The Cartesian Cut-cell Method in Two Dimensional Flood Simulation

Two dimensional dynamic models have been increasingly used for river flood simulation. This commonly uses satellite remote sensor data, recorded on a rectangular (Raster) grid. There are many important features on a flood plain, such as hedges or buildings, which do not follow the grid lines. Irregular meshes can be used to follow these features, but converting raster data to this format involves a loss of detail. The Cartesian cut-cell (CC) method uses a rectangular mesh. The edges of irregular solid bodies are located precisely with sequences of vertex coordinates. Cut-cells, which lie on the edge, are given special treatment. This allows straightforward integration of grid and vector data, potentially within a GIS based framework. This paper introduces the semi permeable internal (SPIn) boundary cut-cell method. This allows the integration of permeable boundaries, such as hedges, into the model. To explore the impact of these features, a small scale river flood event, over a field featuring a hedgerow, is simulate

Two fluid space-time discontinuous Galerkin finite element method. Part I: numerical algorithm

A novel numerical method for two fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local {\it hp}-refinement. The front tracking is incorporated via cut-cell mesh refinement to ensure a sharp interface between the fluids. To compute the interface dynamics the level set method (LSM) is used because of its ability to deal with merging and breakup. Also, the LSM is easy to extend to higher dimensions. Small cells arising from the cut-cell refinement are merged to improve the stability and performance. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative

A conservative and consistent implicit Cartesian cut-cell method for moving geometries with reduced spurious pressure oscillations

A conservative and consistent three-dimensional Cartesian cut-cell method is presented for reducing the spurious pressure oscillations often observed in moving body simulations in sharp-interface Cartesian grid methods. By analysing the potential sources of the oscillation in the cut-cell framework, an improved moving body algorithm is proposed for the cut-cell method for the temporal discontinuity of the solid volume change. Strict conservation of mass and momentum for both fluid and cut cells is enforced through pressure-velocity coupling to reduce local mass conservation errors. A consistent mass and momentum flux computation is employed in the finite volume method. In contrary to the commonly cut-cell methods, an implicit time integration scheme is employed in the present method, which prevents numerical instability without any additional small cut-cell treatment. The effectiveness of the present cut-cell method for reducing spurious pressure oscillations is demonstrated by simulating various two- and three-dimensional benchmark cases (in-line and transversely oscillating cylinder, oscillating and free-falling sphere), with good agreement with previous experimental measurements and other numerical methods available in the literature

A Moving Boundary Flux Stabilization Method for Cartesian Cut-Cell Grids using Directional Operator Splitting

An explicit moving boundary method for the numerical solution of time-dependent hyperbolic conservation laws on grids produced by the intersection of complex geometries with a regular Cartesian grid is presented. As it employs directional operator splitting, implementation of the scheme is rather straightforward. Extending the method for static walls from Klein et al., Phil. Trans. Roy. Soc., A367, no. 1907, 4559-4575 (2009), the scheme calculates fluxes needed for a conservative update of the near-wall cut-cells as linear combinations of standard fluxes from a one-dimensional extended stencil. Here the standard fluxes are those obtained without regard to the small sub-cell problem, and the linear combination weights involve detailed information regarding the cut-cell geometry. This linear combination of standard fluxes stabilizes the updates such that the time-step yielding marginal stability for arbitrarily small cut-cells is of the same order as that for regular cells. Moreover, it renders the approach compatible with a wide range of existing numerical flux-approximation methods. The scheme is extended here to time dependent rigid boundaries by reformulating the linear combination weights of the stabilizing flux stencil to account for the time dependence of cut-cell volume and interface area fractions. The two-dimensional tests discussed include advection in a channel oriented at an oblique angle to the Cartesian computational mesh, cylinders with circular and triangular cross-section passing through a stationary shock wave, a piston moving through an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil profile.Comment: 30 pages, 27 figures, 3 table

A Second Order Linear Discontinuous Cut-Cell Discretization for the SN Equations in RZ Geometry

In this dissertation we detail the development, implementation, and testing of a new cut-cell discretization for the discrete ordinates form of the neutron transport equation. This method provides an alternative to homogenization for problems containing material interfaces that do not coincide with mesh boundaries. A line is used to represent the boundary between the two materials in a mixed-cell converting a rectangular mixed-cell into two non-orthogonal, homogeneous cut-cells. The linear-discontinuous Galerkin finite element method (LDGFEM) spatial discretization is used on all of the rectangular cells as well as the non-orthogonal sub-cells. We have implemented our new cut-cell method in a test code which has been used to evaluate its performance relative to homogenization. We begin by developing the equations and methods associated with the LDGFEM discretization of the transport equation in RZ geometry for a homogenous orthogonal mesh. Next we introduce cut-cell meshes and develop a modification to the LDGFEM equations to account for material interfaces. We also develop methods to account for meshing errors encountered when representing curved material interfaces with linear cell faces. Finally we present test problems including manufactured solutions, fixed source, and eigenvalue problems for geometries with curvilinear material interfaces. The results of these test problems show the new cut-cell discretization to be second-order convergent for the scalar flux removed from singularities in the solution, as well as significantly more computationally efficient than homogenization