Misun Min

A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss-Lobatto-Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge-Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re=400-5000 and flows around an impulsively started cylinder for Re=550-9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.

Fourier spectral simulations and Gegenbauer reconstructions for electromagnetic waves in the presence of a metal nanoparticle

We describe Fourier pseudospectral time-domain simulations, carried out in order to study light interacting with a metallic nanoscale object. The difficulty of using Fourier methods to accurately predict the electromagnetic scattering in such problems arises from the discontinuity in the dielectric function along the surface of the metallic object. Standard Fourier methods lead to oscillatory behavior in approximating solutions that are nonsmooth or that have steep gradients. By applying the Gegenbauer reconstruction technique as a postprocessing method to the Fourier pseudospectral solution, we successfully reduce the oscillations after postprocessing. Our computational results, including comparison with finite-difference time-domain simulations, demonstrate the efficiency and accuracy of the method.

A Spectral Element Method with Transparent Boundary Condition for Periodic Layered Media Scattering

We present a high-order spectral element method for solving layered media scattering problems featuring an operator that can be used to transparently enforce the far-field boundary condition. The incorporation of this Dirichlet-to-Neumann (DtN) map into the spectral element framework is a novel aspect of this work, and the resulting method can accommodate plane-wave radiation of arbitrary angle of incidence. In order to achieve this, the governing Helmholtz equations subject to quasi-periodic boundary conditions are rewritten in terms of periodic unknowns. We construct a spectral element operator to approximate the DtN map, thus ensuring nonreflecting outgoing waves on the artificial boundaries introduced to truncate the computational domain. We present an explicit formula that accurately computes the Fourier coefficients of the solution in the spectral element discretization space projected onto the boundary which is required by the DtN map. Our solutions are represented by the tensor product basis of one-dimensional Legendre–Lagrange interpolation polynomials based on the Gauss–Lobatto–Legendre grids. We study the scattered field in singly and doubly layered media with smooth and nonsmooth interfaces. We consider rectangular, triangular, and sawtooth interfaces that are accurately represented by the body-fitted quadrilateral elements. We use GMRES iteration to solve the resulting linear system, and we validate our results by demonstrating spectral convergence in comparison with exact solutions and the results of an alternative computational method.

An MPI/OpenACC implementation of a high-order electromagnetics solver with GPUDirect communication

We present performance results and an analysis of a message passing interface (MPI)/OpenACC implementation of an electromagnetic solver based on a spectral-element discontinuous Galerkin discretization of the time-dependent Maxwell equations. The OpenACC implementation covers all solution routines, including a highly tuned element-by-element operator evaluation and a GPUDirect gather–scatter kernel to effect nearest neighbor flux exchanges. Modifications are designed to make effective use of vectorization, streaming, and data management. Performance results using up to 16,384 graphics processing units of the Cray XK7 supercomputer Titan show more than 2.5× speedup over central processing unit-only performance on the same number of nodes (262,144 MPI ranks) for problem sizes of up to 6.9 billion grid points. We discuss performance-enhancement strategies and the overall potential of GPU-based computing for this class of problems.

Spectral-element discontinuous Galerkin lattice Boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator

In this paper, a spectral-element discontinuous Galerkin (SEDG) lattice Boltzmann discretization and an exponential time-marching scheme are used to study the flow field past two circular cylinders in tandem arrangement. The basic idea is to discretize the streaming step of the lattice Boltzmann equation by using the SEDG method to get a system of ordinary differential equations (ODEs) whose exact solutions are expressed by using a large matrix exponential. The approximate solution of the resulting ODEs are obtained from a projection method based on a Krylov subspace approximation. This approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension. The exponential time integration scheme is useful especially when computations are carried out at high Courant-Friedrichs-Lewy (CFL) numbers, where most explicit time-marching schemes are inaccurate. Simulations of flow were carried out for a circular cylinder at Re=20 and for two circular cylinders in tandem at Re=40 and a spacing of 2.5D, where D is the diameter of the cylinders. We compare our results with those from a fourth-order Runge-Kutta scheme that is restricted by the CFL number. In addition, important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme. These results are compared with results from our simulation using the RK scheme and with existing benchmark results.

An Efficient High-Order Time Integration Method for Spectral-Element Discontinuous Galerkin Simulations in Electromagnetics

We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell’s equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension

Large-scale electromagnetic modelings based on high-order methods: Nanoscience applications

\kappa \times \kappa

An Operator-Integration-Factor Splitting (OIFS) method for Incompressible Flows in Moving Domains

κ×κ. We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram–Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension