Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers

We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order conservation law. A stabilized unsplit perfectly matching layer (PML) formulation is introduced for the resulting nonlinear flow equations. The proposed PML equations exponentially absorb the difference between the nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic time discretization methods to improve the time step restrictions in small relaxation times. We also introduce a multirate semi-analytic Adams-Bashforth method which preserves efficiency in stiff regimes. Accuracy and performance of the method are tested using distinct cases including isothermal vortex, flow around square cylinder, and wall mounted square cylinder test cases.Comment: 37 pages, 11 figure

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Computational micro-flow with spectral element method and high Reynolds number flow with discontinuous Galerkin Finite Element Method

In this dissertation, two numerical methods with high order accuracy, Spectral Element Method (SEM) and Discontinuous Galerkin Finite Element Method (DG-FEM), are chosen to solve problems in Computational Fluid Dynamics (CFD). The merits of these two methods will be discussed and utilized in different kinds of CFD problems. The simulations of the micro-flow systems with complex geometries and physical applications will be presented by SEM. Moreover, the numerical solutions for the Hyperbolic Flow will be obtained by DG-FEM. By solving problems with these two methods, the differences between them will be discussed as well. Compressible Navier-Stokes equations with Electro-osmosis body force and slip boundary conditions are solved to simulate two independent models. The third order Adams-Bashforth method on time splitting, and up to the eighth order SEM on space analysis are utilized in our cases of the electro-osmosis flow (EOF). To solve the body force caused by EOF, simplification of the Poisson-Boltzmann is discussed in details. Results show that SEM can clearly simulate the electric double layers in EOF. Compared with the finite element method, which uses h-refinement to increase resolution, SEM has obvious advantages by using hp-refinement. The other case for SEM is the simulations of drug delivery through the micro needle. The drug flowing inside the needle is treated as a micro-flow system with complex geometry, while the process of drug fluxing in human skin is developed as in the case of CFD problem in porous media. Incompressible Navier-Stokes equations and Darcy-Brinkman equations are solved to simulate the drug flowing inside the needle and diffusing in human skin, respectively. Results are compared with COMSOL simulation, experimental data, and numerical solutions from Smoothed Particle Hydrodynamics (SPH). The high order DG-FEM method is chosen to do research on Hyperbolic Flow

Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation

We present a high order scheme for approximating kinetic equations with stiff relaxation. The objective is to provide efficient methods for solving the underlying system of conservation laws. The construction is based on several ingredients: (i) a high order implicit upwind Discontinuous Galerkin approximation of the kinetic equations with easy-to-solve triangular linear systems; (ii) a second order asymptotic-preserving time integration based on symmetry arguments; (iii) a palindromic composition of the second order method for achieving higher orders in time. The method is then tested at orders 2, 4 and 6. It is asymptotic-preserving with respect to the stiff relaxation and accepts high CFL numbers

Research in Applied Mathematics, Fluid Mechanics and Computer Science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period October 1, 1998 through March 31, 1999

Lattice Boltzmann modeling for shallow water equations using high performance computing

The aim of this dissertation project is to extend the standard Lattice Boltzmann method (LBM) for shallow water flows in order to deal with three dimensional flow fields. The shallow water and mass transport equations have wide applications in ocean, coastal, and hydraulic engineering, which can benefit from the advantages of the LBM. The LBM has recently become an attractive numerical method to solve various fluid dynamics phenomena; however, it has not been extensively applied to modeling shallow water flow and mass transport. Only a few works can be found on improving the LBM for mass transport in shallow water flows and even fewer on extending it to model three dimensional shallow water flow fields. The application of the LBM to modeling the shallow water and mass transport equations has been limited because it is not clearly understood how the LBM solves the shallow water and mass transport equations. The project first focuses on studying the importance of choosing enhanced collision operators such as the multiple-relaxation-time (MRT) and two-relaxation-time (TRT) over the standard single-relaxation-time (SRT) in LBM. A (MRT) collision operator is chosen for the shallow water equations, while a (TRT) method is used for the advection-dispersion equation. Furthermore, two speed-of-sound techniques are introduced to account for heterogeneous and anisotropic dispersion coefficients. By selecting appropriate equilibrium distribution functions, the standard LBM is extended to solve three-dimensional wind-driven and density-driven circulation by introducing a multi-layer LB model. A MRT-LBM model is used to solve for each layer coupled by the vertical viscosity forcing term. To increase solution stability, an implicit step is suggested to obtain stratified flow velocities. Numerical examples are presented to verify the multi-layer LB model against analytical solutions. The model’s capability of calculating lateral and vertical distributions of the horizontal velocities is demonstrated for wind- and density- driven circulation over non-uniform bathymetry. The parallel performance of the LBM on central processing unit (CPU) based and graphics processing unit (GPU) based high performance computing (HPC) architectures is investigated showing attractive performance in relation to speedup and scalability