## Scalable climate simulations using Trilinos solvers

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Sprinter Lecture Notes

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Predictive simulation of systems comprised of numerous interconnected, tightly coupled com-ponents promises to help solve many problems of scientific and national interest. Howeverpredictive simulation of such systems is extremely challenging due to the coupling of adiverse set of physical and biological length and time scales. This report investigates un-certainty quantification methods for such systems that attempt to exploit their structure togain computational efficiency. The traditional layering of uncertainty quantification aroundnonlinear solution processes is inverted to allow for heterogeneous uncertainty quantificationmethods to be applied to each component in a coupled system. Moreover this approachallows stochastic dimension reduction techniques to be applied at each coupling interface.The mathematical feasibility of these ideas is investigated in this report, and mathematicalformulations for the resulting stochastically coupled nonlinear systems are developed.3

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Singlet oxygen generators are multiphase flow chemical reactors used to generate energetic oxygen to be used as a fuel for chemical oxygen iodine lasers. In this paper, a theoretical model of the generator is presented along with its solutions over ranges of parameter space and oxygen maximizing optimizations. The singlet oxygen generator (SOG) is a low-pressure, multiphase flow chemical reactor that is used to produce molecular oxygen in an electronically excited state, i.e. singlet delta oxygen. The primary product of the reactor, the energetic oxygen, is used in a stage immediately succeeding the SOG to dissociate and energize iodine. The gas mixture including the iodine is accelerated to a supersonic speed and lased. Thus the SOG is the fuel generator for the chemical oxygen iodine laser (COIL). The COIL has important application for both military purposes--it was developed by the US Air Force in the 1970s--and, as the infrared beam is readily absorbed by metals, industrial cutting and drilling. The SOG appears in various configurations, but the one in focus here is a crossflow droplet generator SOG. A gas consisting of molecular chlorine and a diluent, usually helium, is pumped through a roughly rectangular channel. An aqueous solution of hydrogen peroxide and potassium hydroxide is pumped through small holes into the channel and perpendicular to the direction of the gas flow. So doing causes the solution to become aerosolized. Dissociation of the potassium hydroxide draws a proton from the hydrogen peroxide generating an HO{sub 2} radical in the liquid. Chlorine diffuses into the liquid and reacts with the HO{sub 2} ion producing the singlet delta oxygen; some of the oxygen diffuses back into the gas phase. The focus of this work is to generate a predictive multiphase flow model of the SOG in order to optimize its design. The equations solved are the so-called Eulerian-Eulerian form of the multiphase flow Navier-Stokes equations wherein one set of the equations represents the gas phase and another equation set of size m represents the liquid phase. In this case, m is representative of the division of the liquid phase into distinct representations of the various droplet sizes distributed in the reactor. A stabilized Galerkin formulation is used to solve the equation set on a computer. The set of equations is large. There are five equations representing the gas phase: continuity, vector momentum, heat. There are 5m representing the liquid phase: number density, vector momentum, heat. Four mass transfer equations represent the gas phase constituents and there are m advection diffusion equations representing the HO{sub 2} ion concentration in the liquid phase. Thus we are taking advantage of and developing algorithms to harness the power of large parallel computing architectures to solve the steady-state form of these equations numerous times so as to explore the large parameter space of the equations via continuation methods and to maximize the generation of singlet delta oxygen via optimization methods. Presented here will be the set of equations that are solved and the methods we are using to solve them. Solutions of the equations will be presented along with solution paths representing varying aerosol loading-the ratio of liquid to gas mass flow rates-and simple optimizations centered around maximizing the oxygen production and minimizing the amount of entrained liquid in the gas exit stream. Gas-entrained liquid is important to minimize as it can destroy the lenses and mirrors present in the lasing cavity.

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Computers and Mathematics with Applications

We will discuss a parametric study of the solution of the Wigner-Poisson equations for resonant tunneling diodes. These structures exhibit self-sustaining oscillations in certain operating regimes. We will describe the engineering consequences of our study and how it is a significant advance from some previous work, which used much coarser grids. We use LOCA and other packages in the Trilinos framework from Sandia National Laboratory to enable efficient parallelization of the solution methods and to perform bifurcation analysis of this model. We report on the parallel efficiency and scalability of our implementation. © 2006 Elsevier Ltd. All rights reserved.

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3rd M.I.T. Conference on Computational Fluid and Solid Mechanics

The critical Rayleigh number Racr of the Hopf bifurcation that signals the limit of steady flows in a differentially heated 8:1:1 cavity is computed. The two-dimensional analog of this problem was the subject of a comprehensive set of benchmark calculations that included the estimation of Racr [1]. In this work we begin to answer the question of whether the 2D results carry over into 3D models. For the case of the 2D model being extruded for a depth of 1, and no-slip/no-penetration and adiabatic boundary conditions placed at these walls, the steady flow and destabilizing eigenvectors qualitatively match those from the 2D model. A mesh resolution study extending to a 20-million unknown model shows that the presence of these walls delays the first critical Rayleigh number from 3.06 × 105 to 5.13 × 105. © 2005 Elsevier Ltd.

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Proposed for publication in Computer Methods in Applied Mechanics and Engineering.

The solution of the governing steady transport equations for momentum, heat and mass transfer in fluids undergoing non-equilibrium chemical reactions can be extremely challenging. The difficulties arise from both the complexity of the nonlinear solution behavior as well as the nonlinear, coupled, non-symmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this paper, we briefly review progress on developing a stabilized finite element (FE) capability for numerical solution of these challenging problems. The discussion considers the stabilized FE formulation for the low Mach number Navier-Stokes equations with heat and mass transport with non-equilibrium chemical reactions, and the solution methods necessary for detailed analysis of these complex systems. The solution algorithms include robust nonlinear and linear solution schemes, parameter continuation methods, and linear stability analysis techniques. Our discussion considers computational efficiency, scalability, and some implementation issues of the solution methods. Computational results are presented for a CFD benchmark problem as well as for a number of large-scale, 2D and 3D, engineering transport/reaction applications.

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Proposed for publication in Computation Methods in Applied Mechanics and Engineering.

The solution of the governing steady transport equations for momentum, heat and mass transfer in fluids undergoing non-equilibrium chemical reactions can be extremely challenging. The difficulties arise from both the complexity of the nonlinear solution behavior as well as the nonlinear, coupled, non-symmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this paper, we briefly review progress on developing a stabilized finite element ( FE) capability for numerical solution of these challenging problems. The discussion considers the stabilized FE formulation for the low Mach number Navier-Stokes equations with heat and mass transport with non-equilibrium chemical reactions, and the solution methods necessary for detailed analysis of these complex systems. The solution algorithms include robust nonlinear and linear solution schemes, parameter continuation methods, and linear stability analysis techniques. Our discussion considers computational efficiency, scalability, and some implementation issues of the solution methods. Computational results are presented for a CFD benchmark problem as well as for a number of large-scale, 2D and 3D, engineering transport/reaction applications.

Results 101–125 of 135