chieving of better understanding and synergy between these methods. The goal of multiscale finite element methods, as upscaling methods, is to compute coarse scale solutions of the underlying equations as accurately as possible. On the other hand, multigrid methods attempt to solve fine-scale equations rapidly using a hierarchy of coarse spaces. Multigrid methods need “good” coarse scale spaces for their efficiency. The discussions of this workshop partly focused on approximation properties of coarse scale spaces and multigrid convergence. Some other presentations were on upscaling, domain decomposition methods and nonlinear multiscale methods. Some researchers discussed applications of these methods to reservoir simulations, as well as to simulations of filtration, insulating materials, and turbulence

New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries

The field of mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a flourishing area of research. Many such systems arise from mathematical models in material science, fluid dynamics and biology, for example phase separation in alloys, epitaxial growth, dynamics of multiphase fluids, evolution of cell membranes and in industrial processes such as crystal growth. The governing equations for the dynamics of the interfaces in many of these applications involve surface tension expressed in terms of the mean curvature and a driving force. Here the forcing terms depend on variables that are solutions of additional partial differential equations which hold either on the interface itself or in the surrounding bulk regions. Often in applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimization problems with partial differential equation constraints including free boundaries. Because of the maturity of the field of computational free boundary problems it is now timely to consider such control problems

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Computer Science Research Institute 2004 annual report of activities.

This report summarizes the activities of the Computer Science Research Institute (CSRI) at Sandia National Laboratories during the period January 1, 2004 to December 31, 2004. During this period the CSRI hosted 166 visitors representing 81 universities, companies and laboratories. Of these 65 were summer students or faculty. The CSRI partially sponsored 2 workshops and also organized and was the primary host for 4 workshops. These 4 CSRI sponsored workshops had 140 participants--74 from universities, companies and laboratories, and 66 from Sandia. Finally, the CSRI sponsored 14 long-term collaborative research projects and 5 Sabbaticals

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

Development of a Navier-Stokes algorithm for parallel-processing supercomputers

An explicit flow solver, applicable to the hierarchy of model equations ranging from Euler to full Navier-Stokes, is combined with several techniques designed to reduce computational expense. The computational domain consists of local grid refinements embedded in a global coarse mesh, where the locations of these refinements are defined by the physics of the flow. Flow characteristics are also used to determine which set of model equations is appropriate for solution in each region, thereby reducing not only the number of grid points at which the solution must be obtained, but also the computational effort required to get that solution. Acceleration to steady-state is achieved by applying multigrid on each of the subgrids, regardless of the particular model equations being solved. Since each of these components is explicit, advantage can readily be taken of the vector- and parallel-processing capabilities of machines such as the Cray X-MP and Cray-2

Theoretical assessment of the influence of mesa size and shape on the two-dimensional electron gas properties of AlGaN/GaN heterojunctions

AlGaN/GaN heterostructure field-effect transistors (HFETs) are strong candidates for high-power and high-frequency applications. Even in the absence of doping, thanks to high polarization fields, often a two-dimensional electron gas (2DEG) of unprecedented concentrations forms at these heterojunctions. Control over this carrier induction process is crucial in achieving normally-off field-effect transistors (i.e., transistors of zero standby power consumption). One way to achieve this is through polarization engineering. Mesa-isolation geometry seemingly offers interesting avenues to reduce the piezoelectric polarization at the heterointerface, and as a result means for polarization engineering. Using a Poisson-Schrödinger self-consistent solver, the effect of strain on the sheet charge density is investigated in the context of one-, two- and three-dimensional simulations of AlGaN/GaN heterostructures. Properties of the two-dimensional electron gas are detailed and the influences of Aluminum mole fraction, AlGaN barrier thickness, GaN cap layer inclusion are investigated. The carrier confinement in the 2DEG is explored in the case of two-dimensional version of the simulations. Through these studies, the effect of shrinking the size of the mesa on lowering the 2DEG concentration is confirmed. Through performing three-dimensional simulations, the effects of cross-sectional geometry on the average sheet charge density and the threshold voltage are presented. It is shown that as the perimeter-to-area ratio is increased, the carrier concentration decreases, and the threshold voltage becomes less negative. Via these studies, the degree of effectiveness of geometry as means for polarization engineering is, for the first time, theoretically quantified

Matrix-free time-domain methods for general electromagnetic analysis

Many engineering challenges demand an efficient computational solution of large-scale problems. If a computational method can be made free of matrix solutions, then it has a potential of solving very large scale problems. Among existing computational electromagnetic methods, the explicit finite-difference time-domain (FDTD) method is free of matrix solutions. However, it requires a structured orthogonal grid for space discretization. In this work, we develop a new time-domain method that naturally requires no matrix solution, regardless of whether the discretization is a structured grid or an unstructured mesh. No dual mesh, interpolation, projection and mass lumping are needed. Furthermore, a time-marching scheme is developed to ensure the stability for simulating an unsymmetrical numerical system, while preserving the matrix-free merit of the proposed method. This time-marching scheme is then made unconditionally stable, and hence allowing for the use of an arbitrarily large time step without sacrificing the matrix-free property. Extensive numerical experiments have been carried out on a variety of two- and three-dimensional unstructured meshes and even mixed-element meshes. Correlations with analytical solutions and the results obtained from the time-domain finite-element method have validated the accuracy, matrix-free property, stability, and generality of the proposed method.^ In addition to an extensive development of the proposed method in arbitrary 2- and 3-D unstructured meshes, we have also made a connection between the proposed new method and the classical FDTD method. We have found that the proposed matrix-free method naturally reduces to the FDTD method in an orthogonal grid. It also results in a new patch-based single-grid formulation of the FDTD algorithm. This new formulation not only makes the implementation of the original FDTD much easier, but also reveals a natural rank-1 decomposition of the curl-curl operator. Such a representation leads to an efficient extraction of unstable eigenmodes from fine cells only, from which a fast explicit and unconditionally stable FDTD method is developed. In addition, to efficiently handle multiscale structures, we develop an accurate FDTD subgridding algorithm suitable for arbitrary subgridding settings with arbitrary contrast ratios between the normal gird and the subgrid. Although the resulting system matrix is unsymmetric, we develop a time marching method to overcome the stability problem without sacrificing the matrix-free merit of the original FDTD. This method is general, which is also applicable to other subgridding algorithms whose underlying numerical systems are unsymmetric. The proposed FDTD subgridding algorithm is then further made unconditionally stable, thus permitting the use of a time step independent of space step.^ Last but not the least, the framework of the proposed method can be flexibly extended to solve partial differential equations in other disciplines, which we have demonstrated for thermal analysis