Browsing by Subject "equation"
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Item Conservative Deterministic Spectral Boltzmann Solver Near The Grazing Collisions Limit(2012-07) Haack, J. R.; Gamba, I. M.; Haack, Jeffrey R.; Gamba, Irene M.We present new results building on the conservative deterministic spectral method for the space homogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. Within this framework we have extended the formulation to the case of more general case of collision operators with anisotropic scattering mechanisms, which requires a new formulation of the convolution weights. We also derive the grazing collisions limit for the method, and show that it is consistent with the Fokker-Planck-Landau equations as the grazing collisions parameter goes to zero.Item A Conservative Discontinuous Galerkin Scheme With O(N-2) Operations In Computing Boltzmann Collision Weight Matrix(2014-07) Gamba, I. M.; Zhang, C. L.; Gamba, Irene M.; Zhang, ChenglongIn the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of >shifting symmetry> in the weight matrix to reduce the computing complexity from theoretical O(N-3) down to O(N-2), with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpetiMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.Item Discontinuous Galerkin Methods For The Boltzmann-Poisson Systems In Semiconductor Device Simulations(2011-07) Cheng, Y. D.; Gamba, I. M.; Majorana, A.; Shu, C. W.; Cheng, Yingda; Gamba, Irene M.We are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be stable and to have good resolution under coarse meshes. In this paper we consider the discontinuous Galerkin (DG) method, which is a finite element method using discontinuous piecewise polynomials as basis functions and numerical fluxes based on upwinding for stability, for solving the Boltzmann-Poisson system. In many situations, the deterministic DG solver can produce accurate solutions with equal or less CPU time than the traditional DSMC (Direct Simulation Monte Carlo) solvers. Numerical simulation results on a diode and a 2D double-gate MOSFET are given.Item Effect of pore structure on capillary condensation in a porous medium(2009-02) Deinert, M. R.; Parlange, J. Y.; Deinert, M. R.; Deinert, M. R.The Kelvin equation relates the equilibrium vapor pressure of a fluid to the curvature of the fluid-vapor interface and predicts that vapor condensation will occur in pores or irregularities that are sufficiently small. Past analyses of capillary condensation in porous systems with fractal structure have related the phenomenon to the fractal dimension of the pore volume distribution. Recent work, however, suggests that porous systems can exhibit distinct fractal dimensions that are characteristic of both their pore volume and the surfaces of the pores themselves. We show that both fractal dimensions have an effect on the thermodynamics that governs capillary condensation and that previous analyses can be obtained as limiting cases of a more general formulation.Item A Fast Conservative Spectral Solver For The Nonlinear Boltzmann Collision Operator(2014-07) Gamba, I. M.; Haack, J. R.; Hu, J. W.; Gamba, Irene M.; Haack, Jeffrey R.We present a conservative spectral method for the fully nonlinear Boltzmann collision operator based on the weighted convolution structure in Fourier space developed by Gamba and Tharkabhushnanam.. This method can simulate a broad class of collisions, including both elastic and inelastic collisions as well as angularly dependent cross sections in which grazing collisions play a major role. The extension presented in this paper consists of factorizing the convolution weight on quadrature points by exploiting the symmetric nature of the particle interaction law, which reduces the computational cost and memory requirements of the method to O(M(2)N(4)logN) from the O(N-6) complexity of the original spectral method, where N is the number of velocity grid points in each velocity dimension and M is the number of quadrature points in the factorization, which can be taken to be much smaller than N. We present preliminary numerical results.Item Fast Hybrid Algorithms For High Frequency Scattering(2009-03) Engquist, B.; Tran, K.; Ying, L. X.; Engquist, BjörnThis paper deals with numerical methods for high frequency wave scattering. It introduces a new hybrid technique that couples a directional fast multipole method for a subsection of it scattering surface to an asymptotic formulation over the rest of the scattering domain. The directional fast multipole method is new and highly efficient for the solution of the boundary integral formulation of a general scattering problem but it requires at least a few unknowns per wavelength on the boundary. The asymptotic method that was introduced by Bruno and collaborators requires much fewer unknowns. Oil the other hand the scattered field must have a simple structure. Hybridization of these two methods retains their best properties for the solution of the full problem, Numerical examples are given for the solution of the Helmholtz equation in two space dimensions.Item Hamiltonian description of the ideal fluid(1998-04) Morrison, Phillip J.; Morrison, Philip J.The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable when nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods.Item High Performance Computing With A Conservative Spectral Boltzmann Solver(2012-07) Haack, J. R.; Gamba, I. M.; Haack, Jeffrey R.; Gamba, Irene M.We present new results building on the conservative deterministic spectral method for the space inhomogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. We extend this method to second order accuracy in space and time, and explore how to leverage the structure of the collisional formulation for high performance computing environments. The locality in space of the collisional term provides a straightforward memory decomposition, and we perform some initial scaling tests on high performance computing resources. We also use the improved computational power of this method to investigate a boundary-layer generated shock problem that cannot be described by classical hydrodynamics.Item The Physics of Crystallization from Globular Cluster White Dwarf Stars in NGC 6397(2009-03) Winget, D. E.; Kepler, S. O.; Campos, Fabiola; Montgomery, M. H.; Girardi, Leo; Bergeron, P.; Williams, Kurtis; Winget, D. E.; Montgomery, M. H.; Williams, KurtisWe explore the physics of crystallization in the deep interiors of white dwarf (WD) stars using the color-magnitude diagram and luminosity function constructed from proper- motion cleaned Hubble Space Telescope photometry of the globular cluster NGC 6397. We demonstrate that the data are consistent with the theory of crystallization of the ions in the interior of WD stars and provide the first empirical evidence that the phase transition is first order: latent heat is released in the process of crystallization as predicted by van Horn. We outline how these data can be used to observationally constrain the value of Gamma = E(Coulomb)/E(thermal) near the onset of crystallization, the central carbon/oxygen abundance, and the importance of phase separation.Item Pore-scale simulation of fluid flow and solute dispersion in three-dimensional porous media(2014-07) Icardi, Matteo; Boccardo, Gianluca; Marchisio, Daniele L.; Tosco, Tiziana; Sethi, Rajandrea; Icardi, MatteoIn the present work fluid flow and solute transport through porous media are described by solving the governing equations at the pore scale with finite-volume discretization. Instead of solving the simplified Stokes equation (very often employed in this context) the full Navier-Stokes equation is used here. The realistic three-dimensional porous medium is created in this work by packing together, with standard ballistic physics, irregular and polydisperse objects. Emphasis is placed on numerical issues related to mesh generation and spatial discretization, which play an important role in determining the final accuracy of the finite-volume scheme and are often overlooked. The simulations performed are then analyzed in terms of velocity distributions and dispersion rates in a wider range of operating conditions, when compared with other works carried out by solving the Stokes equation. Results show that dispersion within the analyzed porous medium is adequately described by classical power laws obtained by analytic homogenization. Eventually the validity of Fickian diffusion to treat dispersion in porous media is also assessed.