We present a general homogenization method a periodic heterogeneous material with piecewise constants for diffusion, heat conduction, and wave propagation in The method is relevant to the frequently encountered upsca...We present a general homogenization method a periodic heterogeneous material with piecewise constants for diffusion, heat conduction, and wave propagation in The method is relevant to the frequently encountered upscaling issues for heterogeneous materials. The dispersion relation for each problem is first expressed in the general form where the frequency co （or wavenumber k） is expanded in terms of the wavenumber k （or frequency ω）. A general homogenization model can be directly obtained with any given dispersion relation. Next step we study the unit cell of the heterogeneous material and derive the exact dispersion relation. The final homogenized equations include both leading order terms （effective properties） and high order contributions that represent the effect of the microscopic heterogeneity on the macroscopic behavior. That effect can be lumped into a single dimensionless heterogeneity parameter β, which is bounded between -1/12≤β≤ 0 and has a universal expression for all three problems. Numerical examples validate the proposed method and demonstrate a significant computational saving.展开更多
The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectra...The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.展开更多
In this study,we present a new numerical model for crystal growth in a vertical solidification system.This model takes into account the buoyancy induced convective flow and its effect on the crystal growth process.The...In this study,we present a new numerical model for crystal growth in a vertical solidification system.This model takes into account the buoyancy induced convective flow and its effect on the crystal growth process.The evolution of the crystal growth interface is simulated using the phase-field method.A semi-implicit lattice kinetics solver based on the Boltzmann equation is employed to model the unsteady incompressible flow.This model is used to investigate the effect of furnace operational conditions on crystal growth interface profiles and growth velocities.For a simple case of macroscopic radial growth,the phase-field model is validated against an analytical solution.The numerical simulations reveal that for a certain set of temperature boundary conditions,the heat transport in the melt near the phase interface is diffusion dominant and advection is suppressed.展开更多
文摘We present a general homogenization method a periodic heterogeneous material with piecewise constants for diffusion, heat conduction, and wave propagation in The method is relevant to the frequently encountered upscaling issues for heterogeneous materials. The dispersion relation for each problem is first expressed in the general form where the frequency co （or wavenumber k） is expanded in terms of the wavenumber k （or frequency ω）. A general homogenization model can be directly obtained with any given dispersion relation. Next step we study the unit cell of the heterogeneous material and derive the exact dispersion relation. The final homogenized equations include both leading order terms （effective properties） and high order contributions that represent the effect of the microscopic heterogeneity on the macroscopic behavior. That effect can be lumped into a single dimensionless heterogeneity parameter β, which is bounded between -1/12≤β≤ 0 and has a universal expression for all three problems. Numerical examples validate the proposed method and demonstrate a significant computational saving.
基金Research was supported in part by NSF grant DMS-0800612Research was supported by Applied Mathematics program of the US DOE Office of Advanced Scientific Computing ResearchThe Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830
文摘The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.
基金supported by the Nonproliferation Research and Engineering(NA-22)program and the Applied Mathematics program of the US DOE Office of Advanced Scientific Computing Research.Computations were performed using the computational resources of the National Energy Research Scientific Computing Center at Lawrence Berkeley National Laboratory and the William R.Wiley Environmental Molecular Sciences Laboratory(EMSL).
文摘In this study,we present a new numerical model for crystal growth in a vertical solidification system.This model takes into account the buoyancy induced convective flow and its effect on the crystal growth process.The evolution of the crystal growth interface is simulated using the phase-field method.A semi-implicit lattice kinetics solver based on the Boltzmann equation is employed to model the unsteady incompressible flow.This model is used to investigate the effect of furnace operational conditions on crystal growth interface profiles and growth velocities.For a simple case of macroscopic radial growth,the phase-field model is validated against an analytical solution.The numerical simulations reveal that for a certain set of temperature boundary conditions,the heat transport in the melt near the phase interface is diffusion dominant and advection is suppressed.
