Chohong Min

A second order accurate level set method on non-graded adaptive cartesian grids

We present a level set method on non-graded adaptive Cartesian grids, i.e. grids for which the ratio between adjacent cells is not constrained. We use quadtree and octree data structures to represent the grid and a simple algorithm to generate a mesh with the finest resolution at the interface. In particular, we present (1) a locally third order accurate reinitialization scheme that transforms an arbitrary level set function into a signed distance function, (2) a second order accurate semi-Lagrangian methods to evolve the linear level set advection equation under an externally generated velocity field, (3) a second order accurate upwind method to evolve the non-linear level set equation under a normal velocity as well as to extrapolate scalar quantities across an interface in the normal direction, and (4) a semi-implicit scheme to evolve the interface under mean curvature. Combined, we obtain a level set method on adaptive Cartesian grids with a negligible amount of mass loss. We propose numerical examples in two and three spatial dimensions to demonstrate the accuracy of the method.

Geometric integration over irregular domains with application to level-set methods

We present a geometric approach for calculating integrals over irregular domains described by a level-set function. This procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms. This approach produces results that are second-order accurate and robust to the perturbation of the interface location on the grid. Moreover, since we use a cell-wise approach, this procedure can be easily extended to quadtree and octree grids. We demonstrate the second-order accuracy and the robustness of the method in two and three spatial dimensions.

An efficient fluid-solid coupling algorithm for single-phase flows

We present a simple and efficient fluid-solid coupling method in two and three spatial dimensions. In particular, we consider the numerical approximation of the Navier-Stokes equations on irregular domains and propose a novel approach for solving the Hodge projection step on arbitrary shaped domains. This method is straightforward to implement and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity. We demonstrate the accuracy of our method in the L^1 and L^~ norms and present its removing the errors associated with the conventional rasterization-type discretizations. We apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces. We also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate Reynolds number to reproduce the known steady axisymmetric and non-axisymmetric flow regimes. We further apply this algorithm to the coupling of flows with moving rigid bodies.

A second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive grids

We present an unconditionally stable second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive Cartesian grids. We employ quadtree and octree data structures as an efficient means to represent the grid. We use the supra-convergent Poisson solver of C.-H. Min, F. Gibou, H. Ceniceros, A supra-convergent finite difference scheme for the variable coefficient Poisson equation on fully adaptive grids, CAM report 05-29, J. Comput. Phys. (in press)], a second order accurate semi-Lagrangian method to update the momentum equation, an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non-graded grids. We sample all the variables at the grid nodes, producing a scheme that is straightforward to implement. We propose two and three-dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the L1 and L∞ norms.

A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids

We introduce a method for solving the variable coefficient Poisson equation on non-graded Cartesian grids that yields second order accuracy for the solutions and their gradients. We employ quadtree (in 2D) and octree (in 3D) data structures as an efficient means to represent the Cartesian grid, allowing for constraint-free grid generation. The schemes take advantage of sampling the solution at the nodes (vertices) of each cell. In particular, the discretization at one cells node only uses nodes of two (2D) or three (3D) adjacent cells, producing schemes that are straightforward to implement. Numerical results in two and three spatial dimensions demonstrate supra-convergence in the L∞ norm.

A parallel fast sweeping method for the Eikonal equation

We present an algorithm for solving in parallel the Eikonal equation. The efficiency of our approach is rooted in the ordering and distribution of the grid points on the available processors; we utilize a Cuthill-McKee ordering. The advantages of our approach is that (1) the efficiency does not plateau for a large number of threads; we compare our approach to the current state-of-the-art parallel implementation of Zhao (2007) [14] and (2) the total number of iterations needed for convergence is the same as that of a sequential implementation, i.e. our parallel implementation does not increase the complexity of the underlying sequential algorithm. Numerical examples are used to illustrate the efficiency of our approach.

Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions

We present a robust second-order accurate method for discretizing the multi-dimensional Heaviside and the Dirac delta functions on irregular domains. The method is robust in the following ways: (1) integrations of source terms on a co-dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domains boundary; (2) the method depends only on the function value of a level function, not on its derivatives. We present the discretizations in tabulated form to make their implementations straightforward. We present numerical results in two and three spatial dimensions to demonstrate the second-order accuracy in the L^1-norm in the case of the solution of PDEs with singular source terms. In the case of evaluating the contribution of singular source terms on interfaces, the method is also second-order accurate in the L^~-norm.

Efficient symmetric positive definite second-order accurate monolithic solver for fluid/solid interactions

We introduce a robust and efficient method to simulate strongly coupled (monolithic) fluid/rigid-body interactions. We take a fractional step approach, where the intermediate state variables of the fluid and of the solid are solved independently, before their interactions are enforced via a projection step. The projection step produces a symmetric positive definite linear system that can be efficiently solved using the preconditioned conjugate gradient method. In particular, we show how one can use the standard preconditioner used in standard fluid simulations to precondition the linear system associated with the projection step of our fluid/solid algorithm. Overall, the computational time to solve the projection step of our fluid/solid algorithm is similar to the time needed to solve the standard fluid-only projection step. The monolithic treatment results in a stable projection step, i.e. the kinetic energy does not increase in the projection step. Numerical results indicate that the method is second-order accurate in the L^~-norm and demonstrate that its solutions agree quantitatively with experimental results.

A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids

We present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains with Dirichlet boundary conditions. The computational domain is discretized with non-graded Cartesian grids, i.e., grids for which the difference in size between two adjacent cells is not constrained. Refinement criteria is based on proximity to the irregular interface such that cells with the finest resolution is placed on the interface. We sample the solution at the cell vertices (nodes) and use quadtree (in 2D) or octree (in 3D) data structures as efficient means to represent the grids. The boundary of the irregular domain is represented by the zero level set of a signed distance function. For cells cut by the interface, the location of the intersection point is found by a quadratic fitting of the signed distance function, and the Dirichlet boundary value is obtained by quadratic interpolation. Instead of using ghost nodes outside the interface, we use directly this intersection point in the discretization of the variable coefficient Laplacian. These methods can be applied in a dimension-by-dimension fashion, producing schemes that are straightforward to implement. Our method combines the ability of adaptivity on quadtrees/octrees with a quadratic treatment of the Dirichlet boundary condition on the interface. Numerical results in two and three spatial dimensions demonstrate second-order accuracy for both the solution and its gradients in the L1 and L∞ norms.

A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate

We present a level set approach to the numerical simulation of the Stefan problem on non-graded adaptive Cartesian grids, i.e. grids for which the size ratio between adjacent cells is not constrained. We use the quadtree data structure to discretize the computational domain and a simple recursive algorithm to automatically generate the adaptive grids. We use the level set method on quadtree of Min and Gibou [C. Min, F. Gibou, A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys. 225 (2007) 300-321] to keep track of the moving front between the two phases, and the finite difference scheme of Chen et al. [H. Chen, C. Min, F. Gibou, A supra-convergent finite difference scheme for the poisson and heat equations on irregular domains and non-graded adaptive Cartesian grids, J. Sci. Comput. 31 (2007) 19-60] to solve the heat equations in each of the phases, with Dirichlet boundary conditions imposed on the interface. This scheme produces solutions that converge supralinearly (~1.5) in both the L^1 and the L^~ norms, which we demonstrate numerically for both the temperature field and the interface location. Numerical results also indicate that our method can simulate physical effects such as surface tension and crystalline anisotropy. We also present numerical data to quantify the saving in computational resources.