Frédéric Gibou

Simulating water and smoke with an octree data structure

We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric positive definite enabling the use of fast solution methods such as preconditioned conjugate gradients, whereas the standard approximation to the Poisson equation on an octree grid results in a non-symmetric linear system which is more computationally challenging to invert. The semi-Lagrangian characteristic tracing technique is used to advect the velocity, smoke density, and even the level set making implementation on an octree straightforward. In the case of smoke, we have multiple refinement criteria including object boundaries, optical depth, and vorticity concentration. In the case of water, we refine near the interface as determined by the zero isocontour of the level set function.

A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change

In this paper, we describe a sharp interface capturing method for the study of incompressible multiphase flows with phase change. We use the level set method to keep track of the interface between the two phases and a ghost fluid approach to impose the jump conditions at the interface. This work builds on the work of Gibou et al. for the study of Stefan problems [F. Gibou, R. Fedkiw, L.-T. Cheng, M. Kang, A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys. 176 (2002) 205-227] and the work of Nguyen et al. for the simulation of incompressible flames [D. Nguyen, R. Fedkiw, M. Kang, A boundary condition capturing method for incompressible flame discontinuities, J. Comput. Phys. 172 (2001) 71-98]. We compare our numerical results to exact solutions in one spatial dimension and apply this algorithm to the simulation of film boiling in two spatial dimensions.

A Level Set Approach for the Numerical Simulation of Dendritic Growth

In this paper, we present a level set approach for the modeling of dendritic solidification. These simulations exploit a recently developed second order accurate symmetric discretization of the Poisson equation, see [12]. Numerical results indicate that this method can be used successfully on complex interfacial shapes and can simulate many of the physical features of dendritic solidification. We apply this algorithm to the simulation of the dendritic crystallization of a pure melt and find that the dendrite tip velocity and tip shapes are in excellent agreement with solvability theory. Numerical results are presented in both two and three spatial dimensions.

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.

USING THE PARTICLE LEVEL SET METHOD AND A SECOND ORDER ACCURATE PRESSURE BOUNDARY CONDITION FOR FREE SURFACE FLOWS

In this paper, we present an enhanced resolution capturing method for topologically complex two and three dimensional incompressible free surface flows. The method is based upon the level set method of Osher and Sethian to represent the interface combined with two recent advances in the treatment of the interface, a second order accurate discretization of the Dirichlet pressure boundary condition at the free surface (2002, J. Comput. Phys. 176, 205) and the use of massless marker particles to enhance the resolution of the interface through the use of the particle level set method (2002, J. Comput. Phys., 183, 83). Use of these methods allow for the accurate movement of the interface while at the same time preserving the mass of the the liquid, even on coarse computational grids. Also, these methods complement the level set method in its ability to handle changes in interface topology in a robust manner. Surface tension effects can be easily included in our method. The method is presented in three spatial dimensions, with numerical examples in both two and three spatial dimensions.

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 Variational Framework for Multiregion Pairwise-Similarity-Based Image Segmentation

Variational cost functions that are based on pairwise similarity between pixels can be minimized within level set framework resulting in a binary image segmentation. In this paper we extend such cost functions and address multi-region image segmentation problem by employing a multi-phase level set framework. For multi-modal images cost functions become more complicated and relatively difficult to minimize. We extend our previous work, proposed for background/foreground separation, to the segmentation of images in more than two regions. We also demonstrate an efficient implementation of the curve evolution, which reduces the computational time significantly. Finally, we validate the proposed method on the Berkeley segmentation data set by comparing its performance with other segmentation techniques.

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.