Grid Vertices Research Articles

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.

An algorithm for ideal multigrid convergence for the steady Euler equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Flows in two-dimensional channels and over airfoils have been computed. Using Gauss–Seidel relaxation with the grid vertices ordered in the flow direction, ideal multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier–Stokes equations.

On the complexity of string folding

A fold of a finite string S over a given alphabet is an embedding of S in some fixed infinite grid, such as the square or cubic mesh. The score of a fold is the number of pairs of matching string symbols which are embedded at adjacent grid vertices. Folds of strings in two- and three-dimensional meshes are considered, and the corresponding problems of optimizing the score or achieving a given target score are shown to be NP-hard.

Oxygen free radicals and cell death

Efficient and accurate simulation of multiphase flow and transport in porous media plays a key role in optimizing recovery of hydrocarbon resources. Two-Point Flux Approximation (TPFA) is widely used in the industry for discretizing the pressure equation because of its robustness and efficiency despite the fact that it introduces O(1) flux errors on non-K-orthogonal grids. Multi-Point Flux Approximation (MPFA) provides a consistent discretization of the flow equations at the cost of larger flux stencils which leads to a denser, asymmetric system of linear equations requiring more computer memory and work for linear solvers. This drawback of MPFA becomes more pronounced for generally unstructured triangular grids and is even worse in 3D applications. In this work, we develop a two-step finite volume method (TSFVM) that is consistent for the anisotropic pressure equation on triangular and tetrahedral grids with improved efficiency as compared to MPFA methods. During each time step, the pressure equation is solved implicitly and saturation is then updated explicitly (IMPES). To discretize the pressure equation with full permeability tensors, Galerkin finite element method (FEM) is employed in the first step to compute pressure at grid vertices. Pressures at grid vertices are then utilized in the second step to derive a two-point flux stencil with a constant for each grid face. Pressure at cell centers and flux across grid faces can then be solved using a TPFA approach. Finally, the flux field at the new time level is used to update saturations explicitly. Although during each time step our TSFVM needs to solve two systems of linear equations, results of the numerical experiments clearly demonstrate that it is still less expensive than the classical MPFA-O method while providing high-quality numerical solutions matching that of MPFA-O. The gained efficiency is especially significant for 3D tetrahedral grid. As the scale of the problem increases, MPFA becomes less appealing because of its large flux stencils that result in much denser matrices while our TSFVM still remains computationally tractable thanks to the flexibility of FEM in the first step and the two-point flux stencil in the second step. On the other hand, the first step of the TSFVM is reminiscent of the Control Volume Finite Element Method (CVFEM) but with important differences. Unlike CVFEM, our TSFVM does not need to construct a dual control volume mesh around grid vertices on the basis of the primary mesh and the saturation unknowns are not associated with node-centered control volumes but are piece-wise constant on grid cells in which reservoir properties are distributed. This ensures that a saturation value does not straddle across material discontinuities which CVFEM suffer from.