Khosro Shahbazi

Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations

Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The algorithms are based on coupling both p- and h-multigrid (ph-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton-Krylov method. The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton-GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem. It is concluded that the Newton-GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.

Second order accurate volume tracking based on remapping for triangular meshes

This paper presents a second order accurate piecewise linear volume tracking based on remapping for triangular meshes. This approach avoids the complexity of extending unsplit second order volume of fluid algorithms, advection methods, on triangular meshes. The method is based on Lagrangian-Eulerian (LE) methods; therefore, it does not deal with edge fluxes and corner fluxes, flux corrections, as is typical in advection algorithms. The method is constructed of three parts: a Lagrangian phase, a reconstruction phase and a remapping phase. In the Lagrangian phase, the original, Eulerian, grid is projected along trajectories to obtain Lagrangian grids. In practice, this projection is handled through the time integration of velocity field for grid vertices at each time step. The reconstruction is based on truncating the volume material polygon for each Lagrangian mixed grid. Since in piecewise linear approximation, the interface is represented by a segment line, the polygon material truncation is mainly finding the segment interface. Finding the segment interface is calculating the line normal and line constant at each multi-fluid cell. Details of applying two normal calculation methods, differential and geometric least squares (GLS) methods, are given. While the GLS method exhibits second order accurate approximation in reproducing circular interfaces, the differential least squares (DLS) method results in a first order accurate representation of the interface. The last part of the algorithm which is remapping of the volume materials from the Lagrangian grid to the original one is performed by a series of polygon intersection procedures. The behavior of the algorithm is investigated for flow fields with constant interface topology and flow fields inducing large interfacial stretching and tearing. Second order accuracy is obtained if the velocity integration as well as the reconstruction steps are at least second order accurate.

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319-338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC-WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.

Progress in High-Order Discontinuous Galerkin Methods for Aerospace Applications

This paper covers progress in developing and applying high-order accurate discretizations based on the Discontinuous Galerkin approach to aerodynamic applications. The paper concentrates on various specific areas including discretization of diusion terms, shock capturing, and multigrid solution methods. The development of a fully conservative Arbitrary Lagrangian Eulerian formulation which obeys the Geometric Conservation Law while maintaining the design order of accuracy of the static mesh discretization is also presented. Additionally, adjoint-based sensitivity analysis for design optimization as well as error estimation and adaptive control is demonstrated. Finally, prospects for future advancements and eciency gains over traditional finite volume methods are discussed in a concluding section.

Multi-dimensional hybrid Fourier continuation-WENO solvers for conservation laws

We introduce a multi-dimensional point-wise multi-domain hybrid Fourier-Continuation/WENO technique (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and essentially dispersionless, spectral, solution away from discontinuities, as well as mild CFL constraints for explicit time stepping schemes. The hybrid scheme conjugates the expensive, shock-capturing WENO method in small regions containing discontinuities with the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [A. Harten, Adaptive multiresolution schemes for shock computations, J. Comput. Phys. 115 (1994) 319-338]. We consider a WENO scheme of formal order nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws are investigated for problems with both smooth and non-smooth solutions. The Euler equations for gas dynamics are solved for the Mach 3 and Mach 1.25 shock wave interaction with a small, plain, oblique entropy wave using the hybrid FC-WENO, the pure WENO and the hybrid central difference-WENO (CD-WENO) schemes. We demonstrate considerable computational advantages of the new FC-based method over the two alternatives. Moreover, in solving a challenging two-dimensional Richtmyer-Meshkov instability (RMI), the hybrid solver results in seven-fold speedup over the pure WENO scheme. Thanks to the multi-domain formulation of the solver, the scheme is straightforwardly implemented on parallel processors using message passing interface as well as on Graphics Processing Units (GPUs) using CUDA programming language. The performance of the solver on parallel CPUs yields almost perfect scaling, illustrating the minimal communication requirements of the multi-domain strategy. For the same RMI test, the hybrid computations on a single GPU, in double precision arithmetics, displays five- to six-fold speedup over the hybrid computations on a single CPU. The relative speedup of the hybrid computation over the WENO computations on GPUs is similar to that on CPUs, demonstrating the advantage of hybrid schemes technique on both CPUs and GPUs.

CFD Analysis of the Effect of Porosity, Quantity and Emanating Power Variation on Gas Emissions in Block/Panel Cave Mines

Block/panel caving is an underground mining method that uses the gravity for mining massive, deep ore deposits. Since caving is a dynamic process, the design of a ventilation system for block/panel cave mines is a challenging task, especially when the ore body contains uranium-bearing mineralization, where radon gas is a major concern for mining operations. This study utilizes a continuum-based computational fluid dynamics (CFD) approach to investigate the effect of changing cave porosity (O), air quantity (Q), and radon emanating power (B) on radon daughter emissions from a cave in a block/panel cave mine. The aim of the study is to predict radon daughter concentrations in the production drifts based on the quantity supplied to the drift, emanating power of ore and porosity of the cave. The predicted radon daughter concentrations can be useful data for mine ventilation engineers in designing an effective ventilation system for block/panel cave mines.

An Investigation of the Effects of Particle Size, Porosity, and Cave Size on the Airflow Resistance of a Block/Panel Cave

Block/panel caving is a preferred underground mining method for extracting deep seated, low-grade mineral deposits due to its high production rates and low mining costs. In a typical block/panel cave mine, cave zone or cave column is an important part of mine ventilation system, although traditional mine ventilation practices fail to estimate its airflow resistance due to the dynamic nature of caving process. In this study, a combination of flow through porous media and VENTSIM modeling approaches were used first to estimate the airflow resistance of a conduit/column filled with hollow plastic balls and then to predict the airflow resistance of a cave zone/cave column. This study also investigates the effects of broken rock size (particle size), porosity, and cave height and volume on the airflow resistance of a cave zone/cave column in a typical block/panel cave mine using a lab scale physical model.

Robust second-order scheme for multi-phase flow computations

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

Volume Tracking on Triangular Meshes

This paper investigates a volume tracking algorithm on triangular meshes, associated with modeling free surface flow and multi phase flow on complex geometry. The volume tracking algorithm is divided into two main tasks. First, a piecewise linear reconstructing the interface is used to approximate the interface at each interfacial cell with a segment line. The convergence study of this approach reveals first order accuracy in reproducing a circular material distribution. Second, a Lagrangian time integration accompanied by a full remap is carried out in order to advect volume material in time. A circular material distribution is tested under simple translation and rotation. Visualization of results shows that the circular shape is conserved while first order accuracy is observed.Copyright